A Super-Resolution Reconstruction Method for Infrared Polarization Images with Sparse Representation of Over-Complete Basis Sets

Abstract

The spatial resolution of an infrared focal plane polarization detection system is limited by the structure of the detector, resulting in lower resolution than the actual array size. To overcome this limitation and improve imaging resolution, we propose an infrared polarization super-resolution reconstruction model based on sparse representation, optimized using Stokes vector images. This model forms the basis for our method aimed at achieving super-resolution reconstruction of infrared polarization images. In this method, we utilize the proposed model to initially reconstruct low-resolution images in blocks. Subsequently, we perform a division by weight, followed by iterative back projection to enhance details and achieve high-resolution reconstruction results. As a supplement, we establish a near-real-time short-wave infrared time-sharing polarization system for data collection. The dataset was acquired to gather prior knowledge of the over-complete basis set and to generate a series of simulated focal plane images. Simulation experimental results demonstrate the superiority of our method over several advanced methods in objective evaluation indexes, exhibiting strong noise robustness in quantitative experiments. Finally, to validate the practical application of our method, we establish a split-focal plane polarization short-wave infrared system for scene testing. Experimental results confirm the effective processing of actual captured data by our method.

1. Introduction

With advancements in infrared technology, infrared polarization imaging has found growing applications in various domains, such as industrial inspection [1,2,3], medical diagnosis [4,5,6], and astronomical remote sensing [7,8,9]. This technology facilitates the augmentation of information acquired through infrared imaging and enables further exploration of post-image processing algorithms. The infrared focal plane polarization detector has garnered significant interest as the central component of infrared polarization imaging. It consists of multiple groups of micro-linear polarization units and infrared detectors [10]. The micro-linear polarization units are arranged in an ordered structure on the surface of the infrared detector, with each group containing four pixels. Each group records four different necessary intensity response values [11]. However, this design structure decreases the spatial resolution of the infrared focal plane polarization imaging system, resulting in a significant impact on the measurement accuracy of polarization information. To address the issue of limited sampling by the detector and the inability to obtain a full-resolution polarization image, super-resolution reconstruction of the polarization image is necessary. This involves utilizing an effective reconstruction algorithm to enhance image details and clarity, compensating for the loss of resolution caused by the system structure.

Super-resolution reconstruction techniques can be categorized based on their implementation methods into interpolation methods [12,13,14,15,16], reconstruction methods [17,18,19], and learning methods [20,21,22,23,24,25,26,27,28]. The interpolation methods include bilinear and bicubic interpolation methods [12], which were the initial algorithms utilized for super-resolution tasks. Subsequently, new algorithms were proposed, such as gradient-based interpolation [13], Newton polynomial-based interpolation (NP) [14], Edge-Aware Residual Interpolation (EARI) [15], and polarization image demosaicking using Polarization Channel Difference Prior (PCDP) [16], which leverage image geometric information and channel correlation. These interpolation techniques offer the advantage of quick reconstruction and effective edge preservation. However, since the degradation information of the actual pixel is not always accurately known, the models employed are not fixed.

The primary objective of super-resolution algorithms in reconstruction methods is to reverse-engineer the imaging process of an image. This involves establishing an observation model that links the low-resolution image with the high-resolution image. To solve the model, image-specific prior knowledge is introduced, either at a local or global level. As an example, Chen et al. combined the 3D block-matching algorithm with the concept of non-local filtering and introduced the Block-matching 3D Projectile On Convex Sets (BPOCS) algorithm [17]. Zhang et al. proposed the Frequency domain Phase-based Projection On Convex Sets (FPPOCS) algorithm, which combines the Wiener filter with the phase spectrum features of high-resolution images [18]. Similarly, in our previous work, we utilized principal component analysis and soft threshold denoising, using micro-scanning to capture sub-pixel micro-displacement frame sequences without altering the optical system’s structure. We proposed the Micro-scanning Polarization Projection On Convex Sets (MPPOCS) algorithm, designed specifically for infrared focal plane polarization detection systems [19]. The primary focus of the reconstruction method’s super-resolution algorithm is the degradation process of the image. It aims to narrow down the range of potential solutions using an observational model, thereby ultimately improving the preservation of the image’s geometry. However, due to the complex degradation factors encountered during the imaging process, artificial modeling cannot fully establish this process comprehensively.

The underlying principle of learning methods involves establishing correspondence between low-resolution and high-resolution images, utilizing the complementary information within similar blocks to preserve signal characteristics. In 2018, Zhang et al. introduced the Polarization Demosaicing Convolutional Neural Network (PDCNN) [20] to address the challenge of polarization super resolution, which lacked a specialized network for focal plane polarizing images. Subsequently, to mitigate accumulated errors caused by formula computation steps, Zeng et al. proposed an end-to-end fully convolutional neural network, FORK-NET [21]. This network takes the focal plane image as input and generates output intensity, polarization degree, and polarization angle images. An objective evaluation index indicated the superior performance of FORK-NET compared to PDCNN. Consequently, researchers explored alternative network models, such as the Multi-Scale Adaptive Weighted Network (MSAWN) [22], Deep Compressed Sensing (DCS) [23], and the sparsely polarimetric image demosaicing model (Sparse-PDM) [24]. These networks primarily target visible light polarization and necessitate extensive training data, typically sourced from Sony IMX250 series visible light polarization cameras. Articles [24,25] describe the publicly available dataset created by these researchers. The advantage of deep learning methods lies in their ability to achieve excellent reconstruction effects. Nevertheless, these methods also possess certain drawbacks that limit their suitability for specific tasks. Super-resolution performance becomes heavily reliant on computational power and data volume, limiting their deployment in computationally constrained and data-limited environments. Furthermore, the lack of clear interpretation in the structural units is evident. Conversely, shallow learning methods derived from traditional machine learning algorithms offer advantages such as adaptability to small datasets, relatively lower computational costs, and interpretability.

In this paper, we propose an infrared polarization super-resolution reconstruction model based on sparse representation. Our model optimizes the Stokes vector images, eliminating the need for mechanical reconstruction of a multi-channel polarization image. To enhance efficiency and accuracy, we construct three over-complete sets of Stokes vector image dictionaries. These dictionaries streamline the calculation process of the polarization degree and polarization angle while minimizing the potential for error.

Furthermore, we introduce a reconstruction method based on our model. This method utilizes the proposed model to initially reconstruct the low-resolution image in blocks. The image is then divided by the weight coefficients of the overlaps, and the details are refined using the Iterative Back Projection (IBP) technique to achieve high-resolution reconstruction results.

To build the dictionary, we designed a near-real-time short-wave infrared time-sharing polarization system prototype, leveraging our previous error analysis work [29]. This prototype enabled us to acquire the necessary dataset for simulation experiments. Additionally, we established a short-wave infrared focal plane polarization system for capturing real data. The experimental results demonstrate the superiority of our proposed method in both objective and subjective evaluation indexes.

The rest of this paper is organized as follows: Section 2 introduces the reconstruction model and algorithm, including the establishment of an over-complete dictionary set, noise analysis, and the iterative solution method. Section 3 presents the simulation and real experiment results, provides the objective evaluation index, and explains the subjective index. The anti-noise performance of the proposed algorithm is analyzed, and the running time of the algorithm is provided. Next, we discuss the results and introduce future research directions. Finally, Section 4 discusses the advantages of the proposed method, summarizes the novelty of the method and our contributions, and describes future research direction.

2. Super-Resolution Methods

An infrared focal plane detection system can capture essential intensity images from four different angles in a single exposure. Due to limitations imposed by the detector structure, the effective resolution of the four declination angle images is reduced by half of the original pixel size, expressed mathematically as follows:

$$I_{{\theta }_{i=1,2,3,4}}^{\mathrm{M}/2\times \mathrm{N}/2}=I_{\mathrm{out}}^{M\times N}\ast M{\mathrm{ask}}_{\theta }$$

(1)

where $I_{\mathrm{out}}^{M\times N}$ represents the cumulative light intensity recorded by the detector, analogous to an image obtained from a single exposure at a specific angle. $M{\mathrm{ask}}_{\theta }$ is an intensity extraction mask that biases the detection process to isolate the individual angle components within the overall light intensity. This mask is dependent on the pixel arrangement employed by the self-developed detector. Each superpixel within this arrangement consists of a tuple of four pixels. The resulting separated images for each declination angle, denoted as $I_{{\theta }_{i=1,2,3,4}}^{\mathrm{M}/2\times \mathrm{N}/2}$, exhibit a reduced size of half the original dimensions. The Stokes notation is a widely utilized method for representing polarization states in engineering, and its formula is provided below:

$$S=[\begin{array}{cccc}S0& S1& S2& S3\end{array}]$$

(2)

where $S0$, $S1$, $S2$, and $S3$ are parameters that describe different aspects of the incident light. $S0$ represents the total incident light intensity. $S1$ is the difference in intensity between the horizontal and vertical directions. $S2$ is the intensity difference between polarized light at 45 degrees and 90 degrees concerning a reference direction. $S3$ specifically relates to the detection of left-handed versus right-handed circular polarization. In typical projects, the detection of linear polarized light is generally carried out without considering the $S3$ component. By using the above formula, it is possible to solve for the Stokes vector of the incident light inversely. This method allows for the retrieval of information about the polarization degree and polarization angle of the current scene.

$$DP=\frac{\sqrt{S{1}^2+S{2}^2}}{S0}=\frac{\sqrt{{(I_0-I_{90})}^2+{(I_{45}-I_{135})}^2}}{(I_0+I_{45}+I_{90}+I_{135})\times {\displaystyle \frac{1}{2}}}$$

(3)

$$AP=\frac{1}{2}\arctan (\frac{S2}{S1})=\frac{1}{2}\arctan (\frac{I_{45}-I_{135}}{I_0-I_{90}})$$

(4)

When $I_{{\theta }_{i=1,2,3,4}}^{\mathrm{M}/2\times \mathrm{N}/2}$ is directly included in the aforementioned calculation formula, it becomes apparent that the dimensions of $DP$ and $AP$ calculated in this manner are $M/2\times N/2$. Furthermore, due to the unique arrangement structure of the superpixel, only a single declining-angle light intensity of a given pixel can be obtained in a single exposure. This leads to an instantaneous field error in the calculated polarization signal. Hence, it is imperative to conduct super-resolution reconstruction of $I_{{\theta }_{i=1,2,3,4}}^{\mathrm{M}/2\times \mathrm{N}/2}$.

2.1. Infrared Polarization Super-Resolution Reconstruction Model

To obtain the high-resolution image $I_{SR}$, the low-resolution image $I_{LR}$ is used. It is assumed that $I_{SR}$ can be linearly represented as $\alpha$ using the over-complete basis vector space $D_{SR}$. According to the theory of root sparse coding primitives, we can infer that the following formula holds when $\alpha$ is sufficiently sparse:

$${\tilde{I}}_{SR}=D_{SR}\alpha \begin{array}{cc}& \exists \alpha \in R^k with {‖\alpha ‖}_0<<k\end{array}$$

(5)

The over-complete basis vector $D_{SR}$, which can also be referred to as an over-complete dictionary set, is utilized. The objective is to minimize the Euclidean distance between the reconstructed image ${\tilde{I}}_{SR}$ and the actual high-resolution image, while ensuring that $\alpha$ satisfies the sparsity condition. This can be formulated as an optimization problem:

$$\underset{a}{\min }{‖\alpha ‖}_0\begin{array}{cc}& s.t.\end{array}\begin{array}{cc}& {‖D_{SR}\alpha -I_{SR}‖}_2^2<\epsilon \end{array}$$

(6)

Formula (6) satisfies the condition that the square of the 2-norm is smaller than the error $\epsilon$, rather than just the 2-norm being smaller than the error $\epsilon$. This choice is justified by previous research which has shown that in the field of super-resolution reconstruction, penalizing high-frequency errors by using the square of the 2-norm yields better results [26,27,28]. When performing super-resolution tasks with the input of image $I_{LR}$, it is evident that Formula (6) does not have enough known optimization solutions. In this context, the existence of dictionary $D_{LR}$ is assumed. This dictionary shares the sparse representation $\alpha$ with $D_{SR}$, and solving Formula (6) can be equivalently stated as the following equation:

$$\underset{a}{\min }{‖\alpha ‖}_0\begin{array}{cc}& s.t.\end{array}\begin{array}{cc}& {‖G{D}_{LR}\alpha -G{I}_{LR}‖}_2^2<\epsilon \end{array}$$

(7)

In the equation above, $G$ denotes an edge extraction operator, responsible for generating edge features for Formula (7). We observe that the pseudo-polarization information is primarily concentrated in the high-frequency portion. Precisely reconstructing the missing high-frequency content in the targeted high-resolution image holds the utmost significance. Extracting the high-frequency components from low-resolution images proves advantageous for the reconstruction of high-resolution images [30]. Formula (7) corresponds to an L0 problem, which can be approximated through an L1 problem using the Lagrange multiplier method:

$$\underset{a}{\min }{‖G{D}_{LR}\alpha -G{I}_{LR}‖}_2^2+\lambda {‖\alpha ‖}_1$$

(8)

In Formula (8), $\lambda$ is an adjustable parameter that is associated with the noise variance of the input image $I_{LR}$. The derivation process will be conducted later on. With the given image $I_{LR}$ and over-complete dictionary sets $D_{SR}$ and $D_{SR}$, a set of optimal sparse representations ${\alpha }^{\ast }$ is obtained by solving Formula (8). Subsequently, the reconstructed image ${\tilde{I}}_{SR}$ can be obtained by applying it to Formula (5). The resulting image satisfies the constraint stated in Formula (6).

We introduce adjustable parameter $\lambda$ into Formula (8), which is derived and analyzed using the Maximum A Posteriori (MAP) theory. According to the MAP theory, the squared term of the L2 norm in Formula (8) can be interpreted as the maximum likelihood estimate of $\alpha$. Additionally, part of the L1 norm can be considered as the Laplace prior of $\alpha$. Without taking into account the edge extraction operator $G$, this prior can be expressed as follows:

$${\alpha }^{\ast }=\underset{\alpha }{\mathrm{ar}g\max } p(I_{LR} | D_{LR},\alpha )p(\alpha )$$

(9)

The $\alpha$ variable in Formula (9) follows a Laplace distribution with the parameter (0, b). As a result, its probability density function can be expressed as follows:

$$p(\alpha )=\frac{1}{2b}{e}^{-\frac{{\left|\alpha \right|}_1}{b}}$$

(10)

The variable $I_{LR}$ in Formula (9) is distributed according to a Gaussian distribution with the parameter $(D_{L}a,{\delta }^2)$. Thus, its probability density function can be expressed as follows:

$$p(I_{LR} | D_{LR},\alpha )=\frac{1}{\sqrt{2\pi }\delta }{e}^{\frac{1}{-2{\delta }^2}{‖D_{LR}\alpha -I_{LR}‖}_2^2}$$

(11)

By incorporating Formulas (10) and (11) into Formula (9), we derive the resulting Formula (12):

$$\begin{array}{l}{\alpha }^{*}=\underset{\alpha }{\arg \max }\log p(I_{LR} | D_{LR},\alpha )+\log p(\alpha )\\ \hspace{1em}=\underset{\alpha }{\arg \max }{\displaystyle \frac{1}{-2{\delta }^2}}{‖D_{LR}\alpha -I_{LR}‖}_2^2+{\displaystyle \frac{1}{-2b}}{\left|\alpha \right|}_1\\ \hspace{1em}=\underset{\alpha }{\arg \min }{‖D_{LR}\alpha -I_{LR}‖}_2^2+{\displaystyle \frac{{\delta }^2}{b}}{\left|\alpha \right|}_1\end{array}$$

(12)

Equation $\lambda ={\delta }^2/b$ can be observed from the aforementioned formula. Given the sparsity of $\alpha$, it is assumed that the scale parameter $b$ remains relatively constant, while $\lambda$ should be increased in the presence of higher input image noise. For the experiment, a value of 0.2 was assigned to $\lambda$.

2.2. Reconstruction Method

$I_{{\theta }_{i=1,2,3,4}}^{\mathrm{M}/2\times \mathrm{N}/2}$ is a low-resolution image. If $I_{{\theta }_{i=1,2,3,4}}^{\mathrm{M}\times \mathrm{N}}$ in the same scene is solved, the $DP$ and $AP$ of full-resolution images can be solved using Formulas (3) and (4). Consequently, the Stokes vector needs to be calculated. The calculation process involves finding the difference between two groups of orthogonal light intensities. It is important to note that the different image is significantly influenced by the displacement of the target. Based on the experimental findings in [19], displacement-induced pseudo-polarization has a substantial impact on subjective perception. Therefore, our reconstruction method is expressed in the following equation:

$$\begin{array}{l}{\alpha }^{*}=\underset{a}{\arg \min }{‖\alpha ‖}_0\begin{array}{cc}S.T.& {‖G{D}_{LR}\alpha -G{I}_{LR}‖}_2^2<\epsilon \end{array}\\ I_{LR}=\{S{0}_{LR},(S1/S0{)}_{LR},(S2/S0{)}_{LR}\}\\ I_{SR}=D_{SR}{\alpha }^{*}\end{array}$$

(13)

$S{0}_{LR}$ is reconstructed as $S{0}_{SR}$ through the use of super resolution. Subsequently, ${(S1/S0)}_{LR}$ and ${(S2/S0)}_{LR}$ are reconstructed as ${(S1/S0)}_{SR}$ and ${(S2/S0)}_{SR}$, respectively. Finally, the $DP$ and $AP$ images at full resolution are calculated. This reconstruction approach offers several benefits. First, it eliminates the need to calculate the difference between each angular component, thereby avoiding any potential angle errors. Additionally, the reconstruction of ${(S1/S0)}_{SR}$ and ${(S2/S0)}_{SR}$ does not rely on the light intensity of the scene. The light intensity captured by the detector encompasses both natural and fully polarized light components, which can be derived using Malus’ law:

$$I^{\theta }=\frac{1}{2}{I}_N+I\times P\times {\cos }^2(\theta -A)$$

(14)

In the formula above, $I^{\theta }$ represents the intensity of partially polarized light at a given declination angle $\theta$. This can also be interpreted as the light intensity received by the detector at that angle. $I^N$ represents natural light, while $I^N$ represents the total light intensity. $I^N$ denotes the degree of polarization, while $A$ represents the polarization angle. The orthogonal difference of polarization is defined [31] as:

$$I_{\perp }^{\theta }=I_{\perp }^{\theta }-I_{\perp }^{\theta +\frac{\pi }{2}}=I(1-P)\times ({\cos }^2(\theta -A)-{\cos }^2(\frac{\pi }{2}+\theta -A))$$

(15)

In the formula above, when $\theta =0^{°}$, $I_{\perp }^{\theta }=S1$, and $\theta ={45}^{°}$, $I_{\perp }^{\theta }=S2$. The Stokes representation of the total light intensity $I$ is denoted as $S0$. Therefore, ${(S1/S0)}_{LR}$ and ${(S2/S0)}_{LR}$ are influenced by the polarization degree and polarization angle. By reconstructing them, it is possible to achieve better restoration of the $DP$ and $AP$ images at full resolution, instead of directly reconstructing the $DP$ and $AP$ low-resolution images from the high-resolution $S0$, $DP$, and $DP$ images. The proposed method does not take the entire image as input. Instead, a sliding window reconstruction is performed on a low-resolution image. The window size is set at $5\times 5$, with a step length of 1. During the reconstruction process, the number of overlapping pixels is calculated, and the reconstructed ${\tilde{I}}_{SR}$ is divided by the overlap coefficient $W$. The operational details are presented in the figure below.

As shown in Figure 1, the red segment illustrates a pixel-level reconstruction process where each pixel is reconstructed once, using a weight factor of $W=1$. The yellow part represents another reconstruction process. It is important to clarify that this specific depiction of traversing during the second step is not obligatory, as any traversal with a step size of 1 covering the entire image would be sufficient. It is worth noting that there is an overlap between the yellow and red segments. In terms of pixel-level analysis, the overlapping region carries a weight coefficient of $W=2$. The resulting reconstructed $I_{SR}$ is then divided by the overlap coefficient $W$. Additionally, the Iterative Back Projection (IBP) algorithm is employed to enhance image details and minimize the reconstruction error of the reconstructed ${\tilde{I}}_{SR}$. The input for the IBP algorithm is composed of the reconstructed image ${\tilde{I}}_{SR}$ and the reference image obtained through the sampling of $I_{LR}$ using the bicubic interpolation method.

The overall method consists of the following steps:

(1)

The acquired low-resolution image is initially divided into sliding window blocks with a step size of 1 and window size of $5\times 5$.

(2)

The ${\alpha }^{\ast }$ of each component is solved using Formula (8), and in this process, the feature symbol search algorithm proposed in [32] is employed.

(3)

The reconstructed image ${\tilde{I}}_{SR}$ of each component is then divided by the overlap coefficient $W$.

(4)

The Iterative Back Projection (IBP) algorithm is used to further reconstruct each component’s $W$.

(5)

Finally, the Stokes vector image, polarization degree image, and polarization angle image are calculated based on the reconstructed results.

2.3. Establishment of Dictionary Set

Section 2.1 discusses the dictionary, called $D_{LR}$, which is mentioned as having a shared sparse representation with $D_{SR}$. To achieve our desired solution, we need to construct a joint dictionary. The optimization problem for this joint dictionary is represented in the downward form:

$$\underset{D_{LR},D_{SR,}{\alpha }_U}{\min }\frac{1}{{\mathrm{C}}_1}{‖G{D}_{LR}{\alpha }_U-G{T}_{LR}‖}_2^2+\frac{1}{{\mathrm{C}}_2}{‖D_{SR}{\alpha }_U-T_{SR}‖}_2^2+\lambda (\frac{1}{{\mathrm{C}}_1}+\frac{1}{{\mathrm{C}}_2}){\left|{\alpha }_U\right|}_1$$

(16)

The variables ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ in the formula above are used to balance the costs of the Dictionary-based Super-Resolution $D_{SR}$ and Dictionary-based Low-Resolution $D_{LR}$ methods. Since the reconstruction method requires a sliding window, the dictionary set is also built using a sliding window approach with a window size of 5 and step size of 1. However, this practice leads to inconsistent matrix sizes, so we up-sample the low-resolution image before building the dictionary set. The values of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are set to the number of atoms corresponding to $D_{LR}$ and $D_{SR}$, respectively. In our case, ${\mathrm{C}}_2$ is set to 25 and ${\mathrm{C}}_1$ is set to 100, as our edge extraction operator $G$ is defined as the first and second derivatives in the $\mathrm{x}$ and $\mathrm{y}$ directions. Additionally, we further simplify Formula (16):

$$\begin{array}{l}\underset{D_U,{\alpha }_U}{\min }{‖D_U{\alpha }_U-T_U‖}_2^2+\lambda {\left|{\alpha }_U\right|}_1\\ D_U=\left[\begin{array}{l}\frac{1}{10}G{D}_{LR}\\ \frac{1}{5}{D}_{SR}\end{array}\right]& T_U=\left[\begin{array}{l}\frac{1}{10}G{T}_{LR}\\ \frac{1}{5}{T}_{SR}\end{array}\right]\end{array}$$

(17)

According to [33], to optimize the problem of solving Formula (17), it is necessary to constrain $D_U$. This ensures the existence of both ${\mathrm{d}}_{\mathrm{U},\mathrm{i}}^{\mathrm{rank} (D_U)\times 1}$ and ${\alpha }_U$ atomic sets within $D_U$, which leads to consistent results for ${\mathrm{d}}_{\mathrm{U},\mathrm{i}}^{\mathrm{rank} (D_U)\times 1}\cdot {\alpha }_U$ and approaching 0 for ${\left|{\alpha }_U\right|}_1$. As a result, it is crucial to normalize the atomic sets of $D_U$ and impose constraints before training. Representation of the entire dictionary set can be expressed using the following formula, with K being the total number of atomic sets in the dictionary set, set to 512 during training:

$$\begin{array}{cc}{D}_U^{*}=\underset{D_U,{\alpha }_U}{\mathrm{argmin}}{‖D_U{\alpha }_U-T_U‖}_2^2+\lambda {\left|{\alpha }_U\right|}_1& \mathrm{s}.t.\end{array}{‖d_{u.i}^{rank({\alpha }_U)\ast 1}‖}_2^2\le 1,i=1,2,3,\dots K$$

(18)

The formula is not convex on both $D_U$ and ${\alpha }_U$. However, it is convex on one of them. To address this, we follow the approach proposed in [32]. We first initialize $D_U$ and solve ${\alpha }_U$ using the feature symbol search algorithm. Then, we fix ${\alpha }_U$ and solve the duality problem of ${\alpha }_U$ using the conjugate gradient method. Solving the duality problem offers the advantage of generating fewer variables during the optimization process, thus making it faster.

3. Results

To validate the proposed method, we created a prototype near-real-time short-wave infrared time-sharing polarization system for collecting training and simulation data, building upon our previous work. As shown in Figure 2. The system utilized a custom short-wave detector that operated within the 0.95~1.65 μm wavelength range. Additionally, a self-designed zoom lens with a resolution of 120 lp/mm was employed. Real-time communication with the host optical signal was facilitated by the fully electronic design of the rotating target wheel component. The system offered a polarization time resolution of 25 frames. A total of 60 scenes, encompassing indoor and outdoor settings, were captured. Amongst these scenes, 54 were used to form the training dictionary set, while 6 scenes were reserved for simulation testing. The $S0$ images from the simulation test are depicted in Figure A1 in Appendix A, showcasing cars, a window, a wall, a building, a tower, and a tank. Each image has a resolution of 640 × 512 pixels.

In this paper, four interpolation methods were utilized: the Newton polynomial-based interpolation method (NP) [14], the Edge Sensing Residual Interpolation method (EARI) [15], a polarization difference priority-based interpolation algorithm (PCDP) [16], and the Micro-scanning Polarization Projection On Convex Sets (MPPOCS) algorithm [19], which we previously proposed. As the MPPOCS algorithm is a multi-frame super-resolution algorithm, we modified its input. In the simulation experiment, we simulated the micro-scan image input using a single frame image. For the real image test, we directly employed a multi-frame image as the input.

In terms of objective evaluation indicators, three commonly used methods in super-resolution reconstruction tasks were selected: a structural similarity coefficient (SSIM), the Peak Signal-to-Noise Ratio (PSNR), and Root Mean Square Error (RMSE). These evaluation criteria were applied to the normalized images of $S0$, $S1$, $S2$, $DP$, and $AP$, respectively. A higher SSIM indicated that the reconstructed image closely resembled the reference image in terms of intensity, contrast, and structure. The accuracy of each polarization component’s reconstructed image reflected the true polarization characteristics of the object. The noise in the polarization image mainly manifested as false polarization. PSNR, which measures the ratio of peak signal energy to average noise energy, was used to assess the pseudo-polarization level in each component of the reconstructed polarization image. RMSE was used to gauge the degree of similarity between the reconstructed polarization image and the true polarization image in terms of pixel position. A smaller RMSE value indicated a closer match between the reconstructed and true polarization values for the same pixel position. By combining these three objective evaluation measures with subjective criteria, a comprehensive analysis of the proposed reconstruction method was conducted.

3.1. Super-Resolution Performance Test

The experimental setup consisted of a PC host with an i7-4720HQ CPU (Intel Corporation, Santa Clara, CA, USA) and GTX960 GPU (Nvidia Corporation, Santa Clara, CA, USA). The specific parameter settings for both the reconstruction and training parts of the dictionary set are provided and explained in Section 2. It is important to note that dictionary set learning takes place after the image segmentation process. For training purposes, we randomly selected 100,000 windows from 54 training scenes, each with dimensions of 5 × 5. No noise reduction algorithm was employed following the non-uniformity correction and blind element repair of the captured images. Consequently, the resulting images may have contained electronic noise, dark current noise [34], stray light response, and some uncorrected blind elements. The objective evaluation indicators for the simulation results are depicted in

Table 1. The objective evaluation indexes of each algorithm for a tower.

Method	S0	S1	S2	DP	AP
EARI	SSIM = 0.9649	SSIM = 0.9670	SSIM = 0.9751	SSIM = 0.8148	SSIM = 0.7559
	PSNR = 35.6509	PSNR = 41.4362	PSNR = 42.4668	PSNR = 27.9892	PSNR = 19.8982
	RMSE = 0.0165	RMSE = 0.0085	RMSE = 0.0075	RMSE = 0.0399	RMSE = 0.1012
NP	SSIM = 0.9708	SSIM = 0.9807	SSIM = 0.9842	SSIM = 0.8599	SSIM = 0.7749
	PSNR = 36.8702	PSNR = 43.5596	PSNR = 44.0752	PSNR = 29.7688	PSNR = 23.2195
	RMSE = 0.0143	RMSE = 0.0066	RMSE = 0.0063	RMSE = 0.0324	RMSE = 0.0690
PCDP	SSIM = 0.9723	SSIM = 0.9748	SSIM = 0.9827	SSIM = 0.8512	SSIM = 0.7602
	PSNR = 37.1946	PSNR = 42.4574	PSNR = 43.67161	PSNR = 29.2385	PSNR = 22.4482
	RMSE = 0.0138	RMSE = 0.0075	RMSE = 0.0066	RMSE = 0.0345	RMSE = 0.0754
MPPOCS	SSIM = 0.9687	SSIM = 0.9972	SSIM = 0.9952	SSIM = 0.9610	SSIM = 0.9452
	PSNR = 36.1099	PSNR = 54.8412	PSNR = 50.9966	PSNR = 38.9372	PSNR = 38.6978
	RMSE = 0.0156	RMSE = 0.0018	RMSE = 0.0028	RMSE = 0.0113	RMSE = 0.0116
This Work	SSIM = 0.9809	SSIM = 0.9980	SSIM = 0.9972	SSIM = 0.9729	SSIM = 0.9595
	PSNR = 40.0757	PSNR = 56.8644	PSNR = 54.0460	PSNR = 41.8172	PSNR = 41.3630
	RMSE = 0.0099	RMSE = 0.0014	RMSE = 0.0019	RMSE = 0.0081	RMSE = 0.0086

.

The table above clearly demonstrates the excellent reconstruction performance achieved by the proposed method. All five algorithms yielded good results for the $S1$ and $S2$ images. The majority of pixels in the $S1$ and $S2$ images represented the low-frequency component. Despite this, our approach still stood out and achieved the best results. Unlike the other algorithms, our method did not obtain the $S1$ and $S2$ images by detecting the difference in declination angle. On the $S0$ index, the MPPOCS method exhibited a low reconstruction effect. This is because the filtering method used in this algorithm sacrifices some details of the decline-detecting image, reduces the noise of the orthogonal difference image, and neglects the quality of the weighted image. In our reproductions of the other algorithms, some did not align with the original data. We analyzed these replications in our experiments using the publicly available source code provided by the original authors. The primary difference lay in the down-sampling technique. The authors used interlaced pixel extraction for the simulation, resulting in simulated images with minimal noise. In contrast, our simulated image contained significantly more noise. Additionally, our degradation process employs bicubic interpolation to obtain the subsampled image, where the pixels in the neighborhood are double cubic-weighted. This is because we considered that one of the imaging units of the detector does not correspond one-to-one with the real scene, as it contains a weighted component from the neighborhood due to the system’s point spread function. The impact of calculation errors was evident in the $DP$ and $AP$ indexes. If one of the Stokes vector images exhibited unsatisfactory reconstruction, the objective indexes of $DP$ and $AP$ were greatly diminished. In other words, the $DP$ and $AP$ images were more susceptible to noise. Therefore, it was crucial to analyze the noise robustness of the method, and a detailed experimental analysis will be presented in the next section. Figure 3 showcases the simulation reconstruction results of the different algorithms for a tower, where it is evident that the proposed method yielded results closer to the real image. Additionally,

Table A1. Objective evaluation indexes of S0 results of different algorithms on different test images.

Method	Cars	Window	Wall	Building	Tank
EARI	SSIM = 0.4032	SSIM = 0.7799	SSIM = 0.4203	SSIM = 0.5723	SSIM = 0.3943
	PSNR = 13.4202	PSNR = 14.5269	PSNR = 13.4207	PSNR = 21.7906	PSNR = 14.1210
	RMSE = 0.2133	RMSE = 0.1878	RMSE = 0.2133	RMSE = 0.0814	RMSE = 0.1968
NP	SSIM = 0.4374	SSIM = 0.7858	SSIM = 0.4502	SSIM = 0.5519	SSIM = 0.4114
	PSNR = 13.8023	PSNR = 14.6402	PSNR = 12.6379	PSNR = 21.5929	PSNR = 14.6783
	RMSE = 0.2041	RMSE = 0.1854	RMSE = 0.2334	RMSE = 0.0832	RMSE = 0.1845
PCDP	SSIM = 0.4481	SSIM = 0.7918	SSIM = 0.4629	SSIM = 0.5441	SSIM = 0.4148
	PSNR = 13.6557	PSNR = 14.7660	PSNR = 12.8692	PSNR = 19.9623	PSNR = 14.4895
	RMSE = 0.2076	RMSE = 0.1827	RMSE = 0.2273	RMSE = 0.1004	RMSE = 0.1886
MPPOCS	SSIM = 0.6442	SSIM = 0.8353	SSIM = 0.6514	SSIM = 0.8971	SSIM = 0.6470
	PSNR = 17.5827	PSNR = 15.7587	PSNR = 20.3389	PSNR = 36.8555	PSNR = 16.9773
	RMSE = 0.1321	RMSE = 0.1630	RMSE = 0.0962	RMSE = 0.0144	RMSE = 0.1416
This Work	SSIM = 0.6966	SSIM = 0.8318	SSIM = 0.7093	SSIM = 0.9210	SSIM = 0.6933
	PSNR = 18.0517	PSNR = 15.7436	PSNR = 20.8368	PSNR = 38.7827	PSNR = 17.3267
	RMSE = 0.1252	RMSE = 0.1632	RMSE = 0.0908	RMSE = 0.0115	RMSE = 0.1360

in Appendix A displays the objective index of $S0$ after the reconstruction of other test images, while

Table A2. Objective evaluation indexes of S1 results of different algorithms on different test images.

Method	Cars	Window	Wall	Building	Tank
EARI	SSIM = 0.9637	SSIM = 0.9922	SSIM = 0.9195	SSIM = 0.8338	SSIM = 0.9886
	PSNR = 39.6600	PSNR = 39.6600	PSNR = 37.5078	PSNR = 32.5971	PSNR = 46.5985
	RMSE = 0.0104	RMSE = 0.0045	RMSE = 0.0133	RMSE = 0.0235	RMSE = 0.0047
NP	SSIM = 0.9806	SSIM = 0.9958	SSIM = 0.9639	SSIM = 0.8393	SSIM = 0.9931
	PSNR = 42.4280	PSNR = 48.6937	PSNR = 40.3364	PSNR = 33.1142	PSNR = 48.0276
	RMSE = 0.0076	RMSE = 0.0037	RMSE = 0.0096	RMSE = 0.0221	RMSE = 0.0040
PCDP	SSIM = 0.9781	SSIM = 0.9947	SSIM = 0.9498	SSIM = 0.8365	SSIM = 0.9916
	PSNR = 41.7149	PSNR = 47.9808	PSNR = 39.1224	PSNR = 33.0923	PSNR = 47.3843
	RMSE = 0.0082	RMSE = 0.0040	RMSE = 0.0111	RMSE = 0.0222	RMSE = 0.0042
MP	SSIM = 0.9938	SSIM = 0.9989	SSIM = 0.9984	SSIM = 0.9882	SSIM = 0.9986
	PSNR = 48.9114	PSNR = 59.5889	PSNR = 58.3427	PSNR = 48.7383	PSNR = 58.9306
	RMSE = 0.0036	RMSE = 0.0011	RMSE = 0.0012	RMSE = 0.0037	RMSE = 0.0011
This Work	SSIM = 0.9959	SSIM = 0.9991	SSIM = 0.9986	SSIM = 0.9926	SSIM = 0.9989
	PSNR = 51.2608	PSNR = 60.5056	PSNR = 59.0080	PSNR = 51.1248	PSNR = 60.0059
	RMSE = 0.0027	RMSE = 0.0010	RMSE = 0.0011	RMSE = 0.0028	RMSE = 0.0010

presents the objective indicators of $S1$, and

Table A3. Objective evaluation indexes of S2 results of different algorithms on different test images.

Method	Cars	Window	Wall	Building	Tank
EARI	SSIM = 0.9466	SSIM = 0.9909	SSIM = 0.9196	SSIM = 0.9356	SSIM = 0.9894
	PSNR = 36.0382	PSNR = 45.3721	PSNR = 36.9036	PSNR = 37.7624	PSNR = 46.8909
	RMSE = 0.0158	RMSE = 0.0055	RMSE = 0.0143	RMSE = 0.0129	RMSE = 0.0045
NP	SSIM = 0.9582	SSIM = 0.9933	SSIM = 0.9614	SSIM = 0.9518	SSIM = 0.9927
	PSNR = 37.4198	PSNR = 46.3064	PSNR = 39.6867	PSNR = 39.1404	PSNR = 48.1039
	RMSE = 0.01346	RMSE = 0.0048	RMSE = 0.0104	RMSE = 0.0110	RMSE = 0.0039
PCDP	SSIM = 0.9554	SSIM = 0.9930	SSIM = 0.9435	SSIM = 0.9518	SSIM = 0.9920
	PSNR = 36.9204	PSNR = 46.1596	PSNR = 38.2077	PSNR = 38.9414	PSNR = 47.6478
	RMSE = 0.01426	RMSE = 0.0049	RMSE = 0.0123	RMSE = 0.0113	RMSE = 0.0042
MP	SSIM = 0.9918	SSIM = 0.9983	SSIM = 0.9974	SSIM = 0.9706	SSIM = 0.9977
	PSNR = 47.1006	PSNR = 56.4553	PSNR = 55.7543	PSNR = 42.0427	PSNR = 56.1393
	RMSE = 0.0044	RMSE = 0.0015	RMSE = 0.0016	RMSE = 0.0079	RMSE = 0.0016
This Work	SSIM = 0.9948	SSIM = 0.9888	SSIM = 0.9981	SSIM = 0.9857	SSIM = 0.9965
	PSNR = 49.6021	PSNR = 45.8931	PSNR = 57.6130	PSNR = 45.9872	PSNR = 53.0169
	RMSE = 0.0033	RMSE = 0.0051	RMSE = 0.0013	RMSE = 0.0050	RMSE = 0.0022

exhibits the objective indicators of $S2$. Furthermore,

Table A4. Objective evaluation indexes of DP results of different algorithms on different test images.

Method	Cars	Window	Wall	Building	Tank
EARI	SSIM = 0.8003	SSIM = 0.9442	SSIM = 0.6495	SSIM = 0.7586	SSIM = 0.6876
	PSNR = 27.8272	PSNR = 30.8589	PSNR = 21.5937	PSNR = 26.9068	PSNR = 27.2853
	RMSE = 0.0406	RMSE = 0.0287	RMSE = 0.0832	RMSE = 0.0452	RMSE = 0.0432
NP	SSIM = 0.8284	SSIM = 0.9419	SSIM = 0.6942	SSIM = 0.7590	SSIM = 0.7227
	PSNR = 29.3589	PSNR = 30.6841	PSNR = 24.7070	PSNR = 27.8181	PSNR = 28.3058
	RMSE = 0.0340	RMSE = 0.0292	RMSE = 0.0582	RMSE = 0.0407	RMSE = 0.0384
PCDP	SSIM = 0.8216	SSIM = 0.9484	SSIM = 0.6709	SSIM = 0.7798	SSIM = 0.7113
	PSNR = 28.9635	PSNR = 31.2784	PSNR = 22.9016	PSNR = 28.0442	PSNR = 27.7868
	RMSE = 0.0356	RMSE = 0.0273	RMSE = 0.0716	RMSE = 0.0396	RMSE = 0.0408
MP	SSIM = 0.9496	SSIM = 0.9747	SSIM = 0.9682	SSIM = 0.9293	SSIM = 0.9213
	PSNR = 36.8699	PSNR = 40.9748	PSNR = 44.0480	PSNR = 36.1728	PSNR = 37.4231
	RMSE = 0.0143	RMSE = 0.0089	RMSE = 0.0063	RMSE = 0.0155	RMSE = 0.0135
This Work	SSIM = 0.9653	SSIM = 0.9496	SSIM = 0.9775	SSIM = 0.9599	SSIM = 0.9338
	PSNR = 40.6423	PSNR = 31.4468	PSNR = 45.8423	PSNR = 39.6173	PSNR = 35.5246
	RMSE = 0.0093	RMSE = 0.0268	RMSE = 0.0051	RMSE = 0.0105	RMSE = 0.0167

provides the objective index of $DP$, and

Table A5. Objective evaluation indexes of AP results of different algorithms on different test images.

Method	Cars	Window	Wall	Building	Tank
EARI	SSIM = 0.9665	SSIM = 0.9834	SSIM = 0.9616	SSIM = 0.9223	SSIM = 0.9788
	PSNR = 31.4232	PSNR = 41.2151	PSNR = 31.9594	PSNR = 27.5346	PSNR = 41.9182
	RMSE = 0.0268	RMSE = 0.0087	RMSE = 0.0252	RMSE = 0.0420	RMSE = 0.0080
NP	SSIM = 0.9740	SSIM = 0.9848	SSIM = 0.9665	SSIM = 0.9477	SSIM = 0.9834
	PSNR = 32.7683	PSNR = 42.4306	PSNR = 33.1721	PSNR = 29.4241	PSNR = 43.2909
	RMSE = 0.0230	RMSE = 0.0076	RMSE = 0.0220	RMSE = 0.0338	RMSE = 0.0069
PCDP	SSIM = 0.9752	SSIM = 0.9859	SSIM = 0.9696	SSIM = 0.9484	SSIM = 0.9848
	PSNR = 32.8784	PSNR = 42.6716	PSNR = 33.6649	PSNR = 29.4233	PSNR = 43.6527
	RMSE = 0.0227	RMSE = 0.0074	RMSE = 0.0207	RMSE = 0.0338	RMSE = 0.0066
MP	SSIM = 0.9725	SSIM = 0.9852	SSIM = 0.9675	SSIM = 0.9335	SSIM = 0.9818
	PSNR = 32.2656	PSNR = 41.3038	PSNR = 32.7580	PSNR = 27.6061	PSNR = 42.0923
	RMSE = 0.0244	RMSE = 0.0086	RMSE = 0.0230	RMSE = 0.0417	RMSE = 0.0079
This Work	SSIM = 0.9845	SSIM = 0.9892	SSIM = 0.9810	SSIM = 0.9757	SSIM = 0.9918
	PSNR = 34.7240	PSNR = 46.0793	PSNR = 37.4299	PSNR = 33.1091	PSNR = 46.8903
	RMSE = 0.0184	RMSE = 0.0050	RMSE = 0.0134	RMSE = 0.0221	RMSE = 0.0045

presents the objective index of $AP$ after the reconstruction of other test images.

3.2. Noise Robustness

Limited by the manufacturing process of the detector, the noise of a focal plane polarization system is generally higher than that of a time-sharing polarization system. Therefore, it was crucial to analyze the noise robustness of the algorithm. Based on the comparison results presented in the previous section, both the proposed method and MPPOCS outperformed other methods. Consequently, this section compares the noise robustness between these two methods. The test image again was a tower. Gaussian white noise with mean 0 and standard deviation ranging from 0 to 3 was added to the focal plane image synthesized by each bias detection component of the tower. The value of $\lambda$, introduced in this paper, was set to 0.02. The objective indexes for the reconstructed $S0$ and $DP$ images of the two algorithms are depicted in Figure 4, while the reconstructed indexes for the $S1$, $S2$, and $AP$ images are shown in the attached Figure A2 and Figure A3a,b. SSIM is represented on the black axis, PSNR on the intermediate axis, and RMSE on the blue axis. The + line represents the algorithm proposed in this paper, and the × line represents MPPOCS.

In Figure 4a, the objective evaluation index of the proposed algorithm gradually decreased as the level of noise increased, with a decline rate higher than that of MPPOCS. In Figure 4b, when the mean noise was 0.9, the SSIM index of our proposed method started to become lower than that of MPPOCS. When the mean noise values were 2.7 and 2.8, respectively, both PSNR and RMSE were weaker than those of the MPPOCS algorithm. Similar results were observed for the $S1$, $S2$, and $AP$ images. In this set of experiments, we kept the $\lambda$ values unchanged. These findings proved that the proposed algorithm possessed inherent noise resistance, as the reconstruction results were superior to MPPOCS when the noise level was low. Additionally, our algorithm demonstrated better performance in the case of low noise, as shown in Figure 4a. This can be attributed to the fact that the $S0$ image was less sensitive to the noise in each channel compared to the other components, resulting in consistently optimal reconstruction results. We analyzed the results to ascertain the reasons behind the MPPOCS algorithm’s high noise robustness. The inclusion of wavelet threshold noise reduction in the algorithm grants it a natural adaptability to noise, especially in the presence of high levels of noise. However, in the context of weak noise, this can lead to excessively smooth image results, as illustrated in Figure 3.

3.3. Real Image Testing

We established a focal plane polarization detection system operating in the short-wave infrared range to collect real images to test our algorithm. The size of the focal plane images was 320 × 256 pixels. The obtained test results are shown in Figure 5.

The selected test scene had cloudy weather conditions. The drone used as the target was the DJI Spirit 4Pro (DJI, Shenzhen, China). All display results were normalized to the same quantization level. No additional image enhancement algorithms were applied to the presentation results. Figure 5 shows the full-field $S0$ image, revealing a dim target. Among the various algorithms tested, the EARI algorithm produced the fuzziest $S0$ image. In several other channels, it was not possible to identify the target as a drone. However, some areas of the $DP$ image allowed for the calculation of the degree of polarization. The NP and PCDP algorithms did not clearly show the target in the $S1$ image. In the $DP$ image, neither of these two algorithms accurately calculated the polarization degree of the UAV. Man-made objects, being more polarized than the natural background, did not exhibit this phenomenon well in the results. These two algorithms introduced significant errors in the $AP$ images. Due to the drone’s orientation towards the lens, the polarization angles of its left and right sides did not significantly differ. As for MPPOCS, it can be observed that the $S0$ image was overly blurry. The filter utilized in this algorithm reduced noise in the entire image, but it also sacrificed some high-frequency information. The $S1$ and $S2$ images performed better than with the NP and PCDP algorithms. The $DP$ image provided a clearer representation of the target’s polarization information compared to the EARI algorithm. The $AP$ image was slightly more distinct than with the first three algorithms. Ultimately, our proposed $S0$ algorithm yielded the best results for the image. The brackets on the left and right holding the rotors were separated from the drone’s fuselage. The $S1$ and $S2$ images outperformed other algorithms, particularly in the $AP$ image, where our proposed algorithm accurately reconstructed the entire shape of the UAV. With a certain level of prior knowledge, the UAV could be identified as the DJI Genie 4Pro drone even without specialized equipment. Additionally, in the $DP$ image, our reconstruction results displayed the target’s polarization degree. The target’s edges were sharp, and the difference between the target and background was highly noticeable, providing a strong foundation for the recognition of small polarized targets. Finally, the processing time of the proposed algorithm was statistically analyzed and is shown in

Table 2. The real image reconstruction results of each algorithm’s running time.

Method	EARI	NP	PCDP	MPPOCS	This Work
Running Time/s	0.31	0.19	0.03	1.18	18.84

.

The running time of the method proposed in this paper appeared to be relatively long, as shown in Table 2. To analyze the source code, we employed MATLAB’s runtime analysis tool. Our analysis revealed that the feature symbol search algorithm accounted for a significant portion of the computing time, specifically 80.1%. Considering this finding, we are convinced that there is ample scope for improvement. Thus, one of our forthcoming research areas will focus on optimizing the feature symbol search algorithm.

4. Conclusions

This article presents a sparse representation-based model for infrared polarization super-resolution reconstruction. We propose a method to overcome the limitations of the detector’s sampling on the focal plane and achieve satisfactory results without requiring a large training set. Consequently, this approach is suitable for tasks where obtaining a vast amount of data is not feasible. Specifically, instead of independently solving each polarization angle, we solve the model on a designed Stokes vector field. This strategy effectively avoids errors caused by differential calculations during the polarization image-solving process. The simulation results depicted in Figure 3 demonstrate that our method effectively suppresses partial pseudo-polarization phenomena and generates more realistic polarization information compared to other existing methods. Furthermore, the results from a real test scene illustrated in Figure 5 reveal the significant advantages of our reconstruction results in object detection and classification tasks. Our algorithm delivers clearer edges of the targets and a noticeable contrast between the target and background, marking an improvement over the other algorithms discussed in this paper. In addition, to construct the dictionary set, we employ a self-developed detector to create a near-real-time short-wave infrared time-sharing polarization system. This system is equipped with a high-precision grating disk, enabling the continuous acquisition of polarization images. The main findings of our research are described in this article as follows:

(1) We propose a super-resolution reconstruction model for infrared polarization. (2) We design a reconstruction method based on the model. (3) We produce a prototype of a near-real-time short-wave infrared time-sharing polarization system and collect an infrared polarization dataset. (4) We build a short-wave infrared focal plane polarization system and conduct experiments to demonstrate the excellent results achieved by the proposed method in terms of quantitative measurement and visual quality.

In our future research, we intend to optimize the algorithms mentioned in the paper to enhance their speed in a software environment, with a primary focus on accelerating the feature symbol search algorithm. Moreover, we plan to integrate the proposed algorithm into our developed short-wave focal plane polarization detection system. Additionally, our future work involves incorporating convex optimization algorithms and deep learning methods to further improve image quality. Furthermore, we aim to explore the combination of metasurface design and its application in achieving full optical image reconstruction.