Coherent diffraction / digital information reconstruction by iterative phase recovery using a special mask.

Abstract

The totagram is generated by an iterative spectral phase retrieval process that results in complete information recovery without the use of a reference beam and by using special masks. The use of these special masking systems reduces the computation time, the number of masks, and the number of iterations. The special masking systems are: (1) a unity mask with one or more bipolar binary masks with elements equal to 1 and -1; (2) a unity mask with one or more phase masks; (3) a unity mask with one pair or multiple pairs of masks with binary amplitudes of 0 and 1, where the masks in the pair are complementary to each other in amplitude; or (4) one or more pairs of complementary masks with binary amplitudes of 0 and 1 without a unity mask.

Description

Technical Field

[0001] The present invention relates to a phase recovery system and method. More specifically, the amplitude and phase are reconstructed for a coherent wave after measurement of its amplitude in the spectral output.

Background Art

[0002] Information embedded in terms of amplitude and phase, as in coherent wave representation, leads to applications in one or more dimensions, such as in imaging. In such systems, often the phase is more important than the amplitude. In many coherent systems, the phase is lost, because what is measurable is the intensity, which is proportional to the square of the amplitude. Also, the phase may be intentionally lost. Also, in some applications such as speech recognition, blind channel estimation, and blind deconvolution, phase recovery is important for one-dimensional signals. The phase problem dates back to when Rayleigh described it in 1892. Phase recovery has continued to be a well-known problem since then, and the process has accelerated since the 1960s when lasers and other important sources of coherent radiation were discovered.

[0003] In diffraction imaging, which ultimately yields 3D information, there are indirect methods for recovering the phase and thereby achieving, for example, complete information recovery. Holography, discovered by Dennis Gabor, is one such method, achieving 3D imaging by introducing a reference wave. This method has many similarities to the modulation principles used in communications. Other methods are closely related to the Gerchberg-Saxton algorithm (1971-72), also called the original Gerchberg-Saxton algorithm and referred to herein as "GSA," which involves measurements on two related planes: the input plane and the output spectral plane. Advances in several fields of science and technology are related to the GSA, which was published in 1972 (RW Gerchberg, WO Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik, Vol. 35, pp. 237-246, 1972).

[0004] Subsequently, RW Gerchberg made improvements to the GSA by introducing N independent measurement systems on two planes, particularly by using a phase mask. This improvement is referred to herein as “Gerchberg’s Second Method” or “G2.” G2 is published in RW Gerchberg’s “A New Approach to Phase Retrieval of a Wave Front,” Journal of Modern Optics, 49:7, 1185-1196, 2002, which is fully incorporated herein by reference. Further embodiments of G2 are described in U.S. Patents No. 6,369,932, 6,545,790, and 8,040,595, which are fully incorporated herein by reference.

[0005] Unlike holography, G2 does not require a reference wave. Rather, G2 is similar to measuring a quantity in question in N independent ways and then performing averaging among the results. These patents demonstrate how to actually achieve this when using waves. G2 is considered the first method of its kind that uses multiple measurements for reliable phase retrieval. Several other well-known methods for phase retrieval include the ER (Error Reduction) algorithm (J, R, Fienup, 'Reconstruction of an object from its Fourier transform,' Optics Letters, Vol. 3, No. 1, pp. 27-29, July 1978; JR Fienup, 'Phase retrieval algorithms, a comparison,' Applied Optics, Vol. 21, No. 15, pp. 2758-2769, 1 August, 1982), ASR (Averaged Successive Relaxation) (JCH Spence, 'Diffractive (lensless) imaging,' Ch. 19, Science of Microscopy, edited by PW Hawkes, JCH Spence, Springer, 2007), and HPR (Hybrid Projection Reflection) (HH Bauschke, PL Combettes, D. Russell Luke, 'Hybrid projection-reflection method for phase retrieval,' J. Optical Soc. Am. A, Vol. 20, No. 6, pp. 1025-1034, June 2003);Russell Luke, 'Relaxed averaged alternating reflections for diffraction imaging,' Inverse Problems, Vol. 21, pp. 37-50, 2005), OSS(Oversampling Smoothness)(JARodriguez,R. DM (Difference map) (V. Elser, 'Solution of the crystallographic phase problem by iterated projections,' Acta Crystallography. Section A: Foundations Crystallography, Vol. 59, pp. 201-209, 2003). Although relatively recent, several algorithms exist that utilize relatively effective optimization methods such as SO2D and SO4D (Stefano Marchesini, 'Phase retrieval and saddle-point optimization,' J. Optical Soc. Am. A, Vol. 24, No. 10, pp. 3289-3296, October 2007). PhasePack (R. Chandra, T. Goldstein, C. Studer, 'Phasepack: a phase retrieval library,' IEEE 13th international conference on sampling theory and applications, pp. 3289-3296, October 2007)(1-5, 2019) describes a new benchmark study of many common phase acquisition algorithms. This study uses averaging and masking with eight bipolar binary masks on the system input, along with 12 iterative phase retrieval methods, in the same manner as in G2.

[0006] A common theme in all these algorithms is achieving the best phase retrieval by using prior information and constraints. The use of an input mask provides such prior information. Non-negativity, support information, and amplitude information are also commonly used as prior information. Support information is particularly important. This often means that an N×N (complex) image is centered in a window surrounded by zeros in order to generate a total size of 2N×2N. This is also important when using the Fast Fourier Transform (FFT) to approximate the Continuous Fourier Transform in digital implementations.

[0007] Experimental studies have shown that, in the case of a single amplitude measurement in the Fourier domain for complete phase and image recovery, sufficient prior information is usually unavailable. In other words, while recovery results from given data may be better in some cases, the recovery is usually not complete; that is, it is often an approximation without further information. Studies involving multiple measurements using input masks, as outlined above, compensate for this drawback.

[0008] Recently, machine learning, and especially deep learning methods, have often been used to improve the results obtained by conventional phase retrieval methods. For example, two deep neural networks (DNNs) are used in conjunction with the HIO method to improve phase retrieval results (C.I. Sll, FS Oktem, and A. Koc, 'Deep iterative reconstruction for phase retrieval,' Applied Optics, Vol. 58, pp. 5422-5431, 2019).

[0009] First, to improve the reconstruction, a DNN is used iteratively with the HIO method. Then, a second DNN is trained to remove the remaining artifacts.

[0010] Within the research community, there is a growing recognition that multiple measurements are necessary when high-quality phase and image recovery is required. Very recently, several such methods have been published in the literature. Below, we describe several methods with multiple measurements that bear some similarity to Gercberg's G2 method.

[0011] In the phase lift method by Candes et al. (EJ Candes, Y. Eldar, T. Strohmer, V. Voroninski, 'Phase Retrieval via Matrix Completion,' preprint, August 2011; EJ Candes, X. Li, M. Soltanolkotabi, 'Phase Retrieval from Coded Diffraction Patterns,' Stanford University, Technical Report No. 2013-12, December 2013), the initial method is identical to that in Gerchberg's G2 method. In other words, several measurements are obtained by using several masks. They also mention the use of optical gratings, ptychography, and tilt illumination as alternatives to masks. However, masks are the primary mechanism used in their literature. The averaging step in G2 is replaced by a convex optimization method, which is also related to the matrix completion or matrix retrieval problem.

[0012] Furthermore, in the Fourier-weighted projection method developed by Sicairos and Fienup (M. Guizar-Sicairos, JR Fienup, 'Phase Retrieval with Fourier-Weighted Projections,' J. Optical Soc. Am. A, Vol.25, No. 3, pp. 701-709, March 2008), masks are also used to achieve high-quality phase retrieval. They propose different types of masks for this purpose.

[0013] Ptychography is another method that utilizes multiple diffraction intensity measurements (JM Rodenburg, 'Ptychography and Related Imaging Methods,' Advances in Imaging and Electron Physics, Vol. 150, pp. 87-184, 2008). It was first introduced by Hoppe between 1968 and 1973, particularly for X-ray imaging. Instead of relying on a mask, ptychography relies on recording at least two diffraction intensities by shifting the illumination function or aperture function by a known amount in relation to the object being imaged. Thus, there is a moving probe illuminating the object simultaneously. When sufficient overlap exists between different parts of illumination, phase retrieval can be achieved by an iterative phase acquisition algorithm. Recently, another related algorithm has been developed by Sicairos and Fienup based on divergent far-field measurements obtained after translating an object in relation to known lighting patterns (M. Guizar-Sicairos, JR Fienup, 'Phase Retrieval with Transverse Translation Diversity: A Nonlinear Optimization Approach,' Optics Express, Vol.16, No. 10, pp. 7264-7278, 12 May, 2008). Nonlinear optimization is used in this study.

[0014] In short, within the research community, multiple diffraction intensity measurements are currently used to solve phase and image recovery problems associated with diffraction (lensless) imaging, for example (B. Abbey et al, 'Lensless Imaging Using Broadband X-Ray Sources,' Nature Photonics, pp. 420-424, 26 June 2011). This is particularly important in areas such as X-ray and far-infrared imaging where lenses are extremely expensive. [Overview of the Initiative] [Problems that the invention aims to solve]

[0015] Embodiments of the present invention improve upon prior art methods by using a minimum number of masks specially selected for superior spectral phase and thereby complete information recovery. This also results in increased computational speed. According to the method for recovering phase information from an array of points (e.g., pixels), each having amplitude, at least one conversion unit having an input and a spectral output is provided. The array of points may yield optically coherent waves or electronically data. The array may be one-dimensional or more, with two-dimensional applications being relatively common. Amplitude information is recorded at the spectral points. The conversion unit may be a lens system having one or more lenses, or free-space wave propagation, or a digital processing unit. [Means for solving the problem]

[0016] Acting on the input to the conversion unit are at least two specially selected masks. Two masking versions exist. In the first version, one of the masks is a unity mask (all of its elements are equal to 1, also called a transparent mask). In the second version, there is at least one pair of complementary unipolar masks, whose elements are equal to 0 or 1 in amplitude. The input is applied separately to each of the at least two masks to generate a modified input from each of the masks. According to an optical embodiment or similar, the mask is a physical space mask. In such an embodiment, the input is a wave. The mask acting on the wave can be switched from one mask to another so that the input is received sequentially and separately by each of the at least two physical space masks. Such switching can be achieved in real time, for example, by an optical device such as a spatial light modulator or a micromirror array. Alternatively, the input wave could be split so that it is received separately in parallel by each of the physical space masks.

[0017] In any embodiment, it may be advantageous to include an outer boundary surrounding each mask that sets the amplitude of any point that coincides with the boundary to zero.

[0018] According to embodiments of the present invention, the number of masks required can be reduced to two or three. In one embodiment, the mask consists of a unity mask and a phase mask (Figure 1C-I). Specifically, the phase may be accompanied by a quantized phase value. Thus, in a particular embodiment, the phase mask is a bipolar (meaning 1 and -1) binary mask corresponding to a phase value equal to 0 or π (Figure 1C-II). In a further embodiment, the mask consists of at least one pair of a unity mask and a complementary unipolar binary mask (in a switchable state, one mask has elements 1 and 0, and the other has 1 and 0) (Figure 1C-III). Furthermore, the mask may also consist of a pair of masks that are complementary in relation to a unity mask and an amplitude equal to 1.

[0019] Furthermore, in embodiments that do not involve the use of a unity mask, efficient selection of masks can be achieved. According to this further embodiment, there are four masks, each consisting of two pairs of masks (Figure 2). In each pair, the masks are complementary to each other in relation to amplitude. Thus, the unity elements on such masks may further include phase factors, which may have continuous or quantized phase values ​​between 0 and 2π. In even more specific embodiments, the mask consists of two pairs of complementary unipolar binary masks.

[0020] As used herein, the term Generalized Fourier Transform (FT) encompasses transformations performed physically or digitally. The generalized FT is performed by a transformation unit on the modified inputs received from each mask to generate the transformed modified input. The spectral plane (output) is defined as the output (plane) of the generalized FT. Naturally, the generalized FT occurs as a result of coherent wave propagation and / or as the modified input passes through the lens system. This involves further phase factors. A notable example is Fresnel diffraction in coherent optics.

[0021] In the case of a digital processing unit, the conversion unit, the generalized FT may be a generalized fast Fourier transform (FFT). At the spectral output of the conversion unit, amplitude values ​​are recorded in an array of points to generate a phasorgram from each of the converted and modified inputs. In optical embodiments, recording can be performed by an intensity sensor such as a camera in the spectral plane (output) of the lens system. The resulting amplitude information at the spectral output is referred to as a phasorgram.

[0022] The method further includes the step of associating phase values ​​with points on each phasogram in order to form a plurality of complex phasograms. In any embodiment, the phase values ​​may initially be random phase values. The complex phasograms are fed into an iterative process that operates until convergence is achieved to generate a totagram that constitutes a reconstructed input having amplitude and phase information. The totagram contains complete and valuable information that can be used in any number of ways. For example, in any embodiment, the totagram can be used to display a representation of the reconstructed input having amplitude and phase.

[0023] According to one embodiment of performing iterative processing, a plurality of complex phasograms are processed by an inverse generalized Fourier transform and perhaps other optimization steps. A single estimated value of the input is obtained by averaging the complex information at each input point. To obtain a plurality of intermediate arrays of points, the single estimated value of the input is passed through a process of replicating each of the masks. To generate another plurality of complex phasograms, a generalized fast Fourier transform is performed on each of the intermediate arrays, and then the amplitude value at each point in the transformed intermediate array is replaced by the corresponding originally recorded amplitude value. Here, there can also be further optimization steps. The iterative process is repeated by the generated complex phasograms until convergence is achieved, in which case the single estimated value of the input at completion is the totalogram.

[0024] In any of the embodiments, any number of methods can be used to determine convergence. One simple method is to count up to a given number of iterations. Alternatively, instead, convergence is achieved when the absolute difference between successive single estimated values reaches a predefined threshold.

[0025] In any of the optical embodiments, at least one conversion unit includes a low-pass filter having a numerical aperture (NA) of 0.7 or more.

[0026] Any embodiment can generate super-resolution amplitude and phase information of an input wavefront by applying a linear phase modulation to the input wave before passing it through each of at least two physical spatial masks, or by moving an intensity sensor spatially.

[0027] Any embodiment can include the step of performing a preceding generalized Fourier transform (FT) on the input before separately applying the input to each of at least two masks, for example, for lensless imaging of a distant object.

[0028] In any embodiment, at least two physical spatial masks can have elements each having an aperture size of either (i) 8×8 pixels or less or (ii) 16×16 pixels or less for relatively easy implementation. Each element of the mask has an associated constant amplitude and / or phase applied to each pixel or point that passes through that element of the mask.

[0029] The system embodiments of the present invention operate in accordance with one or more of the method embodiments. The system includes a conversion unit, which may be a lens system having one or more lenses, or may be a digital processing unit. The system further includes at least two masks. According to some embodiments, at least two masks include a unity mask. In one embodiment, the mask is composed of a unity mask and a phase mask (FIG. 1C-I). The phase mask can have quantized phase values. Specifically, the mask can be composed of a unity mask and a bipolar binary mask (FIG. 1C-II). In a further embodiment, the mask is composed of a pair of a unity mask and a complementary unipolar binary mask (FIG. 1C-III). More generally, the mask can also be composed of one pair or a plurality of pairs of masks that are complementary in relation to the unity mask and the binary amplitude.

[0030] In a further embodiment, the mask may consist of one or more pairs of masks that are complementary in relation to amplitude without involving a unity mask. Specifically, there may be four masks, each containing two pairs of masks, in which case the masks within each pair are complementary in relation to amplitude. This means that values ​​1 and 0 in one mask become 0 and 1, respectively, in the second mask (Figure 2). The unity points on such masks may further include phase factors. In a more specific embodiment, the mask consists of two pairs of complementary unipolar binary masks.

[0031] In any embodiment, it may be advantageous to include an outer boundary surrounding each mask that sets the amplitude of any point coinciding with the boundary to zero. In fact, according to a further embodiment, the mask consists of a pair of complementary unipolar binary masks, each having an outer boundary that sets the amplitude of any point coinciding with the boundary to zero.

[0032] In an optical embodiment, the mask is a physical space mask positioned in the input plane of an optical lens system. The mask, acting on the wave, is switchable from one mask to another so that the input is received sequentially and separately by each of at least two physical space masks. Such switching can be achieved in real time, for example, by an optical device such as a spatial light modulator or a micromirror array. Alternatively, the input wave can be split by a beam splitter so that it is received in parallel and separately by each of the physical space masks.

[0033] The inputs, individually modified by each mask, are passed through a conversion unit. To generate a phasogram, amplitude values ​​are recorded in an array of points of the converted and modified inputs. In optical embodiments, the recording of amplitude values ​​is performed by at least one sensor system. The sensor system may be an intensity sensor such as a camera.

[0034] The system further includes a processor configured to (1) associate initial phase values ​​with each point on each phasogram in order to form a plurality of complex phasograms, and (2) iteratively process the plurality of complex phasograms until convergence is achieved in order to generate a totagram that constitutes a reconstructed input having amplitude and phase information.

[0035] The features of the above embodiments can be more easily understood by referring to the following detailed description provided with reference to the attached drawings. [Brief explanation of the drawing]

[0036] [Figure 1A] Figure 1A is a schematic diagram of one embodiment of the system according to the present invention.

[0037] [Figure 1B-I] Figure 1B-I is a schematic diagram of another embodiment of the system according to the present invention.

[0038] [Figure 1B-II] Figure 1B-II is a schematic diagram of another embodiment of the system according to the present invention.

[0039] [Figure 1C-I] Figure 1C-I is a schematic diagram of an embodiment of a system having a unity mask and a phase mask according to the present invention.

[0040] [Figure 1C-II]Figure 1C-II is a schematic diagram of one embodiment of a system having a unity mask and a bipolar binary mask according to the present invention.

[0041] [Figure 1C-III] Figure 1C-III is a schematic diagram of one embodiment of a system having a pair of unity masks and unipolar masks according to the present invention.

[0042] [Figure 2] Figure 2 is a schematic diagram of one embodiment of a system having two pairs of unipolar masks according to the present invention.

[0043] [Figure 3A] Figure 3A is a flowchart of a phase recovery method according to an embodiment of the present invention. [Figure 3B] Figure 3B is a flowchart of a phase recovery method according to an embodiment of the present invention.

[0044] [Figure 4] Figure 4 shows the binary space mask when the aperture size is 16x16 pixels.

[0045] [Figure 5] Figure 5 shows the binary space mask when the aperture size is 8x8 pixels.

[0046] [Figure 6] Figure 6 shows the reconstruction results using G2 with one unity mask and one bipolar binary mask.

[0047] [Figure 7] Figure 7 shows the error reduction curves associated with one unity mask and one bipolar binary mask when the aperture size is 16 × 16 pixels.

[0048] [Figure 8]Figure 8 shows the reconstruction results using G2 with one pair of complementary unipolar binary masks.

[0049] [Figure 9A] Figure 9A shows the reconstruction results using two bipolar binary masks with the Fienup iterative phase retrieval method.

[0050] [Figure 9B] Figure 9B shows the reconstruction results using three bipolar binary masks with the Fienup iterative phase retrieval method.

[0051] [Figure 10] Figure 10 shows the reconstruction results of one embodiment of the present invention using one unity mask and one bipolar binary mask by the Fienup iterative phase retrieval method.

[0052] [Figure 11] Figure 11 shows the reconstruction results of one embodiment of the present invention using two pairs of complementary unipolar binary masks by the Fienup iterative phase retrieval method. [Modes for carrying out the invention]

[0053] Definitions: The following terms used in this description and the attached claims shall have the meanings given unless the context requires otherwise.

[0054] The term "totagram" is defined herein as input phase and amplitude information resulting from an iterative spectral phase retrieval process using a mask. The information may be one-dimensional or multi-dimensional. In certain embodiments, the totagram is the reconstructed amplitude and phase of an input coherent wave at a particular wavelength.

[0055] The terms "totagraphy" or "totagraphy method" used herein are defined as the process of obtaining a totagram.

[0056] "Tragraphy imaging" involves recording spectral amplitudes by a sensor / camera on a spectral plane, in contrast to other imaging systems where image information is recorded by a camera on an image plane.

[0057] "Holography" involves the physical recording of interference patterns resulting from the mixing of an object wave and a reference wave that generates a hologram. Totagraphy, on the other hand, replaces the recording of interference patterns between the object wave and the reference wave as in holography, and instead performs several measurements using a special mask that is iteratively processed to generate a totagram by using the methods and systems defined herein.

[0058] In this specification, a "phasogram" is defined as information containing measured or recorded spectral amplitude information after processing an input wave by a transform unit (e.g., a generalized Fourier transform) in relation to a particular input mask. The phasogram has little or no similarity to the input wave because spectral phase information is discarded and spectral amplitude is recorded.

[0059] Introduction Figure 1A is a schematic diagram of one embodiment of a system 100 used in conjunction with the present invention. The system 100 recovers phase and amplitude information from a coherent input wave 102. The input wave 102 can be generated from an object 120, which may be an illuminated object. The system 100 includes at least one conversion unit 110 having a spectral plane (SP) and an input plane (IP), and at least two masks 108. The system is configured such that the input wave 102 is applied sequentially and separately to each of the masks. In optical embodiments, the masks are physical spatial masks. Such physical masks can be implemented to change in real time from one mask to another by an optical device such as a spatial light modulator or a micromirror array. At least two physical spatial masks can be arranged in the IP of at least one conversion unit 110. In some embodiments, the conversion unit may be a lens system having one or more lenses. Also, in any optical embodiment, at least one conversion unit can function as a low-pass filter having a numerical aperture (NA) of 0.7 or greater. In other embodiments, the conversion unit can be implemented in a digital processor.

[0060] Each of at least two masks may include an input window 106 formed from individual opaque boundaries surrounding the mask. Each opaque boundary is configured to block pixels in the input wave that coincide with the boundary, thereby setting the amplitude of those pixels to zero. At least two masks 108 are configured to modify the phase or amplitude of their separately received input waves. At least one transform unit is configured to perform a generalized Fourier transform (FT) on the modified separately received input waves.

[0061] System 100 further includes at least one sensor 112 configured to record amplitude values ​​in an array of points of each converted modified input in SP. At least one sensor generates a phasogram 152 containing measured or recorded spectral amplitude information. The phasogram 152 may have little or no similarity to the input wave 102 because the phase information is discarded. Sensor 112 may be a camera, which is an intensity sensor. The amplitude value is derived directly from the intensity. The intensity should be understood to be linearly proportional to the square of the amplitude.

[0062] System 100 further includes a digital processor 128. The phasogram 152 is iteratively processed by the processor 128 to generate a totagram 158. The processor 128 is configured to iteratively process the multiple complex phasograms to the point where convergence is achieved, so associating phase values ​​with each point on each phasogram to form multiple complex phasograms, and to generate a totagram 158 that constitutes a reconstructed input wave having amplitude and phase information. The spectral phase is recovered to match the recorded amplitude values. The input amplitude and phase can be obtained from the spectral phase and amplitude, as appropriate, through the use of a generalized IFFT. The processor 128 can provide the totagram 158 for further processing 162. Computer processing 162 may include image processing, machine learning, and / or deep learning. The processed result 178 can form an image 172 on a display 170 accessible by a user interface 118.

[0063] Figures 1B-I and 1B-II are schematic diagrams of systems 150A and 150B, respectively, according to embodiments in which inputs are applied separately in parallel to multiple physical space masks. Systems 150A and 150B each recover phase and amplitude information from an input wave 102. The input wave 102 can be generated from an object 120, which may be an illuminated object. Systems 150A and 150B each include at least two physical space masks 108, each disposed in the input plane of the corresponding conversion unit 110.

[0064] Systems 150A and 150B each further include a splitter 130A (also referred to herein as a “beam splitter”) configured to split an input wave 102 into two or more distinct waves. Each of the distinct waves from the splitter passes through at least two corresponding physical space masks 108 to generate a modified wave. At least one transformation unit 110 is configured to perform a generalized Fourier transform (FT) on the modified input wave 102. Systems 150A and 150B include a sensor 112 configured to record spectral amplitude images of the transformed distinct waves in the spectral plane for each transformation unit. Systems 150A and 150B include a processor 128 operating as described above in relation to Figure 1A.

[0065] In addition, as shown in Figure 1B-II, system 150B can perform a preceding generalized Fourier transform (FT) on the input wave 102 via a conversion unit 107 before passing the input wave 102 individually through the splitter 130A. In certain embodiments, the conversion unit 107 is a lens that receives the input wave on its way to the input mask 108. The preceding generalized Fourier transform of the conversion unit 107 transforms the initial input plane image (wave) into a second image (wave). On this plane, the input mask 108 is used together with the second image, along with a generalized Fourier transform via the same conversion unit 110 as before, for illuminating the intensity sensor 112.

[0066] The iterative phase retrieval process includes only the second image (wave). Once phase retrieval is complete, the initial input plane image (wave) is recovered by the final inverse generalized Fourier transform.

[0067] According to embodiments of the present invention, Figures 1C-I, 1C-II, 1C-III, and 2 each show the minimum set of masks that achieve complete phase recovery more quickly and efficiently than systems with many more masks. Figures 1C-I, 1C-II, and 1C-III utilize a unity mask 108t, which may also be referred to as a transparent mask. The input wave passes through the unity mask in an unobstructed state, and thus, physically, the unity mask can be realized by any unobstructed optical path. Optionally, to further improve the efficiency of the computation process, an opaque outer boundary can surround the unity mask, as shown. The boundary sets the amplitude of points on the wave that coincide with the outer boundary to zero. Multiple masks may be added to the system, but according to these embodiments of the present invention, the inclusion of the unity mask allows the desired totagram to be realized with a small number of additional masks, perhaps one or more. When many more masks are used, the final information recovery has even higher quality when one of the masks is the unity mask.

[0068] Figure 1C-I is a schematic diagram of one embodiment of a system having a unity mask 108t and a phase mask 108c. The phase mask 108c imparts a phase shift to points passing through the mask. Each pixel or element (group of pixels) of the mask may be assigned its own specified phase shift, which can vary element by element. In a two-dimensional mask, an element is typically a square consisting of pixels with an aperture size measured by the number of pixels on one side. A phase mask can be simplified advantageously by involving quantized phase values. For example, according to two-level quantization, the quantized phase values ​​are 0 and π, resulting in a phase factor equal to 1 or -1. Such a phase mask is referred to as a bipolar binary mask.

[0069] As shown in Figure 1C-II, the unity mask 108t and the bipolar binary mask 108b can consist of only two masks according to one embodiment of the present invention. Even these two simple masks efficiently generate a totagram. The unity mask is essentially all 1s, and the bipolar binary mask is a "checkerboard" of randomly distributed 1s and -1s. To further improve efficiency, an opaque outer boundary can surround the masks.

[0070] Figure 1C-III shows one embodiment of the present invention utilizing a unipolar binary mask 108u. Pixels or elements of the unipolar binary mask are either open (amplitude 1, meaning pass-through) or closed (amplitude 0, meaning non-pass-through). The pixels or elements are arranged in a random pattern. The unipolar binary masks are used as pairs of masks that are complementary to each other in relation to amplitude. This means that for the location of an element in one mask that has an amplitude of 1, the corresponding element in the other mask has an amplitude of 0. The minimum mask configuration for efficiently generating a totagram includes a complementary pair of unitary mask 108t and unipolar binary mask 108u.

[0071] Figure 2 is a schematic diagram of an alternative embodiment of the present invention that uses two pairs of complementary binary masks 108v, in which case a unitary mask is not required. According to this embodiment, the totagram can be determined using at least four masks. The four masks include two pairs of masks, in which case the masks within one pair are complementary to each other in relation to amplitude. Specifically, all four masks may be unipolar binary masks. The amplitude of the mask elements is 1 or 0, while it is also possible to include a random phase factor within the element that complexifies the mask.

[0072] In a further embodiment of the present invention, the mask 108v can be reduced to only one pair of complementary binary masks. Such a mask configuration may present difficulties in generating a totagram when the input has the entire range of phase variation from 0 to 2π. However, for inputs limited to a narrower phase range such as 0 to π, one pair of complementary masks may suffice. In this case as well, the masks in the pair are complementary to each other in relation to amplitude. Specifically, both masks may be unipolar binary masks. This case would be further improved if an opaque boundary surrounds the masks. Thus, for certain applications, a single pair of these complementary binary masks can be used instead of two pairs.

[0073] The phase retrieval system used with the mask of the embodiment of the present invention will be described in more detail below in relation to Figures 3A and 3B. The input 102 to the system is an array of points, each point having an amplitude. The array may be one-dimensional or more. In an optical environment, the input is a coherent wave captured as a two-dimensional array of pixels (points). The system can receive its input from an object 120 via an input image (wave). Object 120 may be an illuminated object. Several phase retrieval systems can be configured to operate in parallel to recover full-color wave information. For example, each system can operate for its own coherent wave for one of three primary colors (wavelengths). This can be generalized to multispectral and hyperspectral images (waves) having more than three wavelengths.

[0074] The inputs must be presented separately for each mask. This is easily accomplished in digital embodiments where the input array is processed separately through each of the multiple masks. In optical embodiments, a splitter 130B can be used to duplicate the input point array for each mask. Alternatively, the input masks can be continuously switched by a spatial light modulator or a micromirror array, as described in relation to Figure 1A.

[0075] The system is composed of multiple masks 188 according to any of the embodiments described above in relation to Figures 1C-I, 1C-II, 1C-III, and 2. Further masks may also be used, but it is advantageous to minimize the number of masks and, therefore, the computational complexity and implementation difficulty (Ddifficulty). Optionally, an outer boundary surrounding the masks can be used to further enhance the efficiency and accuracy of the iterative computation process. The input modified by each mask is passed through a transform unit 110 to perform a generalized Fourier transform 182. In optical embodiments, the transform unit may be a lens or a system of lenses. In digital embodiments, the generalized Fourier transform 182 is computed. The generalized Fourier transform may be a generalized FFT.

[0076] Each converted and modified input is supplied to a sensor 112, which records the amplitude value in the spectral array of each converted and modified input point. The array of amplitude values ​​is referred to as a phasogram. Sensor 112 is not sensitive to phase. Therefore, any phase aberration that can be modeled as a phase variation on the spectral plane (output) is removed by the sensor. In optical embodiments, sensor 112 may be an intensity sensor such as a camera. The intensity is linearly proportional to the square of the amplitude.

[0077] The method further includes the step of associating phase values ​​196 with each point on each phasogram in order to form multiple complex phasograms within the digital processor 128. In a preferred embodiment, randomly selected phase values ​​are associated with each point. The inclusion of phases results in a complex phasogram.

[0078] The complex phasogram enters an iterative process. Several methods are known in the art. One such process is G2. Other methods are shown, for example, in Phasepack. Depending on the processes implemented in the system, each complex phasogram can optionally pass through an optimization process 198 (which may be located at the input and / or output of the iterative system). Each complex phasogram is then processed through an inverse generalized Fourier transform 186. In the case of the FFT, the inverse is the IFFT, and the inverse is also true.

[0079] The output of the inverse generalized Fourier transform is, at will, optimized according to the implemented process, and then the complex information at each corresponding point is averaged to generate a single estimate of the input.178 At will, another optimization process may be included on the output side of the iterative process. Each time a single estimate is obtained, the process determines whether convergence has occurred.172 According to one convergence test, the process continues until the difference between consecutive single estimates reaches a predetermined threshold. According to another method, convergence is assumed to have been reached after a given number of iterations for determining a single estimate have been completed. According to some embodiments, the predetermined threshold is reached when the fractional error, i.e., the sum of squared errors (SSE) across all N image outputs from the inverse generalized Fourier transform divided by the squared amplitude (total energy) across all N images between two consecutive iterations, is less than a value such as 0.0001, without limitation. The SSE represents the squared difference between the N current waveforms and the last estimate. Alternatively, SSE can be defined in terms of the current and final estimates after averaging. Once convergence is achieved, the final estimates of the input amplitude and phase constitute a totagram.

[0080] Figure 3B further illustrates the iterative process. A single estimate is passed through process 189, which duplicates each of the input masks to obtain multiple intermediate arrays, one from each mask. In other words, the phase shift and / or amplitude factor of each element in the corresponding mask modifies the input to result in the intermediate arrays. A generalized fast Fourier transform 183 is performed on each of the intermediate arrays. At each point in the transformed intermediate array, the amplitude value is replaced by the corresponding amplitude value originally recorded by the sensor 112 170, thereby generating another iteration of the complex phasogram. The complex phasogram is optimized if applicable to the implemented iterative process. Then, as in the first iteration, each complex phasogram is processed through the inverse generalized Fourier transform 186. For the FFT, the inverse is the IFFT, and the inverse is also true. The output of the inverse generalized Fourier transform is, by choice, optimized according to the implemented process, and then the complex information at each corresponding point is averaged to produce a single estimate of the input.178 The process continues iteratively until convergence occurs.172

[0081] Optionally, the phase recovery method can generate super-resolution amplitude and phase information from the input wave by performing linear phase modulation on the input wave several times before passing it through each of at least two physical spatial masks, or by spatially moving the intensity sensor after passing the input wave through each of at least two physical spatial masks several times. This can also be achieved by moving the location of the spectral output several times.

[0082] In any embodiment of at least two masks, each element of the non-unity mask may have an aperture size of 8x8 pixels or less. In any embodiment consisting of at least two masks, the non-unity mask may have an aperture size of 16x16 pixels or less. Each element of the non-unity mask has a corresponding constant amplitude and / or phase applied to each pixel or point that passes through the element of that mask.

[0083] Any embodiment may include a step of processing a totagram to provide a solution to a task. These tasks may include microscopy, encoding, signal processing, wavefront detection, and / or optical calculations. The information in the totagram can be converted into a hologram using recovered amplitude and / or phase information. The result is referred to as a digital hologram or computer-generated hologram. The 3D information of the totagram can also be visualized in other ways by digital techniques such as computer graphics, stereoscopic displays, virtual reality, augmented reality, or mixed reality. Any embodiment may include a step of displaying the result or representation of the solution on a display.

[0084] The effectiveness of the system and method of the present invention is demonstrated for various inputs. When the input has zero phase, this means that the input has only amplitude variation. This is the simplest case. The most common case is when the input phase varies between 0 and 2π radians.

[0085] There are two main categories of suitable mask combinations according to embodiments of the present invention. In the first category, the first mask is a unity mask (clear, transparent, when all elements are equal to +1). The second mask may be (1) a phase mask that varies between 0 and 2π radians, (2) a quantized phase mask having elements equal to the quantized phase value, (3) a bipolar binary mask having elements equal to +1 and -1 corresponding to the quantized phase selected as 0 and π radians, or (4) a pair of complementary masks, where one mask has 0 and exp(jθ1), where θ1 is the quantized or continuous phase, and the second mask has a corresponding element equal to exp(jθ2), where θ2 is the quantized or continuous phase and 0, respectively. In other words, the masks are complementary in relation to amplitude. If an element of one mask has a value of 0, then the corresponding element in the other mask of the pair has an amplitude of 1 and an associated phase factor. In certain cases, when θ1 and θ2 are selected to be equal to 0, the mask becomes a complementary pair of unipolar binary masks with elements equal to 0 and 1. In other specific cases, when θ1 and θ2 are limited to 0 or π, the mask becomes a complementary pair of binary masks with elements equal to 0 and ±1. Binary refers to two amplitude values ​​that are either 0 or 1.

[0086] In the second category, a transparent mask is not required; rather, there are preferably two or more pairs of complementary masks. Specifically, two pairs of complementary unipolar (+1 and 0) binary masks can be used in practice. When even more masks are used, the number of phase retrieval iterations is usually reduced.

[0087] In all cases described in categories 1 and 2, it is possible to use an outer boundary filled with zeros. For example, using a boundary by doubling the mask size and filling the outer boundary of the mask with zeros usually results in a relatively accurate reconstruction or a reduced number of phase retrieval iterations.

[0088] Coherent phase / amplitude recovery by G2 The primary application of coherent phase / amplitude recovery is imaging, which may be 2D, 3D, or even higher dimensions. To achieve multidimensional imaging, it is necessary to have complete wave information consisting of amplitude and phase. Below, G2 is described as one example of several candidate methods for coherent phase / amplitude recovery.

[0089] By assuming a constant z (in the longitudinal direction), a coherent spatial wave can be described as follows:

number

[0090] At this point, the inventors assume that the wave is subjected to a generalized Fourier transform. In the digital implementation, this means that the wave is processed by a generalized FFT. In the optical implementation, the wave passes through a lens system having a focal length F. As a result, the initial wave is assumed to be at z = -F. The spectral plane is at z = F. On the spectral plane, it is known that the wave is proportional to the Fourier transform of the input wave (OK Ersoy, Diffraction, Fourier Optics and Imaging, J. Wiley, November 2006), and this document is fully incorporated herein by reference. This is a case described later.

[0091] On the spectral plane, the corresponding wave can be described by the following equation:

number

number

number

number

number

number

[0092] By assuming that the sensor is positioned on the spectral plane, or intentionally, the spectral phase is lost, and the spectral amplitude is given by the spectral intensity as follows:

number

number

[0093] In subsequent iterations performed by the computer,

number

[0094] The following will further explain the details of digital processing involving the Discrete Fourier Transform (DFT) and its inverse (IDFT), as well as their fast algorithms, the Fast Fourier Transform (FFT) and the Inverse Fast Fourier Transform (IFFT). The following will be defined.

number

[0095] θ i This is randomly selected within the range [0, 2π] in the first iteration during phase retrieval.

[0096] The initial transformation in the first iteration between the input space and the output space is given by the following equation:

number

number

number

[0097] Next, equations (6-10) in the current iteration are repeated. The iteration is,

number

[0098] The DFT and inverse DFT in the 1D case are given by the following equations.

number

[0099] Equations (11) and (12) can be easily extended to the 2D case.

[0100] Design for digital / optical mounting configurations The digital implementation of the iterative phase recovery method can be performed within a computer system.

[0101] Furthermore, the digital / optical implementation of the iterative phase recovery method can also be carried out by manufacturing an optical system that is coupled with a digital system supplied by the output of a digital sensor / camera for subsequent iterative processing.

[0102] In digital / optical implementations, spectral imaging with real-time electronic phase / amplitude masks, such as high-resolution cameras and spatial light modulators, is used. Subsequent digital processing is performed by high-precision computer systems. The FFT technique requires its own sampling interval, which must be matched to the pixel interval along with the camera.

[0103] Once amplitude and phase recovery are complete in the optical / digital system, the information is referred to as a totagram.

[0104] Any embodiment may include a step of processing a totagram to provide a solution for a task. These tasks may include microscopy, encoding, signal processing, wavefront detection, and / or optical computation. The information in the totagram can be converted into a hologram by using the recovered amplitude and phase information. The result is referred to as a digital hologram or computer-generated hologram. The 3D information of the totagram can be visualized in other ways by digital techniques such as computer graphics, stereoscopic displays, virtual reality, augmented reality, or mixed reality.

[0105] Experimental results in optical / digital systems may not be as complete as results from purely digital implementations. Machine learning (ML) and deep learning (DL) techniques can be used to improve results and compensate for these differences. Such techniques have been reported to support phase recovery and diffraction imaging (Y. Rivenson, Y. Zhang, H. Gunaydin, Da Teng and A. Ozcan, 'Phase Recovery and Holographic Image Reconstruction Using Deep Learning in Neural Networks,' Light: Science & Applications, Vol. 7, 17141, 2018; G. Barbastatis, A. Ozcan, G. Situ, 'On the Use of Deep Learning for Computational Imaging,' Optica, Vol. 6, No. 8, pp. 921-943, August 2019), and all of these publications are incorporated herein by reference. ML and DL utilize very large databases of images. For example, the input images to the system can be experimentally realized, and the desired output images should ideally be what they should be. It has been reported that ML and DL methods can achieve good results by training with a very large database of such images.

[0106] Iterative phase retrieval method with diffraction-limited optical components The conversion unit 110 may be a coherent optical system controlled by at least a diffraction-limited coherent transfer function (CTF), which is a point spreading function and its Fourier transform. The system functions as an ideal low-pass filter with a cutoff frequency controlled by the numerical aperture NA of the lens system. In this section, the inventors show and assert that, with a sufficiently large NA (~0.7), iterative phase recovery is not hindered by diffraction.

[0107] A diffraction-limited lens system is a coherent transfer function, which is the Fourier transform of the point spread function h(x,y) and h(x,y).

number

number

number

number

number

[0108] Let's assume coherent wave illumination on a 3D object. This can be achieved by a laser or a high-quality light-emitting diode (LED). For example, a He-Ne laser has a wavelength of 0.6386 microns (μ=10⁻¹⁶). -6 The LED operates at a wavelength λ equal to m), and the LED operates at approximately λ = 0.5 μm.

[0109] Some of the quantities in question are as follows:

number

[0110] Due to diffraction, the optical imaging system has a cutoff frequency given by the following equation.

number

[0111] The sampling frequency on the spectral plane is described by the following equation:

number

number

number

[0112] Next, the coherent transfer function is given by the following equation:

number

number

[0113] The inventors conducted experiments to discover an NA value that allows for complete reconstruction. It was determined that when NA = 0.7 or higher, the reconstructed image was visually as good as the original.

[0114] aberration Aberration is the deviation of the ideal wave in the exit pupil of a lens system from its ideal form. In a coherent imaging system, this can be modeled as the multiplication of the optical transfer function by a phase factor. In this section, it is shown and asserted that phase aberration does not adversely affect the performance of the iterative phase recovery method.

[0115] A diffraction-limited system means that the wave of interest is perfect in the exit pupil, and the only imperfection is a finite aperture size. Aberration is the deviation of the ideal wave in the exit pupil from its ideal form. To include phase aberration, the exit pupil function can be modified as follows:

number

number

[0116] Phase function

number

number

number

number

[0117] Higher-order terms can be added to this function. The right-hand terms in equation (19) represent the following:

number

[0118] Zernicke polynomial Furthermore, phase aberrations present in optical systems can also be expressed in terms of orthogonal and normalized Zernicke polynomials within a circle of unit radius (VN Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Engineering and Laboratory Notes, RR Shannon, editor, supplement to Applied Optics, pp. 8121-8124, December 1994). In this process, the phase function

number

number

[0119] Each Zernike polynomial is typically expressed in the following form:

number

number

number

number

number

[0120] The coefficient Anm is determined for finite values ​​of n and m using least squares.

number

number

number

number

[0121] Note that the Zernike representation of aberrations is valid when the exit pupil is circular. Otherwise, the Zernike polynomials are not orthogonal.

[0122] Coherent optical systems exhibit aberrations. These are typically modeled as phase factors on the system's spectral plane. For example, such modeling can be performed in terms of polynomials representing phases resulting from aberrations such as Seidel aberration and Zernicke polynomials. On the spectral plane, intensity is measured, and all phases are lost. This includes phases resulting from aberrations. The camera removes all phases, and as a result, the phase aberrations that can be expressed as phase factors on the spectral plane do not adversely affect the performance of the iterative phase recovery method shown in Figures 3A and 3B.

[0123] Ultra-resolution associated with iterative phase recovery method In previous sections, complete phase reconstruction was achieved for applications such as 3D imaging. This was made possible by high NA diffraction-limited lens systems and high-resolution cameras with high dynamic range. In this section, a system including linear phase modulation of object waves and iterative phase recovery methods is described to improve a given lens system having low NA, a low field of view, and aberrations.

[0124] Low NA means that high spatial frequencies on the spectral plane are filtered out. Low field of view means a small area of ​​detection by the camera. Aberrations can be modeled as phase modulation on the spectral plane, as described in a previous section. To bypass these problems and / or achieve a resolution higher than that possible with a given lens system and camera, the inventors consider methods similar to those used in synthetic aperture microscopes (Terry M. Turpin, Leslie H. Gesell, Jeffrey Lapides, Craig H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, Advanced Imaging Technologies and Commercial Applications, doi:10.1117 / 12.217378, 23 August 1995) and Fourier ptychographic imaging (G. Zheng, R. Horstmeyer, C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics, pp. 739-745, Vol. 7, September 2013). For this purpose, the input object wave will be modulated (multiplied) by several plane waves assigned by the following equation.

number

[0125] This can be achieved in several ways. For example, an LED matrix array can illuminate a 3D object with angle-varying plane waves. Alternatively, a spatial light modulator (SLM) can be used to generate a real-time reconfigurable array of diffraction gratings (S. Ahderom, M. Raisi, K. Lo, KE Alameh, R. Mavaddah, “Applications of Liquid crystal light modulators in optical communications,” Proceedings of 5th IEEE International Conference on High Speed ​​Networks and Multimedia Communications, Jeju Island, Korea, 2002).

[0126] For each m, the linear imaging system has an output image assigned by the following equation.

number

[0127] By assuming a diffraction-limited imaging system, the coherent transfer function controls the imaging, and equation (24) in the spectral domain becomes as follows:

number

[0128] For example, when the wavelength λ is 0.5 microns, the wavenumber is given by the following equation.

number

[0129] The cutoff frequency for CTF is given by the following formula:

number

[0130] When NA=0.1 and the DFT (image) size is 256×256, the 32×32 window of the DFT spectral points is:

number

number

[0131] Similarly, when NA=0.2 and the DFT (image) size is 256×256, the 64×64 window of the DFT spectral points is:

number

number

[0132] Recent high-resolution cameras, such as 8K cameras, support significantly larger pixel counts, such as 8192 x 4320 pixels (https: / / www.usa.canon.com / internet / portal / us / home / products / details / cameras / eos-dslr-and-mirrorless-cameras / dslr / eos-5ds-r).

[0133] Since the FFT works best with powers of 2, let's assume a size of 4096 x 4096 pixels. When NA=2 and the DFT (image) size is 16384 x 16384, the 4096 x 4096 window of DFT spectral points is:

number

number

[0134] mask A key consideration is the number of masks required for the iterative phase recovery method to achieve acceptable performance. Since each mask represents a different set of measurements, fewer masks are better. In addition, the masks used have a significant impact on the quality of information reconstruction. Information recovery can be considered in terms of amplitude recovery, phase recovery, or preferably both, of the input image. This is different from phase recovery within the spectral domain. In other words, recovered phase within the spectral domain may provide correct amplitude recovery of the input image, but it is not necessarily correct phase recovery of the input image; i.e., it is incomplete phase recovery of the input image. What is usually reported in the literature is input (wave) amplitude recovery. There is a high possibility that the recovered input (wave) phase is not sufficiently correct. According to embodiments of the present invention, not only complete input image amplitude recovery but also input phase recovery is required.

[0135] Another consideration is the type of mask used. According to embodiments of the present invention, a reduction in the number of masks is achievable. As described above, in the first category, the first mask is a unity mask (clear, transparent when all elements are equal to +1). The second mask may be a pair of complementary masks, where (1) a phase mask having a phase that varies from 0 to 2π radians, (2) a quantized phase mask having elements equal to the quantized phase value, (3) a bipolar binary mask having elements equal to +1 and -1 corresponding to the quantized phase selected as 0 and π radians, or (4) a pair of complementary masks, where the corresponding elements of each mask in the pair are complementary in relation to the amplitude. In the second category, a transparent mask is not required; rather, there are preferably two or more pairs of complementary binary masks. Specifically, two pairs of complementary unipolar (+1 and 0) binary masks can be used. The number of phase retrieval iterations can be reduced when a relatively large number of masks are used.

[0136] Embodiments of the present invention implement iterative phase recovery having one or more additional masks in addition to a unity mask (version 1) or a pair of unipolar masks complementary in relation to amplitude (version 2).

[0137] A unipolar binary mask is no longer a phase mask, but a binary amplitude mask. According to conventional thinking, amplitude masks generally do not work. Unipolar binary masks, on the other hand, are desirable in many applications because they are relatively easier to implement. According to embodiments of the present invention, unipolar binary masks are generated as pairs. The second mask is the complement of the first mask. In other words, 0s and 1s are swapped in all components of the first mask to generate the second mask. This also applies to pairs of unipolar binary masks in which 1s are substituted by phase factors whose amplitude is equal to 1.

[0138] Each element in the mask has a finite size. Therefore, especially in optical packaging configurations, it is important that elements with finite sizes do not degrade performance. The inventors argue that the iterative phase recovery method functions equally well with finite element sizes when they are sufficiently small. Sufficiently small means 16×16 in the binary bipolar case and 8×8 in the unipolar binary case. Figure 4 shows a binary mask for an aperture size equal to 16×16. Figure 5 shows a binary mask with an aperture size equal to 8×8.

[0139] Experimental results associated with complex waves using the proposed mask Figures 6, 7, and 8 provide some experimental results associated with complex waves having amplitude (image) and phase (image). Figure 6 shows the reconstruction results associated with complex waves using G2 when one unity mask and one bipolar binary mask were used. The original amplitude image is shown in (a). The reconstructed amplitude image is shown in (b). The original phase image is shown in (c). The reconstructed phase image is shown in (d). Figure 7 shows the corresponding error reduction curves over iteration.

[0140] Figure 8 shows the reconstruction results for the same complex wave using G2 when one pair of complementary unipolar binary masks is used. The original amplitude image is shown in (a). The reconstructed amplitude image is shown in (b). The original phase image is shown in (c). The reconstructed phase image is shown in (d).

[0141] Imaging of distant objects In this section, the inventors describe iterative phase retrieval for coherent imaging of an object considerably far from the imaging lens system. In such cases, the field at the input of the imaging lens system is directly related to the Fourier transform of the wave arriving from the thin object being imaged. This is particularly applicable to the Fraunhofer approximation for distant wave propagation and can be extended to propagation at less distant locations associated with the Fresnel approximation in the following references (A. Eguchi, J. Brewer, TD Milster, “Optimization of random phase diversity for adaptive optics using an LCoS spatial light modulator,” Optics Letters, Vol. 44, No. 21, 1 November 2019, pp. 6834-6840 and A. Eguchi, TD Milster, “Single shot phase retrieval with complex diversity,” Optics Letters, Vol. 44, No. 21, 1 November 2019, pp. 5108-5111). In the previous section, the input to the lens system was a complex image. Now, this is essentially a spectral image. The inventors can consider passing the input wave through an input mask, as previously done, followed by another generalized Fourier transform by the lens, for example, which will result in an inverted object image. As a result, the camera will record an image. In this configuration, the system is the opposite of the previous system, meaning that the image plane and the spectral plane are swapped. Unfortunately, iterative phase retrieval may not work well under these conditions. The Fourier transform of the object image is usually concentrated at very low frequencies, and the rest of the Fourier plane information is noise-like with small components, which makes the input mask ineffective.

[0142] To address these challenges, the system shown in Figure 1B-II can be used for coherent imaging of thin, distant objects. As illustrated, the lens system can be designed to provide two Fourier transforms instead of one. The first Fourier transform 107 transforms the input image (wave) into another image (wave). The input is then passed through one or more splitters whose outputs are sent to the mask as before. The second Fourier transform 110 regenerates the modified spectral information resulting from masking to illuminate the image sensor 112. Thus, the portion of the lens system including the mask, represented by the image plane and the image sensor plane, is identical to that of the previously used system.

[0143] Iterative phase recovery method with masking Several iterative phase recovery methods are used in conjunction with masking. According to embodiments of the present invention, using a unity (clear) mask as one of the masks significantly improves the performance of the iterative phase recovery method compared to using a bipolar binary mask or a phase mask. Furthermore, the use of complementary binary mask pairs (perhaps without a unity mask) within such a system is highly effective.

[0144] The effectiveness of one embodiment of the present invention was digitally evaluated. A simple FFT system with a digitally implemented mask without boundaries was used. Without loss of generality, the coherent input was the amplitude image only, meaning that the input phase was assumed to be zero at each pixel. Figure 9A shows the image recovery associated with the Fienup method when two bipolar binary masks were used. It is observed that the image is not recovered. The corresponding results with all other methods were identical. The results are slightly improved when three bipolar binary masks are used, as shown in Figure 9B with the Fienup method. However, the results are still not satisfactory. Replacing one bipolar binary mask with a clear mask results in a dramatic improvement, as shown in Figure 10, in which case one clear mask and one bipolar binary mask result in image recovery.

[0145] The results associated with pairs of complementary unipolar binary masks indicate that they are self-sufficient in the absence of a unity mask. Figure 11 shows the image recovery results associated with two pairs of complementary unipolar binary masks using the Fienup method. It is observed that two pairs of complementary masks produced better results than one pair of complementary masks. For more than two pairs of masks, further improvement in results is minimal.

[0146] The performance was very similar for other iterative phase recovery methods. However, such methods include, but are not limited to, those listed in Table 1. [Table 1]

[0147] Table 2 shows the mean squared error performance of all methods using three bipolar binary masks versus one clear mask and two bipolar binary masks. It is observed that replacing one bipolar binary mask with a clear mask significantly improves the error performance (Fineup, G2, TAF, Wirtflow) with large MSE errors that do not have a clear mask.

[0148] Table 3 shows how the number of iterations and computation time vary as a function of, for example, the number of pairs of complementary unipolar binary masks having the RAF method. It is observed that as the number of mask pairs increases up to 3 for the method, the performance improves considerably in terms of computation speed.

[0149] Table 4 shows the optimal number of complementary unipolar mask pairs for best visual performance. This number is 2 (in most cases) or 3. [Table 2] [Table 3] [Table 4]

[0150] Conclusion The iterative phase retrieval method can be implemented digitally, for example, within a digital processor such as a computer. The input may be, for example, a pre-recorded image or an array of other points. In this case, generalized FFT and inverse generalized FFT (IFFT) can be used. Furthermore, by using the term "optical" in a general sense to encompass all waves, the iterative phase retrieval method can also be implemented by a coherent optical system or a coherent optical / digital system. In these cases, the initial Fourier transform operation and amplitude detection are usually performed by a lens / camera system. In the case of a coherent optical / digital system, the wave amplitude information obtained by the lens / camera system is input to a computer system to perform iterations of FFT and IFFT according to the iterative phase retrieval method. This may be followed by other possible operations, such as the generation of a 3D image.

[0151] In digital implementations, the input mask can be generated within a computer, possibly along with complex input information. In coherent optics or coherent optics / digital implementations, these can be implemented in real time by optical devices such as spatial light modulators and micromirror arrays.

[0152] A coherent optical system is at least diffraction-limited. This means that the lens system functions as a low-pass filter characterized by the numerical aperture (NA). The iterative phase retrieval function requires that the system's NA be sufficiently large. According to embodiments of the present invention, an NA ≥ 0.7 has been found to be sufficient.

[0153] Coherent optical systems exhibit aberrations. These are typically modeled as phase factors on the system's spectral plane. For example, such modeling can be performed in terms of polynomials representing phases resulting from aberrations such as Seidel aberration and Zernecke polynomials. In coherent systems, aberration phase factors appear as additional phases added to the input spectral phase on the Fourier plane. The camera is sensitive only to amplitude, thereby eliminating all aberrations that can be modeled as phase fluctuations on the spectral plane. Therefore, spectral phase aberrations do not adversely affect the performance of the spectral iterative phase recovery method.

[0154] Optical systems with limited numerical apertures (NAs) and aberrations can be used to achieve superresolution by employing an iterative phase retrieval method, which includes linear phase modulation with several input signals. The linear phase modulation portion is similar to that performed in synthetic aperture microscopy and Fourier ptychography imaging. Iterative phase retrieval operates with the spectral amplitude obtained from all linear phase-modulated portions of the input signal by each mask, resulting in superresolution amplitude and phase information. Similar results can be achieved by spatially moving the intensity sensor instead of linear phase modulation after passing the input wave through at least two physical spatial masks several times.

[0155] An input mask can be generated using elements such as elements of a finite size, for example, if the size is sufficiently small. In the case of a unipolar binary mask, 8x8 elements or less have resulted in satisfactory performance in digital experiments. In the case of a bipolar binary mask, 16x16 elements or less have resulted in satisfactory performance in digital experiments. Therefore, bipolar binary masks are more tolerant than unipolar binary masks. In either case, the use of finite-sized elements means a relatively simpler implementation.

[0156] Iterative phase recovery works well in noisy images. Images severely damaged by noise can be recovered because they appear within the noise. Further noise reduction can be used to generate clear images.

[0157] Coherent imaging of distant objects can be performed by iterative phase retrieval. In this case, the input image (wave) may have already been Fourier transformed and compressed due to coherent wave propagation. As a result, another Fourier transform generates decompressed image (wave) information. The rest of the system is the same as that previously used by the inventors in conjunction with the iterative mask and phase retrieval processes.

[0158] When the input consists only of an amplitude image, the phase is zero at each input point. As a result, excellent results can be achieved without a zero-based boundary region surrounding the input window.

[0159] The performance of the iterative phase recovery method is greatly improved by using the claimed method and system to reduce computation time, reduce the number of masks, reduce the number of iterations, improve reconstruction quality, and improve ease of implementation, by using (1) a unity mask with one or more bipolar binary masks having elements equal to 1 and -1, or (2) a unity mask with one or more phase masks, or (3) a unity mask with one or more pairs of masks having binary amplitudes of 0 and 1 where the masks in the pair are complementary to each other in relation to amplitude, or (4) one or more pairs of complementary masks having binary amplitudes of 0 and 1 without requiring a unity mask. In all cases, it is possible to use an outer boundary filled with zeros. For example, using a boundary by doubling the mask size and filling the outer boundary of the mask with zeros can improve the results. The use of any of these combinations of specially selected masks can improve reconstruction quality and simplify implementation.

[0160] Potential Claims Various embodiments of the present invention may feature the potential claims enumerated in the paragraphs following this paragraph (and prior to the actual claims provided at the end of this application). These potential claims form part of the description provided for this application. Accordingly, the subject matter of the following potential claims may be presented as actual claims in subsequent proceedings relating to this application or any application claiming priority based thereon. Such inclusion of potential claims should not be construed as meaning that the actual claims do not cover the subject matter of the potential claims. Accordingly, a decision not to present these potential claims in subsequent proceedings should not be construed as a donation of subject matter to the public.

[0161] Without limitation, potential subject matter that can be claimed (preceded by the letter "P" to avoid confusion with the actual claims attached) includes:

[0162] P1. A method for recovering phase information from an array of points, wherein each point has an amplitude, and the method is A step of providing at least one conversion unit having an input and a spectral output, and at least two masks, one of which is a unity mask, wherein each of the at least two masks is configured to operate on the input of at least one conversion unit. The steps include applying the input separately to each of at least two masks in order to generate a modified input from each of the masks, The process involves performing a generalized Fourier transform on each modified input to generate a transformed modified input using at least one transformation unit, To generate a phasorgram, the steps include recording the amplitude values ​​in the array of points for each transformed and modified input, To form multiple complex phasorgrams, the steps involve associating phase values ​​with each point on each phasorgram, The steps include iteratively processing multiple complex phasorgrams until convergence is achieved in order to generate a totagram that constitutes a reconstructed input having amplitude and phase information, It has.

[0163] The method is P2.P1, where the array of points is a coherent light wave.

[0164] In the P3.P1 method, the array of points is digital data.

[0165] P4. The method of P1 or P3, wherein at least one transformation unit is a generalized Fourier transform process operating on a digital processor.

[0166] P5. The method of P1, P3, or P4, wherein at least two masks are implemented on a digital processor.

[0167] P6. The method of P1 or P2, wherein at least one conversion unit is a lens system.

[0168] P7. The method of P1, P2, or P6, wherein the mask is a physical space mask, and A step of switching from one of at least two physical space masks to another of at least two physical space masks so that the input is received sequentially and individually by each of at least two physical space masks, The steps include: splitting the input wave so that it is received individually and in parallel by at least two physical space masks, It further has at least one of the following.

[0169] P8. The method of P1, P2, P6, or P7, wherein at least two masks have a physical spatial mask that is implemented in real time by an optical device including any of a spatial optical modulator and a micromirror array.

[0170] P9. The method of P1, P2, P6, P7, or P8, wherein recording is performed by an intensity sensor.

[0171] P10. The methods of P1, P2, P6, P7, P8, or P9, further comprising the steps of (1) performing linear phase modulation on an input wave before passing the input wave through each of at least two physical space masks several times, or (2) spatially moving an intensity sensor after passing the input wave through each of at least two physical space masks several times to generate super-resolution amplitude and phase information of the input wave.

[0172] A method of P11.P1, P2, P6, P7, P8, P9, or P10, further comprising the step of performing a generalized Fourier transform on the input through a lens before separately applying the input to each of at least two masks.

[0173] A method of any one of P12.P1 to P11, wherein each mask includes an outer boundary that sets the amplitude of points that coincide with the outer boundary to zero.

[0174] A method of any one of P13.P1 to P12, wherein at least two masks are composed of a unity mask and a complex phase mask.

[0175] A method of P14.P13, wherein the complex phase mask has a bipolar binary mask.

[0176] A method of any one of the potential claims of P15.P1 to P12, wherein at least two masks are composed of a unity mask and a pair of masks, and the masks within the pair are complementary to each other in relation to an amplitude equal to 0 or 1.

[0177] A method of P16.P15, wherein the pair of masks is a complementary unipolar binary mask.

[0178] A method of P17.P15, wherein the pair of masks includes a unity element having a phase factor.

[0179] A method of any one of P18.P1 to P17, further comprising the step of using a totagram to generate a representation of the information embedded within the reconstructed amplitude and phase of the input after completion.

[0180] A method of any one of P19.P1 to P18, wherein the step of iteratively processing a plurality of complex phasograms (a) The step of processing multiple complex phasograms to obtain a single estimate of the input by performing an inverse generalized Fourier transform on the complex phasogram and averaging the complex information at each point of the input plane, and possibly several other further optimization steps, (b) A step of passing a single estimate of the input through a process that replicates each of the masks in order to obtain multiple intermediate arrays, (c) The steps of performing a generalized fast Fourier transform on each of the intermediate arrays to generate another set of complex phasograms, and replacing the amplitude values ​​at each point in the transformed intermediate array with the corresponding recorded amplitude values, and possibly several other further optimization steps, (d) A step of repeating step (a) for several other complex phasograms followed by steps (b) and (c) until convergence is achieved, wherein upon completion the single estimate of the input is a totagram, It has.

[0181] The method described on pages 20 and 19 is such that convergence is determined by (1) when the squared difference between consecutive single estimates reaches a predetermined threshold and (2) when a given number of iterations of step (a) have been completed.

[0182] P21. A method for recovering phase information from an array of points, wherein each point has an amplitude, and the method is A step of providing at least two masks, each comprising at least one pair of masks that are binary in amplitude, wherein the masks in the pair are complementary to each other in relation to amplitude, and each of the at least two masks is configured to operate on the input of at least one conversion unit. The steps include applying the input separately to each of at least two masks in order to generate a modified input from each of the masks, The process involves performing a generalized Fourier transform on each modified input to generate a transformed modified input using at least one transformation unit, To generate a phasorgram, the steps include recording the amplitude values ​​in the array of points for each transformed and modified input, The process involves associating phase values ​​with each point on each phasorgram to form multiple complex phasorgrams, as well as possibly several other further optimization steps. The steps include iteratively processing multiple complex phasorgrams until convergence is achieved in order to generate a totagram that constitutes a reconstructed input having amplitude and phase information, It has.

[0183] The method described on pages P22 and P21 is such that the array of points is a coherent light wave.

[0184] The method described on pages P23 and P21 is used, where the array of points is digital data.

[0185] The method of P24, P21, or P23, wherein at least one transformation unit is a generalized Fourier transform process operating on a digital processor.

[0186] The method described in P25.P21, P23, or P24, wherein at least two masks are implemented on a digital processor.

[0187] The method of P26, P21, or P22, wherein at least one conversion unit is a lens system.

[0188] The method of P27, P21, P22, or P26, wherein the mask is a physical space mask, and A step of switching from one of at least two physical spatial masks to another of the at least two physical spatial masks such that the input is successively received individually by each of the at least two physical spatial masks, A step of splitting an input wave such that the input wave is received individually and in parallel by each of the at least two physical spatial masks, further having at least one of.

[0189] A method of P28.P21, P22, P26, or P27, wherein at least two masks have physical spatial masks implemented in real time by an optical device including any of a spatial light modulator and a micromirror array.

[0190] A method of P29.P21, P22, P26, P27, P28, or P29, wherein the recording is performed by an intensity sensor.

[0191] A method of P30.P21, P22, P26, P27, P28, or P29, further having (1) a step of performing a linear phase modulation on the input wave before passing the input wave through each of the at least two physical spatial masks several times or (2) a step of spatially moving an intensity sensor after passing the input wave through each of the at least two physical spatial masks several times to generate super-resolution amplitude and phase information of the input wave.

[0192] A method of P31.P21, P22, P26, P27, P28, P29, or P30, further having a step of performing a generalized Fourier transform on the input through a lens before applying the input separately to each of the at least two masks.

[0193] A method of P32.P1 to any one of P31, wherein each mask includes an outer boundary that sets the phase and amplitude of points that coincide with the outer boundary to zero.

[0194] The method described in any one of the sections on pages P33, P21, and P32, wherein at least two masks consist of a pair of complementary unipolar binary masks.

[0195] The method described in any one of the sections P34, P21-P32, wherein the pair of masks includes a unity element having a phase factor.

[0196] The method described in any one of the sections on pages P35, P21-P32, requires that at least two masks consist of two pairs of complementary unipolar binary masks.

[0197] A method according to any one of the items in P36.P1 to P36, further comprising the step of using a totagram to generate a representation of the information embedded within the reconstructed amplitude and phase of the input after completion.

[0198] The method described in any one of the items P1 to P36, wherein the step of iteratively processing multiple complex phasograms is: (a) The step of processing multiple complex phasograms to obtain a simple estimate of the input by performing an inverse generalized Fourier transform on the complex phasogram and averaging the complex information at each point in the corresponding locations, and possibly several other further optimization steps, (b) A step of passing a single estimate of the input through a process that replicates each of the masks in order to obtain multiple intermediate arrays, (c) The steps of performing a generalized fast Fourier transform on each of the intermediate arrays to generate another set of complex phasograms, and replacing the amplitude values ​​at each point in the transformed intermediate array with the corresponding recorded amplitude values, and possibly several other further optimization steps, (d) A step in which step (a) is repeated for several other complex phasograms followed by steps (b) and (c) until convergence is achieved, wherein upon completion, the single estimate of the input is a totagram, It holds.

[0199] The method described on pages 38 and 37 is such that convergence is determined by (1) when the squared difference between consecutive single estimates reaches a predetermined threshold and (2) when a given number of iterations of step (a) have been completed.

[0200] P39. A system for recovering phase information from an input wave, An optical lens system having input and spectral output, At least two physical space masks, each of which is positioned at the input of an optical lens system to receive an input wave, At least two physical space masks are configured to modify the input wave independently, and the optical lens system implements a generalized Fourier transform on the independently modified wave to generate the transformed wave, To generate a phasogram, at least one sensor system configured to record the amplitude values ​​in an array of points of each transformed wave in the spectral plane, In order to form multiple complex phasorgrams, the phase values ​​are associated with each point on each phasorgram, and To generate a totagram that constitutes a reconstructed input wave with amplitude and phase information, multiple complex phasorgrams are processed iteratively until convergence is achieved. The configured digital processor, It holds.

[0201] The P40.P39 system further comprises a beam splitter configured to provide input waves to each of at least two physical space masks in parallel.

[0202] The system of P41.P39 further comprises a spatial light modulator configured to implement at least two physical spatial masks that switch from one of the masks to another of the at least two physical spatial masks, such that the input is received sequentially and individually by each of the at least two physical spatial masks.

[0203] The system of P42.P39 further comprises a micromirror array configured to implement at least two physical space masks, which switch from one mask to another of the at least two physical space masks, such that an input is received sequentially and individually by each of the at least two physical space masks.

[0204] In any one of the systems described in P43, P39, and P42, at least one sensor system is an intensity sensor.

[0205] The system described in P44, P39-P43, further comprises a second lens positioned to receive input waves on the way to at least two physical space masks.

[0206] A system according to any one of the items on pages P45, P39, and P44, wherein each mask includes an outer boundary that sets the phase and amplitude of the points that coincide with the outer boundary to zero.

[0207] A system according to any one of the items on pages P46, P39-P45, wherein at least two masks include a unity mask.

[0208] The system described on pages P47 and P46 consists of at least two masks, one being a unity mask and the other a complex phase mask.

[0209] The system is P48.P47, and the complex phase mask has a bipolar binary mask.

[0210] P49. A system according to any one of the latent claims P39 to P46, wherein at least two masks consist of a unity mask and a pair of masks, the masks in the pair being complementary to each other in relation to amplitude.

[0211] The system is P50.P49, and the mask pair is a complementary unipolar binary mask.

[0212] The system is as described on P51 and P49, where the mask pair includes a unity element with a phase factor.

[0213] P52. A system according to any one of the potential claims P39 to P45, wherein at least two masks have at least one pair of masks, and the masks in the pair are complementary to each other in relation to amplitude.

[0214] The system is P53.P52, and the mask pair is a complementary unipolar binary mask.

[0215] The P54.P53 system consists of at least two masks made up of a pair of complementary unipolar binary masks.

[0216] The system is as described in P55.P52, where the mask pair includes a unity element having a phase factor.

[0217] The system described in any one of the terms on pages P56, P39, and P55, wherein the step of iteratively processing multiple complex phasograms is: (a) Processing multiple complex phasograms to obtain a single estimate of the input waveform by performing an inverse generalized Fourier transform on the complex phasograms, thereby averaging the complex information at each point in the corresponding locations, and possibly through several other further optimization steps. (b) A step of passing a single estimate of the input through a process that replicates each of the masks in order to obtain multiple intermediate arrays, (c) The steps of performing a generalized fast Fourier transform on each of the intermediate arrays to generate another set of complex phasograms, and replacing the amplitude values ​​at each point in the transformed intermediate array with the corresponding recorded amplitude values, and possibly several other further optimization steps, (d) A step in which step (a) is repeated for several other complex phasograms followed by steps (b) and (c) until convergence is achieved, wherein upon completion, the single estimate of the input is a totagram, It holds.

[0218] The system described on pages 57 and 56 is characterized by convergence being determined by either (1) when the squared difference between consecutive single estimates reaches a predetermined threshold, or (2) when a given number of iterations of step (a) have been completed.

[0219] The method described on pages 58 and 15 is used, where the mask pair is a complementary bipolar binary mask.

[0220] The method described in any one of the sections on pages P59, P21 to P32, wherein at least two masks consist of a pair of complementary bipolar binary masks.

[0221] The system is P60.P49, and the mask pair is a complementary bipolar binary mask.

[0222] The embodiments of the present invention described above are for illustrative purposes only, and numerous modifications and variations will become apparent to those skilled in the art. All such modifications and variations should be construed as falling within the scope of the present invention as defined in any appended embodiment. The inventions disclosed herein include the following: [Aspect 1] A method for recovering phase information from an array of points, wherein each point has an amplitude, in the method, A step of providing at least one conversion unit having an input and a spectral output, and at least two masks, one of which is a unity mask, wherein each of the at least two masks is configured to operate on the input of the at least one conversion unit; The steps include applying the array of points in the input separately to each of the at least two masks in order to generate a modified input from each of the masks, The steps include: performing a generalized Fourier transform on each modified input using at least one of the transformation units to generate a transformed modified input from each modified input; To generate a phasorgram, the steps include recording the amplitude values ​​in the array of points for each transformed and modified input, To form multiple complex phasorgrams, the steps involve associating phase values ​​with each point on each phasorgram, The steps include iteratively processing the plurality of complex phasorgrams until convergence is achieved in order to generate a totagram that constitutes a reconstructed input having amplitude and phase information, A method of having. [Aspect 2] The method according to embodiment 1, wherein each mask includes an outer boundary, and the phase and amplitude of the points that coincide with the outer boundary are set to zero. [Aspect 3] The method according to embodiment 1, wherein the input is a wave, and the mask is a physical space mask, and further comprises the step of switching from one of the at least two physical space masks to another of the at least two physical space masks so that the input is received sequentially and individually in time by each of the at least two physical space masks. [Aspect 4] The method according to embodiment 1, further comprising the step of dividing the input wave so that the input is a wave and the mask is a physical space mask, and the input wave is received individually in parallel by each of the at least two physical space masks. [Aspect 5] The method according to embodiment 1, wherein the at least two masks consist of the unity mask and the complex phase mask. [Aspect 6] The method according to embodiment 5, wherein the complex phase mask is a bipolar binary mask. [Aspect 7] The method according to embodiment 1, wherein the at least two masks consist of a pair of a unity mask and a complementary unipolar binary mask. [Aspect 8] The method according to embodiment 1, wherein the input has a wave, and the at least two masks are physical spatial masks implemented in real time by an optical device including any of a spatial light modulator and a micromirror array. [Aspect 9] The method according to embodiment 8, further comprising the steps of (1) performing linear phase modulation on the input wave before passing the input wave through each of the at least two physical space masks several times, or (2) spatially moving the intensity sensor after passing the input wave through each of the at least two physical space masks several times. [Aspect 10] The method according to embodiment 8, wherein recording is performed by an intensity sensor, and the at least one conversion unit has a lens system. [Aspect 11] The method according to embodiment 1, further comprising the step of performing a generalized Fourier transform on the input before applying the input separately to each of the at least two masks. [Aspect 12] The method according to embodiment 1, further comprising the step of using the totagram to generate a representation of the information embedded within the reconstructed amplitude and phase of the input after completion. [Aspect 13] A system for recovering phase information from an input wave, An optical lens system having an input plane and a spectral output, One of these is a unity mask, which comprises at least two physical space masks, each of which is arranged in the input plane of the optical lens system to receive the input wave. The physical space masks are configured to modify the input wave separately, and the optical lens system implements a generalized Fourier transform for each of the separately modified waves in order to generate a wave converted from each of the separately modified waves. To generate a phasogram from each transformed wave, at least one sensor system configured to record the amplitude values ​​in an array of points for each transformed wave in the spectral plane, In order to form multiple complex phasorgrams, the phase values ​​are associated with each point on each phasorgram, and To generate a totagram that constitutes a reconstructed input wave having amplitude and phase information, the plurality of complex phasorgrams are processed iteratively until convergence is achieved. The configured digital processor, A system that has [Aspect 14] The system according to embodiment 13, wherein the at least two physical space masks consist of a unity mask and a bipolar binary mask. [Aspect 15] The system according to embodiment 13, wherein the at least two physical space masks consist of a pair of a unity mask and a complementary unipolar binary mask. [Aspect 16] The system according to embodiment 13, wherein each physical space mask is surrounded by an outer boundary that blocks the wave, thereby setting the phase and amplitude of the points coinciding with the boundary to zero. [Aspect 17] The step of iteratively processing the plurality of complex phasograms is: (a) Processing the multiple complex phasograms to obtain a single estimate of the input wave by performing an inverse generalized Fourier transform on the complex phasograms and averaging the complex information at each point in the corresponding location; (b) The step of passing the single estimate of the input wave through a process of replicating each of the physical space masks in order to obtain a plurality of intermediate arrays, (c) To generate a further complex phasorgram, a generalized fast Fourier transform is performed on each of the intermediate arrays, and the amplitude values ​​at each point in the transformed intermediate array are replaced with the corresponding recorded amplitude values. (d) A step of repeating step (a) for the other multiple complex phasograms followed by steps (b) and (c) until convergence is achieved, wherein upon completion, the single estimate of the input wave is the totagram, The system according to embodiment 13, having the following characteristics. [Aspect 18] The system according to embodiment 17, wherein convergence is determined by either (1) when the difference between a series of single estimates reaches a predetermined threshold, or (2) when a given number of iterations of step (a) have been completed. [Aspect 19] The system according to embodiment 13, wherein the at least two physical space masks are configured to switch from one of the at least two physical space masks to another of the at least two physical space masks, such that the input is received sequentially and individually by each of the at least two physical space masks. [Aspect 20] The system according to embodiment 13, further comprising a beam splitter configured to split the input wave so that it is received individually and in parallel by each of the at least two physical space masks. [Aspect 21] A method for recovering phase information from an array of points, wherein each point has an amplitude, in the method, A step of providing four masks, each including two pairs of at least one conversion unit having an input and a spectral output, wherein the masks in each pair are complementary to each other in relation to amplitude. The steps include applying the input separately to each of the four masks in order to generate a modified input from each of the masks, The steps include: performing a generalized Fourier transform on each modified input in order to generate a transformed modified input using the at least one transformation unit; To generate a phasorgram, the steps include recording the amplitude values ​​in the array of points for each transformed and modified input, To form multiple complex phasorgrams, the steps involve associating phase values ​​with each point on each phasorgram, The steps include iteratively processing the plurality of complex phasorgrams until convergence is achieved in order to generate a totagram that constitutes a reconstructed input having amplitude and phase information, A method of having. [Aspect 22] The method according to embodiment 21, wherein each mask includes an outer boundary, and the phase and amplitude of the points that coincide with the outer boundary are set to zero. [Aspect 23] Each pair of masks is a complementary unipolar binary mask, according to the method of embodiment 21.

Claims

1. A method for recovering phase information from an array of points, wherein each point has an amplitude, in the method, The steps include: applying separately the array of points in the input to each of at least two masks, each of which has an input and a spectral output, and one of which is a unity mask, so that each of the at least two masks generates a modified input; The steps include: an optical lens system having an input plane and a spectral output performing a generalized Fourier transform on each modified input in order to generate a modified input converted from each modified input by at least one of the conversion units; To generate a phasorgram, at least one sensor system records the amplitude values ​​in an array of points for each converted and modified input, To form multiple complex phasorgrams, the digital processor performs the steps of associating phase values ​​with each point on each phasorgram, To generate a totagram comprising a reconstructed input having amplitude and phase information, the digital processor iteratively processes the plurality of complex phasorgrams until convergence is achieved, It has, Each mask is surrounded by an opaque outer boundary, the outer boundary having its phase and amplitude set to zero at points that coincide with the outer boundary.

2. The method according to claim 1, wherein the input is a wave, and the mask is a physical space mask, and at least two physical space masks are configured to switch from one of the at least two physical space masks to another of the at least two physical space masks, such that the input is received sequentially and individually by each of the at least two physical space masks.

3. The method according to claim 1, wherein the input is a wave, and the mask is a physical space mask, and the beam splitter further comprises the step of splitting an input wave so that it is received individually in parallel by each of at least two physical space masks.

4. The method according to claim 1, wherein the at least two masks are comprised of the unity mask and the complex phase mask.

5. The method according to claim 4, wherein the complex phase mask is a bipolar binary mask.

6. The method according to claim 1, wherein the at least two masks consist of a pair of a unity mask and a complementary unipolar binary mask.

7. The method according to claim 1, wherein the input has a wave, and the at least two masks are physical spatial masks implemented in real time by an optical device including any of a spatial light modulator and a micromirror array.

8. The method according to claim 7, wherein the at least two physical space masks further comprises the steps of (1) performing linear phase modulation on the input wave before passing the input wave through each of the at least two physical space masks several times, or (2) spatially moving an intensity sensor after passing the input wave through each of the at least two physical space masks several times.

9. The method according to claim 7, wherein an intensity sensor performs recording, and the at least one conversion unit has a lens system.

10. The method according to claim 1, further comprising the step of performing a generalized Fourier transform on the input before applying the input separately to each of the at least two masks.

11. The method according to claim 1, further comprising the step of the digital processor using the totagram to generate a representation of the information embedded in the reconstructed amplitude and phase of the input after completion.

12. A system for recovering phase information from an input wave, An optical lens system having an input plane and a spectral output, At least two physical space masks, one of which is a unity mask, each of the at least two physical space masks is arranged in the input plane of the optical lens system to receive the input wave. The physical space masks are configured to modify the input wave separately, and the optical lens system implements a generalized Fourier transform for each of the separately modified waves in order to generate a wave converted from each of the separately modified waves. To generate a phasogram from each transformed wave, at least one sensor system configured to record the amplitude values ​​in an array of points for each transformed wave in the spectral plane, In order to form multiple complex phasorgrams, the phase values ​​are associated with each point on each phasorgram, and To generate a totagram that constitutes a reconstructed input wave having amplitude and phase information, the plurality of complex phasorgrams are processed iteratively until convergence is achieved. The configured digital processor, It has, A system in which each physical space mask is surrounded by an opaque outer boundary, and the phase and amplitude of the points that coincide with the outer boundary are set to zero.

13. The system according to claim 12, wherein the at least two physical spatial masks consist of a unity mask and a bipolar binary mask.

14. The system according to claim 12, wherein the at least two physical spatial masks consist of a pair of a unity mask and a complementary unipolar binary mask.

15. The step of iteratively processing the plurality of complex phasograms is: (a) Processing the multiple complex phasograms to obtain a single estimate of the input wave by performing an inverse generalized Fourier transform on the complex phasograms and averaging the complex information at each point in the corresponding location; (b) The step of passing the single estimate of the input wave through a process of replicating each of the physical space masks in order to obtain a plurality of intermediate arrays, (c) To generate a plurality of other complex phasograms, a generalized fast Fourier transform is performed on each of the intermediate arrays, and the amplitude values ​​at each point in the transformed intermediate array are replaced with the corresponding recorded amplitude values. (d) A step of repeating step (a) for the other plurality of complex phasograms followed by steps (b) and (c) until convergence is achieved, wherein upon completion, the single estimate of the input wave is the totagram, The system according to claim 12, having the following features.

16. The system according to claim 15, wherein convergence is determined by (1) when the difference between a series of single estimates reaches a predetermined threshold and (2) when a given number of iterations of step (a) have been completed, either of the above.

17. The system according to claim 12, wherein the at least two physical space masks are configured to switch from one of the at least two physical space masks to another of the at least two physical space masks, such that the input is received sequentially and individually by each of the at least two physical space masks.

18. The system according to claim 12, further comprising a beam splitter configured to split the input wave so that it is received individually and in parallel by each of the at least two physical space masks.

19. A method for recovering phase information from an array of points, wherein each point has an amplitude, in the method, The steps include applying the input separately to each of four masks, each of which includes at least one conversion unit having an input and a spectral output, and two pairs of masks, so that each of the four masks generates a modified input, The steps include: an optical lens system having an input plane and a spectral output performing a generalized Fourier transform on each modified input in order to generate a transformed modified input by the at least one transformation unit; To generate a phasorgram, at least one sensor system records the amplitude values ​​in an array of points for each converted and modified input, To form multiple complex phasorgrams, the digital processor performs the steps of associating phase values ​​with each point on each phasorgram, To generate a totagram comprising a reconstructed input having amplitude and phase information, the digital processor iteratively processes the plurality of complex phasorgrams until convergence is achieved, It has, Each mask is surrounded by an opaque outer boundary, the outer boundary having its phase and amplitude set to zero at points that coincide with the outer boundary.

20. The method according to claim 19, wherein each pair of masks includes a complementary unipolar binary mask.

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