## Abstract

The depth-of-field of imaging systems can be enhanced by placing a phase mask in their aperture stop and deconvolving the image. In general, the mask is optimized using a closed-form image quality criterion assuming deconvolution with a Wiener filter. However, nonlinear deconvolution algorithms may have better performance, and the question remains as to whether a better co-designed system could be obtained from optimization with a criterion based on such algorithms. To investigate this issue, we compare optimization of phase masks with criteria based on the Wiener filter and on a nonlinear algorithm regularized by total variation. We show that the obtained optimal masks are identical, and propose a conjecture to explain this fact. This result is important since it supports the frequent co-design practice consisting of optimizing a system with a closed-form criterion based on linear deconvolution and deconvolving with a nonlinear algorithm.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nowadays, nearly all imaging systems include digital processing algorithms to improve image quality. It is thus natural to jointly optimize the optical system and the processing algorithms. Optical-digital joint design is part of the field of computational imaging [1], and the mathematical and conceptual foundations of this approach for a complete imaging system were laid by Stork and Robinson [2]. This “co-optimization” approach can be formulated in a rigorous way by defining the optimization criterion as the mean square difference between an ideally sharp image and the image delivered by the system after digital deconvolution [3–5]. This approach has been used to successfully co-optimize different types of imaging systems operating in the visible and infrared ranges [6–11]. In particular, it has been applied to design phase masks aimed at extending the depth-of-field (DoF) of camera lenses [12–16], which have been implemented with success in several real-world imagers [11,17].

Most of these works have used the Wiener filter as the image deconvolution algorithm since in this case, the optimization criterion is closed-form [4,15,18], which considerably accelerates the optimization process. Of course, it is possible to use nonlinear algorithms to restore images from systems optimized with a Wiener filter-based criterion [19]. This is often what is done in practice since it is well known that nonlinear deconvolution algorithms yield better image quality than the Wiener filter. It is also possible to take into account a nonlinear deconvolution algorithm directly in the optimization criterion. This was done for example in a recent work reporting joint optimization of binary DoF enhancing masks with a deconvolution algorithm based on neural network [20]. However, this comes at the price of more complex and time consuming optimization process. The question is whether this price is worth paying. In other words, it is important to determine whether better imaging performance can be obtained with an optical system optimized using a nonlinear deconvolution algorithm instead of a linear one. The purpose of this article is to address this issue, which has, to the best of our knowledge, never been addressed in detail.

In addressing this issue, a major difficulty is that there exists many different types of nonlinear deconvolution algorithms and it is impossible to be exhaustive. In consequence, we have chosen to focus on a single type of nonlinear deconvolution algorithm and to concentrate on the interpretation of the results. We chose TV-based deconvolution since it is a well known, well documented, and an efficient approach widely used for edge preservation purpose [21]. Moreover, we restrict our attention to a precise co-optimization problem, namely, the design of binary annular phase masks for DoF extension. This type of masks is easier to manufacture than masks with continuous phase variation, such as the cubic mask, and have been shown to have similar DoF enhancing ability [22]. Within this framework, we demonstrate that the optimal masks obtained with criteria based on the Wiener filter and on the TV-based algorithm are identical. We propose a conjecture to interpret and generalize this result. This finding is important since it supports the frequent co-design practice consisting of optimizing a system with a simple criterion and deconvolving images with a more complex algorithm.

## 2. Depth-of-field extension with phase masks

There are several reasons for an imaging system to be defocused in practice. It may be because the objects in the observed scene are located at different distances, so that the system cannot be focused on all of them [11]. Another frequent reason is variation of temperature, which induces a variation of the focal length of the lens so that the image plane is no longer on the sensor plane [23]. To get a sharp image in these cases, the DoF of the imaging system has to be improved. It is well known that placing a binary annular phase mask in the aperture stop of the lens makes it possible to increase its DoF [14,15]. Such masks are composed of concentric rings [15,24] [see Fig. 1(a)]. Each ring is an annular region that induces phase modulation alternatively equal to 0 or $\pi$ radian at a nominal wavelength $\lambda$. Under monochromatic illumination, simple binary etching in a dielectric substrate is sufficient to guarantee a $\pi$-phase step at $\lambda$, but under polychromatic illumination, this condition is not necessarily met. However, it has been shown that phase masks optimized under monochromatic illumination assumption are quite robust to their use under wide spectrum illumination [25]. Moreover, some approaches have been proposed to improve binary phase masks in the presence of wide spectrum and color imaging. It is for example possible to superpose two dielectric substrates to force the ring phase shift to be close to $\pi$ at three chosen RGB wavelengths [26]. An $N$-ring mask is defined by the set of radii of $N-1$ inner rings, that is, by the parameter vector $\boldsymbol {\rho }=(\rho _1, \ldots , \rho _{N-1})$.

Such masks are placed in the aperture stop of the imaging lens in order to extend its DoF. Let us assume that one observes an object at a certain distance $z_{\textrm {o}}$ of an imaging system with effective focal length $f$, with a sensor located at a fixed imaging distance $z_{\textrm {i}}$. The amplitude of the defocusing is classically defined by the following defocus parameter, expressed in units of wavelength:

An image observed by a system equipped with a binary phase mask with parameter set $\boldsymbol {\rho }$ in the presence of a defocus parameter $\psi$ can be modeled as

where the symbol $*$ denotes convolution, $\boldsymbol {y}\in \mathbb {R}^{N}$ the observed $N$-pixel image, $\boldsymbol {x}_0\in \mathbb {R}^{N}$ the truth-unknown image, $\boldsymbol {n}\in \mathbb {R}^{N}$ a white stationary zero mean noise, and $\boldsymbol {h}_{\boldsymbol {\rho },\psi }\in \mathbb {R}^{N\times N}$ the point spread function (PSF) of the optical system, that depends on $\boldsymbol {\rho }$ and $\psi$. The estimate $\boldsymbol {\hat {x}}$ of $\boldsymbol {x}_0$ is estimated from $\boldsymbol {y}$ by using a deconvolution algorithm.Many different criteria have been proposed to optimize the parameters $\boldsymbol {\rho }$ of a DoF enhancing binary annular phase mask. Many of them are based on the properties of the modulation transfer function (MTF), that is, the Fourier transform of $\boldsymbol {h}_{\boldsymbol {\rho },\psi }$. For example, in Ref. [14], the criterion consists in maximizing the effective cutoff frequency, defined as the largest frequency for which the value of the MTF is larger than a certain threshold. A Wiener filter is used for restoring the image. In the present article, we used an alternative approach that consists in taking directly the quality of the restored image as the optimization criterion [7,15]. This quality is defined as the mean square error (MSE) between the estimated image $\boldsymbol {\hat {x}}$ and the truth-unknown image $\boldsymbol {x}_0$:

The MSE criterion defined in Eq. (3) depends on the mask parameters $\boldsymbol {\rho }$ and on the value of the defocus $\psi$. We define the global image quality as the maximum value of the MSE over the defocus range:

In practice, maximization over $\psi$ in Eq. (4) is done over a discrete number $N_\psi$ of defocus values $\psi _i$ uniformly distributed in the interval $[0,\psi _{\textrm {max}}]$. Finally, the optimal mask parameters $\boldsymbol {\rho }_{\textrm {opt}}$ are found by minimizing the MMSE:However, it is well known that nonlinear algorithms usually yield better deconvolution performance than the Wiener filter. In consequence, considering that the estimated image $\boldsymbol {\hat {x}}$ in the MSE criterion of Eq. (3) is obtained with a nonlinear algorithm, it could lead to better co-optimized imaging systems. In order to verify this hypothesis, we consider a nonlinear deconvolution algorithm based on a quadratic data fitting term and a TV-based regularization term:

**$\rho$**, the estimation of the regularization parameter in Eq. (10) cannot be implemented in practice. The results obtained with this method will thus be an upper bound on the performance that can be actually achieved with the deconvolution algorithm defined in Eq. (9).

## 3. Comparison of the masks optimized with linear and nonlinear deconvolution algorithms

In this section we optimize masks with the criteria based on the averaged Wiener filter and on the TV algorithm. Then, we compare the parameters and the performance of the optimal masks obtained with these two criteria.

For this purpose, we consider three different values of targeted DoF range, namely $\psi _{\textrm {max}}=\{1\lambda ,1.5\lambda ,2\lambda \}$. It was shown in Ref. [16] that these values of the DoF range can be reached with masks having only 2 to 4 rings. Moreover, it is clear that the result of mask optimization depends on the considered truth-unknown image $\boldsymbol {x}_0$. We have performed mask optimizations with several examples of image $\boldsymbol {x}_0$ with different characteristics, and have obtained in all cases the same optimal masks for the two criteria. In this section, the optimization results will thus be first presented and analyzed in the case where $\boldsymbol {x}_0$ is the *Butterfly* image displayed in Fig. 1(b), and will then be generalized to other examples of images. In all cases, the input SNR is set to $34$ dB, which corresponds to a noise standard deviation $\sigma =2.4~.10^{-3}$ on the *Butterfly* image (the pixel values of $\boldsymbol {x}_0$ are assumed to range from 0 to 1).

In order to find the optimal parameters $\boldsymbol {\rho }_{\textrm {opt}}$ of the phase masks, we perform an exhaustive exploration of the parameter space. This deterministic approach avoids any potential problem of getting stuck in local minima, that could be encountered with an iterative optimization algorithm. Our exploration step size for each radius $\rho _n$ is $\Delta \rho _n = 5~.10^{-2}$. We refine this step to $\Delta \rho _n = 5~.10^{-3}$ around the position of the global minimum to estimate its position with better precision. Moreover, in the case of TV-based criterion, the best regularization parameter $\hat {\mu }$, as defined in Eq. (10), is estimated for each value of $\boldsymbol {\rho }$ and $\psi$ with a local optimization algorithm based on the Nelder-Mead simplex algorithm [32]. Since this exhaustive optimization process is time consuming, we used parallel computing to accelerate computations.

Let us first consider optimization of a 2-ring mask defined by the radius $\rho _1$ of its inner ring. Figure 2(a) represents the value of the MMSE criterion defined in Eq. (4) as a function of $\rho _1$ when the Wiener filter is used. The three curves correspond to the three considered values of the DoF range: $\psi _{\textrm {max}}=\{1\lambda ,1.5\lambda ,2\lambda \}$. Figure 2(b) represents the results obtained with the TV-based deconvolution algorithm. By comparing these two graphs, it is observed that for all the considered values of $\psi _{\textrm {max}}$, the curves obtained with the two criteria have similar shapes and reach their minima for the same values of $\rho _1$: the optimal masks obtained with the criteria based on the Wiener filter and on the TV deconvolution algorithm are thus identical.

Let us now consider a 3-ring mask, which is defined by two parameters. Figures 2(c) and 2(d) represent the variations of the Wiener and TV-based criteria as a function of these two parameters for $\psi _{\textrm {max}}=2 \lambda$. It is seen that in this case again, the shapes of the maps are very similar for the two criteria. The best value of the criterion is obtained for the same parameter set $(\rho _1, \rho _2)=(0.55,0.75)$. Thus, the optimal masks are identical. We have also performed the optimization of a 4-ring mask, which is defined by 3 parameters. Here again, we found that the optimal masks for both criteria are identical.

These results have been obtained with the *Butterfly* image [see Fig. 1(b)] as the truth-unknown image $\boldsymbol {x}_0$. In order to confirm this result, we performed the same optimizations on many other images of different types. We observed that in all cases, one obtains the same optimal masks for the two criteria. For example, we present in Fig. 3 and Fig. 4 the optimization results obtained with two other original images $\boldsymbol {x}_0$ with different characteristics. It is seen that the optimal 2-ring and 3-ring masks obtain with both criteria are identical.

In order to summarize these results, we have represented in Fig. 5(a) the profiles of the optimal masks (which are identical for both criteria) for different numbers of rings and different values $\psi _{\textrm {max}}$ of the targeted DoF range. In addition, Fig. 5(b) plots the values of the image quality criterion MMSE$(\boldsymbol {\rho }_{\textrm {opt}})$ of Eq. (4) obtained with these optimal masks. Each curve corresponds to one of the two considered deconvolution algorithms and one of the 3 defocus ranges. It is observed on these curves that the TV deconvolution algorithm always yields smaller MMSE values than the Wiener filter. This corroborates a well known result: TV-based deconvolution algorithms work better than the Wiener filter on natural images. As a conclusion, we have shown that the optimization criteria based on Wiener filter and on TV deconvolution yield the same optimal DoF enhancing masks, even if when using this optimal mask, the final image quality depends on the deconvolution algorithm. To generalize this conclusion, we have performed optimization of a 2-ring phase mask for the targeted DoF range $\psi _{\textrm {max}}=1\lambda$ and two additional noise levels. Note that according to Fig. 5(b), 2-ring masks are sufficient to reach minimal MMSE when $\psi _{\textrm {max}}=1\lambda$. The two selected noise levels, 24 dB and 9 dB, respectively correspond to noise standard deviations $\sigma =8.0~.10^{-3}$ and $\sigma =4.3~.10^{-2}$ on the *Butterfly* image. The results are shown in Table 1, where we have represented, for the three considered noise levels and three different images, the optimal mask parameter $\rho _1^{\textrm {opt}}$ and the corresponding MMSE. It is clearly seen that optimization criteria based on Wiener filter and TV-based algorithm yield quasi-identical optimal masks whatever the noise level.

This result is important in practice since optimization of the Wiener filter-based criteria is much faster, thanks to the closed-form expression of the MSE in Eq. (8). Note that in practice, it is preferable to optimize a single phase mask over a large set of images relevant to the application at hand. When using the Wiener filter, this is easy since it mainly boils down to using a prior power spectral density $S_{xx}$ for the whole set of images in the expression of the Wiener filter and of the MSE [see Eqs. (7) and (8)]. In contrast, when using the TV regularized algorithm, this is more time consuming since one needs to deconvolve each image in order to compute an averaged MSE. Since this has to be done at each iteration of the mask optimization algorithm, the optimization time is proportional to the size of the image set.

## 4. Interpretation and discussion

How is it possible to explain this equivalence of the two criteria for optimizing DoF enhancing masks? A first interpretation can be found with the help of Fig. 6, where we have represented the variation of the PSF profile with the defocus $\psi$ (note that since this PSF has circular symmetry, a cross-section along any line passing through its center is sufficient to represent its properties). The upper graph of Fig. 6 represents the profile of the PSFs obtained without mask, for $\psi \in [0,1\lambda ]$, and the lower graph represents the profile obtained with a 3-ring mask optimal for $\psi _{\textrm {max}} = 1 \lambda$. As expected, it is observed that the PSF obtained with the phase mask is much more invariant to defocus. This invariance is an essential property to ensure good image restoration. It is equally important for Wiener and TV deconvolution, since the expressions of both algorithms depend on the PSF (or the MTF) averaged over the values of $\psi$ within the targeted DoF range, namely, ${1}/{N_\psi }\sum _{i=1}^{N_\psi }h_{\boldsymbol {\rho },\psi _i}$[see Eqs. (7) and (9)]. For deconvolution to be efficient, this averaged PSF has to be representative of all the PSFs within the defocus range, and thus, these PSFs have to be similar. Indeed, it has been shown in Ref. [15] that DoF enhancing mask optimization is a tradeoff between maximizing similarity of the PSFs at the different defocus values $\psi$ and minimizing the width of these PSFs (or maximizing the level of the corresponding MTF) to limit noise enhancement by deconvolution.

To give more insight on the behavior of the deconvolution algorithms as a function of the spatial frequency content of the image, we performed a quantitative comparison between the results obtained with the Wiener and the TV deconvolution algorithm on two regions of the *Butterfly* image having very different graylevel variations. These regions are represented in Fig. 1(c): the zoom-in A1 contains many details, such as highly contrasted edges, whereas the zoom-in A2 has smooth spatial variations. We assume that the lens is equipped with a 3-ring binary phase mask optimized for $\psi _{\textrm {max}}=1\lambda$ and that the image is acquired under a defocus value of $\psi =0.75\lambda$. The bar chart of Fig. 7(a) displays the root mean square error, defined as ${\rm{RMSE}} = [\textrm {MSE}(\psi ,\boldsymbol {\rho }_{\textrm {opt}})]^{1/2}$ [see Eq. (3)] obtained with the Wiener and the TV deconvolution algorithms on the whole image A and on the two selected zoom-ins A1 and A2. It is seen that the RMSE obtained on the whole image A and on the “smooth” zoom-in A2 are smaller when using the TV algorithm. On the other hand, on the zoom-in A1, the RMSE is roughly equal for both algorithms. It is even slightly smaller with the Wiener filter. To illustrate the generality of this conclusion, we have also evaluated the image quality with two other metrics [33]: the L1 norm [see Fig. 7(b)], and the differential structural similarity index (DSSIM) [see Fig. 7(c)], defined as $\operatorname {\textrm {DSSIM}}=(1-\operatorname {\textrm {SSIM}})/2$, where SSIM is defined in Ref. [34]. It is seen that these two graphs lead to equivalent conclusions as the RMSE: on the whole image A and on the “smooth” zoom-in A2, the image quality is better when using the TV algorithm, whereas on the zoom-in A1, the image quality is roughly equal for both algorithms.

These results are visually illustrated on examples of deconvolved images in Fig. 8. The first column of Fig. 8 represents the original sharp image of the whole scene A and of the zoom-ins A1 and A2. The second column represents the blurred images obtained for a defocus $\psi =0.75\lambda$, when using the optimal 3-ring mask for $\psi _{\textrm {max}}=1 \lambda$ defocus range. The third column represents the images deconvolved with the Wiener filter, and the fourth column with the TV algorithm. These results show that the zoom-in A1 is restored with similar accuracy by the TV and the Wiener algorithms. The edges seem a little “sharper” in the TV image, due to the well-known “cartoon” effect of this type of algorithms. However, they are not more accurate: as shown in Fig. 7, the values of the RMSE yielded by the two algorithms are similar, the RMSE of the Wiener algorithm being even a little smaller. On the other hand, TV restoration yields much better results on the zoom-in A2, that mostly contains low spatial frequencies. This algorithm thus reaches a better compromise between recovery of sharp edges (where it equals the Wiener filter) and restoration of smooth regions, where it clearly outperforms the Wiener filter. Since this type of smooth regions represent most of the image area, the TV algorithm yields a smaller global RMSE on the whole image.

Based on these observations, we can propose the following conjecture. Whatever the deconvolution method, the same characteristics of the optical system are required to recover the details of the image. In the case of DoF extension, this characteristic is the invariance of the PSF with defocus. As a consequence, the optimal mask parameters, which govern the characteristics of the PSF, are common to both algorithms. On the other hand, since TV regularization is more adapted to real-world images than the quadratic regularization used implicitly in the Wiener filter, it yields much better restoration of smooth regions.

## 5. Conclusion

Considering the problem of optimization of DoF enhancing binary annular phase masks, we have shown that co-design procedures based on linear and nonlinear deconvolution methods yield identical optimal masks. This has been demonstrated on masks with different numbers of rings, different targeted defocus ranges, and for different types of observed images. We have proposed a conjecture to explain this fact: the same PSF characteristics are required to recover the details of the image whatever the deconvolution method. Of course, even with the same masks, the TV-based deconvolution algorithm yields better image reconstruction performance, in particular in the smooth regions of the image, since TV regularization is more adapted to real-world images. This result is important since it supports the frequent co-design practice consisting of optimizing a system with a simple closed-form criterion and deconvolving with a nonlinear algorithm to get better image quality. This work has many perspectives. The main one is to verify the generalization of this conjecture to other nonlinear deconvolution algorithms, other phase mask structures and other co-design problems.

## Disclosures

The authors declare no conflicts of interest.

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