# Nonlinear Reconstruction of Images from Patterns Generated by Deterministic or Random Optical Masks—Concepts and Review of Research

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## Abstract

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## 1. Introduction

_{O}= PSF⊗O + N, where ‘⊗’ is a 2D convolutional operator, PSF is the point spread function and N is a noise function. During deconvolution, the reconstructed image is given as ${I}_{R}={\mathcal{F}}^{-1}\left[\mathcal{F}\left({I}_{O}\right)/\mathcal{F}\left(\mathrm{PSF}\right)\right]$, where $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ are the Fourier transform and inverse Fourier transform operators, respectively. Substituting for I

_{O}in the above equation, we obtain ${I}_{R}=O+{\mathcal{F}}^{-1}\left[\mathcal{F}\left(N\right)/\mathcal{F}\left(\mathrm{PSF}\right)\right]$. As seen from the last expression, the noise distribution in the reconstructed image might be amplified. In general, the Fourier transform of the scattered PSF might have values smaller than a noise distribution and usually has many nulls. This was the main drawback of CAI with a random array of pinholes.

## 2. Methodology

_{1}is a complex constant and $R=\sqrt{{x}^{2}+{y}^{2}}$. The complex amplitude after the optical modulator is given as

_{t}is the refractive index [42]. For vortex-beam generation, the optical modulator is a spiral Fresnel lens with a phase of $\mathrm{exp}\left[-j\left\{L\theta +\left(\pi {R}^{2}/\lambda \right)\left(1/u+1/v\right)\right\}\right]$, where L is the topological charge and θ is the azimuthal angle given as $\mathsf{\theta}={\mathrm{tan}}^{-1}\left(y/x\right)$ [43,44]. For the generation of a scattered beam, the optical modulator is a quasi-random lens with a phase given as $\mathrm{exp}\left[-j\left\{\left(\pi {R}^{2}/\lambda \right)\left(1/u+1/v\right)+{\Phi}_{R}\right\}\right]$, where Φ

_{R}is the random phase matrix with a particular scattering degree synthesized using Gerchberg–Saxton algorithm (GSA) [2,15,26]. For the generation of accelerating Airy beams, the optical modulator is a cubic phase mask with a phase given as $\mathrm{exp}\left[-j\left(2\pi /\lambda \right)\zeta \left({x}^{3}+{y}^{3}\right)\right]$ [45]. For the generation of the FINCH hologram, the optical modulator has a phase function given as $M\mathrm{exp}\left[-j\left(\pi {R}^{2}/\lambda \right)\left(1/u+2/v\right)\right]+\left(1-M\right)\mathrm{exp}\left[-j\left(\pi {R}^{2}/\lambda \right)\left(1/u\right)\right]$, where M is a binary {0,1} quasi-random matrix and so (1 − M) is its antimask, which is mutually exclusive to M. The intensity pattern observed at a distance of v from the modulator is given as the magnitude square of a convolution of the complex amplitude beyond the modulator with the quadratic-phase function Q(1/v),

_{2}is a complex constant. The sensor intensity for a 2D object O can be expressed as I

_{O}= O⊗I

_{PSF}. Unlike a coherent source, where the complex amplitude is convolved with PSF, here only the intensity distribution is convolved. Therefore, the object intensity pattern I

_{O}is formed by the replacement of every object point by I

_{PSF}followed by their summation. Consequently, there is no role for the phase profiles of the optical beams in this indirect imaging framework, and only the intensity distribution is considered. For example, a ring pattern generated by a lens-axicon pair [46,47] and a higher-order Laguerre–Gaussian beam will have the same imaging characteristics. The image reconstruction is carried out using NLR and optimized using the values of α and β. The imaging resolution in direct imaging mode is the diffraction-limited spot size ~1.22λf/D. The speckles formed by scattering have an average size of the diffraction-limited spot size. During autocorrelation, a peak with a width of twice the diffraction-limited spot is generated, which is equal to the diffraction-limited spot size when NLR was applied [27]. Therefore, there are two resolutions, namely optical and computational, and the computational resolution of NLR is usually higher than other computational reconstruction methods. The performance of the NLR in the case of various optical fields is studied in the following.

## 3. Simulation Results

_{PSF}*I

_{PSF}|. The width of the autocorrelation function is approximately the lateral resolution of the indirect imaging system. Recalling the expression for reconstruction, ${I}_{R}={\mathrm{I}}_{\mathrm{PSF}}{\ast \mathrm{I}}_{\mathrm{PSF}}\otimes O+N$, the autocorrelation function is the fundamental building block of the reconstructed image. Row 4: The modulation transfer function (MTF), which in direct imaging is $\mathrm{MTF}=c\left|\mathcal{F}\left({\mathrm{I}}_{\mathrm{PSF}}\right)\right|$, and in indirect imaging framework is $\mathrm{MTF}={c}^{\prime}\left|\mathcal{F}\left({\mathrm{I}}_{\mathrm{PSF}}\ast {\mathrm{I}}_{\mathrm{PSF}}\right)\right|$, where c and c’ are constants that guarantee the MTFs are normalized. Row 5: The NLR of a single point. Row 6: The MTF of the systems with NLR. Although the NLR violates the linearity of the imaging system, we define the MTF of such a system as the normalized magnitude of the Fourier transform of the point image. The reconstructed image due to the NLR is

_{o}= I

_{PSF}, and recall that ${\tilde{I}}_{PSF}=H\ast H,$ where $H=\mathrm{exp}\left(-j{\Phi}_{\mathrm{OM}}\right)$ is the transfer function of the modulator, the MTF of the systems with NLR is $\mathrm{MTF}={\left|H\ast H\right|}^{\alpha +\beta}$. Comparing the various rows of Figure 2, it is clear that the NLR of a point is sharper than the conventional autocorrelation function and the MTF of the NLR is wider than the conventional MTF in all the cases of different modulators. According to these observations, it is expected that the image resolution of the NLR is superior to the conventional techniques, although the numerical aperture is identical for the entire optical modulators and techniques. Note that in all previous studies, the image resolution of NLR was found to be higher than the other tested methods [28,29].

_{R,p}= (I

_{R})

^{p}, which suppresses the background information. In the Fourier domain, the above operation can be expressed as a convolution resulting in an increase in the bandwidth. For p = 2, if G is the MTF corresponding to I

_{R}, then G

_{p}$=\mathcal{F}\left[{\mathcal{F}}^{-1}\left(G\right)\times {\mathcal{F}}^{-1}\left(G\right)\right]=G\otimes G$. Therefore, with each increase in p, the bandwidth increases by the bandwidth of G. The influence of this process on imaging is examined in the following. This process is suitable only for objects with binary values and is detrimental for objects with greyscale values. In fact, most, if not all, of the previous applications of NLR to stochastic—as well as deterministic—optical fields involved only binary objects such as standard-resolution targets [2,27,30,31,32,34]. The reconstructed point and normalized MTF after the application of the above method for p = 2 with NLR for eight cases of Figure 2 are shown in Figure 3a. As seen from the results, raising the image to the power of 2 improves the MTF. The variation in the greyscale values when p was varied from 1 to 5 is shown in Figure 3b. As seen in Figure 3b, with an increase in the value of p, the greyscale profile changes from linear to nonlinear.

## 4. Experimental Results

#### 4.1. Lensless I-COACH

#### 4.2. Random Array of Pinholes

_{c}= 617 nm, FWHM = 18 nm) was used for illumination. The PSF (pinhole φ = 100 μm) was recorded when the distances between the object and the mask containing a random array of pinholes and between the mask and the sensor plane (DCU223M, 1024 × 768 pixels, pixel size = 4.65 μm) were both 10 cm. The intensity patterns of the PSF, the object’s response and reconstruction results of NLR (α = 0, β = 0.6) for p = 1, 2 and 3 are shown in Figure 6a–e, respectively.

#### 4.3. QRL Fabricated Using Electron-Beam Lithography

^{TWO}) with a diameter of 5 mm and focal length of 5 cm with a binary-phase profile [51], as shown in Figure 7a. The same LED source as in the previous section was used for illumination. The distance between the pinhole (φ = 100 μm) and the QRL was 10 cm. The image sensor was located at a distance of 10 cm from the QRL. The intensity patterns of the PSF, the object’s response and the reconstruction results of NLR (α = 0, β = 0.6) for p = 1, 2 and 3 are shown in Figure 7b–f, respectively.

#### 4.4. QRL Fabricated by Grinding Lens

#### 4.5. Photon-Sieve Axicon

#### 4.6. Diffractive Lens

#### 4.7. Spiral Fresnel Lens

#### 4.8. Lens–Axicon Pair

#### 4.9. FINCH with Polarization Multiplexing

#### 4.10. FINCH with Spatial Random Multiplexing

^{TWO}) with focal lengths of 5 cm and 10 cm and a diameter of 5 mm. A pinhole with a size of 20 μm was mounted at 5 cm from the diffractive element. Around 50% of the light was collected and focused at 5 cm from the diffractive element by the lens with a focal length of 10 cm, and the remaining was collimated. An image sensor (Thorlabs DCU223M, 1024 pixels × 768 pixels, pixel size = 4.65 μm) was used for recording the hologram at a distance of 10 cm from the diffractive element. The optical microscope image of the diffractive element is shown in Figure 15a. The PSF, object’s intensity response, and reconstruction results of NLR (α = 0, β = 0.6) for p = 1, 2 and 3 are shown in Figure 14b–f, respectively [30].

#### 4.11. Double-Helix Beam with Rotating PSF

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Optical configuration of imaging systems. The optical modulator can be a bifocal lens (FINCH), regular lens (direct imaging), spiral phase plate (vortex beam), an axicon (Bessel beam) or a random pinhole array (scattered beam).

**Figure 2.**Comparison between different phase modulators according to functions and distributions that related to image reconstruction and resolution.

**Figure 3.**(

**a**) Autocorrelation with NLR followed by raising the image to the power of p = 2 and their respective MTF profiles. (

**b**) The influence of p on a grayscale slope.

**Figure 4.**Simulated intensity distribution for a test object and the reconstruction results for p = 1, 2 and 3.

**Figure 5.**(

**a**) Recorded PSF and (

**b**) object intensity response. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for LI-COACH with a QRL.

**Figure 6.**(

**a**) Recorded PSF, (

**b**) object’s response pattern. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for random array of pinholes.

**Figure 7.**(

**a**) Optical microscope image of the central part of the QRL fabricated using electron-beam lithography. (

**b**) Recorded PSF, (

**c**) object’s response pattern. Reconstruction results using (

**d**) NLR (p = 1), (

**e**) NLR (p = 2) and (

**f**) NLR (p = 3).

**Figure 8.**(

**a**) Image of the QRL fabricated using lens grinding with sandpaper. (

**b**) Recorded PSF, (

**c**) object’s response pattern. Reconstruction results using (

**d**) NLR (p = 1), (

**e**) NLR (p = 2) and (

**f**) NLR (p = 3).

**Figure 9.**(

**a**) Image of the photon-sieve axicon fabricated using femtosecond ablation. (

**b**) Recorded PSF, (

**c**) object’s response pattern. Reconstruction results using (

**d**) NLR (p = 1), (

**e**) NLR (p = 2) and (

**f**) NLR (p = 3).

**Figure 10.**(

**a**) Recorded PSF, and (

**b**) object’s response pattern. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for a diffractive lens.

**Figure 11.**(

**a**) Recorded PSF, (

**b**) object’s response pattern. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for a spiral Fresnel zone lens with L = 1.

**Figure 12.**(

**a**) Recorded PSF, (

**b**) object’s response pattern. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for a spiral Fresnel zone lens with L = 5.

**Figure 13.**(

**a**) Recorded PSF, (

**b**) object’s response pattern. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for lens–axicon pair.

**Figure 14.**(

**a**) Recorded PSF, (

**b**) object’s response pattern. Reconstruction results using (

**c**) NLR (p = 1), (

**d**) NLR (p = 2) and (

**e**) NLR (p = 3) for FINCH with random multiplexing configuration.

**Figure 15.**(

**a**) Optical microscope image of the randomly multiplexed bifocal diffractive lenses. (

**b**) Recorded PSF, (

**c**) object’s response pattern. Reconstruction results using (

**d**) NLR (p = 1), (

**e**) NLR (p = 2) and (

**f**) NLR (p = 3) for FINCH with spatial random multiplexing configuration.

**Figure 16.**(

**a**) Phase image of the multifunctional DOE. (

**b**) Recorded PSF, (

**c**) object’s response pattern. Reconstruction results using (

**d**) NLR (p = 1), (

**e**) NLR (p = 2) and (

**f**) NLR (p = 3) for double-helix beam with rotating PSF.

Peak-to-Background Ratio | Lens | Axicon | Lens–Axicon Pair | Spiral Fresnel Zone Lens L = 1 | Spiral Fresnel Zone Lens L = 5 | Cubic Phase Mask | Quasi-random Lens | Randomly Multiplexed Lenses |

Autocorrelation | 2518 | 8 | 99 | 925 | 262 | 61 | 23 | 10 |

NLR | 5957 | 5258 | 3472 | 5739 | 4068 | 18,147 | 7565 | 5977 |

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Smith, D.; Gopinath, S.; Arockiaraj, F.G.; Reddy, A.N.K.; Balasubramani, V.; Kumar, R.; Dubey, N.; Ng, S.H.; Katkus, T.; Selva, S.J.;
et al. Nonlinear Reconstruction of Images from Patterns Generated by Deterministic or Random Optical Masks—Concepts and Review of Research. *J. Imaging* **2022**, *8*, 174.
https://doi.org/10.3390/jimaging8060174

**AMA Style**

Smith D, Gopinath S, Arockiaraj FG, Reddy ANK, Balasubramani V, Kumar R, Dubey N, Ng SH, Katkus T, Selva SJ,
et al. Nonlinear Reconstruction of Images from Patterns Generated by Deterministic or Random Optical Masks—Concepts and Review of Research. *Journal of Imaging*. 2022; 8(6):174.
https://doi.org/10.3390/jimaging8060174

**Chicago/Turabian Style**

Smith, Daniel, Shivasubramanian Gopinath, Francis Gracy Arockiaraj, Andra Naresh Kumar Reddy, Vinoth Balasubramani, Ravi Kumar, Nitin Dubey, Soon Hock Ng, Tomas Katkus, Shakina Jothi Selva,
and et al. 2022. "Nonlinear Reconstruction of Images from Patterns Generated by Deterministic or Random Optical Masks—Concepts and Review of Research" *Journal of Imaging* 8, no. 6: 174.
https://doi.org/10.3390/jimaging8060174