Multiphoton quantum imaging using natural light

Abstract

A method for imaging an object illuminated with natural light including arranging an optical filter in an optical path of at least a portion of the natural light that is reflected and / or refracted or scattered from the object; arranging an optical detector in an optical path of light at least one of reflected from and / or transmitted through the optical filter; detecting, using the detector, individual photons for a time period to provide a times series of individual photon events; segmenting the time series into a plurality of bins; determining a number of detected photons within each bin to provide a time series of photon counts per bin; replacing and / or reconfiguring the optical filter a plurality of times followed by detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin; and forming an image of the object based on the plurality of times series.

Description

Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTMULTIPHOTON QUANTUM IMAGING USING NATURAL LIGHTCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The present patent application claims priority benefit to U. S. Provisional Patent Application No. 63 / 638,236 filed on April 24, 2024, the entire content of which is herein incorporated by reference.GOVERNMENT LICENSE RIGHTS

[0002] This invention was made with government support under 2225986 awarded by the United States National Science Foundation (NSF). The government has certain rights in the invention.BACKGROUND1. Technical Field

[0003] The presently claimed embodiments of the current invention relate to imaging systems, and methods of imaging, and more particularly to quantum imaging systems and methods of imaging using natural light.2. Discussion of Related Art

[0004] The use of nonclassical correlations of photons to produce optical images in a nonlocal fashion gave birth to the field of quantum imaging almost three decades ago. Interestingly, it was then discovered that exploiting the quantum properties of the light field enables improving the resolution of optical instruments beyond the diffraction limit. It was also shown that schemes for quantum imaging allow for the formation of images with sub-shot noise levels of precision. These features have been exploited to demonstrate the formation of few-photon images with high contrast. Furthermore, the compatibility of quantum imaging techniques with protocols for quantum cryptography have cast interest in the development of schemes for quantum-secured imaging. Despite the enormous potential of quantum imaging for microscopy, remote sensing, and astronomy, schemes for quantum imaging remain fragile against realistic conditions of lossVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTand noise. Unfortunately, these limitations render the realistic application of quantum imaging unfeasible.

[0005] Sharing similarities with other quantum technologies, existing techniques for quantum imaging rely on the use of nonclassical states of light. However, the brightness of available quantum light sources is generally low. For example, existing sources of nonclassical light allow for the preparation of few-photon states that exhibit fragile quantum correlations. This situation leads to common scenarios where environmental noise is typically larger than the signal of photons produced by processes of spontaneous parametric down-conversion or four-wave mixing. Unfortunately, it is not feasible to produce brighter quantum light sources to overcome these limitations. Moreover, losses and noise cannot be avoided in realistic scenarios. Thus, any robust protocol for quantum imaging must rely on ubiquitous natural sources of light, such as thermal light.

[0006] However, there remains a need to discern features from noise within an image using quantum imaging under natural light.SUMMARY OF THE DISCLOSURE

[0007] An aspect of the present invention is to provide a method for imaging an object illuminated with natural light. The method includes arranging an optical filter in an optical path of at least a portion of the natural light at least one of reflected, refracted or scattered from the object; arranging an optical detector in an optical path of light at least one of reflected from or transmitted through the optical filter; detecting, using the optical detector, individual photons for a measurement time period to provide a times series of individual photon events; segmenting the time series into a plurality of time bins; determining a number of detected photons within each time bin of the plurality of time bins to provide a time series of photon counts per time bin; at least one of replacing or reconfiguring the optical filter a plurality of times follow ed by corresponding of pluralities of the detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin: forming an image of the object based on the plurality of times series of photon counts per bin. The optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions. Each replaced or reconfigured optical filter has a two-dimensional structured pattern of at least tw o of reflecting, scattering, transmitting,Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTor refraction portions such that each the optical filter has a two-dimensional structured pattern that differs from all other two-dimensional structured patterns of the other the optical filters.

[0008] Another aspect of the present invention is to provide a system for imaging an object illuminated with natural light. The system includes an optical filter configured arranged to be in an optical path of at least a portion of the natural light at least one of reflected, refracted or scattered from the object; an optical detector arranged to be in an optical path of light at least one of reflected from or transmitted through the optical filter; and an image processor configured to communicate with the optical detector. The optical detector is configured to detect individual photons for a measurement time period to provide a times series of individual photon events and to communicate the time series to the image processor. The image processor is configured to segment the time series into a plurality of time bins, and determine a number of detected photons within each time bin of the plurality' of time bins to provide a time series of photon counts per time bin. The optical filter is configured to be at least one of replaceable or reconfigurable a plurality of times to be followed by corresponding of pluralities of the detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin. The image processor is configured to form an image of the object based on the plurality of times series of photon counts per bin. The optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions, and each replaced or reconfigured optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions such that each the optical filter has a two-dimensional structured pattern that differs from all other two-dimensional structured patterns of the other the optical filters.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] The present disclosure, as well as the methods of operation and functions of the related elements of structure and the combination of parts and economies of manufacture, will become more apparent upon consideration of the following description and the appended claims with reference to the accompanying drawings, all of which form a part of this specification, wherein like reference numerals designate corresponding parts in the various figures. It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only and are not intended as a definition of the limits of the invention. All references cited anywhereVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTin this specification, including the Background and Detailed Description sections, are incorporated by reference as if each had been individually incorporated.

[0010] FIG. 1 is a schematic diagram of a quantum imaging system for performing quantum imaging under natural or thermal light to extract quantum image features from classical noisy images, according to an embodiment of the present invention;

[0011] FIG. 2 is a schematic diagram of a quantum imaging system used for multiphoton quantum imaging using natural sources of light, according to another embodiment of the present invention;

[0012] FIG. 3 shows ajoint photon-number distribution of the natural source, according to an embodiment of the present invention;

[0013] FIG. 4 shows the image reconstructed or the projection of thermal light scattered by a target object into its constituent multi photon subsystems and the formation of high-contrast quantum images, according to an embodiment of the present invention;

[0014] FIG. 5A shows the extraction of quantum images from a classical compressive sensing (CS) reconstruction, using a single-pixel camera for classical thermal light, according to an embodiment of the present invention;

[0015] FIG. 5B is a plot of signal-to-noise ratio (SNR) versus a number of photons showing improvement in the signal-to-noise ratio (SNR) with the number of photons, according to an embodiment of the present invention;

[0016] FIG. 6A show a characterization of noise in bunched and anti-bunched extracted light, according to an embodiment of the present invention;

[0017] FIG. 6B shows various image panels depicting the effect of noise in a bunched four-photon system, zero-photon system, eight-photon system, and seven-photon system, according to an embodiment of the present invention;

[0018] FIG. 7 shows photon-subtracted multiphoton quantum imaging according to an embodiment of the present invention; andVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCT

[0019] FIG. 8 is a graph of the signal-to-noise (SNR) ratio versus the subtracted photon number showing the performance of photon-subtracted multiphoton quantum imaging.DETAILED DESCRIPTION

[0020] Some embodiments of the current invention are discussed in detail below. In describing embodiments, specific terminology is employed for the sake of clarity. However, the invention is not intended to be limited to the specific terminology so selected. A person skilled in the relevant art will recognize that other equivalent components can be employed, and other methods developed, without departing from the broad concepts of the present invention. All references cited anywhere in this specification are incorporated by reference as if each had been individually incorporated.

[0021] As used herein, the term “light” is intended to have a broad meaning to regions of the electromagnetic spectrum that are both visible and not visible to the human eye. For example, the term light is intended to include, but is not limited to, visible light, infrared light (IR) and ultraviolet light (UV).

[0022] The term “natural light” is used herein to mean photons emitted by spontaneous emission (e.g., light from sunlight, light from a fluorescent bulb, light from light emitting diodes, light from a fluorescent molecule, etc.) as opposed to stimulated coherent emission (e.g.. laser beam of light). However, as it can be appreciated, the coherent light from a laser can also be rendered incoherent by introducing phase shifting to simulate “natural light” from a laser.

[0023] FIG. 1 is a schematic diagram of a quantum imaging system 100 for performing quantum imaging under natural or thermal light to extract quantum image features from classical noisy images, according to an embodiment of the present invention. Specifically, the quantum imaging system 100 isolates multiphoton subsystems of thermal light sources that can dramatically improve the signal-to-noise ratio of imaging instruments. This robust protocol for quantum imaging is demonstrated through the implementation of a novel single-pixel camera with photon-number resolving capabilities. The quantum imaging system enables the extraction of information from the vacuum-fluctuation components of thermal light sources to produce quantum images with improved contrast. This technique shows a remarkable exponential improvement in the contrast of quantum images. We also demonstrate the possibility of usingVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTcorrelated multiphoton subsystems to form high-contrast quantum images from images in which the background noise is comparable to the signal of thermal light sources. These surprising results can only be explained using quantum physics. The present quantum imaging system has the potential of combining natural light with nonclassical detection schemes for the development of robust quantum technologies that opens a new paradigm in the field of quantum imaging.

[0024] As shown in FIG. 1, the quantum imaging system 100 includes a digital micromirror devices (DMD) 102, an optical detector or photodetector (e.g., a photodiode) 104, a plurality of lenses 106 and 108, and a computing device 110. A DMD 102 has on its surface several hundred thousand microscopic mirrors arranged in a rectangular array. The mirrors and supporting mechanical structures are constructed using surface micromachining. The mirrors can be individually rotated (e g., ±10-12°), to an on or off state. In the on state, light is reflected into a lens making the pixel appear bright on a screen. In the off state, the light is directed elsewhere (usually onto a heatsink), making the pixel appear dark on the screen. Although a DMD 102 is used in the quantum imaging system 100, the imaging system 100 is not limited to DMD 102 but can include any optical filter that can reflect, refract or scatter a portion of the of the natural light reflected by the object 120.

[0025] Light from an object 120 illuminated with natural light 101 is collected by a first lens 106 and transmitted towards the DMD 102. Some mirrors within the matrix of mirrors of the DMD 102 reflect light towards a second lens 108 in a pattern 112. Various matrix patterns 112 are generated and varied through time by turning ”ON" and “OFF” some mirrors within the DMD 102. For example, in the matrix patterns 112, the white areas in the matrix represent mirrors within the DMD 102 that are oriented to reflect light towards the second lens 108 while the dark areas in the matrix represent mirrors within the DMD 102 that do not direct light towards the second lens 108 (for example reflect light away from the lens 108). The second lens 108 focusses the light reflected by the mirrors within the DMD 102 towards the photodetector 104. The photodetector 104 detects a number of photons received (photon number resolving or PNR).

[0026] A signal representing the number of photons is sent to a computing device 110 (image processor) in communication with the optical detector or photodetector 104. The optical detector or photodetector 104 is arranged to be in an optical path of light at least one of reflectedVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTfrom or transmitted through the DMD 102 (optical filter). The optical detector or photodetector 104 is configured to detect individual photons for a measurement time period to provide a times series of individual photon events and to communicate the time series to the computing device 110 (image processor).

[0027] The computing device 110 (image processor) is configured to segment the time series into a plurality of time bins, and determine a number of detected photons within each time bin of the plurality of time bins to provide a time series of photon counts per time bin. The computing device 110 (image processor) is configured to form an image 122 of the object 120 based on the plurality of times series of photon counts per bin. In an embodiment, the computing device 110 reconstructs the image 122 of the object 120 using a compressive sensing process. The computing device 110 reconstructs an image 122 of the object 120 based on the changing matrix pattern of mirrors within the DMD 102 and based on the light from the object 120. In an embodiment, the computing device 110 controls the DMD 102 and varies the matrix pattern of mirrors 112 of the DMD 102 over time.

[0028] The compressive sensing process includes using the following expression to reconstruct the image 122 of the object 120.II*"

[0029] where O’ represents the image 122, A represents the matrix patern of mirrors 112 of the DMD 102, Y represents a vector corresponding to a time series of photon counts per time from object 120.

[0030] FIG. 2 is a schematic diagram of a quantum imaging system 200 used for multiphoton quantum imaging using natural sources of light 202, according to another embodiment of the present invention. Examples of natural sources of light 202 are provided in the above paragraphs. As show n in FIG. 2, a portion of light from the natural source of light 202 is reflected by a target object 204. Thermal light field 205 reflected off the target object 204 is projected into a series of random binary matrices 206 of a DMD 208 and then coupled via a lens 210 into a single-mode fiber (SMF) 212. The binary sensing matrices 206 are displayed andVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTvaried within the DMD 208. In an embodiment, the thermal light field 205 from the object 204 is coupled into the SMF 212 and then split by a 50:50 fiber coupler 214 and measured by two photodetectors 216A and 216B (e.g., PNR detectors). Two optical fibers 218A and 218B are coupled to a fiber coupler 214 and are configured to direct each portion of light to corresponding optical detector or photodetectors 216A and 216B. Although two photodetectors 216A and 216B are shown used in the present system 200, one or more photodetectors can be used. For example, a single photodetector can be used, as shown in FIG. 1. Signals corresponding to a number of photons detected by each of the photodetectors 216A and 216B are received by a computing device 220 (image processor).

[0031] Although DMD 208 is shown used in the quantum imaging system 200, the imaging system 300 is not limited to DMD 208 but can include any optical filter that can reflect, refract or scatter a portion of the of the natural light reflected by the object 120.

[0032] The optical detectors or photodetectors 216A and 216B are configured to detect individual photons for a measurement time period to provide a times series of individual photon events and to communicate the time series to the computing device 220 (image processor).

[0033] The computing device 220, similar to the computing device 110 shown in FIG. 1, reconstructs an image 222 of the object 202 based on the changing matrix pattern of mirrors 206 within the DMD 208 and based light from the object 204. In an embodiment, the computing device 220 controls the DMD 208 and varies the matrix pattern of mirrors 206 of the DMD 208 over time. The computing device 220 (image processor) is configured to segment the time series into a plurality' of time bins, and determine a number of detected photons w ithin each time bin of the plurality of time bins to provide a time series of photon counts per time bin.

[0034] The thermal photons from the DMD 208 are collected by the single-mode fiber (SMF) 212 and then probabilistically split by the fiber coupler 214. The photons in each fiber 218A and 218B are measured by two photon-number-resolving (PNR) detectors 216A and 216B. The random sensing matrices of mirrors 206 displayed on the DMD 208 are used to implement a single-pixel camera. Therefore, the DMD 208 (generally an optical filter) is configured to be at least one of replaceable or reconfigurable a plurality' of times to be follow edVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTby corresponding of pluralities of detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin.

[0035] The detectors 216A and 216B detect a number of photons and the computing device 220 counts the number of photons. The photon counting scheme enables to project the coupled thermal light field 205 into its constituent multiphoton subsystems. In the embodiment shown in FIG. 2, the optical fiber 218B corresponds to the multiphoton detection signal arm and the optical fiber and 218A corresponds to the conditional multiphoton detection arm. The conditional multiphoton detection arm is used as a threshold setting for counting photons in the multiphoton detection signal.

[0036] In an embodiment, a percentage of 4096 (i.e., 64x64) unique random matrices, for example, are displayed on the DMD 208, with the measurement time for each matrix fixed at one second. In other embodiments, different size matrices can also be used (e.g., 2x2, 4x4. 8x8, 16x16, 32x32,..., nxn, where n is an integer multiple of 2). For example, the reconstructions may use 1025 projective measurements, which corresponds to 25% (or any desired percentage) of the 4096 measurements that our compressive sensing (CS) algorithm can use. Generally, the measurement time bins have a width determined according to a coherence length of the natural light 202. For example, the width is determined to be substantially equal to a coherence length of the natural light 202.

[0037] The images are obtained with 12% of the measurements, which means 512 binary matrices. A complete realization of the experiment using all 4096 frames may take approximately 10 hours. However, this time can be drastically reduced by pre-programming all matrices in the DMD. To reconstruct the image, we apply the method described in the following paragraphs. A pseudo-thermal light source 202 is used. Due to the long coherence time of the pseudo-thermal light source, each one-second measurement is binned into Iqs intervals. The number of detections in each bin is counted to provide a photon-number resolving event. For each event, the photon-counts in the two arms asand n2is determined. To achieve / V-photon subtraction, we isolate the events where n2= N. In other words, we filter n1by only considering the events where n2= N in the second arm (conditional arm 218A). The reconstruction process is then performed with this conditional dataset (based on the conditional arm 218A) to obtain an enhanced image quality. Conversely, for post-selection on n and mVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTphotons, we compute the probability that n = n and n2= m. Then, we perform image reconstruction using this probability to obtain an improvement in image quality.

[0038] FIG. 3 shows a joint photon-number distribution 300 of the natural source 202, according to an embodiment of the present invention. The x-axis represents the conditional arm photon number, the y-axis represents the signal arm photon number, and the z-axis represents a joint probability photon-number distribution. The classical nature of the source 202 is certified by the degree of second-order coherence g(2)which is equal to 1.97 in the experiment. Each element in this joint probability distribution represents a multiphoton subsystem can be isolated through the implementation of projective measurements. This measurement approach lies at the heart of the protocol for multiphoton quantum imaging. The x-axis represents a number of photons detected from multiphoton detection signal arm (optical fiber 218B). The y-axis represents the number of photons detected from conditional multiphoton detection arm (optical fiber 218A). The z-axis (vertical axis) represents a probability distribution of the number of photons. As shown, in FIG. 3, the probability of the distribution of photons is the highest at zero photon in the signal arm and zero photon in the conditional arm which corresponds to noise.

[0039] FIG. 4 shows the image 222 reconstructed or the projection of thermal light scattered by a target object 204 into its constituent multi photon subsystems and the formation of high-contrast quantum images 402, according to an embodiment of the present invention. The present method enables extracting quantum images of the target object 204, even when environmental noise prevents the formation of its classical image through intensity measurements.

[0040] As shown in FIG. 4, our ability to measure the multiphoton subsystems, represented by the elements of the joint photon-number distribution of the thermal source, enables us to demonstrate quantum imaging even in situations in which noise prevents the formation of the classical image of the object. Specifically, the environmental noise shown in FIG. 4 forbids the imaging of the character h. However, the projection of the thermal light field into its vacuum component reveals the presence of the object. Remarkably, the projection into larger multiphoton subsystems enables the extraction of quantum images of the object that was not visible in the classical image.Venable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCT

[0041] In the following paragraphs, the quantum multiphoton processes that make this effect possible will be described in detail. For the sake of simplicity, we assume the uniform illumination of the object s0by a thermal light field. As depicted in FIG. 2, the projection of the object 204 into random sensing matrices 206, represented by the covector Qt. enables us to discretize the object into X pixels. The label t indexes the different sensing matrices. All such matrices can be represented by the M x X matrix Q =xQt, where M is the number of sensing matrix configurations. Then, each filtering configuration results in a thermal state with a mean photon number given by nt= Qt• s0. The multiphoton state after the fiber coupler can be written in terms of the Glauber-Sudarshan P function as equation (1).1 l«l2d2a — — ent (1) t=i X|acos (0), icrsin (0))(crcos (0), icrsin (0)|a b(1)

[0042] The indices a and b denote the output modes of the fiber coupler. Furthermore, the parameter 0 describes the splitting ratio between the two output ports 218A and 218B.

[0043] FIG. 5A shows the extraction of quantum images from a classical CS reconstruction, according to an embodiment of the present invention. The reconstructed image using our singlepixel camera for classical thermal light is shown in panel a of FIG. 5A. In this case, environmental noise is higher than the signal and consequently the reconstructed image shows a low contrast that prevents the observation the object. Surprisingly, the projection of the light field into its vacuum component boosts the contrast of the image, this is reported in panel b of FIG. 5A. Naturally, the formation of this image cannot be understood using classical optics. As demonstrated in c, the projection of the thermal source of light into three-photon events enables the extraction of a quantum image with an improved signal-to-noise ratio (SNR). Remarkably, the projection of the detected thermal field into seven-particle subsystems leads to the formation of the high-contrast quantum image in panel d of FIG. 5 A.

[0044] FIG. 5B is a plot of signal-to-noise ratio (SNR) versus a number of photons showing improvement in the signal-to-noise ratio (SNR) with the number of photons, according to an embodiment of the present invention. As reported in FIG. 5B, and in agreement with Equation (3), the improvement in the SNR is exponential with the number of projected photons. TheseVenable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTresults were obtained using 25% of the total number of measurements that can be used in our CS algorithm. Furthermore, the mean photon number ntof the thermal light source is 0.8. Since we extracted the quantum images from a classical CS reconstruction, all of them possessed the same acquisition time, which is 1 second. The error bars represent the standard deviation of the SNR of 15 different reconstructions, where each reconstructions use millions of PNR measurements.

[0045] Next, we describe the signal-to-noise ratio (SNR) and how this quantity is modified by projecting the thermal field into its constituent multiphoton subsystems. To account for noise, we consider photocounting with quantum efficienciesa / band noise counts va / b. Specifically, the joint photon-number distribution reported in FIG. 3, can be mathematically described as © / (Pana+ va)11, I: - j - e 0 ml t=l '(2) where na / bis the photon number operator, and: •: represents the normal ordering prescription. We write the tthcomponent of this vector as PQit(n, m). Additionally, when there is no signalv<f and only noise is measured, we will have the probability distribution pn,i(k) = e-vvik / k! in eacharm, where i = a, b. The joint probability distribution in this case is then given by pn(k, l) = pn,a(k)pn,b(l)

[0046] The two-mode multiphoton system described by Equation (2) enables two schemes for projective measurements that lead to different scaling factors for the SNR of quantum images. First, we project one of the arms into a particular multiphoton subsystem. In other words, we ignore arm b and implement a photon-number-projective measurement in arm a. For such post-selection on a multiphoton subsystem with N photons, the SNR scales as shown in following equation (3).Zm=0 PQ(N, m) pQ(N)SNR post (3)Pn,aW Pn.aW)Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCT

[0047] FIGS. 6A and 6B show a characterization of noise in bunched and anti-bunched extracted light, according to an embodiment of the present invention. As shown in FIG. 6A. the conditional detection of a thermal beam enables us to extract bunched and anti-bunched multiphoton light with different levels of noise. These results are obtained by using two detectors 216A and 216B as depicted in FIG. 2. Multi photon subsystems with nonclassical subshot noise properties are characterized by a g̃(2)(0) below one, whereas classical bunched multiphoton light exhibits a g̃(2)(0) larger than one. The noise is localized in specific multi photon subsystems.

[0048] This feature becomes evident if we consider the noise in the bunched four-photon system reported in FIG. 6B. FIG. 6B shows various image panels depicting the effect of noise in a bunched four-photon system, zero-photon system, eight-photon system, and seven-photon system, according to an embodiment of the present invention. The noise in this multiphoton subsystem forbids the observation of the object, as shown in panel b of FIG. 6B. Notably, the correlated measurement of the vacuum component of the thermal field leads to the formation of the image in panel c of FIG. 6B. The contrast of this image can be further improved by measuring the bunched eight-photon system reported in panel d of FIG. 6B. Notably, for the first time, the present method for imaging enables exploiting the quantum features of anti-bunched multiphoton subsystems to produce quantum images with attenuated levels of noise. As demonstrated in panel e of FIG. 6B, this can be achieved by projecting the classical thermal light beam into a quantum anti-bunched seven-photon subsystem. These results were obtained using 25% of the total number of measurements that can be used in our CS algorithm.

[0049] Remarkably, this expression follows an exponentially increasing trend with respect to N, meaning that postselection can significantly reduce the noise of a quantum image.

[0050] The second approach relies on the subtraction of N photons from the thermal multiphoton system in Equation (2). This procedure entails measuring photon events in arm a conditioned on the detection of N photons in arm b. Using Equation (2), the intensity in arm a is then given by ⟨n̂a⟩N= ⊕Mt=0(∑∞k=0kpQ,t(k,N)) / (∑∞k=0pQ,t(k,N)). Additionally, the photon-subtracted noise can be written as {na)N0= ®^-o(^k-okPn(.k’ / (^k-oPn(.k’ - This scheme leads to the following expression for the SNR in equation (4).Venable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCT⟨n̂a⟩N / ⟨n̂a⟩N,0The quantum enhancement for the SNR in this case is linearly increasing with respect to N. Therefore, photonsubtraction is also an effective means for noise-reduction.

[0051] The series of spatial projective measurements described by the vector Qtenables implementing a single-pixel camera with photon-number resolving capabilities via compressive sensing (CS). This technique permits the reconstruction of multiphoton quantum images described by s' via the minimization of the following quantity with respect to s' in equation (5).X∑Xi=0‖∇s'i‖l+ (μ / 2)‖Qs' − ⟨n̂⟩‖l1 = 0

[0052] As described above, ⟨n̂⟩ could be either p⃗Q(N) or ⟨n̂a⟩N. Moreover, the 1- and 2-norm are denoted by ‖·‖land ‖·‖l, respectively. The discrete gradient operator is described by V, and the penalty factor by p. The maximum number of measurements that our CS algorithm can use is equal to the total number of pixels in the target image. In this case, 4096 measurements, for example. However, by optimizing this algorithm, we can reduce the total number of measurements necessary to reliably reconstruct the image.

[0053] We now discuss the experimental process of quantum image extraction from classical images. This was implemented using one PNR detector. In panel a of FIG. 5A, we show the CS reconstruction of a classical image for a situation in which environmental noise is comparable to the signal. In this case, the level of noise forbids the observation of the object. The mean photon number ntof the thermal light source is 0.8. Surprisingly, the projection of the thermal signal into its vacuum component reveals the presence of the object. As such, the quantum image in panel b of FIG. 5A is formed by the vacuum-fluctuation component of the electromagnetic field and cannot be explained using classical physics. This nonclasical reconstruction, obtained from the measurement of vacuum events, demonstrates that the process of projecting the thermal light signal into one of its constituent quantum subsystems, such as the vacuum, modifies the SNR as established by Equation (3). As suggested by the reconstruction in panel c of FIG. 5 A, the postselection on higher multiphoton events leads to quantum images with an improved contrast. Interestingly, the projection of the thermal light signal into seven-photon subsystems leads to aVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTdramatic improvement of the contrast of the image. This effect becomes evident in the quantum image shown in panel d of FIG. 5 A. Remarkably, the exponential growth of the SNR with the number of projected multiphoton subsystems is summarized in FIG. 5B. These results demonstrate that our single-pixel camera with PNR capabilities enables the extraction of quantum multiphoton images with unprecedented degrees of contrast.

[0054] To gain a deeper insight into the unique features of our technique, it is essential to delve into the noise present within the measured thermal light field. As depicted in FIG. 2, this task requires the use of two correlated detectors 216A and 216B. We describe these measurements through the multiphoton degree of second-order coherence for the subsystems that form the classical light beam. This quantity is defined as g̃(2)(n, m) = pQ,t(n,m) / [(∑∞M=0pQ,t(n,M))(∑∞N=0pQ,t(N,m))] and reported in FIG. 6A. This measurement uncovers the presence of different kinds of light, characterized by distinct levels of noise and correlations, to perform multiphoton imaging. Specifically, multiphoton subsystems exhibiting a g̃(2)(n, m) below one correspond to nonclassical anti-bunched light, whereas those with a g^\n, m) above one denote classical bunched light. It is interesting to note that noise is localized in specific multiphoton subsystems reported in FIG. 6A. Specifically, the noise associated with the bunched four-photon system in panel b of FIG. 6B hinders the observation of the object. However, as demonstrated in panel c of FIG. 6B, the correlated measurement of the vacuum component of the thermal light field reveals an image with attenuated noise. Notably, the contrast of this image can be further improved through the measurement of the strongly correlated eight-photon system in panel d of FIG. 6B. Remarkably, our technique enables the extraction of nonclassical antibunched light from classical thermal beams to achieve high-contrast quantum imaging. This is reported in panel e of FIG. 6B. It is worth noting that while classical imaging using photon counting has been extensively investigated in the last decades all of the observed features can be classically described. Furthermore, quantum schemes for imaging have been limited to two-photon dynamics. However, our scheme for multiphoton imaging enables the implementation of either classical or quantum imaging from classical light sources.

[0055] While the projection of thermal light into its constituent multi photon subsystems enables the extraction of quantum images with high contrast, it is also possible to correlate photon events to improve the SNR of a quantum imaging protocol. This feature also enables usVenable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTto perform quantum imaging at low light levels. We now experimentally demonstrate this possibility by implementing a scheme for photon subtraction on our single-pixel camera with PNR capabilities. In this case, the mean photon number ntis equal to 0.08. one order of magnitude lower than the brightness of the source used for the experiment discussed in FIGS. 5 A and 5B. As illustrated in FIG. 2, this quantum imaging scheme utilizes two PNR detectors.

[0056] FIG. 7 shows photon-subtracted multiphoton quantum imaging, according to an embodiment of the present invention. The noise accompanying a signal reflected off a target object produces the classical image reported in a. Here, it is not possible to identify the object of interest with a classical single-pixel camera. The mean photon number ntof our thermal light source is 0.08. Interestingly, the subtraction of one photon improves the contrast of the image leading to the CS reconstruction in panel b of FIG. 7. Furthermore, our single-pixel camera with PNR capabilities enables multiphoton subtraction to produce the quantum images shown in panels c and d of FIG. 7. In these cases, we subtracted two and three photons, respectively. These images were produced using only 12% of the total number of measurements that can be used in our CS algorithm. The advantage provided by our protocol for photon-subtracted quantum imaging can be understood through the photon-number distributions reported from panel e to panel h of FIG. 7. The unconditional detection of the weak thermal light signal produces the histogram in panel e. This histogram unveils the overwhelming presence of vacuum and single-photon events used to reconstruct the image in a. Furthermore, as shown in panel f of FIG. 7, the process of one-photon subtraction increases the mean photon number of the thermal signal while reducing its degree of second-order coherenceThe subtraction of two-photon events leads to a stronger signal characterized by the histogram in panel g of FIG. 7. This conditional signal produces the enhanced image of the object in panel c of FIG. 7. Notably, the implementation of three-photon subtraction leads to the optical signal with nearly coherent statistics reported in panel h of FIG. 7. This boosted signal enables the reconstruction of the high-contrast image in panel d of FIG. 7. The error bars represent the standard deviation of 512 datasets. Each dataset consists of millions PNR measurements.

[0057] First, we use the noisy thermal signal to reconstruct the classical image shown in FIG. 7, panel a. Here, the large levels of noise forbid the observation of the target object.Remarkably, the subtraction of one photon from the thermal noisy signal reveals the presence of the object in FIG. 7, panel b. As predicted by Equation (4), the process of multiphotonVenable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTsubtraction leads to enhanced quantum images. Specifically, two-photon subtraction leads to the improved image in FIG. 7, panel c. Furthermore, the CS reconstruction of the three-photon subtracted quantum image reported in FIG. 7, panel d shows a significant improvement of the contrast with respect to the classical image in FIG. 7, panel a. The physics behind our scheme for quantum imaging can be understood through the increasing mean photon number that characterizes the histograms shown from panels e to h in FIG. 7. Moreover, the thermal fluctuations of the detected field are reduced by subtracting photons. This effect is indicated by the decreasing degree of secondorder coherencecorresponding to the photon-number distributions in FIG. 7.

[0058] FIG. 8 is a graph of the signal-to-noise (SNR) ratio versus the subtracted photon number showing the performance of photon-subtracted multiphoton quantum imaging. The SNR of the photon-subtracted quantum images shows a linear dependence on the number of subtracted photons. This behavior is in good agreement with Equation (4). Interestingly, the collection of larger sets of data leads to faster improvements of the SNR for our multiphoton quantum imaging scheme. The percentages represent the number of CS measurements with respect to the total number of pixels in the image. The error bars represent the standard deviation of the SNR of image reconstructions using five different datasets, where each dataset contains millions of PNR measurements.

[0059] The improvement in the SNR of the experimental photon-subtracted quantum images is quantified in FIG. 8. In agreement with Equation (4), the contrast of the filtered images, as a function of the number of subtracted photons, follows a linear dependence. Although, the benefits of our photon-subtracted scheme for multiphoton quantum imaging are evident for small and incomplete sets of data, the rate at which the SNR increases can be further amplified by collecting larger sets of data. It is worth noting that the exponential and linear mechanisms, reported in Fig. 5B and FIG. 8, for improving the SNR of weak and noisy imaging signals have the potential to enable the realistic application of robust quantum cameras with PNR capabilities. As such, these findings could lead to novel quantum techniques for multiphoton microscopy and remote sensing.

[0060] Quantum imaging schemes have been demonstrated to be fragile against realistic environments in which the background is comparable to the nonclassical signal of the imagingVenable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTphotons. This issue prevents the realistic application of quantum imaging techniques for microscopy, remote sensing, and astronomy. In this work, we overcome this paradigmatic limitation by developing a multiphoton quantum imaging scheme that relies on the use of natural sources of light. This is demonstrated through the implementation of a single-pixel camera with photon number resolving capabilities that enables the projection of classical thermal light fields into its constituent multiphoton subsystems. This kind of quantum measurement enables us to extract high-contrast quantum images from noisy classical images of target objects. Our technique shows a remarkable exponential enhancement in the contrast of quantum images. Surprisingly, we demonstrated the formation of quantum images produced by the vacuumfluctuation components of thermal light sources. We also demonstrate the possibility of using correlated multiphoton subsystems to form high-contrast quantum images from images in which the background noise is comparable to the signal of thermal light sources. Thus, we believe that our scheme opens a new paradigm in the field of quantum imaging. Furthermore, it unveils the potential of combining natural light sources with nonclassical detection schemes for the development of robust quantum technologies.

[0061] Experimental Setup: Our proof-of-principle quantum imaging setup utilizes pseudo-thermal light, which has the same properties of coherence as natural sources of light. In an embodiment, this source is generated by passing the coherent light from a continuous wave laser at 633 nm through a rotating ground glass. The thermal light is then collected into a singlemode fiber and collimated with a lens ( / = 5 cm) to illuminate the target object. In an example embodiment, the target object "h" is generated using a digital micro-mirror device DMD (DLP6500 DLP ®DMD). Then, the reflected " h " is projected onto a second DMD with a 4-f system including two lenses, each with a focal length of 10 cm. This second DMD facilitates compressive sensing by displaying a series of random binary matrices. Next, the reflected light from the DMD is imaged using another 4-f system including two lenses, with focal lengths of 25 and 10 cm. Then, we couple the reflected light into a 1 x 2 50:50 fiber beam splitter (Thorlabs TW630R5F1) using a Rochester lens (f = 4.5 mm). The split beams are detected by two fiber-coupled avalanche photodiodes (APDs, Excelitas SPCM-AQRH-13-FC), where photon-number-resolving detection is implemented. Finally, these detection events are recorded by a time tagger (PicoQuant MultiHarp 150) and analyzed. This experimental setup allows us to accurately measure the joint photon-number distribution at both outputs of the fiber beam-Venable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTsplitter. This enables us to perform photon subtraction and post selection for image reconstruction.

[0062] Data Analysis and Image Reconstruction: In an embodiment, in the compressive sensing (CS) process, we sequentially display some percentage of 4096 unique random matrices on the second DMD. with the measurement time for each matrix fixed at one second. For example, the reconstructions for FIGS. 5A-5B and 6A-6B utilized 1025 projective measurements, which corresponds to 25% of the 4096 measurements that our CS algorithm can use. The images reported on FIG. 7 were obtained w ith 12% of the measurements, which means 512 binary matrices. A complete realization of the experiment using all 4096 frames takes approximately 10 hours. We note that this time is very long since it includes the overhead by uploading and displaying the matrix on the DMD, and taking brief pauses between each measurement. However, this time can be drastically reduced by pre-programming all matrices in the DMD beforehand and syncing the display on the DMD with the APDs. To reconstruct the image, we apply the TVAL3 algorithm. Due to the long coherence time of our pseudo-thermal light source, we bin each one-second measurement into Iqs intervals. Then we count the number of detections in each bin, and this defines a photon-number resolving event. For each event, we then denote the photon-counts in the two arms asand n2. To achieve / V-photon subtraction, we isolate the events where n2= N. In other words, we filter n by only considering the events where n2= N in the second arm. We then perform the reconstruction process with this conditional dataset to obtain an enhanced image quality. Conversely, for postselection on n and m photons, we compute the probability that= n and n2= m. Then, we perform image reconstruction using this probability to obtain an improvement in image quality.

[0063] As it can be appreciated from the above paragraphs, there is provided a method for imaging an object illuminated with natural light 101, 202. The method includes arranging an optical filter (e.g., DMD 102 or DMD 208) in an optical path of at least a portion of the natural light 101, 202 at least one of reflected, refracted or scattered from the object 120, 204. The method further includes arranging an optical detector or photodetector 104, 216A, 216B in an optical path of light at least one of reflected from or transmitted through the optical filter (e.g., DMD 102 or DMD 208). The method further includes detecting, using the optical detector or photodetector 104, 216A, 216B, individual photons for a measurement time period to provide a times series of individual photon events. The method also includes segmenting the time seriesVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTinto a plurality of time bins, and determining a number of detected photons within each time bin of the plurality of time bins to provide a time series of photon counts per time bin. The method further includes at least one of replacing or reconfiguring the optical filter (e.g., DMD 102 or DMD 208) a plurality of times followed by corresponding of pluralities of the detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin. The method includes forming an image 122, 222 of the object 120, 204 based on the plurality of times series of photon counts per bin.

[0064] The optical filter (e.g., DMD 102 or DMD 208) has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions. Each replaced or reconfigured optical filter (e.g., DMD 102 or DMD 208) has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions such that each optical filter (e.g., DMD 102 or DMD 208) has a two-dimensional structured pattern that differs from all other two-dimensional structured patterns of the other optical filters (e.g., DMD 102 or DMD 208).

[0065] In an embodiment, forming the image 122, 222 is based on time bins within the plurality of times series of photon counts per bin that have a number n of photon counts per bin, and wherein n is a non-negative integer. For example, n is an integer greater than or equal to 0. For example, n is an integer greater than 1, or n is an integer greater than 6.

[0066] In an embodiment, each of the optical filters (e.g., DMD 102 or DMD 208) provides a two-dimensional, randomly arranged pixilated array.

[0067] In an embodiment, the time bins have a width determined according to a coherence length of the natural light 101, 202. In an embodiment, the time bins have a width determined to be substantially equal to a coherence length of the natural light 101, 202.

[0068] In an embodiment, forming the image uses an optimization process to reduce a number of the plurality’ of times series used to form the image 122, 222. In an embodiment, the forming of the image 122, 222 uses a compressive sensing process. The compressive sensing process includes using a matrix corresponding to the two-dimensional structured pattern and a vector corresponding to the time series of photon counts per time bin to form the image.Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCT

[0069] The compressive sensing process includes using the following expression to reconstruct the image 122 of the object 120.4- ||Y -where O' represents the image 122, A represents the matrix pattern of mirrors 112 of the DMD 102, Y represents a vector corresponding to a time series of photon counts per time from object 120.

[0070] In an embodiment, the method includes arranging a first optical detector 216A and a second optical detector 216B in an optical path of light at least one of reflected from or transmitted through the optical filter (e.g., DMD 208), as shown in FIG. 2, for example. The method further includes detecting, using the second optical detector 216B, individual photons for a measurement time period to provide a times series of individual photon events. The method includes segmenting the time series into a plurality of time bins. The method further includes determining a number of detected photons within each time bin of the plurality of time bins to provide a time series of photon counts per time bin. The method includes during the at least one of replacing or reconfiguring the optical filter (e.g., DMD 208) a plurality of times perform corresponding of pluralities of the detecting, segmenting and determining with the second optical detector 21 B to obtain a second plurality of times series of photon counts per bin, and forming the image 222 of the object 204 based on the first-mentioned and the second plurality of times series of photon counts per bin.

[0071] While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not limitation. Thus, the breadth and scope of the present invention should not be limited by any of the abovedescribed illustrative embodiments, but should instead be defined only in accordance with the following claims and their equivalents.

[0072] The embodiments illustrated and discussed in this specification are intended only to teach those skilled in the art how to make and use the invention. In describing embodiments of the disclosure, specific terminology is employed for the sake of clarity. However, the disclosureVenable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTis not intended to be limited to the specific terminology7so selected. The above-described embodiments of the disclosure may be modified or varied, without departing from the invention, as appreciated by those skilled in the art in light of the above teachings. It is therefore to be understood that, within the scope of the claims and their equivalents, the invention may be practiced otherwise than as specifically described. For example, it is to be understood that the present disclosure contemplates that, to the extent possible, one or more features of any embodiment can be combined with one or more features of any other embodiment.

Claims

Venable Ref.: 144240.609437LSU Ref: LSU-2024-023-02 PCTWE CLAIM:

1. A method for imaging an object illuminated with natural light, comprising:arranging an optical filter in an optical path of at least a portion of said natural light, said at least the portion of said natural light being reflected, refracted or scattered from said object;arranging an optical detector in an optical path of light at least one of reflected from or transmitted through said optical filter;detecting, using said optical detector, individual photons for a measurement time period to provide a times series of individual photon events;segmenting said time series into a plurality of time bins;determining a number of detected photons within each time bin of said plurality of time bins to provide a time series of photon counts per time bin;at least one of replacing or reconfiguring said optical filter a plurality of times followed by corresponding of pluralities of said detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin; andforming an image of said object based on said plurality of times series of photon counts per bin,wherein said optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions, andwherein each replaced or reconfigured optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions such that each said replaced or reconfigured optical filter has a two-dimensional structured pattern that differs from all other two-dimensional structured patterns of other replaced or reconfigured optical filters.

2. The method according to claim 1, wherein said forming the image is based on time bins within said plurality of times series of photon counts per bin that have a number n of photon counts per bin, andwherein n is a non-negative integer.

3. The method according to claim 2, wherein n is an integer greater than 1.Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCT4. The method according to claim 3, wherein n is an integer greater than 6.

5. The method according to any one of claims 1 to 4. wherein said optical filter provides a two-dimensional, randomly arranged pixilated array.

6. The method according to any one of claims 1 to 5, wherein said optical filter is a configuration of a digital micromirror device (DMD).

7. The method according to any one of claims 1 to 6, wherein said time bins have a width determined according to a coherence length of said natural light.

8. The method according to any one of claims 1 to 6. wherein said time bins have a width determined to be substantially equal to a coherence length of said natural light.

9. The method according to any one of claims 1 to 8. wherein said forming said image uses an optimization process to reduce a number of said plurality’ of times series used to form said image.

10. The method according to any one of claims 1 to 8, wherein said forming said image uses a compressive sensing process.

11. The method according to claim 10, wherein the compressive sensing process includes using a matrix corresponding to the two-dimensional structured pattern and a vector corresponding to said time series of photon counts per time bin to form said image.

12. The method according to any one of claims 1 to 11, further comprising:arranging a second optical detector in an optical path of light at least one of reflected from or transmitted through said optical fdter;detecting, using said second optical detector, individual photons for a measurement time period to provide a times series of individual photon events;segmenting said time series into a plurality' of time bins;Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTdetermining a number of detected photons within each time bin of said plurality of time bins to provide a time series of photon counts per time bin;during said at least one of replacing or reconfiguring said optical filter a plurality of times perform corresponding of pluralities of said detecting, segmenting and determining with said second optical detector to obtain a second plurality of times series of photon counts per bin; andforming an image of said object based on a first plurality of time series corresponding to said time series and based on said second plurality of times series of photon counts per bin.

13. A system for imaging an object illuminated with natural light, comprising:an optical filter configured arranged to be in an optical path of at least a portion of said natural light at least one of reflected, refracted or scattered from said object;an optical detector arranged to be in an optical path of light at least one of reflected from or transmitted through said optical filter; andan image processor configured to communicate with said optical detector, wherein said optical detector is configured to detect individual photons for a measurement time period to provide a times series of individual photon events and to communicate said time series to said image processor,wherein said image processor is configured to:segment said time series into a plurality of time bins, anddetermine a number of detected photons within each time bin of said plurality of time bins to provide a time series of photon counts per time bin,wherein said optical filter is configured to be at least one of replaceable or reconfigurable a plurality of times to be followed by corresponding of pluralities of the detecting, segmenting and determining to obtain a plurality of times series of photon counts per bin,wherein said image processor is configured to form an image of said object based on said plurality of times series of photon counts per bin,wherein said optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions, andwherein each replaced or reconfigured optical filter has a two-dimensional structured pattern of at least two of reflecting, scattering, transmitting, or refraction portions such that each said replaced or reconfigured optical filter has a two-dimensional structured pattern that differsVenable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCTfrom all other two-dimensional structured patterns of other replaced or reconfigured optical filters.

14. The system according to claim 13, wherein said forming said image is based on time bins within said plurality of times series of photon counts per bin that have a number n of photon counts per bin, andwherein n is a non-negative integer.

15. The system according to claim 14, wherein n is an integer greater than 1.

16. The system according to claim 15, wherein n is an integer greater than 6.

17. The system according to any one of claims 13 to 16, wherein said optical filter is a two-dimensional, randomly arranged pixilated array.

18. The system according to any one of claims 13 to 17. wherein said optical filter is a digital micromirror device (DMD).

19. The system according to any one of claims 13-18, wherein said time bins have a width determined according to a coherence length of said natural light.

20. The system according to any one of claims 13-18, wherein said time bins have a width determined to be substantially equal to a coherence length of said natural light.

21. The system according to any one of claims 13 to 20. wherein said image processor is further configured to form said image using an optimization process to reduce a number of said plurality of times series used to form said image.

22. The system according to any one of claims 13 to 20. wherein said image processor is further configured to form said image using a compressive sensing process.Venable Ref.: 144240.609437LSU Ref.: LSU-2024-023-02 PCT23. The system according to claim 22, wherein the compressive sensing process includes using a matrix corresponding to the two-dimensional structured pattern and a vector corresponding to said time series of photon counts per time bin to form said image.

24. The system according to any one of claims 13 to 23, further comprising:a second optical detector arranged to be in an optical path of light at least one of reflected from or transmitted through said optical filter, the second optical detector detecting individual photons for a measurement time period to provide a second plurality of times series of individual photon events to said image processor,said image processor being further configured to:segment said time series into a plurality of time bins,determining a number of detected photons within each time bin of said plurality of time bins to provide a second time series of photon counts per time bin, andform an image of said object based on a first plurality of time series corresponding to said time series and based on said second plurality of time series of photon counts per bin.

25. A non-transitory computer-readable medium storing instructions that when executed by a computer processor cause the computer processor to perform the method of claims 1-12.

Images