A method for performing on-axis holographic imaging

Abstract

The invention relates to a method for performing on-axis holographic imaging with a holographic imaging system (HIS), where a complex phase modulator (PM) represents a spatial transfer function H being positioned in an optical path between a front of the lens and the image sensor. Processing is performed for holographic rendering of the object. The processing is performed using the conjugated spatial transfer function H* corresponding to the complex phase modulator (PM), where the spatial transfer function H is selected so as to minimize, preferably eliminate, the influence of the cross-interfering term (O*R) and / or self-interfering term (OO*) in the holographic equation of the holographic imaging system (HIS) for improved image quality. Careful selection of the complex phase modulator, and the corresponding processing circumvents on-axis holographic artifacts from cross- and self-interfering terms that can make it difficult to reconstruct for example the original 3D scene of potentially partly or fully transparent objects with sufficient fidelity.

Description

[0001]84026PC01 1 AMETHOD FOR PERFORMING ON-AXIS HOLOGRAPHIC IMAGINGFIELD OF THE INVENTIONThe present invention relates to a method for performing on-axis holographicimaging, such as holographic microscopy, and a corresponding system for performing on-axis holographic imaging. The invention also relates to variousapplications for such on-axis holographic imaging, such as imaging of biologicalcells, for example blood cells. BACKGROUND OF THE INVENTIONHolography has found uses in numerous areas of academia and industry, includingmicroscopy, light-shaping, particle trapping and manipulation, cryptography, etc. Interestingly, while holography’s foundational principles were laid by Denis Gabor, he faced the persistent challenge of the so-called twin-image problem. At thetime, there was no analytical solution to quantify or eliminate this twin-imageissue, leading Gabor to effectively abandon his on-axis holography research after exploring various optical setups. This decision stands in contrast to the subsequent proliferation and significance of holography in diverse fields. The ability to regain both amplitude and phase information of an optical wavefrontafter its recording by an intensity-sensitive detector is a key factor of itspopularity. The holographic equation, which reveals this property of holographic image capture, can be written as the intensity of the superposition of two incidentwavefronts - the object O and reference R. The interference pattern recorded bythe image sensor can thus be described by the intensity distribution I(x,y): (1) I(x,y) = (O + R)(O + R)* = OO* + RR* + OR* + O*Rwhere O is the complex amplitude distribution of the object light, i.e., thescattered light from the object generated when the object is illuminated by thelight, and R is the complex amplitude distribution of the reference light, i.e., theunobstructed light from the light beam, x and y denotes the x- and y-coordinateson the image sensor and therefore correspond to the discretized pixel coordinates 84026PC01 2and * denotes the complex conjugate. The x- and y-coordinates of O and R, areomitted for convenience.The four terms comprising the holographic equation (1) are then:- OO*; the scattered object wave interfering with itself, thus denoted here as“self-interference” term,- RR*; the directly transmitted and un-scattered reference wavecorresponding to the background, and -OR* and O*R; real and virtual images, respectively, of the original objectwavefront, scaled by the reference wave, containing both amplitude andphase information of the object.Typically, due to the complex conjugation, the term O*R is often denoted as theso-called twin-image of the object. Now, to extract the amplitude and phase ofthe original object wavefront, the term OR* - i.e., the real image, must beisolated to the greatest extent.In off-axis holography, in which the illuminating light is split into an object beamand an angled reference beam, thus introducing a spatial carrier wave, the fourterms of the holographic equation (1) can “simply” be spatially separated andfiltered-out in Fourier space. However, when discussing on-axis holography, theobject illuminating light source serves as both the object and reference wave. It isconceptually divided into the two waves, but originates from the same light beamdirection. Thus, both wavefronts propagate axially in line, not allowing for spatialseparation of the terms. It should be noted that on-axis holography can also beembodied by using an independent reference wave – again taken from the samelight source due to holographic coherence requirements - co-propagating in linewith the object wave. This is particularly relevant when the object is semi-opaque, opaque or holographically illuminated in a reflection geometry. The reconstruction of the object beam in on-axis holography is typicallyundertaken with iterative numerical methods, usually from derivations of the well-known Gerchberg-Saxton algorithm. While good results can be achieved, theseiterative phase retrieval algorithms are inherently non-convex, not ensuring 84026PC01 3 convergence to the global minimum. In addition, the iterative nature limits thereconstruction speed for most practical applications.In the last decade, machine learning algorithms using e.g., deep neural networksare also tackling the same problem of wavefront retrieval and is showingpromising results. While inference can be performed at high reconstruction speed,the training of the deep neural networks requires both long training time andmany thousand training samples, cf. for example the recent review article “On- axis digital holographic microscopy: Current trends and algorithms”, OpticsComm. 537, (2023) 129458, by A. E. G. Madsen et al. (incl. the presentinventor). Hence, an improved method for holographic reconstruction in on-axis holographywould be advantageous, and in particular a more efficient and / or reliable methodwould be advantageous. OBJECT OF THE INVENTION It is a further object of the present invention to provide an alternative to the prior art. In particular, it may be seen as an object of the present invention to provide animproved method for holography and a corresponding holographic system thatsolves the aforementioned problems of the prior art with on-axis holographicreconstruction, especially handling the influence of the twin-image (O*R) and / orthe self-interference (OO*) term.SUMMARY OF THE INVENTIONThus, the above-described object and several other objects are intended to beobtained in a first aspect of the invention by providing a method for performing on-axis holographic imaging with a holographic imaging system (HIS), preferably a digital holographic microscope, the HIS comprising:- a lens (OL), such as an infinity-corrected objective lens, for imaging of anassociated object, such as a biological sample, 84026PC01 4- an image sensor positioned in an optical path of the lens arranged forcapturing an intensity (I) pattern representing an image of said object forholographic imaging,- an optional intermediate lens (IL), such as a tube lens, positioned in saidoptical path between the objective lens and image sensor,- a complex phase modulator (PM) representing a spatial transfer function H,preferably an adjustable transfer function, the complex phase modulator being positioned in said optical path between a front of the lens and the image sensor, optionally the intermediate lens, preferably between the front and a rear of said lens, and- a processor operably connected to the image sensor for reconstructing aholographic rendering of the object,the method comprising:- capturing a spatial signal indicative of an intensity (I) pattern representingthe object for holographic imaging through the lens,- transmitting said spatial signal through the lens and transmitting thespatial signal through the complex phase modulator and the optional intermediatelens,- detecting a spatial signal at the image sensor, and- processing the detected spatial signal so as to obtain a reconstructedholographic rendering of the object,wherein the processing of the detected spatial signal is performed using theconjugated spatial transfer function H* corresponding to the said complex phasemodulator (PM), said spatial transfer function H being selected so as to minimize,preferably eliminate, the influence of the cross-interfering term (O*R) and / or self- interfering term (OO*) in the holographic equation of the holographic imagingsystem (HIS) for improved image quality.The invention is particularly, but not exclusively, advantageous for obtaining animproved method for holography that solves the above-mentioned problems ofthe prior art with on-axis holographic reconstruction, especially handling the influence of the twin-image (O*R) and / or the self-interference (OO*) term. Thisinvention is based on a combined hardware with careful selection of the complex 84026PC01 5phase modulator, and a processing approach with a corresponding conjugatedspatial transfer function to circumvent the aforementioned on-axis holographicartifacts from cross- and self-interfering terms that can make it difficult toreconstruct for example the original 3D scene of potentially partly or fullytransparent objects with sufficient fidelity. All the advantages of an on-axisholographic approach, especially for digital holography, are maintained by thepresent invention so that it is industrially attractive to apply the variousembodiments. Moreover, a key advantage of on-axis digital holography is the useof commercially available image sensors that would not be sufficient pixel-pitchwise for off-axis holography e.g., in situations where high spatial frequencyfringes will be inherently under-sampled, creating e.g., aliasing, noise anddistortions. The electromagnetic components or the ‘hardware’ part of theinvention will not add substantially to the complexity of future instruments andsystems based on this invention due to the large computational part that makes itfor example possible to real-time adapt for inherent imperfections of digitally recorded holograms.Alternatively in one embodiment of the present invention for reconstructing thevirtual image instead of the real image, one may apply the non-conjugated spatialtransfer function H again for the processing of the detected spatial signal beingselected so as to minimize, preferably eliminate, the influence of the conjugate cross-interfering term (OR*) and / or self-interfering term (OO*). This dual-use ofthe same spatial transfer function H is e.g., advantageous for a bi-directionalhardware-only embodiment of the invention. Thus, in this embodiment of theinvention, the processing of the detected spatial signal is performed using thenon-conjugated spatial transfer function H corresponding to the said complexphase modulator (PM), said spatial transfer function H being selected so as to minimize, preferably eliminate, the influence of the term (OR*) i.e. the “realimage” and / or term (O*O) in the holographic equation of the holographic imagingsystem (HIS) for improved image quality. Alternatively in one embodiment of the present invention instead of transmitting said spatial signal through the lens, it may be a reflection in a curved mirror as the skilled person will readily understand. 84026PC01 6 Alternatively in one embodiment of the present invention instead oftransmitting the spatial signal through the said complex phase modulator (PM), itcan be advantageous to use a PM in reflection mode. In general, all components can be embodied in reflection mode, transmission mode or partial transmissionand reflection mode. The same applies for the object to be in-line or on-axisholographed. If the object is partially or fully reflecting it is advantageous toembody so-called object support. This can be e.g., a flat reflecting surfacepositioned in conjunction with the object and being larger than the object so as to obtain an in-line or on-axis reference for interference with the object reflection on the image sensor. The invention is further advantageous in having a so-called modulated (optionally complex) phase-only transfer function in an in line or on-axis holographic configuration in combination with a relatively simple computational light wavepropagation will make it possible to dynamically self-calibrate and adapt digitallyrecorded 3D holographic scenes so that they are not suffering from theaforementioned limitations. In particular, it will be useful for reconstructing semi-transparent biological cells or micro-organisms in a volume solution and withmuch higher fidelity than is possible today. Advantageously, it is contemplatedthat the invention can be applied for performing blood cell analyses, such asdynamically differentiating the five most common types of white blood cells.Moreover, in a holographic reflection geometry embodiment the invention offers all the advantages of an on-axis geometry even in situations where the reference wave is independently controllable as is typically the case for an in line holographic recording geometry. The use of a partial or fully reflecting background surface having an area larger than a partial or fully reflecting object can be embodied as so-called object support. Advantageously, it is contemplated that the invention can be applied for highly accurate surface topology metrology using an in line holographic geometry. Advantageously, it is also contemplated that the invention can be embodied in a hardware-only configuration so that both holographic recording (in reflection or transmission or both) and holographic reconstruction is applied via the same hardware and hence with no need for a software reconstruction. It is further contemplated that the invention can offer advantageous visual fidelity with a hardware-only configuration as 3D dynamic holographic display system. In a hardware-only embodiment it is further optional 84026PC01 7 to replace the electronic image sensor by a dynamic intensity dependent recording material such as a photorefractive crystal or a dynamic metasurface optical element. Moreover, for quantum technology applications and for quantum optics it is further contemplated that the invention can offer key advantages due to the applied in-line common-path configuration creating increased robustness and coherence relaxation amongst others. In the context of the present application, it is to be understood that the holographic imaging system (HIS) comprises or cooperates with a light sourcearranged to generate a semi-coherent or coherent input beam such as a laserbeam. The degree of spatial and temporal coherence of the input beam may depend on the application. Thus, a temporally semi-coherent beam from a spatially coherent LED, a super-luminescent diode or a semi-coherent laser source may be sufficient. The input beam may also originate from a pulsed light source, such as a femto-second laser. In the context of the present application, it is to be understood that the present invention may be applied in the full electromagnetic spectrum E / M from radio frequencies, micro-waves, Tera Hertz (THz), infrared, near-infrared (NIR), visible range, UV, EUV, soft X-ray, X-Ray, Gamma etc. Thus, the skilled person will immediately understand that the principle and teaching from optical holography can be readily understood to apply in for example X-ray holography. In the context of the present application, it is to be further understood that the present invention may be applied for subject-matter waves, for example electronbeam or electron holography the invention can be applied. An electron microscopeis a microscope that uses a beam of electrons as a source of illumination. Onemay also use electron optics that are analogous to the glass lenses of an opticallight microscope to control the electron beam, for instance focusing them toproduce magnified images or electron diffraction patterns. The electron beamenablement can also use electron optics for the complex phased modulator (PM). For example, in the paper “Axicon Lens for Electrons Using a Magnetic Vortex: TheEfficient Generation of a Bessel Beam”, Phys. Rev. Letters 19, 174801 (2017), theauthors demonstrate an efficient electron axicon lens using a magnetic vortex. They show that naturally occurring magnetic vortices with circular magnetic 84026PC01 8 moment distributions in a soft-magnetic thin film can create conical phase shifts for fast electrons. Such radially symmetric linear phase ramps are equivalent toideal light optical axicons and highlights the potential for using magneticnanostructures as highly efficient and flexible phase plates for crafting desired electron beam shapes. In the context of the present application, it is to be understood that the lens ofthe holographic system may have a convergent transformation onto the beam,however, in some embodiments, it is to be understood that the beam illuminating the object may be convergent when entering the system, and therefore the HISmay comprise a lens-less embodiment of the invention using e.g., a converginglight wave for illuminating the objects for holographic recording. In the context of the present application, it is to be understood that theholographic imaging system (HIS) is arranged for performing on-axis holographicimaging as opposed to off-axis holographic imaging. For the on-axis holographicimaging process the object illuminating light source effectively functions as both the object wave and reference wave. It is conceptually divided into the twowaves but originates from the same directional light beam. It should be noted thaton-axis holography can also be embodied by using an independent reference wave– again taken from the same light source due to holographic coherencerequirements - co-propagating in line with the object wave. This is particularlyrelevant when the object is semi-opaque, opaque or holographically illuminated in a reflection geometry.In one embodiment, said spatial transfer function H may be chosen as so tominimize the influence on the signal to noise ratio (SNR) from the sum of thereconstructed cross-interfering term (O*R) and self-interfering term (OO*).Additionally or alternatively, said spatial transfer function H may be chosen as soto maximize the signal to noise ratio (SNR) of the sum of the reconstructed object(OR*) and optionally background (RR*) to improve the image quality of theobjected, which will be demonstrated in more detail below.In particular advantageous embodiments, the influence of the cross- and / or self-interfering terms may be minimized by displacing the cross-interfering term (O*R) 84026PC01 9 and / or self-interfering term (OO*) away from a signal of interest (SoI) in thereconstructed space. These embodiments have turned out to be very usefulbecause holographic imaging is significantly improved in this way. Preferably, said signal of interest (SoI) in the reconstructed space may comprise the real image (OR*). In other advantageous embodiments, an expression of the spatial transferfunction H of the complex phase modulator (PM) may be estimated using theprinciple of the stationary phase method, where a local spatial frequency of thecomplex phase modulator (PM) is estimated as a first-order spatial derivative of its spatial phase, and said estimated expression of the spatial transfer function H is applied in said processing of the detected optical signal using the conjugatedspatial transfer function H* corresponding to the said complex phase modulator(PM). Preferably, a local spatial frequency, f, of a phase element may thereby beestimated as a spatial derivative of its spatial phase in cartesian or polar coordinates, respectively as: ^ ^^(^,^) ^ ^^(^,^) ^^= ^^ ^^ ^^= ^^ ^^ , or ^ ^^^^=(^,^)^^^ ^,^^^ ^^ ^^=( )^^ ^^ ,where φ is the phase in cartesian coordinates or polar coordinates, which may beadvantageous to implement.In some embodiments, said lens may not be comprised in the HIS, the intensitypattern representing the object being already convergent, or divergent, into the HIS onto the complex phase modulator (PM), thus, the skilled person will readily understand that the effect or function of the lens may be provided outside of the holographic imaging system and therefore the lens may be omitted in the HIS. In particular beneficial embodiments, the complex phase modulator (PM)representing a spatial transfer function H may have a multiplexing of phasecontrast and / or dark field filtering. Thus, this may be advantageous forreconstructing invisible phase-only objects in hardware-only where processing is 84026PC01 10performed partially or fully in optical parts or components. In other embodiments,it may be advantageous to apply the same hardware configuration for bothholographic recording and holographic reconstruction (potentially in reverse).In some embodiments, the lens may be positioned in the optical path so as toperform an optical Fourier transformation of the object with the complex phase modulator (PM) being positioned near, or at, the back focal plane (BF_OL) of thelens. Thus, for example with quadratic phase compensation of the conjugatedspatial transfer function H*, the compensation can be fixed later in the optical processing, which may increase robustness.In some advantageous embodiments, the intermediate lens may be positioned inthe optical path so as to perform an inverse Fourier transformation of the object -relative to the said objective lens – with the image sensor being positioned near,or at, the back focal plane (BF_IL) of the intermediate lens.In other advantageous embodiments, the complex phase modulator (PM) may bean active optical component, preferably being an adjustable phase-only phasemodulator, such as a spatial light modulator (SLM) or other modulators readilyavailable for the skilled person in optics.In alternative embodiments, the complex phase modulator (PM) may be a passiveoptical component, preferably being a lens, more preferably an axicon lens or a diffractive, or reflective, axicon lens with a radial grating period (d), or other phase modulators readily available for the skilled person in optics.In beneficial embodiments, the complex phase modulator may be chosen so thatthe spatial transfer function, H, can be expressed in a mathematical form of the type: H= A_1 ∙ exp (-i ∙A_2 ∙ r k)where r is a radial coordinate, A_1 is a constant, and A_2 is another constantdepending on the wavelength of the light and the phase modulator, and 84026PC01 11wherein the exponent k is chosen as k≠2, preferably k < 2, more preferably k <1.3, and wherein the complex phase modulator is further chosen so that theinfluence of the cross- and / or self-interfering terms is minimized by displacing thereconstructed cross-interfering term (O*R) and / or self-interfering term (OO*) away from a signal of interest (SoI) in the reconstructed phase space.As it will be demonstrated below, the variation of k i.e. by a so-called k-sweepseparation convincingly demonstrates the effect of the present invention.In yet another embodiment of the present invention, the processing of thedetected spatial signal may be performed using the non-conjugated spatialtransfer function H corresponding to the said complex phase modulator (PM), saidspatial transfer function H being selected so as to minimize, preferably eliminate, the influence of the term (OR*) representing the real image and / or the term (O*O) in the holographic equation of the holographic imaging system (HIS) for improved image quality. Thus, in a sense this is an opposite variant of the invention relative to the first aspect, but the skilled person in optics will readily understand that this embodiment is also part of the present invention, and may also be advantageous for holographic imaging.In a second aspect, the invention relates to a holographic imaging system forperforming on-axis holographic imaging, preferably a digital holographic microscope, the HIS comprising:- a lens (L), such as an infinity corrected objective lens, for imaging of anassociated object, such as a biological sample,- an image sensor positioned in an optical path of the lens arranged forcapturing an intensity (I) pattern representing an image of said object for holographic imaging,- an optional intermediate lens (IL), such as a tube lens, positioned in saidoptical path between the objective lens and image sensor,- a complex phase modulator (PM) representing a spatial transfer function H,preferably an adjustable transfer function, the complex phase modulator being positioned in said optical path between a front of the lens and the image sensor, optionally the intermediate lens, preferably between the front and a rear of said lens, and 84026PC01 12- a processor operably connected to the image sensor for reconstructing aholographic rendering of the object, wherein the system is arranged for:- capturing a spatial signal indicative of an intensity (I) pattern representingthe object for holographic imaging through the lens,- transmitting said spatial signal through the lens and transmitting thespatial signal through the complex phase modulator and the optional intermediate lens,- detecting a spatial signal at the image sensor,- processing the detected spatial signal so as to obtain a reconstructedholographic rendering of the object, wherein the processor is arranged for processing the detected spatial signal byusing the conjugated spatial transfer function H* corresponding to the saidcomplex phase modulator (PM), said spatial transfer function H being selected soas to minimize, preferably eliminate, the influence of the cross-interfering term (O*R) and / or self-interfering term (OO*) in the holographic equation of the HIS for improved image quality.In another aspect, the invention relates to a computer program product beingadapted to enable a computer system comprising at least one computer having data storage means in connection therewith to control holographic imagining system apparatus according to the second aspect of the invention, such as a computer program product comprising instructions which, when the program is executed by a computer, cause the computer to carry out [the steps of] the method of first aspect of the invention. This aspect of the invention is particularly, but not exclusively, advantageous in that the present invention may be computer-implemented and / or be accomplished by a computer program product enabling a computer system to carry out theoperations of the holographic imaging system of the first aspect of the inventionwhen down- or uploaded into the computer system. Such a computer programproduct may be provided on any kind of computer readable medium, or through a network. 84026PC01 13The individual aspects of the present invention may each be combined with any ofthe other aspects. These and other aspects of the invention will be apparent from the following description with reference to the described embodiments. BRIEF DESCRIPTION OF THE FIGURES The invention will now be described in more detail with regard to the accompanying figures. The figures show one way of implementing the present invention and is not to be construed as being limiting to other possible embodiments falling within the scope of the attached claim set. Figure 1 is a schematic flow-chart of the method according to the present invention, Figure 2 is a schematic out-line of a holographic imaging system (HIS) according to the present invention, Figure 3 shows schematically two embodiments of a holographic imaging system (HIS) according to the present invention, Figure 4 shows schematically two embodiments of a holographic imaging system (HIS) with different positions of the complex phase modulator (PM) according to the present invention, Figure 5 shows schematically two other embodiments of a holographic imaging system (HIS) according to the present invention, Figure 6 shows schematically another embodiment of a holographic imaging system (HIS) according to the present invention, Figure 7 shows simulations with a quadratic power lens, i.e. k=2,Figure 8 shows similar simulations to Figure 7 but with various values of k lowerthan 2, 84026PC01 14 Figure 9 shows further simulations with an axicon phase mask,Figure 10 shows a Fourier phase mask and the simulated corresponding CCDintensity pattern on an image sensor,Figure 11 and Figure 12 shows reconstructed images from numerical experimentsfor verifying the present invention, Figure 13 schematically shows a lens-less embodiments, andFigure 14 is a schematic system-chart representing an outline of the operations ofa method according to the invention.DETAILED DESCRIPTION OF AN EMBODIMENT Figure 1 is a schematic flow-chart of the method according to the present invention. Thus, initially there is performed a capturing of an optical signal indicative of an optical image of the object for holographic imaging through the lens in the upper so-called hardware branch and transmitting said optical signal through the lens and transmitting the optical signal through the complex phase modulator with a corresponding spatial transfer function H. In the final part of thehardware branch, there is performed detection of an optical signal at an imagesensor (not shown here). In the lower software branch, there is performed aprocessing of the detected optical signal so as to obtain a holographic image of the object by the processing of the detected optical signal, which is performedusing the conjugated spatial transfer function H* corresponding to the saidcomplex phase modulator (PM), the spatial transfer function H being selected soas to minimize, preferably eliminate, the influence of the cross-interfering term (O*R) and / or self-interfering term (OO*) in the holographic equation of the HIS for improved image quality when it is reconstructed in the final step. Thus, in the upper branch there is performed various optical processes using hardware, such as complex phase modulation using for example a spatial lightmodulator (SLM) with a well-controlled and possibly dynamic spatial transfer 84026PC01 15function H, and in the lower branch there is performed various processing stepstypically using a computer with appropriate software with a correspondingconjugated spatial transfer function H* for the reconstruction of the image.However, for some alternative embodiments, parts or all of the computerprocessing may alternatively be performed in or by corresponding opticalcomponents, for example where the transfer function is well-known andunchanged. For a mathematical formulation, the invention may be formulated as follows:Detected Intensity – I – by Hardware:^ = 1 + ^ ⊗ ℎ + (^ ⊗ ℎ)∗ + |^ ⊗ ℎ|^Computational – C – reconstruction typically by software: The Aim 1. Compute a spatial Transfer Function – H – such that one minimize the SNRinfluence of: ^^^[ ℑ^^(ℑ(^)∗^∗^∗) + |^ ⊗ ℎ|^ ⊗ ℎ∗ ]2. while one maximizes the SNR influence of the computationallyreconstructed object and optionally background: ^^^[ 1 + ^ ]In order to enhance the visibility or contract or SNR of the detected intensity, I, onthe image sensor it can be advantageous to multiplex zero-order centred phase contrast or dark field filtering or both in the spatial transfer function, H. Moreover,such multiplexed phase contrast or dark field filtering or both can be advantageousfor reconstructing invisible phase-only objects in hardware-only embodiments 84026PC01 16 exploring the same hardware configuration for both holographic recording and holographic reconstruction. Holographic terms from the intensity-dependent image sensor (O: Object light, R: Reference light): I= (O + R) ∙ (O + R)* = OO* + RR* + OR* + O*RThe sparse input object (being transformed hereinafter), O, on homogeneousbackground (R = 1) is Fourier Transformed and multiplied with a (complex phase-only) spatial Transfer Function, H, (typically passing the homogeneousbackground unhindered or using the aforementioned multiplexed zero-ordercentred phase contrast or dark field filtering): The detected intensity is then given by:^ = (ℑ^^(ℑ(o)^) + 1)(ℑ^^(ℑ(o)^) + 1)∗or written out as convolutional terms: ^= (^ ⊗ ℎ + 1)(^ ⊗ ℎ + 1)∗ = 1 + ^ ⊗ ℎ + (^ ⊗ ℎ)∗ + |^ ⊗ ℎ|^Computationally reversing this process by the conjugate (complex phase-only) spatial Transfer Function described by the corresponding conjugate impulse response function, gives: ^= 1 + ^ + (^ ⊗ ℎ)∗ ⊗ ℎ∗ + |^ ⊗ ℎ|^ ⊗ ℎ∗or explicitly written out using the complex phase-only Transfer Function for the undesired twin-term: ^= 1 + ^ + ℑ^^(ℑ(^)∗^∗^∗) + |^ ⊗ ℎ|^ ⊗ ℎ∗where it is implicitly understood that: 84026PC01 17 Figure 2 is a schematic out-line of a holographic imaging system (HIS) according to the present invention. Thus, generally there is:- a lens, such as an infinity corrected objective lens, for imaging of anassociated object, such as a biological sample,- an image sensor positioned in an optical path (indicated as the central linethrough the system) of the lens arranged for capturing an intensity (I)representing an image of said object for holographic imaging,- an optional intermediate lens, such as a tube lens as shown here,positioned in said optical path between the objective lens and image sensor,- a complex phase modulator, here shown as a phase mask, representing aspatial transfer function H, the complex phase modulator being positioned in saidoptical path between a front of the lens and the image sensor, here shown behind a rear of said lens, and- a processor (not shown) operably connected to the image sensor forreconstructing a holographic rendering of the object.For holographic imaging a light source, such as a coherent light source, e.g. a laser or similar, is also conventionally used, but not shown here for clarity. Thus, the hardware part of the holographic imaging system comprises:• An infinity-corrected Objective lens which performs an optionally perfectoptical Fourier Transform from its front focal plane to its back focal plane indicated by fobj• A computed phase mask of Transfer Function H, which is inserted at ornear the back focal plane of the Objective lens (here shown just to the right of theback focal plane indicated with vertical dotted line)• A tube lens which performs a perfect or near-perfect optical FourierTransform corresponding to an inverse Fourier Transform with inverted coordinates (with image upside-down). Magnification is then given by ftube / fobj• An image sensor, which is detecting output intensity I located in the backfocal plane of tube lens as shown to the right in Figure 2. 84026PC01 18 Figures 3-6 show various holographic imaging systems (HIS) according to the present invention with different relative positions and variants of the optical components for implementing the invention: Figure 3 shows schematically two embodiments of a holographic imaging system (HIS) according to the present invention.Thus, in Figure 3A, a lens L, here an infinity corrected objective lens, is arrangedfor imaging of an associated object O, such as a biological sample, with variousoptical components including laser light LL entering at the top of the figure and being merged in an optical combiner unit OC into the optical path OP, the systemcomprising for example capturing lens and connecting optical fibers, e.g., imagingfiber bundles, from the object O toward the lens L and the phase modulator PM.In the embodiment the complex phase modulator PM is positioned outside and behind the rear lens of the lens L and at the back focal plane. The phase modulator is an axicon in this embodiment, but various other phase modulators are readily available for the skilled person as will be illustrated below. An imagesensor IS positioned in an optical path of the lens L is arranged for capturing animage of said object for holographic imaging. In this embodiment, an intermediatelens IL i.e. a tube lens is positioned in the optical path between the objective lensL and image sensor IS. The phase modulator (PM) having a spatial transferfunction H is thus positioned in said optical path between the lens L and theintermediate lens IL. In Figure 3B, the laser light is not shown for clarity and the figure is similar to Figure 3B, but here the phase modulator (also an axicon) is positioned inside thelens L being an infinity corrected objective lens, i.e., behind the front lens of thelens and in front of the rear lens of the lens L. Figure 4 shows schematically two embodiments of a holographic imaging system (HIS) with different positions of the complex phase modulator (PM) according to the present invention. Thus, similar to Figures 3A and 3B, the phase modulator i.e. an axicon in this embodiment can be positioned inside the lens L as in Figure 84026PC01 19 4A or outside the lens L as in Figure 4B i.e. between the lens L and the intermediate lens IL. Figure 5 shows schematically two other embodiments of a holographic imagingsystem (HIS) according to the present invention, where there is no intermediatelens and the optical path continues from the lens L to the phase modulator PM andimage sensor IS. In Figure 5B, the complex phase modulator is inside the lens Lhaving a front lens and a rear lens as indicated schematically by the two double arrows. Figure 6 shows schematically another embodiment of a holographic imagingsystem (HIS) according to the present invention similar to Figure 5B, but wherethe phase modulator PM is a diffractive axicon with a certain radial grating periodd and centre phase ^^. Simulations with this phase modulator will also bepresented below. Below some preferred embodiments will be explained based on the mathematical formalism described above in connection with Figures 1 and 2: Embodiment 1: Embodiment 3: ∗ ^^^(^^^^^ / ^^) ()( ( ( ∗ ^^^^^^^ / ^^) ℑ(^) ~ (^^^^^ / ^^)^exp^^^ ^^ ^ − 1^ ℑ ^) ) ~^^^(^^^^^ / ^^)^^^^^^^(^^)^^^^So that one gets for a phase-disk object, analytically that: Embodiment 4: 84026PC01 20Digital (and analog) free-space propagation using a Fresnel convolution kernel: When using so-called digital free-space propagation as Transfer Function H in thehardware i.e., the optical components, one may have to optimize the image sensorcontrast by dampening the reference DC light so that: ^(~0,0) ≪ 1.Also one should keep in mind that any natural / analog free-space propagation before the microscope objective has to be dealt with according to hardware total TransferFunction then being: ^^^^^^ = ^^^^^^^^^^^^^^^ where all H describes quadratic phasefunctions according to the above analog or digital z-distance dependence. So effectively one will simply have to add the analog and the digital z-values to havean effective total z for the overall quadratic phase Transfer Function.As an embodiment, for a microscopic phase disk one will thus have the followingcomplex field - falling on the image sensor - to contrast optimize:^(^; ^) = ^(~0,0) + where one can fully trust its validity for Fresnel number ^F < 1 / 8, but possiblyalso valid for larger Fresnel numbers. ^(~0,0) ∝ [1 − exp(^^^F)]For this Embodiment 4 there is an obvious limit on Total (analog + digital) z givenby the Jinc-shaped spill over from neighboring diffracted phase disks on the imagesensor creating undesired cross-interference effects. Advantageously, thisembodiment 4 can apply a thin lens having a similar spatial Transfer Function asthe quadratic phase H.Embodiment 5: Inserting a phase ramp Transfer Function - with ^(~0,0) = 1 - onehas that: 84026PC01 21 ^∗^∗ = exp^−2^^(^^^ + ^^^)^ => ℎ∗ ⊗ ℎ∗ = ^(−^ − ^, −^ − ^)Embodiment 6: Inserting a convex lens with focal length, f, corresponding toembodiment 4: Embodiment 7: Inserting a mathematic “k-power lens” phase Transfer Functionwith virtual focal length, f, corresponding to: In Figure 7, simulations with a quadratic power lens (k = 2) are given. In sectionA, the input free space phase is shown, section B shows the “CCD” intensity on theimage sensor. Section C shows reconstructed free space amplitude together withthe free space phase in section D. Sections E and F show reconstructed amplitudeand phase, respectively. It is seen in Sections E and F that merging or interferencebetween the four original input points is taking place.Embodiment 8: Inserting an “axicon lens” with virtual focal length, f,corresponding to: where one finds the conjugated axicon phase and corresponding convolutionkernel as: ^∗ = exp(^2^^^^^) ⇒ ℎ∗ = ^(^ − ^^)Embodiment: Axicon-based or diffractive or A diffractive axicon with radial grating period, d, and center phase, ^^, as in Figure 6 can be expressed by: 84026PC01 22 where one has the relation:^^ ^^= ^Binary-^ diffractive axicon gives superposed ±1. order:^ ^^= ^^exp The conjugated diffractive axicon is given by: Giving rise to the corresponding conjugate convolution kernel: ℎ∗ = exp(−^^^) ^(^ − ^^)Re-calling terms from the reconstruction: where inserting ℎ∗gives while ignoring the constant exp(−^^^): ^= 1 + ^ + exp(−2^^^) ^∗ ⊗ ^(^ − 2^^) + exp(−^^^) |^∗ ⊗ ^(^ − ^^)|^ ⊗ ^(^ − ^^)that can be simplified to: ^= 1 + ^ + [exp(−2^^^) ^∗ + exp(−^^^) |^|^] ⊗ ^(^ − 2^^)Alternatively, by using the original axicon (e.g., in hardware) one can recreate the conjugate object: ^= 1 + ^∗ + [exp(2^^^) ^ + exp(^^^) |^|^] ⊗ ^(^ − 2^^) 84026PC01 23 Image sensor read-out can be verified as: ^= 1 + (^∗ ⊗ ℎ∗)∗ + (^∗ ⊗ ℎ∗) + |^∗ ⊗ ℎ∗|^ ⇒ which mathematically is describing “ring co-location” of all holographic terms onthe image sensor i.e., enabling that the influence of the cross-interfering term(O*R) and self-interfering term (OO*) in the holographic equation can be minimal or eliminated resulting in improved image quality for holographic imaging and microscopy. Axicon cone-tilt reconstruction analysisOne has the conjugate convolution kernel and conjugate transfer function fromabove, respectively: The corresponding convolution kernel and transfer function can therefore be written as: ℎ= exp(^^^) ^(^ + ^^)^ = exp(^^^) exp ^−^2^^^^^ ^^ ^ From before, one can now write the image sensor read-out using direct(unphysical) mathematical terms:^ = 1 + (^ ⊗ ℎ) + (^∗ ⊗ ℎ∗) + |^∗ ⊗ ℎ∗|^ ⇒^ = 1 + exp(^^^) ^ ⊗ ^(^ + ^^) + exp(−^^^) ^∗ ⊗ ^(^ − ^^) + |^|^ ⊗ ^(^ − ^^)Reconstructing with conjugate kernel ℎ^∗ = exp(−^^^^) ^(^ − ^^̃), one obtains: 84026PC01 24 Writing primed kernel radius as a function of actual kernel radius; ^^̃ = ^^^, onehas: So, for the matching conditions where k = 1, one gets: or simply: Figure 8 shows similar simulations to Figure 7 but with various values of k lowerthan 2 of reconstructed phases ; i.e., with section A having k=1.50, section B withk=1.40, section C with k=1.30, and section D with k=1.27. In the last two section with k=1.27 and k=1.30, it is evident that the cross-interfering term (O*R) and self-interfering term (OO*) are displaced to a periphery of the space, which can minimize or remove their influence on the reconstructed image by using thecentre part of the space. The process depicted in Figure 8 may also be describedas a k-sweep i.e. a gradual change of the k-value to see more clearly the effect ofthis parameter. Figure 9 shows further simulations with an axicon phase mask similar to the simulations in Figure 7, where again it is clear that the present invention providesa separation to the periphery of undesirable self- and cross-interfering terms. 84026PC01 25 Diffractive prism-based approachA diffractive x-directed prism with ramp-grating period, d, and center phase, ^^, -as in Figure 6 - is given by: where one has the relation: Binary-^ diffractive x-directed prism gives superposed ±1. order:^ ^ = ^^exp The conjugated diffractive x-directed prism is given by: Giving rise to the corresponding conjugate convolution kernel: ℎ∗ = exp(−^^^) ^(|^| − ^^)Re-calling terms from the reconstruction: ^= 1 + ^ + (^ ⊗ ℎ)∗ ⊗ ℎ∗ + |^ ⊗ ℎ|^ ⊗ ℎ∗ ⇒^ = 1 + ^ + ^∗ ⊗ (ℎ∗ ⊗ ℎ∗) + |^∗ ⊗ ℎ∗|^ ⊗ ℎ∗where inserting ℎ∗gives while ignoring the constant exp(−^^^): ^= 1 + ^ + ^∗ ⊗ ^(|^| − 2^^) + |^∗ ⊗ ^(|^| − ^^)|^ ⊗ ^(|^| − ^^)that can be simplified to: ^= 1 + ^ + [^∗ + |^|^] ⊗ ^(|^| − 2^^) 84026PC01 26 Image sensor read-out can also be verified as: ^= 1 + (^∗ ⊗ ℎ∗)∗ + (^∗ ⊗ ℎ∗) + |^∗ ⊗ ℎ∗|^ ⇒ Mathematically describing conjugated “co-location” of the holographic terms onthe image sensor. Thus, when performing on-axis holographic imaging accordingto the present invention, the influence of the cross- and / or self-interfering termscan be minimized by displacing the reconstructed cross-interfering term (O*R) and / or self-interfering term (OO*) away from a signal of interest (SoI) in the reconstructed phase space. For example, a signal of interest (SoI) in the reconstructed phase space can comprise the real image (OR*). General reconstruction analysis Using the principle of the stationary phase method one can alternatively find thelocal spatial frequency of an optical wavefront modulating element as a spatialderivative of its spatial phase in cartesian or polar coordinates: Using these expressions makes it possible to conveniently design arbitraryconvolution kernels using the so-called Eikonal Equations.One particularly simple embodiment is the aforementioned axicon-phase that with high accuracy leads to a simple circular ring as convolution kernel: Figure 10 shows a Fourier phase mask and the simulated corresponding CCDintensity pattern on an image sensor. Again, it is evident in the CCD image thatthe invention provides a separation of noise or interference terms to the periphery, which can be exploited for minimize or eliminate the influence of these 84026PC01 27noise terms. The skilled person will readily understand the image sensor canalternatively be a CMOS based image sensor, or other suitable image sensorsavailable in connection with optical imaging, in particular holographic imaging.Figure 11 and Figure 12 shows reconstructed images from numerical experimentsfor verifying the present invention. The numerical experiments convincinglydemonstrate that it is possible to minimize the influence of the cross- and / or self-interfering terms in the holographic equation for on-axis holographic imaging in a very advantageous way.Figure 13 is a schematic of a lens-less embodiment of the invention using aconverging light wave for illuminating the objects for holographic recording similarto Figures 3-6. The lens-less embodiment can also be adjusted for both transmission, reflection or hybrid geometries. Figure 14 is a schematic system-chart representing an out-line of the operationsof a method according to the invention for performing on-axis holographic imagingwith a holographic imaging system (HIS) as shown in Figures 1 and 2, the HIScomprising:- a lens (L), such as an infinity corrected objective lens, for imaging of anassociated object, such as a biological sample,- an image sensor (IS) positioned in an optical path of the lens arranged forcapturing an intensity pattern (I) representing an image of said object for holographic imaging,- an optional intermediate lens (IL), such as a tube lens, positioned in saidoptical path between the objective lens and image sensor,- a complex phase modulator (PM) representing a spatial transfer function H,preferably an adjustable transfer function, the complex phase modulator being positioned in said optical path between a front of the lens and the image sensor, optionally the intermediate lens, preferably between the front and a rear of said lens, and- a processor operably connected to the image sensor for reconstructing aholographic rendering of the object, the method comprising: 84026PC01 28S1 capturing a spatial signal indicative of an intensity pattern (I) representingthe object for holographic imaging through the lens,S2 transmitting said spatial signal through the lens and transmitting the spatialsignal through the complex phase modulator PM and the optional intermediate lens,S3 detecting a spatial signal at the image sensor, andS4 processing the detected spatial signal so as to obtain a reconstructedholographic rendering of the object,S5 processing of the detected spatial signal using the conjugated spatial transferfunction H* corresponding to the said complex phase modulator (PM), said spatialtransfer function H being selected so as to minimize, preferably eliminate, theinfluence of the cross-interfering term (O*R) and / or self-interfering term (OO*) inthe holographic equation of the holographic imaging system (HIS) for improvedimage quality, cf. Figure 12 for some advantageous results of the presentinvention.In short, the invention relates to a method for performing on-axis holographicimaging with a holographic imaging system (HIS), where a complex phasemodulator (PM) represents a spatial transfer function H being positioned in anoptical path between a front of the lens and the image sensor, cf. Figure 2. Processing is performed for holographic rendering of the object. The processing isperformed using the conjugated spatial transfer function H* corresponding to thecomplex phase modulator (PM), where the spatial transfer function H is selected so as to minimize, preferably eliminate, the influence of the cross-interfering term(O*R) and / or self-interfering term (OO*) in the holographic equation of theholographic imaging system (HIS) for improved image quality. Careful selection of the complex phase modulator, and the corresponding processing circumvents on-axis holographic artifacts from cross- and self-interfering terms that can make itdifficult to reconstruct for example the original 3D scene of potentially partly or fully transparent objects with sufficient fidelity.The invention can be implemented by means of hardware, software, firmware orany combination of these. The invention or some of the features thereof can also be implemented as software running on one or more data processors and / or 84026PC01 29 digital signal processors. In particular the invention may be computer- implemented as a method with one or more steps being implemented on a computer with corresponding software. The individual elements of an embodiment of the invention may be physically,functionally and logically implemented in any suitable way such as in a single unit,in a plurality of units or as part of separate functional units. The invention may be implemented in a single unit, or be both physically and functionally distributed between different units and processors. Although the present invention has been described in connection with the specified embodiments, it should not be construed as being in any way limited to the presented examples. The scope of the present invention is to be interpreted inthe light of the accompanying claim set. In the context of the claims, the terms“comprising” or “comprises” do not exclude other possible elements or steps. Also,the mentioning of references such as “a” or “an” etc. should not be construed as excluding a plurality. The use of reference signs in the claims with respect to elements indicated in the figures shall also not be construed as limiting the scopeof the invention. Furthermore, individual features mentioned in different claims,may possibly be advantageously combined, and the mentioning of these features in different claims does not exclude that a combination of features is not possible and advantageous.

Claims

84026PC01 30 Claims1. A method for performing on-axis holographic imaging with a holographicimaging system (HIS), preferably a digital holographic microscope, the HIScomprising:- a lens (L), such as an infinity corrected objective lens, for imaging of anassociated object, such as a biological sample,- an image sensor (IS) positioned in an optical path of the lens arranged forcapturing an intensity pattern (I) representing an image of said object forholographic imaging,- an optional intermediate lens (IL), such as a tube lens, positioned in saidoptical path between the objective lens and image sensor,- a complex phase modulator (PM) representing a spatial transfer function H,preferably an adjustable transfer function, the complex phase modulator beingpositioned in said optical path between a front of the lens and the image sensor,optionally the intermediate lens, preferably between the front and a rear of saidlens, and- a processor operably connected to the image sensor for reconstructing aholographic rendering of the object,the method comprising:- capturing a spatial signal indicative of an intensity pattern (I) representingthe object for holographic imaging through the lens,- transmitting said spatial signal through the lens and transmitting thespatial signal through the complex phase modulator and the optional intermediatelens,- detecting a spatial signal at the image sensor, and- processing the detected spatial signal so as to obtain a reconstructedholographic rendering of the object,wherein the processing of the detected spatial signal is performed using theconjugated spatial transfer function H* corresponding to the said complex phasemodulator (PM), said spatial transfer function H being selected so as to minimize,preferably eliminate, the influence of the cross-interfering term (O*R) and / or self-interfering term (OO*) in the holographic equation of the holographic imaging84026PC01 31system (HIS) for improved image quality.

2. A method for performing on-axis holographic imaging according to claim 1,wherein said spatial transfer function H is chosen as so to minimize the influenceon the signal to noise ratio (SNR) from the sum of the reconstructed cross-interfering term (O*R) and self-interfering term (OO*).

3. A method for performing on-axis holographic imaging according to claim 1 or 2,wherein said spatial transfer function H is chosen as so to maximize the signal tonoise ratio (SNR) of the sum of the reconstructed object (OR*) and optionallybackground (RR*).

4. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein the influence of the cross- and / or self-interfering termsis minimized by displacing the cross-interfering term (O*R) and / or self-interferingterm (OO*) away from a signal of interest (SoI) in the reconstructed space.

5. A method for performing on-axis holographic imaging according to claim 4, wherein said signal of interest (SoI) in the reconstructed space comprises the realimage (OR*).

6. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein an expression of the spatial transfer function H of thecomplex phase modulator (PM) is estimated using the principle of the stationaryphase method, where a local spatial frequency of the complex phase modulator(PM) is estimated as a first-order spatial derivative of its spatial phase, and saidestimated expression of the spatial transfer function H is applied in saidprocessing of the detected optical signal using the conjugated spatial transferfunction H* corresponding to the said complex phase modulator (PM).

7. A method for performing on-axis holographic imaging according to claim 6,wherein a local spatial frequency, f, of a phase element is estimated as a spatialderivative of its spatial phase in cartesian or polar coordinates, respectively as:84026PC01 32where φ is the phase in cartesian coordinates or polar coordinates.

8. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein said lens is not comprised in the HIS, the intensitypattern representing the object being already convergent, or divergent, into theHIS onto the complex phase modulator (PM).

9. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein the complex phase modulator (PM) representing a spatialtransfer function H, said spatial transfer function, H, having a multiplexing of phasecontrast and / or dark field filtering.

10. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein the lens (L) is positioned in the optical path so as toperform an optical Fourier transformation of the object with the complex phasemodulator (PM) being positioned near, or at, the back focal plane (BF_OL) of the lens.

11. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein the intermediate lens (IL) is positioned in the opticalpath so as to perform an inverse Fourier transformation of the object - relative tothe said objective lens – with the image sensor being positioned near, or at, theback focal plane (BF_IL) of the intermediate lens.

12. A method for performing on-axis holographic imaging according to any ofpreceding claims, wherein the complex phase modulator (PM) is an active opticalcomponent, preferably being an adjustable phase-only phase modulator, such asa spatial light modulator (SLM).

13. A method for performing on-axis holographic imaging according to any of theclaims 1-11, wherein the complex phase modulator (PM) is a passive optical84026PC01 33component, preferably being a lens, more preferably an axicon lens or adiffractive, or reflective, axicon lens with a radial grating period (d).

14. A method for performing on-axis holographic imaging according to any of thepreceding claims, wherein the complex phase modulator is chosen so that thespatial transfer function, H, can be expressed in a mathematical form of the type:H = A_1 ∙ exp (-i ∙A_2 ∙ r k)where r is a radial coordinate, A_1 is a constant, and A_2 is another constantdepending on the wavelength of the light and the phase modulator, andwherein the exponent k is chosen as k≠2, preferably k < 2, more preferably k <1.3, andwherein the complex phase modulator is further chosen so that the influence ofthe cross- and / or self-interfering terms is minimized by displacing thereconstructed cross-interfering term (O*R) and / or self-interfering term (OO*) away from a signal of interest (SoI) in the reconstructed phase space.

15. A method for performing on-axis holographic imaging according to any of the preceding claims, wherein the processing of the detected spatial signal isperformed using the non-conjugated spatial transfer function H corresponding tothe said complex phase modulator (PM), said spatial transfer function H being selected so as to minimize, preferably eliminate, the influence of the term (OR*)representing the real image and / or term (O*O) in the holographic equation of theholographic imaging system (HIS) for improved image quality.84026PC01 3416. A holographic imaging system (HIS) for performing on-axis holographicimaging, preferably a digital holographic microscope, the HIS comprising:- a lens (L), such as an infinity corrected objective lens, for imaging of anassociated object, such as a biological sample,- an image sensor (IS) positioned in an optical path of the lens arranged forcapturing an intensity (I) pattern representing an image of said object for holographic imaging,- an optional intermediate lens (IL), such as a tube lens, positioned in saidoptical path between the objective lens and image sensor,- a complex phase modulator (PM) representing a spatial transfer function H,preferably an adjustable transfer function, the complex phase modulator being positioned in said optical path between a front of the lens and the image sensor, optionally the intermediate lens, preferably between the front and a rear of said lens, and- a processor operably connected to the image sensor for reconstructing aholographic rendering of the object,wherein the system is arranged for:- capturing a spatial signal indicative of an intensity (I) pattern representingthe object for holographic imaging through the lens,- transmitting said spatial signal through the lens and transmitting thespatial signal through the complex phase modulator and the optional intermediatelens,- detecting a spatial signal at the image sensor,- processing the detected spatial signal so as to obtain a reconstructedholographic rendering of the object,wherein the processor is arranged for processing the detected spatial signal byusing the conjugated spatial transfer function H* corresponding to the saidcomplex phase modulator (PM), said spatial transfer function H being selected soas to minimize, preferably eliminate, the influence of the cross-interfering term (O*R) and / or self-interfering term (OO*) in the holographic equation of the HIS for improved image quality.84026PC01 3517. Use of the holographic imaging system according to claim 16 for determiningan object featuring one or more of:- micro-plastics suspended in a liquid,- biological cells, such as blood cells, or platelets in bodily fluids,- aerosols or particles in air,- plankton or algae in underwater sampling, and- reflecting or transparent specimens with rough or micro-structured surfaces.

18. A computer-program being adapted to enable a computer system comprising at least one computer having data storage means in connection therewith to control a holographic imaging system according to claim 16.

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