Abstract

Interferenceless coded aperture correlation holography (I-COACH) is an incoherent opto-digital technique for imaging 3D objects. In I-COACH, the light scattered from an object is modulated by a coded phase mask (CPM) and then recorded by a digital camera as an object digital hologram. To reconstruct the image, the object hologram is cross-correlated with the point spread function (PSF)—the intensity response to a point at the same object’s axial location recorded with the same CPM. So far in I-COACH systems, the light from each object point has scattered over the whole camera area. Hence, the signal-to-noise ratio per camera pixel is lower in comparison to the direct imaging in which each point is imaged to a single image point. In this work, we consider the midway between the camera responses of a single point and of a continuous pattern over the entire camera area. The light in this study is focused onto a set of dots randomly distributed over the camera plane. With this technique, we show that there is a PSF with a best number of dots, yielding an image with a maximum product of the signal-to-noise ratio and the image visibility and a maximum value of structural similarity.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical imaging can be roughly classified to direct and indirect imaging [1,2]. In direct imaging the object is replicated directly on the sensing medium, which can be the retina of the eye, the photograph plate, the sensing matrix of the digital camera, or any other device that can sense the intensity distribution of light. In the other less common class of indirect imaging, the image of the object is not recorded directly by the camera, but instead, some other intensity distribution is recorded and processed to yield the final image. Digital holography [3,4], coded aperture imaging [5–10], integral imaging [11,12], synthetic aperture imaging [13,14] and light-field imaging [15,16] are just a few examples of indirect imaging techniques. Interferenceless coded aperture correlation holography (I-COACH) [17,18] is another indirect imaging technique with several advantages over other imaging methods. I-COACH can image 3D scene by a single camera shot [19,20], or through a partial aperture [21], or by a synthetic aperture [22], or with an enhanced image resolution [23,24], or with an enhanced field of view [25]. I-COACH can even image 3D objects through a scattering layer [26,27]. All these benefits justify a further intensive research and development of this new imaging concept, as is indeed done in the present study.

To understand what should further be improved in I-COACH, let us briefly summarize its main principles. The I-COACH system is first calibrated by a series of object points (guide- stars in the technical jargon) distributed along the Z axis. Each point emits spherical wave that propagates through a coded phase mask (CPM). The light from the CPM is recorded by a digital camera and stored as a library of digital point spread functions (PSFs). Following the one-time calibration stage, an incoherently illuminated object is positioned inside the axial range of the calibration stage. The object light propagates through the same CPM and is recorded as an object hologram. The 3D image is reconstructed by a series of 2D cross-correlations between the object hologram and the corresponding PSFs from the PSF library.

One of the weaknesses of I-COACH is the low signal-to-noise ratio (SNR) of the reconstructed images in comparison to the conventional direct imaging. Several techniques of noise reduction in the cross-correlation process were proposed [28–31]. However, further improvements with less sacrifices might make this technology more practical. In the present study we present a new method for increasing the SNR that can be applied to the other previously proposed methods [19,21,22,25]. To understand the proposed strategy, let us compare the camera responses of direct and indirect imaging for a single point object. In direct imaging, each object point is imaged to a corresponding point on the camera plane. On the other hand, in the previous I-COACH systems [17,18], the detected intensity pattern induced from any object point spreads over the entire area of the camera, or over a high percentage of this area. Consequently, the signal per camera pixel is much lower than in the case of direct imaging, and for a given noise level of the camera, the SNR is thus lower than that of the direct imaging. Between the two extreme responses of a single point of the direct imaging and the continuous chaotic pattern of I-COACH, we propose herein the sparse response of isolated randomly distributed dots. In other words, the method in this study is actually CPM engineering with the goal to design a sparse response, or more precisely, a sparse PSF. How much the PSF can be sparse? Enhancing the sparsity reduces the number of dots in the PSFs, and as we see in the following, PSFs with less dots increase the background noise of the reconstructed image. Therefore, there is a best sparsity which keeps the two types of noise as low as possible. This study presents the modified I-COACH, dubbed sparse I-COACH (SI-COACH), with engineered CPMs to obtain the best PSF. This PSF is designed as a set of isolated dots randomly distributed over a limited area of the camera, where the density of the dots is determined to yield images with maximum SNR and sharpness. The PSF sparsity is defined such that between any two PSF dots having a diffraction limited size, there is at least one zero valued pixel.

To understand which of the other features of I-COACH are sacrificed for the improved image quality, we realize that the PSF is a set of focused dots obtained as a response for a point source located at a specific Z plane. Light from source points outside this specific axial plane are out of focus on the camera plane. The PSFs from out-of-focus points cannot reconstruct images with acceptable quality, because the autocorrelation of such PSF is far from the desired sharp peak. Therefore, the proposed SI-COACH is 2D and the feature of 3D imaging of the classical I-COACH is sacrificed for the higher SNR. However, it should be emphasized that the advantages of I-COACH over direct imaging mentioned in [19–27] exist also in its 2D version. Therefore, on one hand, these unique advantages justify the trade-off of 3D imaging for higher SNR. On the other hand, the proposed SI-COACH can be easily extended for 3D imaging by multiplexing several CPMs with different diffractive lenses, as is demonstrated in experimental section. The next section describes the method for engineering the CPMs. The experiments are discussed in the third section, followed by the conclusion in the last section.

2. Methodology

The optical configuration of SI-COACH system shown in Fig. 1 is identical to the systems proposed in [20,23]. The only difference between these and the current system is the CPM displayed on the spatial light modulator (SLM). The CPM constructed in the proposed method modulates the light such that all the light originated from a point source is focused into a set of randomly distributed dots on the camera plane. The pattern of isolated dots is in contrast to the pattern of scattered intensity distribution of the classical I-COACH [18,20,23].

 

Fig. 1 Optical configuration of SI-COACH. L0, L1 – Refractive lenses: P- Polarizer; SLM - Spatial light modulator; DL – Diffractive lens; CPM - Coded phase mask.

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The operation principle of the optical configuration of Fig. 1 is described as follows. Light from the incoherent source critically illuminates the object using the refractive lens L0. The light diffracted from the object is collimated by lens L1 located a distance of zs from the object. The collimation of the light is achieved by choosing the distance zs to be equal to the focal length f of L1. The collimated light beyond L1 is incident on the SLM on which the combination of CPM and a diffractive lens (DL) is displayed. The image sensor is positioned at the focal plane of the DL, such that without the CPM, light from an object point is focused on the camera plane.

The optical system is calibrated once, by recording the PSF of the system. The CPMs for the proposed method are generated by a modified version of the Gerchberg-Saxton algorithm (GSA) [32]. The schematic of the CPM generation is shown in Fig. 2. The iterative algorithm starts with an initial random phase and a uniform amplitude at the CPM plane. The complex matrix is Fourier transformed to the Fourier (camera) plane, in which the amplitude of the transformed function is replaced by predefined randomly distributed dots. The dots are distributed on a defined area on the camera plane and their phase values are left unchanged for the next iteration. The algorithm is run till the difference between two successive matrices is negligible.

 

Fig. 2 Modified Gerchberg-Saxton algorithm for synthesizing the CPMs.

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Based on our previous studies [19], the intensity response to a point at (r¯s,zs)=(xs,ys,f) is the magnitude square of the scaled Fourier transform of the CPM multiplied by a linear phase function L(r¯s/f)=exp[i2πr¯sr¯/λf], as follows,

IPSF(r¯o;r¯s)=|ν[1λzh]F{C0L(r¯sf)exp[iΦ(r¯)]}|2=IPSF(r¯ozhfr¯s;0),
where C0 is a complex constant, Φ(r¯)is the phase distribution of the CPM synthesized using the modified GSA and zh is the focal length of the diffractive lens attached to the CPM.F{}is the operator of 2D Fourier transform and ν[·] is the scaling operator defined by the relation ν[α]f(x) = f(αx). The intensity on the camera plane is a shifted (by r¯szh/f) version of the intensity response to a point object located on the optical axis (r¯s=0). The PSF for a point located at (0,0,-zs = -f1) is a scaled Fourier transform of the CPM, and hence the Fourier condition of the GSA is satisfied in the experimental setup.

A 2D object illuminated by a spatially incoherent light and located at the same distance zs = f from the lens L1 can be considered as a collection of N uncorrelated object points given as,

o(r¯s)=jNajδ(r¯r¯j).
Since the system is linear, the intensity distribution for the 2D object on the camera plane is a sum of all the shifted point responses, given by,
IOBJ(r¯o)=jajIPSF(r¯ozhfr¯j;0).
Both IOBJ and IPSF are real positive functions. The recovery of the image can be performed by a cross-correlation between them. However, a cross-correlation between two real positive yields undesired background distribution around the recovered image [6]. Based on previous studies [17–27], it is known that the background noise of the reconstructed image is much lower when the reconstruction is done between two bipolar functions. The bipolar PSF (HPSF) is created by recording two responses with two independent CPMs and subtracting one intensity response from the other. The bipolar object response (HOBJ) is created similarly with the same CPMs. The goal of subtracting one pattern from the other is to decrease the bias terms which are the major contributor of the background noise around the reconstructed image. Hence the bipolar distribution for the point object and for the object are,
HPSF(r¯o)=IPSF,1(r¯o)IPSF,2(r¯o).HOBJ(r¯o)=IOBJ,1(r¯o)IOBJ,2(r¯o).
The object reconstruction is done by cross-correlating HOBJ with the phase-only filtered version of HPSF given by HPSF(r¯o)=F1{exp[iargF{HPSF(r¯o)}]}in order to remove the background noise further [18]. Cross-correlating HOBJ with the phase-only filtered version of the HPSF sharpens the correlation peaks used in our method as the points of the reconstructed image [18]. The reconstruction of the object can be written as
IREC=HOBJHPSF=F1{F{HOBJHPSF}}=F1{H˜OBJH˜PSF*}=F1{H˜OBJexp(iarg{H˜PSF})},
where H˜OBJ,H˜PSFand H˜PSFare the 2D Fourier transforms of HOBJ,HPSFand HPSF, respectively. The symbol indicates cross-correlation, defined as fg=f(x,y)g*(xx',yy')dxdy,where in the case f = g this integral becomes autocorrelation.

As mentioned above, the proposed PSF in this work is a randomly distributed collection of bipolar dots. To quantify the performances of the system with the proposed PSF we consider the SNR of the reconstructed image for a point object at the origin of the system input. Assuming both the object and the PSF are accompanied by a random noise, the reconstructed image is formulated as,

IREC=(HOBJ+n1)(HPSF+n2)=(HPSF+n1)(HPSF+n2)HPSFHPSF+HPSFn2+HPSFn1,
Where n1 and n2 are different random noise distributions obtained on the camera plane, and it is assumed that the cross-correlation between them is low enough to justify a neglection. The SNR of the reconstructed image is defined as the ratio between the correlation peak of the reconstructed image point and the maximum value of cross-correlations outside the origin,
SNR=HPSFHPSF|(x,y)=(0,0)Max(x,y)(0,0){HPSFHPSF+HPSFn2+HPSFn1},
The numerator of Eq. (7) is not dependent on the shape of the PSF because the CPMs for all the PSFs are phase-only functions and based on Parseval’s theorem the peak value of the autocorrelation of the entire PSFs is the same. The difference between the SNRs are in the denominator. Under relatively low illumination power and for a continuous chaotic PSF over the camera window, the dominant term of the denominator is Max{HPSFn2+HPSFn1},the maximal value of the cross-correlation with the noise n1 and n2. Moreover, this value increases linearly with the width of the camera window. However, when the same illumination power is distributed between reduced number of sparse dots, we can neglect the cross-correlation with the noise and consider only the term Max(x,y)(0,0){HPSFHPSF} as the source of the background noise. Assuming the reconstruction point image is an autocorrelation of randomly distributed bipolar dots, and magnitude of the entire dots are approximately the same, the SNR is the ratio between the total number of dots and the maximal number of overlaps, in sign and in location, between dots outside the origin. For example, if the PSF is only 2 dots, somewhere outside the origin there are two events of a cross-correlation between two dots and hence the SNR is 2. For higher number of dots, the number of overlaps is random and hence the SNR cannot be predicted in advance but can be measured experimentally as is described in the next section. However, between the SNR = 2 of two dots and the SNR of the continuous chaotic PSF which is sensitive to the noise of the camera and to the size of the camera window, we can predict that higher SNR than these two extremes, with relatively poor SNRs, is expected.

3. Experiments and results

The experimental verification of the proposed technique was carried out using a conventional I-COACH setup shown in Fig. 3. The experimental setup consists of illumination with LEDs (Thorlabs LED635L, 170 mW, λ = 635 nm, Δλ = 15 nm). Lens L0A and L0B were used to critically illuminate the objects. In channel 1, a pinhole with a size of 10 μm was mounted for capturing the PSF. In channel 2, element 18 of National Bureau of Standards (NBS) resolution target was illuminated using the second LED. The object and the pinhole were positioned at the front focal plane of lens L1 at a distance of 17.5 cm from L1. The distance between the lens L1 and the phase-only reflective SLM (Holoeye PLUTO, 1920 × 1080 pixels, 8 μm pixel pitch, phase-only modulation) was 20.5 cm. The distance between the lens L1 and the beam splitter BS2 was 15 cm. The distance between the SLM and the digital camera (Thorlabs 8051-M-USB, 3296 × 2472 pixels, 5.5 μm pixel pitch, monochrome) was zh = 40 cm.

 

Fig. 3 Experimental setup. BS1 and BS2 – Beam splitters; SLM – Spatial light modulator; NBS – National Bureau of Standards ; L0A, L0B and L1 – Refractive lenses; LED1 and LED2 – Identical light emitting diodes; CPM – Coded phase mask; DL – Diffractive lens; BPF – Band pass filter (λc = 632.8 nm and Δλ = 5 nm); P- Polarizer; - Polarization orientation perpendicular to the plane of the page.

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The displayed phase pattern was constructed by modulo-2π phase addition of the CPM with the DL of focal length zh = 40 cm. The CPM was constructed using the GSA and the DL was combined to satisfy the Fourier relation of GSA between the SLM and camera plane. CPMs had the size of 1080 × 1080 px and the random dots were distributed over the window size of 500 × 500 px. An important parameter to control the SNR of the process is the dot density ψ defined as the ratio between the actual number of dots and the maximal number of dots in some defined area. CPMs were constructed with 28 values of dot number between 11 to 201600 in variable differences, indicating a dot density between ψ = 44·10−6 to 0.08064. CPMs, PSFs, HPSF, object responses and HOBJ for the density ψ = 0.000164 are shown in Fig. 4. In order to compare SI-COACH with the conventional I-COACH, CPMs generating a continuous chaotic response over a camera window of 500 × 500 px was also constructed. For each of the above described cases, two uncorrelated CPMs were computed and the intensity responses to them were recorded. Bipolar HPSF for the pinhole, and HOBJ for the object were calculated according to Eq. (4). For comparison purposes, the exposure times of the camera for the entire pinhole and object responses were set to 800 and 300 μs, respectively. The Object reconstruction was done by cross-correlating the HOBJ and HPSF using Phase-only filter.

 

Fig. 4 (a-b) Images of the CPMs, (c-d) intensity PSFs and (e) bipolar PSF, (f-g) object intensity responses and (h) bipolar HOBJ for dot density of ψ = 0.000164.

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In order to quantify the reconstruction results for different dot densities, visibility and SNR have been calculated. Visibility was calculated as ν=(ImaxImin)/(Imax+Imin),where Imax and Imin are the maximum and minimum intensity values of an averaged line profile along the gratings of the reconstructed object. In the SNR calculation, the signal was calculated by averaging over the reconstructed object, whereas the noise was calculated by averaging the background around the reconstructed object. Finally, to determine the best dot density ψo a normalized product of the visibility and SNR as function of ψ was calculated as ξ(ψ)=[ν(ψ)SNR(ψ)]/(max{ν}max{SNR}) and the ψ having the highest ξ(ψ) has been considered as the best dot density. Another parameter that had been considered for determination of the best dot density ψo was structural similarity (SSIM) [33]. SSIM was calculated between the reconstructed image of object and direct imaging. SSIM has been added to the study in order to support the conclusions obtained based on the behavior ofξ(ψ). The whole analysis was done for three different levels of the LED optical power to demonstrate the behavior of the proposed technique in different illumination conditions. The levels of the optical power illuminating the object were 3.5mW (low power), 7 mW (mid power) and 10mW (high power). The experimental results and their analysis for each case are discussed below.

Visibility, SNR, ξ(ψ) and SSIM plots of SI-COACH for different dot densities in case of low power illumination are shown in Figs. 5(a)–5(d) by the blue curves, whereas the results of the conventional full windowed I-COACH are plotted by the red lines. The maximum visibility of 0.395 is noted for dot density of 0.000244, whereas the maximum SNR of 7.7 is obtained for dot density of 0.000204. In case of the full windowed I-COACH, the visibility and SNR are only 0.2194 and 0.9594, respectively. According to Figs. 5(c) and 5(d), the best dot density is 0.000164. Reconstruction results for the best SI-COACH case and for the full window I-COACH are shown in Figs. 5(e) and 5(f), respectively.

 

Fig. 5 (a) Visibility, (b) SNR, (c) ξ(ψ), (d) SSIM, (e) Reconstructed image of SI-COACH for the best dot density and (f) Reconstructed image for full window I-COACH in case of low power illumination of the object.

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Similar results in case of mid power illumination are shown in Fig. 6. The maximum visibility of 0.372 is obtained for dot density of 0.000244, whereas the maximum SNR of 7.92 is noted for dot density of 0.00224. In full windowed I-COACH, the visibility and SNR are only 0.1723 and 1.5936, respectively. The best dot density was calculated from Figs. 6(c) and 6(d) at 0.000164. Reconstruction results for the best dot density of SI-COACH and for full window I-COACH are shown in Figs. 6(e) and 6(f), respectively.

 

Fig. 6 (a) Visibility, (b) SNR, (c) ξ(ψ), (d) SSIM, (e) Reconstructed image of SI-COACH for best dot density and (f) Reconstructed image for full window I-COACH in case of mid power illumination of the object.

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Visibility, SNR, ξ(ψ)and SSIM plots for different dot densities in case of high power illumination are shown in Figs. 7(a)–7(d) by the blue curves. As before, the continuous full windowed I-COACH is plotted by the red lines. The maximum visibility of 0.295 is obtained for dot density of 0.000244 as in the previous two cases, whereas the maximum SNR of 7.8 happens for dot density of 0.000204. In the full windowed I-COACH, the visibility and SNR are 0.1631 and 1.7542 respectively, values which are below the compared values of the proposed technique. As in the two previous cases, the best dot density of SI-COACH obtained from Figs. 7(a)–7(d) is at dot density of 0.000164. Reconstruction result of SI-COACH for the best parameters and full window I-COACH are shown in Figs. 7(e) and 7(f), respectively.

 

Fig. 7 (a) Visibility, (b) SNR, (c) ξ(ψ), (d) SSIM, (e) Reconstructed image of SI-COACH for the best dot density and (f) Reconstructed image for full window I-COACH in case of high power illumination of the object.

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To summarize, in each of the above cases, the best dot density is 0.000164, although we suspect that there is nothing special in this value and probably for different object size and other parameters the best density value would be different. Comparing with the conventional I-COACH which uses the full window of 500 × 500 px resulting in total of 250,000 points, in the proposed method with dot density of 0.000164, the PSF has only 41 dots. Theoretically, the average power per pixel is increased by a factor of (500·500/41) = 6097 in the proposed technique. In the experiment, the full window CPM generates an intensity response of size of 2090 × 2090 camera pixels for the PSF. In the SI-COACH case, the pinhole generates a dot pattern on the camera, where each dot has a diameter of 5.13 px. Hence, the increment in the average power per pixel is by a factor of (2090·2090/(41·π·2.572)) ≈5140. Unfortunately, this increment in the signal power is not directly translated to an improvement in the SNR of the reconstructed images. The improvement in the SNR of the reconstructed images is less than 10 in all three tested cases. This moderate improvement can be explained as a compromise between two conflicting processes. On one hand, the signal per pixel of the SI-COACH’s PSF is three order of magnitude more intense and thus the SNR of the intensity responses are higher by the same amount. But on the other hand, because of the sparsity, the complexity of the PSF of SI-COACH is lower than that of I-COACH. Therefore, the peak-to-sidelobes ratio (sidelobes here are the distribution outside the location of the correlation peak) of the PSF autocorrelation in the SI-COACH is lower, and consequently the background noise of the reconstructed images is higher. Fortunately, the net outcome between these contradictory phenomena is the moderate improvement of less than 10 times in the SNR for SI-COACH in comparison to I-COACH.

Last but not least, the 3D imaging capability of the method has been demonstrated. In this experiment, one object (USAF target Group 3) is mounted in each channel of the experimental setup at different axial location, where the gap between them is 15 cm. Best dot density for each axial location are measured beforehand and PSF library for each axial location are recorded. For the object response, two CPMs of the best density and their corresponding DLs for different axial locations were multiplexed in a chessboard pattern on the SLM, where each chessboard cell consists of 10 SLM pixels. An object hologram for the two-plane object is captured. Object reconstruction for different planes is done by cross-correlation of the object hologram with its corresponding PSF. Object reconstructions for the two planes are shown in Fig. 8.

 

Fig. 8 Reconstruction results of object in (a) plane 1, (b) plane 2 separated by a distance of 15 cm.

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4. Summary and conclusions

In the proposed system, a modified Gerchberg–Saxton algorithm has been used to improve the power efficiency of recorded intensity responses. The increase in the power efficiency leads to some suppression of the noise in the reconstructed image. The power efficiency has been improved by projecting all the light from the point source to a series of randomly distributed dots on the camera plane, and thus increasing the power per pixel on the camera. CPM with dot density between 0.000044 to 0.08064 with various step sizes has been constructed, and the corresponding object and PSFs were recorded and reconstructed. To determine the spot density which yielding the best reconstruction, visibility, SNR and SSIM of the reconstructed image have been calculated. To determine the best dot density, a normalized multiplication of visibility and SNR values are considered and the dot density having the highest product number is considered as the best choice.

The experiment for the proposed technique was done for three different illumination powers of 3.5, 7 and 10 mW to demonstrate the power efficiency capabilities of the system and to validate that the proposed technique is valid for broad range of power illumination. In all cases the dot density of 0.000164 is the best value. Experimentally, the power intensity per pixel is improved by the factor of about 5000. However, the experiments have shown that the SNR improvement factor of the reconstructed image is only 8 for low power case, 5 for mid power case and 4.5 for the high power case. To conclude, the proposed technique presents noise suppression in general, and probably it performs best in the low power illumination cases. Therefore, the proposed technique can be very useful in partial [21] and synthetic [22] aperture techniques, where inherent low power efficiency demands new techniques to increase the SNR.

Funding

Israel Science Foundation (ISF) (1669/16); Israel Ministry of Science and Technology (MOST).

Acknowledgment

This study was done during a research stay of JR at the Alfried Krupp Wissenschaftskolleg Greifswald.

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26. S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018). [CrossRef]   [PubMed]  

27. S. Mukherjee and J. Rosen, “Imaging through scattering medium by adaptive non-linear digital processing,” Sci. Rep. 8(1), 10517 (2018). [CrossRef]   [PubMed]  

28. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23(6), 812 (1984). [CrossRef]   [PubMed]  

29. G.-G. Mu, X.-M. Wang, and Z.-Q. Wang, “Amplitude-compensated matched filtering,” Appl. Opt. 27(16), 3461–3463 (1988). [CrossRef]   [PubMed]  

30. S. Vallmitjana, A. Carnicer, E. Martín-Badosa, and I. Juvells, “Nonlinear filtering in object and Fourier space in a joint transform optical correlator: comparison and experimental realization,” Appl. Opt. 34(20), 3942–3949 (1995). [CrossRef]   [PubMed]  

31. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017). [CrossRef]   [PubMed]  

32. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

33. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004). [CrossRef]   [PubMed]  

References

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  20. M. R. Rai, A. Vijayakumar, and J. Rosen, “Non-linear adaptive three-dimensional imaging with interferenceless coded aperture correlation holography (I-COACH),” Opt. Express 26(14), 18143–18154 (2018).
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    [Crossref]
  23. M. R. Rai, A. Vijayakumar, Y. Ogura, and J. Rosen, “Resolution enhancement in nonlinear interferenceless COACH with point response of subdiffraction limit patterns,” Opt. Express 27(2), 391–403 (2019).
    [Crossref] [PubMed]
  24. M. R. Rai, A. Vijayakumar, and J. Rosen, “Superresolution beyond the diffraction limit using phase spatial light modulator between incoherently illuminated objects and the entrance of an imaging system,” Opt. Lett. 44(7), 1572–1575 (2019).
    [Crossref] [PubMed]
  25. M. R. Rai, A. Vijayakumar, and J. Rosen, “Extending the field of view by a scattering window in an I-COACH system,” Opt. Lett. 43(5), 1043–1046 (2018).
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  26. S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018).
    [Crossref] [PubMed]
  27. S. Mukherjee and J. Rosen, “Imaging through scattering medium by adaptive non-linear digital processing,” Sci. Rep. 8(1), 10517 (2018).
    [Crossref] [PubMed]
  28. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23(6), 812 (1984).
    [Crossref] [PubMed]
  29. G.-G. Mu, X.-M. Wang, and Z.-Q. Wang, “Amplitude-compensated matched filtering,” Appl. Opt. 27(16), 3461–3463 (1988).
    [Crossref] [PubMed]
  30. S. Vallmitjana, A. Carnicer, E. Martín-Badosa, and I. Juvells, “Nonlinear filtering in object and Fourier space in a joint transform optical correlator: comparison and experimental realization,” Appl. Opt. 34(20), 3942–3949 (1995).
    [Crossref] [PubMed]
  31. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017).
    [Crossref] [PubMed]
  32. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).
  33. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
    [Crossref] [PubMed]

2019 (2)

2018 (7)

M. R. Rai, A. Vijayakumar, and J. Rosen, “Extending the field of view by a scattering window in an I-COACH system,” Opt. Lett. 43(5), 1043–1046 (2018).
[Crossref] [PubMed]

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018).
[Crossref] [PubMed]

S. Mukherjee and J. Rosen, “Imaging through scattering medium by adaptive non-linear digital processing,” Sci. Rep. 8(1), 10517 (2018).
[Crossref] [PubMed]

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, “Incoherent digital holography: A review,” Appl. Sci. (Basel) 8(1), 143 (2018).
[Crossref]

M. R. Rai, A. Vijayakumar, and J. Rosen, “Non-linear adaptive three-dimensional imaging with interferenceless coded aperture correlation holography (I-COACH),” Opt. Express 26(14), 18143–18154 (2018).
[Crossref] [PubMed]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Superresolution far-field imaging by coded phase reflectors distributed only along the boundary of synthetic apertures,” Optica 5(12), 1607–1616 (2018).
[Crossref]

2017 (5)

2016 (1)

M. N. Lakshmanan, J. A. Greenberg, E. Samei, and A. J. Kapadia, “Design and implementation of coded aperture coherent scatter spectral imaging of cancerous and healthy breast tissue samples,” J. Med. Imaging (Bellingham) 3(1), 013505 (2016).
[Crossref] [PubMed]

2015 (3)

J. A. Greenberg, M. N. Lakshmanan, D. J. Brady, and A. J. Kapadia, “Optimization of a coded aperture coherent scatter spectral imaging system for medical imaging,” Proc. SPIE 9412, 94125E (2015).

M. J. DeWeert and B. P. Farm, “Lensless coded-aperture imaging with separable Doubly-Toeplitz masks,” Opt. Eng. 54(2), 023102 (2015).
[Crossref]

Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23(16), 20941–20960 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (1)

D. S. Kittle, D. L. Marks, and D. J. Brady, “Design and fabrication of an ultraviolet-visible coded aperture snapshot spectral imager,” Opt. Eng. 51(7), 071403 (2012).
[Crossref]

2011 (2)

2009 (1)

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

1995 (1)

1988 (1)

1984 (1)

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Athale, R. A.

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Brady, D. J.

J. A. Greenberg, M. N. Lakshmanan, D. J. Brady, and A. J. Kapadia, “Optimization of a coded aperture coherent scatter spectral imaging system for medical imaging,” Proc. SPIE 9412, 94125E (2015).

D. S. Kittle, D. L. Marks, and D. J. Brady, “Design and fabrication of an ultraviolet-visible coded aperture snapshot spectral imager,” Opt. Eng. 51(7), 071403 (2012).
[Crossref]

Bulbul, A.

Cannon, T. M.

Carnicer, A.

Chi, W.

DeWeert, M. J.

M. J. DeWeert and B. P. Farm, “Lensless coded-aperture imaging with separable Doubly-Toeplitz masks,” Opt. Eng. 54(2), 023102 (2015).
[Crossref]

Euliss, G. W.

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Farm, B. P.

M. J. DeWeert and B. P. Farm, “Lensless coded-aperture imaging with separable Doubly-Toeplitz masks,” Opt. Eng. 54(2), 023102 (2015).
[Crossref]

Fenimore, E. E.

Fienup, J. R.

George, N.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Gianino, P. D.

Greenberg, J. A.

M. N. Lakshmanan, J. A. Greenberg, E. Samei, and A. J. Kapadia, “Design and implementation of coded aperture coherent scatter spectral imaging of cancerous and healthy breast tissue samples,” J. Med. Imaging (Bellingham) 3(1), 013505 (2016).
[Crossref] [PubMed]

J. A. Greenberg, M. N. Lakshmanan, D. J. Brady, and A. J. Kapadia, “Optimization of a coded aperture coherent scatter spectral imaging system for medical imaging,” Proc. SPIE 9412, 94125E (2015).

Hayasaki, Y.

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, “Incoherent digital holography: A review,” Appl. Sci. (Basel) 8(1), 143 (2018).
[Crossref]

Hong, K.

Horner, J. L.

Javidi, B.

Juvells, I.

Kapadia, A. J.

M. N. Lakshmanan, J. A. Greenberg, E. Samei, and A. J. Kapadia, “Design and implementation of coded aperture coherent scatter spectral imaging of cancerous and healthy breast tissue samples,” J. Med. Imaging (Bellingham) 3(1), 013505 (2016).
[Crossref] [PubMed]

J. A. Greenberg, M. N. Lakshmanan, D. J. Brady, and A. J. Kapadia, “Optimization of a coded aperture coherent scatter spectral imaging system for medical imaging,” Proc. SPIE 9412, 94125E (2015).

Kashter, Y.

Kelner, R.

Kittle, D. S.

D. S. Kittle, D. L. Marks, and D. J. Brady, “Design and fabrication of an ultraviolet-visible coded aperture snapshot spectral imager,” Opt. Eng. 51(7), 071403 (2012).
[Crossref]

Kumar, A.

Kumar, M.

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018).
[Crossref] [PubMed]

M. Kumar, A. Vijayakumar, and J. Rosen, “Incoherent digital holograms acquired by interferenceless coded aperture correlation holography system without refractive lenses,” Sci. Rep. 7(1), 11555 (2017).
[Crossref] [PubMed]

Lakshmanan, M. N.

M. N. Lakshmanan, J. A. Greenberg, E. Samei, and A. J. Kapadia, “Design and implementation of coded aperture coherent scatter spectral imaging of cancerous and healthy breast tissue samples,” J. Med. Imaging (Bellingham) 3(1), 013505 (2016).
[Crossref] [PubMed]

J. A. Greenberg, M. N. Lakshmanan, D. J. Brady, and A. J. Kapadia, “Optimization of a coded aperture coherent scatter spectral imaging system for medical imaging,” Proc. SPIE 9412, 94125E (2015).

Lee, B.

Liu, J.-P.

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, “Incoherent digital holography: A review,” Appl. Sci. (Basel) 8(1), 143 (2018).
[Crossref]

Mait, J. N.

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Marks, D. L.

D. S. Kittle, D. L. Marks, and D. J. Brady, “Design and fabrication of an ultraviolet-visible coded aperture snapshot spectral imager,” Opt. Eng. 51(7), 071403 (2012).
[Crossref]

Martín-Badosa, E.

Martinez-Corral, M.

Mu, G.-G.

Mukherjee, S.

S. Mukherjee and J. Rosen, “Imaging through scattering medium by adaptive non-linear digital processing,” Sci. Rep. 8(1), 10517 (2018).
[Crossref] [PubMed]

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018).
[Crossref] [PubMed]

Ogura, Y.

Park, J.-H.

Poon, T.-C.

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, “Incoherent digital holography: A review,” Appl. Sci. (Basel) 8(1), 143 (2018).
[Crossref]

Rai, M. R.

Ratnam Rai, M.

Rivenson, Y.

Rosen, J.

M. R. Rai, A. Vijayakumar, and J. Rosen, “Superresolution beyond the diffraction limit using phase spatial light modulator between incoherently illuminated objects and the entrance of an imaging system,” Opt. Lett. 44(7), 1572–1575 (2019).
[Crossref] [PubMed]

M. R. Rai, A. Vijayakumar, Y. Ogura, and J. Rosen, “Resolution enhancement in nonlinear interferenceless COACH with point response of subdiffraction limit patterns,” Opt. Express 27(2), 391–403 (2019).
[Crossref] [PubMed]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Superresolution far-field imaging by coded phase reflectors distributed only along the boundary of synthetic apertures,” Optica 5(12), 1607–1616 (2018).
[Crossref]

M. R. Rai, A. Vijayakumar, and J. Rosen, “Extending the field of view by a scattering window in an I-COACH system,” Opt. Lett. 43(5), 1043–1046 (2018).
[Crossref] [PubMed]

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018).
[Crossref] [PubMed]

M. R. Rai, A. Vijayakumar, and J. Rosen, “Non-linear adaptive three-dimensional imaging with interferenceless coded aperture correlation holography (I-COACH),” Opt. Express 26(14), 18143–18154 (2018).
[Crossref] [PubMed]

S. Mukherjee and J. Rosen, “Imaging through scattering medium by adaptive non-linear digital processing,” Sci. Rep. 8(1), 10517 (2018).
[Crossref] [PubMed]

A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017).
[Crossref] [PubMed]

M. Ratnam Rai, A. Vijayakumar, and J. Rosen, “Single camera shot interferenceless coded aperture correlation holography,” Opt. Lett. 42(19), 3992–3995 (2017).
[Crossref] [PubMed]

A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25(12), 13883–13896 (2017).
[Crossref] [PubMed]

M. Kumar, A. Vijayakumar, and J. Rosen, “Incoherent digital holograms acquired by interferenceless coded aperture correlation holography system without refractive lenses,” Sci. Rep. 7(1), 11555 (2017).
[Crossref] [PubMed]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25(26), 33315–33329 (2017).
[Crossref]

Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23(16), 20941–20960 (2015).
[Crossref] [PubMed]

Samei, E.

M. N. Lakshmanan, J. A. Greenberg, E. Samei, and A. J. Kapadia, “Design and implementation of coded aperture coherent scatter spectral imaging of cancerous and healthy breast tissue samples,” J. Med. Imaging (Bellingham) 3(1), 013505 (2016).
[Crossref] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Stern, A.

Tahara, T.

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, “Incoherent digital holography: A review,” Appl. Sci. (Basel) 8(1), 143 (2018).
[Crossref]

Tippie, A. E.

Vallmitjana, S.

Vijayakumar, A.

M. R. Rai, A. Vijayakumar, Y. Ogura, and J. Rosen, “Resolution enhancement in nonlinear interferenceless COACH with point response of subdiffraction limit patterns,” Opt. Express 27(2), 391–403 (2019).
[Crossref] [PubMed]

M. R. Rai, A. Vijayakumar, and J. Rosen, “Superresolution beyond the diffraction limit using phase spatial light modulator between incoherently illuminated objects and the entrance of an imaging system,” Opt. Lett. 44(7), 1572–1575 (2019).
[Crossref] [PubMed]

S. Mukherjee, A. Vijayakumar, M. Kumar, and J. Rosen, “3D imaging through scatterers with interferenceless optical system,” Sci. Rep. 8(1), 1134 (2018).
[Crossref] [PubMed]

M. R. Rai, A. Vijayakumar, and J. Rosen, “Extending the field of view by a scattering window in an I-COACH system,” Opt. Lett. 43(5), 1043–1046 (2018).
[Crossref] [PubMed]

M. R. Rai, A. Vijayakumar, and J. Rosen, “Non-linear adaptive three-dimensional imaging with interferenceless coded aperture correlation holography (I-COACH),” Opt. Express 26(14), 18143–18154 (2018).
[Crossref] [PubMed]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Superresolution far-field imaging by coded phase reflectors distributed only along the boundary of synthetic apertures,” Optica 5(12), 1607–1616 (2018).
[Crossref]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25(26), 33315–33329 (2017).
[Crossref]

A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25(12), 13883–13896 (2017).
[Crossref] [PubMed]

M. Kumar, A. Vijayakumar, and J. Rosen, “Incoherent digital holograms acquired by interferenceless coded aperture correlation holography system without refractive lenses,” Sci. Rep. 7(1), 11555 (2017).
[Crossref] [PubMed]

M. Ratnam Rai, A. Vijayakumar, and J. Rosen, “Single camera shot interferenceless coded aperture correlation holography,” Opt. Lett. 42(19), 3992–3995 (2017).
[Crossref] [PubMed]

A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56(13), F67–F77 (2017).
[Crossref] [PubMed]

Wang, X.-M.

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Wang, Z.-Q.

Xiao, X.

Adv. Opt. Photonics (1)

J. N. Mait, G. W. Euliss, and R. A. Athale, “Computational imaging,” Adv. Opt. Photonics 10(2), 409–483 (2018).
[Crossref]

Appl. Opt. (7)

Appl. Sci. (Basel) (1)

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, “Incoherent digital holography: A review,” Appl. Sci. (Basel) 8(1), 143 (2018).
[Crossref]

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

J. Med. Imaging (Bellingham) (1)

M. N. Lakshmanan, J. A. Greenberg, E. Samei, and A. J. Kapadia, “Design and implementation of coded aperture coherent scatter spectral imaging of cancerous and healthy breast tissue samples,” J. Med. Imaging (Bellingham) 3(1), 013505 (2016).
[Crossref] [PubMed]

Opt. Eng. (2)

M. J. DeWeert and B. P. Farm, “Lensless coded-aperture imaging with separable Doubly-Toeplitz masks,” Opt. Eng. 54(2), 023102 (2015).
[Crossref]

D. S. Kittle, D. L. Marks, and D. J. Brady, “Design and fabrication of an ultraviolet-visible coded aperture snapshot spectral imager,” Opt. Eng. 51(7), 071403 (2012).
[Crossref]

Opt. Express (7)

Opt. Lett. (3)

Optica (1)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Proc. SPIE (1)

J. A. Greenberg, M. N. Lakshmanan, D. J. Brady, and A. J. Kapadia, “Optimization of a coded aperture coherent scatter spectral imaging system for medical imaging,” Proc. SPIE 9412, 94125E (2015).

Sci. Rep. (3)

M. Kumar, A. Vijayakumar, and J. Rosen, “Incoherent digital holograms acquired by interferenceless coded aperture correlation holography system without refractive lenses,” Sci. Rep. 7(1), 11555 (2017).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Optical configuration of SI-COACH. L0, L1 – Refractive lenses: P- Polarizer; SLM - Spatial light modulator; DL – Diffractive lens; CPM - Coded phase mask.
Fig. 2
Fig. 2 Modified Gerchberg-Saxton algorithm for synthesizing the CPMs.
Fig. 3
Fig. 3 Experimental setup. BS1 and BS2 – Beam splitters; SLM – Spatial light modulator; NBS – National Bureau of Standards ; L0A, L0B and L1 – Refractive lenses; LED1 and LED2 – Identical light emitting diodes; CPM – Coded phase mask; DL – Diffractive lens; BPF – Band pass filter (λc = 632.8 nm and Δλ = 5 nm); P- Polarizer; - Polarization orientation perpendicular to the plane of the page.
Fig. 4
Fig. 4 (a-b) Images of the CPMs, (c-d) intensity PSFs and (e) bipolar PSF, (f-g) object intensity responses and (h) bipolar HOBJ for dot density of ψ = 0.000164.
Fig. 5
Fig. 5 (a) Visibility, (b) SNR, (c) ξ( ψ ), (d) SSIM, (e) Reconstructed image of SI-COACH for the best dot density and (f) Reconstructed image for full window I-COACH in case of low power illumination of the object.
Fig. 6
Fig. 6 (a) Visibility, (b) SNR, (c) ξ( ψ ), (d) SSIM, (e) Reconstructed image of SI-COACH for best dot density and (f) Reconstructed image for full window I-COACH in case of mid power illumination of the object.
Fig. 7
Fig. 7 (a) Visibility, (b) SNR, (c) ξ( ψ ), (d) SSIM, (e) Reconstructed image of SI-COACH for the best dot density and (f) Reconstructed image for full window I-COACH in case of high power illumination of the object.
Fig. 8
Fig. 8 Reconstruction results of object in (a) plane 1, (b) plane 2 separated by a distance of 15 cm.

Equations (7)

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I PSF ( r ¯ o ; r ¯ s )= | ν[ 1 λ z h ]F{ C 0 L( r ¯ s f )exp[ iΦ( r ¯ ) ] } | 2 = I PSF ( r ¯ o z h f r ¯ s ;0 ),
o( r ¯ s )= j N a j δ( r ¯ r ¯ j ) .
I OBJ ( r ¯ o )= j a j I PSF ( r ¯ o z h f r ¯ j ;0 ) .
H PSF ( r ¯ o )= I PSF,1 ( r ¯ o ) I PSF,2 ( r ¯ o ). H OBJ ( r ¯ o )= I OBJ,1 ( r ¯ o ) I OBJ,2 ( r ¯ o ).
I REC = H OBJ H PSF = F 1 { F{ H OBJ H PSF } } = F 1 { H ˜ OBJ H ˜ PSF * }= F 1 { H ˜ OBJ exp( iarg{ H ˜ PSF } ) },
I REC =( H OBJ + n 1 )( H PSF + n 2 )=( H PSF + n 1 )( H PSF + n 2 ) H PSF H PSF + H PSF n 2 + H PSF n 1 ,
SNR= H PSF H PSF | ( x,y )=( 0,0 ) Max ( x,y )( 0,0 ) { H PSF H PSF + H PSF n 2 + H PSF n 1 } ,

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