# Single-Shot 3D Incoherent Imaging Using Deterministic and Random Optical Fields with Lucy–Richardson–Rosen Algorithm

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## Abstract

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^{2}A), a recently developed computational reconstruction method, was found to perform better than its predecessors, such as matched filter, inverse filter, phase-only filter, Lucy–Richardson algorithm, and non-linear reconstruction (NLR), for certain apertures when the point spread function (PSF) is a real and symmetric function. For other cases of PSF, NLR performed better than the rest of the methods. In this tutorial, LR

^{2}A has been presented as a generalized approach for any optical field when the PSF is known along with MATLAB codes for reconstruction. The common problems and pitfalls in using LR

^{2}A have been discussed. Simulation and experimental studies for common optical fields such as spherical, Bessel, vortex beams, and exotic optical fields such as Airy, scattered, and self-rotating beams have been presented. From this study, it can be seen that it is possible to transfer the 3D imaging characteristics from non-imaging-type exotic fields to indirect imaging systems faithfully using LR

^{2}A. The application of LR

^{2}A to medical images such as colonoscopy images and cone beam computed tomography images with synthetic PSF has been demonstrated. We believe that the tutorial will provide a deeper understanding of computational reconstruction using LR

^{2}A.

## 1. Introduction

^{2}A) was developed by combining the well-known Lucy–Richardson algorithm (LRA) with NLR [34,35,36]. The performance of LR

^{2}A was found to be significantly better than LRA and NLR in many studies [37,38,39]. In this tutorial, we present the possibilities of using LR

^{2}A as a generalized reconstruction method for deterministic as well as random optical fields for incoherent 3D imaging applications. The physics of optical fields when switching from a coherent source to an incoherent source is different [40,41]. Many optical fields, such as Bessel, Airy, and vortex beam, have unique spatio-temporal characteristics that are useful for many applications. However, they cannot be used for imaging applications, as the above fields do not have a point focus. In this tutorial, we discuss the procedure for transferring the exotic 3D characteristics of the above special beams for imaging applications using LR

^{2}A. For the first time, the optimized computational MATLAB codes for implementing LR

^{2}A are also provided (see Supplementary Materials). The manuscript consists of six sections. The methodology is described in the second section. The third section contains the simulation studies. The experimental studies are presented in the fourth section. In the fifth section, an interesting method to tune axial resolution is discussed. The conclusion and future perspectives of the study are presented in the final section.

## 2. Materials and Methods

^{2}A. There have been many recent studies that have achieved this in the framework of I-COACH [42,43]. In the above studies, NLR has been used for reconstruction. It performs best with scattered distributions, and so in both studies, the deterministic fields’ axial characteristics have been transferred to the imaging system with additional sparsely random encoding. The optical configuration of the proposed system is shown in Figure 1. Light from an object is incident on a phase mask and recorded by an image sensor. The point spread function (PSF) is pre-recorded and used as the reconstructing function to obtain the object information. There are numerous methods available to process the object intensity distribution with the PSF; these include matched filter, phase-only filter, inverse filter, NLR, and LR

^{2}A. In this study, LR

^{2}A has been used, and it has a better performance than other methods when deterministic optical fields are considered.

_{1}is a complex constant. In the SLM, the phase mask for the generation of an optical field is displayed. The phase function ${\psi}_{PM}$ modulates the incoming light and the complex amplitude after the SLM is given as $\sqrt{{I}_{s}}{C}_{1}L\left(\frac{{\stackrel{\u0304}{r}}_{s}}{{z}_{s}}\right)Q\left(\frac{1}{{z}_{s}}\right){\psi}_{PM}$. At the image sensor, the recorded intensity distribution is given as

_{T}= z

_{h}/z

_{s}. The lateral and axial resolution limits of the system for a typical lens are 1.22λz

_{s}/D and 8λ(z

_{s}/D)

^{2}, where D is the diameter of the SLM.

_{PSF}can be expressed as

^{2}A, where the (n + 1)th reconstructed image is given as

_{O}, which is a distorted version of O. In the first step, I

_{O}and I

_{PSF}are convolved to obtain the corresponding object intensity I

_{O}′ if I

_{O}is O. In the next step, the ratio between the original object intensity I

_{O}and obtained object intensity I

_{O}′ is calculated. This ‘Ratio’ is correlated with I

_{PSF}using NLR and the result is called the Residue. This ‘Residue’ is multiplied by the previous solution, which is I

_{O}after the first iteration and ${I}_{R}^{n}$ after the second iteration. The result gradually moves from I

_{O}towards O with every iteration. This process is iterated until an optimal reconstruction is obtained. There is a forward convolution ${I}_{R}^{n}\otimes {I}_{PSF}$ and the ratio between this and ${I}_{O}$ is non-linearly correlated with ${I}_{PSF}$. The better estimation from NLR enables a rapid convergence.

## 3. Simulation Results

^{2}A, it was shown that the performance of LR

^{2}A is best when the I

_{PSF}is a symmetric distribution along the x and y directions [39,44]. The shift variance in LR

^{2}A was demonstrated [44]. Some solutions in the form of post-processing to address the problems associated with the asymmetry of the I

_{PSF}have been discussed in [39]. However, all the above effects were due to the fact that LR

^{2}A was not generalized. In this study, LR

^{2}A has been generalized to all shapes of I

_{PSF}s and both real and complex cases. The simulation space has been constructed in MATLAB with the following specifications: Matrix size = 500 × 500 pixels, pixel size Δ = 10 μm, wavelength λ = 650 nm, object distance z

_{s}= 0.4 m, recording distance z

_{h}= 0.4 m, and focal length of diffractive lens f = 0.2 m. For the diffractive lens, to prevent the direct imaging mode, the z

_{h}value was modified to 0.2 m. The phase masks of a diffractive lens, spiral lens, axicon, and spiral axicon are given as $\mathit{exp}\left[-i\pi ({\lambda f)}^{-1}{R}^{2}\right]$, $\mathit{exp}\left[-i\pi ({\lambda f)}^{-1}{R}^{2}\right]\times \mathit{exp}\left[iL\theta \right]$, $\mathit{exp}\left[-i2\pi {\Lambda}^{-1}R\right]$, $\mathit{exp}\left[-i2\pi {\Lambda}^{-1}R\right]\times \mathit{exp}\left[iL\theta \right]$, respectively, where L is the topological charge and Λ is the period of the axicon. The optical configuration in this study is simple, consisting of three steps, namely free space propagation, interaction, and again a free space propagation. The equivalent mathematical operations for the above three steps, propagation, interaction, and propagation, are convolution, product, and convolution, respectively. The first mathematical operation is quite direct, as a single point is considered; this constitutes a Kronecker Delta function. Any function convolved with a Delta function creates a replica of the function. Therefore, the complex amplitude obtained after the first operation is equivalent to $\mathit{exp}\left[i\pi ({\lambda {z}_{s})}^{-1}{R}^{2}\right]$. In the next step, the above complex amplitude is multiplied by ${\psi}_{PM}$ by an element-wise multiplication operation. In the final step, the resulting complex amplitude is propagated to the sensor plane by a convolution operation expressed as three Fourier transforms ${\mathcal{F}}^{-1}\left[\mathcal{F}\left\{\left.\mathit{exp}\left[i\pi ({\lambda {z}_{s})}^{-1}{R}^{2}\right]\times {\psi}_{PM}\right\}\right.\times \mathcal{F}\left\{\mathit{exp}\left[i\pi ({\lambda {z}_{h})}^{-1}{R}^{2}\right]\right\}\right]$. The intensity distribution I

_{PSF}can be obtained by squaring the absolute value of the complex amplitude matrix obtained at the sensor plane. The steps in the MATLAB software (Version 9.12.0.1884302 (R2022a)) can be found in the supplementary files of [45].

_{s}from 0.2 to 0.6 m in steps of 4 mm, and the I

_{PSF}(z

_{s}) was accumulated into a cube matrix (x, y, z, I). The images of the phase masks for diffractive lens, spiral lens (L = 1, 3, and 5), diffractive axicon, and spiral axicon (L = 1, 3, and 5) are shown in row 1 of Figure 2. The 3D axial intensity distributions for different diffractive elements, such as diffractive lens, spiral lens (L = 1, 3 and 5), diffractive axicon, and spiral axicon (L = 1, 3, and 5), are shown in row 2 of Figure 2. The diffractive lens does not have a focal point, as the imaging condition was not satisfied within this range of z

_{s}when z

_{h}is 0.2 m. For the spiral lens, a focused ring is obtained at the recording plane corresponding to z

_{s}= 0.4 m and the ring blurs and expands, similar to the case of a diffractive lens. For a diffractive axicon, a Bessel distribution is obtained in the recording plane and it is invariant with changes in z

_{s}[46,47]. A similar behavior is seen in Higher-Order Bessel Beams (HOBBs) [48]. As seen in the axial intensity distributions, none of the beams can be directly used for imaging applications. It may be argued that Bessel beams can be used, but as is known, the non-changing intensity distribution comes at a price, which is the loss of higher spatial frequencies [33]. Consequently, imaging using Bessel beams results in low-resolution images.

_{PSF}but $\left|{I}_{PSF}\u229a{I}_{PSF}\right|$, where ‘$\u229a$’ is the LR

^{2}A operator with n iterations, which is equivalent to the autocorrelation function. It must be noted that what is performed in LR

^{2}A is not a regular correlation as in a matched filter. The autocorrelation function was calculated for the above z

_{s}variation for the different cases of phase masks and accumulated in a cube matrix, as shown in the third row of Figure 2. As seen, the autocorrelation function is a cylinder with uniform radius for all values of z

_{s}. This is the holographic 3D PSF from whose profile it seems that it is possible to reconstruct the object information with a high resolution for all the object planes. To understand the axial imaging characteristics in the holographic domain, the I

_{PSF}(z

_{s}) was cross-correlated with I

_{PSF}of a reference plane, which in this case has been set at z

_{s}= 0.4 m. Once again, this cross-correlation is the nearest equivalent term in LR

^{2}A, which is calculated as $\left|{I}_{PSF}\left({z}_{s}\right)\u229a{I}_{PSF}({z}_{s}=0.4\mathrm{m})\right|$. The cube data obtained for different phase masks are shown in the fourth row from the top in Figure 3. As seen in the figure, the axial characteristics have been faithfully transferred to the imaging system except that now it is possible to perform imaging at any one or multiple planes of interest simultaneously. With a diffractive lens, a focal point is seen at a particular plane and blurred in other planes with respect to z

_{s}. The same behavior can be observed about the spiral lenses which contains the diffractive lens function. The case of the axicon shows a long focal depth, and a similar behavior is seen for the cases of the spiral axicon. With the indirect imaging concept and LR

^{2}A, it is possible to faithfully transfer the exotic axial characteristics of special beams to an imaging system. It must be noted that the cross-correlation was carried out with fixed values of α, β, and n for every case. In some of the planes, there is some scattering seen, indicating either a non-optimal reconstruction condition or slightly lower performance of LR

^{2}A.

_{PSF}s that are radially symmetric. When considering asymmetric cases such as Airy beams, self-rotating beams, and speckle patterns, NLR performed better than LR

^{2}A, as it was not optimized for asymmetric cases. In this study, LR

^{2}A has been generalized for asymmetric shapes. Three phase masks—a cubic phase mask with a phase function $\mathit{exp}\left[-i\left(2\pi /\lambda \right)\left({ax}^{3}+{by}^{3}\right)\right]$, where a = b~1000 [28,40], a diffractive lens with azimuthally varying focal length (DL-AVF) $exp\left\{\frac{-i2{\pi}^{2}{R}^{2}}{\lambda \left(2\pi {f}_{0}+\Delta f\theta \right)}\right\}$ [29,49], and a quasi-random lens with a scattering ratio of 0.04, obtained using the Gerchberg–Saxton algorithm [3] are investigated. The images of the phase masks for the above three cases are shown in column 1 in Figure 3. In this case, to show the curved path of the Airy pattern, the axial range was extended for z

_{s}from 0.4 m to 0.6 m. The 3D intensity distribution in the direct imaging mode for the three cases is shown in column 2 of Figure 3. The 3D autocorrelation distribution is shown in column 3 of Figure 3 for the three cases. The 3D cross-correlation distribution obtained by LR

^{2}A for the three cases is shown in column 4 of Figure 3. As seen from columns 2 and 4, the axial characteristics of the exotic beams have been faithfully transferred to the imaging system. The quasi-random lens or any scattering mask behaves exactly like a diffractive lens in the holographic domain. Comparing the results in Figure 3 and the previous results [39,44], a significant improvement is seen.

_{s}= 0.4 m and 0.5 m, respectively, is presented next. The images of the I

_{PSF}s for z

_{s}= 0.4 m and 0.5 m, the object intensity pattern I

_{O}obtained by convolution of object “CIPHR” with I

_{PSF}(z

_{s}= 0.4 m) and convolution of “TARTU” with I

_{PSF}(z

_{s}= 0.5 m) followed by a summation and the reconstructions I

_{R}corresponding to the two planes using LR

^{2}A for a diffractive lens, spiral lens (L = 5), spiral axicon (L = 3), cubic phase mask, DL-AVF, and quasi-random lens are shown in Figure 4. As seen from the results in Figure 4, the cases of the diffractive lens and quasi-random lens appear similar with respect to the axial behavior, i.e., when a particular plane information is reconstructed, only that plane information is focused and enhanced, while the other plane information is blurred and weak. However, for the elements such as the spiral lens and spiral axicon, the other plane information consists of “hot spots”, which are prominent and sometimes even stronger than the information in the reconstructed plane. This is due to the fact that there is similarity between the two I

_{PSF}s. This is one of the pitfalls of using such deterministic optical fields. The problem with such hotspots is that it is not possible to discriminate if the hotspot corresponds to any useful information related to the object or blurring due to a different plane if the object is not known prior.

_{PSF}from a special function is the sampling. Except for the cases of the quasi-random lens and diffractive lens, the other cases have distorted the object information in some ways. In the case of the Airy beam, the I

_{PSF}consists of periodic dot patterns along the x and y directions, which, when it samples the object information, the curves are sampled into square shapes. A ring-shaped I

_{PSF}has shaped the object information in this fashion, which again raises concerns about the reliability of the measurement, when the object information is not already known. In all the simulation studies, the optimal reconstructions were obtained for the following values: $8\le n\le 20$, $\alpha =0$, and $0.6\le \beta \le 0.8$.

## 4. Experiments

_{PSF}. The object is critically illuminated using a refractive lens (Lens2) (f = 5 cm). The light from the object is collimated by another refractive lens (Lens3) (f =5 cm) and passed through the beam splitter and incident on the SLM. On the SLM, phase masks of deterministic and random optical fields were displayed one after another, and the I

_{PSF}and I

_{O}were recorded by the image sensor.

^{2}A are presented in Figure 7. In Figure 7, the phase masks of the deterministic and random optical fields are shown in column 1 and their corresponding I

_{PSF}s and I

_{O}s (z

_{s}= 5 cm) are shown in columns 2 and 3, respectively. The reconstruction results of matched filter, phase-only filter, NLR, and LR

^{2}A are shown in columns 4 to 7 respectively. From Figure 7, the better performance of LR

^{2}A is evident. The structural similarity index measure (SSIM) and root mean square error (RMSE) are calculated for different optical fields and different reconstruction methods and the values are compared in a bar graph, as shown in Figure 8. As seen from Figure 8, we can conclude that the performance of LR

^{2}A is better than other reconstruction methods.

_{O}values for two objects are recorded at two different depths (z

_{s}= 5 cm and z

_{s}= 5.6 cm) and summed to demonstrate 3D imaging. The 3D experimental results are presented in Figure 9. The I

_{PSF}s for z

_{s}= 5 cm and z

_{s}= 5.6 cm are shown in columns 1 and 2, respectively, in Figure 9. I

_{O}is shown in column 3 and I

_{R}by LR

^{2}A for z

_{s}= 5 cm and z

_{s}= 5.6 cm are shown in column 4 and column 5, respectively. Once again, it can be seen that the 3D characteristics of the beams have been faithfully transferred to the indirect imaging system. In all the experimental studies, the optimal reconstructions were obtained for the following values: $5\le n\le 10$, $0\le \alpha \le 0.6$, and $\beta =1$.

^{2}A to real applications in day-to-day life, medical images were obtained from surgeons. A male 65-year-old patient with a known case of prostrate cancer had radiotherapy about a year ago. He developed bleeding and mucous discharge from the anus. The colonoscopy finding shows mucosal pallor, telangiectasias, edema, spontaneous hemorrhage, and friable mucosa. It can be noted that the mucosa was congested with ulceration stricture. The images were captured using colonoscope Olympus system and CaptureITPro medical imaging software. The direct images obtained using the colonoscope are shown in Figure 10. The I

_{PSF}can be synthesized in a computer or an isolated dot can be taken from Figure 10, padded with zeros and used as the reconstructing function. The red region indicates telangiectasias and the black region shows necrosis, which is the death of body tissue. The yellow region shows mucosal sloughing, and the magnified region shows the necrotic area with dead mucosa. In this study, the I

_{PSF}was synthetic [38,39]. The different color channels were extracted from the image and processed separately using synthetic I

_{PSF}and different types of filters, such as matched filter, phase-only filter, NLR, and LR

^{2}A, and then combined as discussed in [38]. The NLR was carried out for α = 0 and β = 0.8 and LR

^{2}A was carried out for α = 0.2 and β = 1 with n = 20 iterations. The reconstructed images are shown in Figure 9, which shows that LR

^{2}A has a better performance compared to other methods.

_{PSF}and LR

^{2}A were used to improve the resolution and contrast of the images. The images were reconstructed using different types of filters, such as matched filter, phase-only filter, NLR, and LR

^{2}A. The reconstructed images have a better resolution and contrast compared to the direct images. The NLR was carried out for α = 0 and β = 0.7 and LR

^{2}A was carried out for α = 0.3 and β = 1 with n = 20 iterations. The reconstructed images are shown in Figure 11, which shows that LR

^{2}A has a better performance compared to other methods.

## 5. Discussion

^{2}A and the indirect imaging concept have been used as a tool to transfer the 3D imaging characteristics faithfully from the beam to the imaging system [42,43,51]. In our recent study [51], the possibility of tuning axial resolution of an imaging system after completing the recording process has been demonstrated by post hybridization methods. The same can be achieved with the common and exotic beams. A hybrid PSF I

_{HPSF}can be formed by summing the pure I

_{PSF}s with appropriate weights w in the following form: ${I}_{HPSF}\left(z\right)=\sum _{k=1}^{m}{w}_{k}{I}_{PSF}(k,z)$. The object intensity distribution can be hybridized by the same weights as used for creating the hybrid PSFs as ${I}_{HO}=\sum _{k=1}^{m}{w}_{k}{I}_{O}\left(k\right)$. The object information can be reconstructed with the effective imaging characteristics using LR

^{2}A. Two cases are simulated here. In the first case, hybridization has been achieved between the Airy pattern and self-rotating beam, and in the second case, hybridization has been achieved between the Airy pattern and quasi-random lens. The 4D intensity distributions generated for the hybrid PSFs are shown in Figure 12a,c for cases 1 and 2, respectively. The reconstructed 4D patterns using LR

^{2}A for cases 1 and 2 are shown in Figure 12b,d, respectively. As expected, the focal depth decreased in the second case Figure 12d more than the first case Figure 12b, as the second ingredient in the second case has a high axial resolution. This study is not limited to only two ingredients, but any type of ensemble with m number of beams can be constructed easily post-recording.

## 6. Conclusions

^{2}A has been presented as a generalized computational reconstruction method for different types of deterministic fields and scattered patterns. The algorithm was also tested for a wide range of beams and variations, and it was always possible to obtain a high-quality reconstruction result by tuning the parameters α, β, and n. In all the cases, the axial characteristics have been faithfully transferred from the non-imaging beam to the imaging system using LR

^{2}A. Further, medical images recorded directly using a colonoscope and CBCT have been processed using synthetic I

_{PSF}and LR

^{2}A and an enhancement in resolution and contrast was observed. The method can be extended for aberration correction as well by choosing an aberrated dot from the recorded images. The above high-quality reconstructions in both laboratories set up with wide range of deterministic optical fields and scattered field and medical images demonstrate the wide applicability of LR

^{2}A, making it a universal computational reconstruction method.

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Concept figure: recording images with different phase masks—diffractive lens, spiral lens, spiral axicon and axicon, and reconstruction using LR

^{2}A by processing object intensity and PSF. OTF—Optical transfer function; n—number of iterations; ⊗—2D convolutional operator; $\mathfrak{I}*$—complex conjugate following a Fourier transform; ${\mathfrak{I}}^{-1}$—Inverse Fourier transform; R

^{n}is the nth solution and n is an integer, when n = 1, R

^{n}= I; ML—Maximum Likelihood; α and β are varied from −1 to 1.

**Figure 2.**Row 1: Phase masks of a diffractive lens, spiral lens (L = 1, 3, and 5), diffractive axicon, and spiral axicon (L = 1, 3, and 5). Row 2: Cube data of axial intensity distributions obtained at the recording plane for a diffractive lens, spiral lens (L = 1, 3, and 5), diffractive axicon, and spiral axicon (L = 1, 3, and 5). Row 3: Cube data of the intensity of the autocorrelation function for the different cases of phase masks. Row 4: Cube data of the cross-correlation function for the different cases of phase masks, when z

_{s}is varied from 0.2 to 0.6 m.

**Figure 3.**Column 1: Phase images of cubic phase mask, DL-AVF, and quasi-random lens. Column 2: Cube data of axial intensity distributions obtained at the recording plane for cubic phase mask, DL-AVF, and quasi-random lens. Column 3: Cube data of the intensity of the autocorrelation function for the different cases of phase masks. Column 4: Cube data of the cross-correlation function for the different cases of phase masks, when z

_{s}is varied from 0.1 to 0.7 m.

**Figure 4.**Column 1: Images of I

_{PSF}(z

_{s}= 0.4 m), column 2: images of I

_{PSF}(z

_{s}= 0.5 m), column 3: object intensity distributions I

_{O}for the two-plane object consisting of “CIPHR” and “TARTU”, column 4: reconstruction results I

_{R}using column 1, column 5: reconstruction results I

_{R}using column 2, for diffractive lens (row 1), spiral lens (L = 5) (row 2), spiral axicon (L = 3) (row 3), cubic phase mask, DL-AVF (row 5), and quasi-random lens (row 6). Scale bar—1 mm.

**Figure 6.**Photograph of experimental setup: (1) LED, (2) LED power controller, (3) iris(I1), (4) diffuser, (5) refractive lens (L1), (6) polarizer, (7) refractive lens (L2), (8) object/pinhole, (9) refractive lens (L3), (10) iris (I2), (11) beam splitter, (12) SLM, (13) image sensor.

**Figure 7.**Experimental results of 2D imaging. Column 1: Images of phase masks, column 2: images of I

_{PSF}(z

_{s}= 5 cm), column 3: images of object intensity distributions I

_{O}of the object at depth I

_{O}(z

_{s}= 5 cm). Reconstruction results by matched filter, phase-only filter, NLR, and LR

^{2}A in column 4, column 5, column 6, and column 7, respectively. Scale bar—50 μm.

**Figure 8.**Bar graph of SSIM and RMSE for different reconstruction methods such as matched filter, phase-only filter, NLR, and LR

^{2}A.

**Figure 9.**Experimental results of 3D imaging. Column 1: images of I

_{PSF}(z

_{s}= 5 cm), column 2: images of I

_{PSF}(z

_{s}= 5.6 cm), column 3: images of summed object intensity distributions I

_{O}of the two objects at different depths I

_{O}(z

_{s}= 5 cm) and I

_{O}(z

_{s}= 5.6 cm), column 4: Reconstruction results I

_{R}(z

_{s}= 5 cm) using corresponding I

_{PSF}(z

_{s}= 5 cm) in column 1, column 5: Reconstruction results I

_{R}(z

_{s}= 5.6 cm) using corresponding I

_{PSF}(z

_{s}= 5.6 cm) in column 2. Scale bar—50 μm.

**Figure 10.**Experimental colonoscopy results. Red region—telangiectasias; black region—necrosis, which is the death of body tissue; yellow region—mucosal sloughing; magnified region—necrotic area (dead mucosa). Reconstruction by matched filter, phase-only filter, NLR, and LR

^{2}A. The scale bar is 6 cm. The dotted areas are magnified in the subsequent row.

**Figure 11.**Experimental cone beam computed tomography results of four objects. The saturated region indicates metal artifacts. The size of each square is 8 cm × 8 cm. Reconstruction by matched filter, phase-only filter, NLR, and LR

^{2}A. Scale bar is 2 cm.

**Figure 12.**Simulation results of post-hybridization: 4D distribution of hybrid beam obtained by combining (

**a**) Airy beam and self-rotating beam and (

**b**) its reconstruction using LR

^{2}A; 4D distribution of hybrid beam obtained by combining (

**c**) Airy beam and scattered beam and (

**d**) its reconstruction using LR

^{2}A.

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## Share and Cite

**MDPI and ACS Style**

Ignatius Xavier, A.P.; Arockiaraj, F.G.; Gopinath, S.; John Francis Rajeswary, A.S.; Reddy, A.N.K.; Ganeev, R.A.; Singh, M.S.A.; Tania, S.D.M.; Anand, V.
Single-Shot 3D Incoherent Imaging Using Deterministic and Random Optical Fields with Lucy–Richardson–Rosen Algorithm. *Photonics* **2023**, *10*, 987.
https://doi.org/10.3390/photonics10090987

**AMA Style**

Ignatius Xavier AP, Arockiaraj FG, Gopinath S, John Francis Rajeswary AS, Reddy ANK, Ganeev RA, Singh MSA, Tania SDM, Anand V.
Single-Shot 3D Incoherent Imaging Using Deterministic and Random Optical Fields with Lucy–Richardson–Rosen Algorithm. *Photonics*. 2023; 10(9):987.
https://doi.org/10.3390/photonics10090987

**Chicago/Turabian Style**

Ignatius Xavier, Agnes Pristy, Francis Gracy Arockiaraj, Shivasubramanian Gopinath, Aravind Simon John Francis Rajeswary, Andra Naresh Kumar Reddy, Rashid A. Ganeev, M. Scott Arockia Singh, S. D. Milling Tania, and Vijayakumar Anand.
2023. "Single-Shot 3D Incoherent Imaging Using Deterministic and Random Optical Fields with Lucy–Richardson–Rosen Algorithm" *Photonics* 10, no. 9: 987.
https://doi.org/10.3390/photonics10090987