Research on Photon-Integrated Interferometric Remote Sensing Image Reconstruction Based on Compressed Sensing

Abstract

Achieving high-resolution remote sensing images is an important goal in the field of space exploration. However, the quality of remote sensing images is low after the use of traditional compressed sensing with the orthogonal matching pursuit (OMP) algorithm. This involves the reconstruction of the sparse signals collected by photon-integrated interferometric imaging detectors, which limits the development of detection and imaging technology for photon-integrated interferometric remote sensing. We improved the OMP algorithm and proposed a threshold limited-generalized orthogonal matching pursuit (TL-GOMP) algorithm. In the comparison simulation involving the TL-GOMP and OMP algorithms of the same series, the peak signal-to-noise ratio value (P_SNR) of the reconstructed image increased by 18.02%, while the mean square error (M_SE) decreased the most by 53.62%. The TL-GOMP algorithm can achieve high-quality image reconstruction and has great application potential in photonic integrated interferometric remote sensing detection and imaging.

1. Introduction

With the increasingly mature manufacturing process of photonic integrated devices and interference detection technology, the segmented planar imaging detector for electro-optical reconnaissance (SPIDER), which has photonic integrated interference imaging as its core technology, has attracted a lot of attention from researchers in the field of astronomical observation or remote sensing detection. It has been used to replace traditional optical telescopes with large volume, weight, and energy consumption [1] in the detection of targets. For example, the Hubble Telescope is 13.3 m long and weighs 27,000 pounds [2].

Interferometry is an important technology used in photonic integrated interferometric imaging systems. It uses electromagnetic wave superposition to extract the wave source information and provides technical support for the reconstruction of high-resolution images. Optical interferometry unifies the light from many lens pairs on a photonic integrated chip (PIC) and then reconstructs the remote sensing image from the optical signal obtained by interferometry. Optical interferometer arrays are the preferred instruments for high-resolution imaging. Such interferometer arrays include the CHARA array [3,4], larger telescope interferometer [5], and navy precision optical interferometer [6]. These systems use far-field spatial coherence measurements to form intensity images of light source targets [7]. In our previous publication [8,9], we discussed the definition of a small-scale interferometric imager, which we called a planar photoelectric detection imaging detector (SPIDER). The SPIDER imager [8] comprises one-dimensional interferometric arms arranged along the azimuth angles in multiple directions. Each interference arm has the same design structure. Any two lenses on the interference arm form the interference baseline; the collected optical signals are coupled in the PIC and interfered in the multi-mode interferometer (MMIs), while the fringe data are read by the two-dimensional detector array. Because the interference arms in the PIC are distributed along the azimuth angle [0, 2$\mathsf{\pi }$], and since interference baselines of any length on the interference arms correspond to the spatial frequency information in the two-dimensional Fourier Transform domain, the PIC can obtain optical frequency information through sparse sampling in all directions. We can use the compressing sensing (CS) theory algorithm to reconstruct sparse optical signal data in order to obtain the content information of detection targets. The CS theory can be applied in the field of photonic integrated interference imaging to meet our needs in life, production, and scientific exploration.

In recent years, CS has attracted increasing attention in signal processing. Donoho et al. proposed this theory in 2006. The traditional Nyquist sampling theorem [10] requires that the sampling frequency of the signal be greater than or equal to twice the signal frequency. The proposed compressed sensing theory overcomes the limitations of traditional sampling theorems. If the collected signals are sufficiently sparse, the original signals can be reconstructed by projection onto random vectors. More specifically, the original signals could be reconstructed at low speeds. Therefore, this innovative theory of improving sampling efficiency has been of great interest in the fields of digital signal processing [11], optical imaging [12], medical imaging [13], radio communication [14], radar imaging [15], and pattern recognition [16]. The research conducted on compressed sensing comprises three main areas: (1) the sparse representation of original signals, with commonly used sparse transform methods such as the Fourier Transform (FT) [17], Discrete Cosine Transform (DCT) [18], and Wavelet Transform (DWT) [19]; (2) the design of the measurement matrix, including the random measurement matrix [20,21] and deterministic measurement matrix [22,23]; (3) reconstruction algorithms, such as the basis pursuit (BP) algorithm [24,25], matching pursuit (MP) algorithm [26], and orthogonal matching pursuit algorithm [27,28,29,30].

The compressed sensing OMP algorithm is one of the most representative greedy algorithms; it is simple, stable, has low computational complexity, and has been widely studied by researchers. In contrast, the traditional OMP algorithm produces Gaussian noise when reconstructing an image, which significantly affects the quality of the reconstructed image. Consequently, the traditional OMP algorithm has continuously been improved over time, and enhanced algorithms such as stagewise orthogonal matching pursuit (STOMP), generalized orthogonal matching pursuit (GOMP), and stagewise weak orthogonal matching pursuit (SWOMP) have been generated to improve the quality of the reconstructed image. To further solve the above-mentioned problems, we improved the threshold limited-generalized orthogonal matching tracing algorithm using the traditional OMP algorithm.

The main contributions of this paper can be summarized as follows:

(1)

We improved the traditional OMP algorithm and proposed the TL-GOMP algorithm, which was used to reconstruct the sparse spatial frequency information collected by the PIC and recover the content information of the detected target. In the simulation, we compared the TL-GOMP algorithm with the other improved OMP image reconstruction algorithm from the same series and the non-OMP image reconstruction algorithm, and subsequently verified its superiority in image reconstruction.

(2)

Simultaneously, we used this algorithm to reconstruct and simulate the sparse signals collected by photonic integrated chips at different distances. The simulation results showed that the TL-GOMP algorithm can be applied in the field of photon-integrated interferometric remote sensing detection and imaging to recover the content information of unknown targets.

2. Related Work

Image reconstruction is based on sparse original signals from the target or image, and the content and feature information of the target or image are restored and reproduced by designing reconstruction algorithms. At present, the compressed sensing reconstruction algorithm has become the mainstream in the field of image reconstruction, mainly because the image signal has two characteristics: high dimension and can be sparse. The research on compressed sensing theory mainly includes three aspects: sparse signal representation, measurement matrix design, and reconstruction algorithm design.

2.1. Sparse Signal Representation

The sparse representation of signals is an important premise and foundation of compressed sensing theory. When a signal can become approximately sparse under the action of a change domain, it is said to have sparsity or compressibility, which can achieve the purpose of reducing signal storage space and effectively compressed sampling. If the length of a signal is N, and the number of non-zero value elements is no more than k after representation by the sparse basis matrix, we can define it as a k-sparse signal. The sparsity k of the sparse signal directly affects the accuracy of the reconstructed signal; that is, the higher the sparsity, the higher the accuracy of the reconstructed signal. Based on the above reasons, the reasonable selection of the sparse basis matrix is very important. The commonly used transform bases are as follows: Fourier Transform basis [17], Discrete Cosine Transform basis [18], Discrete Wavelet Transform basis [19], Contourlet Transform basis [31], and the K-singular value decomposition method based on matrix decomposition [32].

2.2. Design of Measurement Matrix

In compressed sensing theory, the measurement matrix has the function of sampling the original signal, and its selection is very important. It can project the signal from a high-dimensional space to a low-dimensional space to obtain the corresponding measurement value. In order to obtain an accurate sparse representation through measurement values, an uncorrelated relationship between the observed matrix and the sparse basis matrix was required to satisfy the Restricted Isometry Property (RIP), which guaranteed that the original space and the sparse space could be mapped one-to-one. At the same time, the matrix formed by arbitrarily extracting the number of column vectors that was equal to the number of observed values is non-singular. Commonly used measurement matrices are as follows: Gaussian random matrix [33], measurement matrix constructed based on equilibrium Gold sequence [34], partial Fourier matrix [35], and partial Hadamard matrix [36]. Wang Xia proposed a deterministic random sequence measurement matrix [37] and verified its effectiveness through experiments.

2.3. Design of Reconstruction Algorithm

In recent years, remarkable achievements have been made in the research on compressed sensing reconstruction algorithms, which can be divided into the traditional iterative compressed sensing reconstruction algorithm and the deep compressed sensing network-based reconstruction algorithm.

2.3.1. Traditional Iterative Compressed Sensing Reconstruction Algorithm

The purpose of compressed sensing is to find the sparsest original signal to meet the demands of measurement, which can be understood as the inverse problem of minimizing the norm of l0. Specific methods for achieving this are as follows: (1) convex relaxation method, which converts the minimum l0-norm problem into the minimum l1 norm problem under certain conditions, that is, the non-convex problem is converted into a convex problem such as the basis pursuit algorithm [38] and the Gradient Projection for Sparse Reconstruction (GPSR) [39]; (2) greedy matching tracking algorithms such as the matching pursuit algorithm [26] and the orthogonal matching pursuit algorithm [27,28]; (3) non-convex optimization methods, including the Bayesian Compressed Sensing algorithm (BCS) [40]; and (4) model-based optimization algorithms, the first three of which are based on the sparsity of original signals; these may not be valid for ordinary signals, such as the improved Total Variation-based algorithm (TV) [41].

2.3.2. Reconstruction Algorithm Based on Deep Compressed Sensing Network

As the use of deep learning in various research fields has increased, it has gradually been introduced into the research on compressed sensing image reconstruction algorithms. Ali Mousavi et al. proposed a Stacked Denoising Autoencode (SDA) algorithm, which mainly realized the end-to-end mapping between measured values and reconstructed images and adopted an unsupervised learning method. Kulkarni et al. proposed ReconNet [42], a non-iterative framework based on convolutional neural networks, and applied convolutional neural networks to compressed sensing reconstruction for the first time. The network structure consisted of a fully connected layer and six convolutional layers. Yao and Dai et al. combined the idea of residual learning with ReconNet and proposed a Deep Residual Reconstruction Network (DR2-Net) for compressed image perception reconstruction [43]; the network was cascaded. Kulkarni K, Lohit S et al. [44] further deepened the network structure of ReconNet and used the network structure of a full connection layer to replace the original Gaussian matrix in order to realize the image sampling. This kind of network is called adaptive sampling ReconNet. Xuemei Xie et al. [45] made some improvements to the sampling process of compressed sensing and also used full connection and deconvolution methods to optimize the compressed sensing network. Nie and Fu et al. not only used the convolutional neural network for image reconstruction, but also added image denoising into the network. The ResConv network they proposed [46] has these two characteristics. The CSnet network proposed by Shi et al. [47] redesigned the sampling process, which, as with the previous algorithms, does not only realize image reconstruction, but also puts forward a novel sampling mechanism to match the reconstructed network.

3. Methods

The theoretical framework of compressed sensing consists of three main aspects: the sparse representation of the original signal vector $\overrightarrow{x}$; the measurement matrix designed to change the high-dimensional original signal into a low-dimensional measurement vector $\overrightarrow{y}$; and the algorithm designed to obtain the approximate sparse representation $\overrightarrow{\widehat{\theta }}$ in order to recover the original signal.

3.1. The Reconstruction Principle of the OMP Algorithm Based on Compressed Sensing

Figure 1 shows a schematic for solving sparse representations in compressed sensing. Here, we consider the compressed sensing theory as a linear model:

$$\left[\overrightarrow{y_1},\overrightarrow{y_2},\cdots \cdots ,\overrightarrow{y_n}\right]=B_{m\times n}\times \left[\overrightarrow{x_1},\overrightarrow{x_2},\cdots \cdots ,\overrightarrow{x_n}\right]$$

(1)

where $\overrightarrow{y}\in R^m$ and $\overrightarrow{x}\in R^n$ represent the column vectors in the observation data and unknown image, respectively, and the measurement matrix $B\in R^{m\times n}$ arranged in the order of the column vectors is known. We chose the unknown image $D\in R^{n\times n}$, which can be represented by $D=\left[\overrightarrow{x_1},\overrightarrow{x_2},\cdots \cdots ,\overrightarrow{x_n}\right]$.

Because the original signal $\overrightarrow{x}$ is not absolutely sparse, to transform it into a compressible signal, a sparse basis matrix $\Psi \in R^{n\times n}$ is adopted, which transforms the original signal into a sparse domain and forms a sparse representation vector $\overrightarrow{\theta }\in R^{n\times 1}$. The number of non-zero values in the sparse representation vector $\overrightarrow{\theta }=[k_1,k_2\cdots \cdots k_{n-1},0,0]$ is $k\ll n$, and thus the vector $\overrightarrow{\theta }$ is called the k-sparse representation. The measured data of the target can be written as follows:

$$\begin{array}{cc}\hfill \left[\overrightarrow{y_1},\overrightarrow{y_2},\cdots \cdots ,\overrightarrow{y_n}\right]& =B\times \Psi \times \left[\overrightarrow{{\theta }_1},\overrightarrow{{\theta }_2},\cdots \cdots ,\overrightarrow{{\theta }_n}\right]\hfill \\ & =\left(\begin{array}{ccc}{b}_{11}& \dots & b_{1n}\\ ⋮& \ddots & ⋮\\ b_{m1}& \cdots & b_{mn}\end{array}\right)\times \left(\begin{array}{ccc}{\psi }_{11}& \dots & {\psi }_{1n}\\ ⋮& \ddots & ⋮\\ {\psi }_{m1}& \cdots & {\psi }_{mn}\end{array}\right)\times \left[\overrightarrow{{\theta }_1},\overrightarrow{{\theta }_2},\cdots \cdots ,\overrightarrow{{\theta }_n}\right]\hfill \end{array}$$

(2)

Here, we define the sensor matrix $A$, whose function is to establish the linear relationship between the sparse representation $\overrightarrow{\theta }$ and the measured value $\overrightarrow{y}$.

The measurement data of the target can then be expressed as

$$\left[\overrightarrow{y_1},\overrightarrow{y_2},\cdots \cdots ,\overrightarrow{y_n}\right]=A\times \left[\overrightarrow{{\theta }_1},\overrightarrow{{\theta }_2},\cdots \cdots ,\overrightarrow{{\theta }_n}\right]$$

(3)

Here, we use the most commonly used OMP algorithm [30,31] to illustrate the approximate solution $\overrightarrow{\widehat{\theta }}$ of the sparse representation. A column vector $\overrightarrow{y}\in \left[\overrightarrow{y_1},\overrightarrow{y_2},\cdots \cdots ,\overrightarrow{y_n}\right]$ of the measurement data is selected, and ${\overrightarrow{r}}^{(k)}$ is used to represent the residual value after the $k_{TH}$ iteration. The initial value of the residual is set as ${\overrightarrow{r}}^{(k)}={\overrightarrow{y}}^{(0)}$. ${\Lambda }_k$ represents the matrix used to store the column vector ${\overrightarrow{a}}_k$ of the sensor matrix after the $k_{TH}$ iteration. The initial value of the matrix is represented by ${\Lambda }_0$. The sensor matrix is defined as follows:

$$A={\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right)}_{m\times n}=\left[{\overrightarrow{a}}_1,{\overrightarrow{a}}_2\cdots \cdots {\overrightarrow{a}}_n\right]$$

(4)

After multiplying the transposed form ${[{\overrightarrow{a}}_1,{\overrightarrow{a}}_2\cdots \cdots {\overrightarrow{a}}_n]}^T$ of the sensor matrix A with the initial residual value ${\overrightarrow{r}}^{(0)}$, $\overrightarrow{b}$ can be expressed as

$$\overrightarrow{b}={\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1m}\\ ⋮& \ddots & ⋮\\ a_{n1}& \cdots & a_{nm}\end{array}\right)}_{n\times m}\times \overrightarrow{y}={\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1m}\\ ⋮& \ddots & ⋮\\ a_{n1}& \cdots & a_{nm}\end{array}\right)}_{n\times m}\times {\left[\begin{array}{l}{y}_1\\ ⋮\\ y_m\end{array}\right]}_{m\times 1}={\left[\begin{array}{l}{b}_1\\ ⋮\\ b_m\end{array}\right]}_{m\times 1}$$

(5)

Here, each element in the vector ${\overrightarrow{b}}^T=[b_1,b_2\cdots \cdots b_m]$ represents the inner product of each row vector in ${[{\overrightarrow{a}}_1,{\overrightarrow{a}}_2\cdots \cdots {\overrightarrow{a}}_n]}^T$ with ${\overrightarrow{r}}^{(0)}$, that is, $b_i={\overrightarrow{a}}_j^T\times {\overrightarrow{r}}^{(0)}$ ($i=1,2\cdots \cdots m$; $j=1,2\cdots \cdots n$). The corresponding column vector ${\overrightarrow{a}}_j$ in the sensor matrix is selected according to the maximum inner product value $b_i$, and ${\overrightarrow{a}}_j$ is stored in the $\Lambda$ matrix. The least-squares method is used to obtain the minimum residual value ${\overrightarrow{c}}^{(k)}={({\overrightarrow{a}}_j^T\times {\overrightarrow{a}}_j)}^{-1}\times {\overrightarrow{a}}_j^T\times {\overrightarrow{y}}^{(k-1)}$. The residual value ${\overrightarrow{r}}^{(k)}$ after the $k_{TH}$ iteration is

$${\overrightarrow{r}}^{(k)}={\overrightarrow{y}}^{(k-1)}-{\overrightarrow{a}}_j^{(k)}\times {\overrightarrow{c}}^{(k)}$$

(6)

where ${\overrightarrow{a}}_j^{(k)}$ represents the column vector selected from the sensor matrix during the $k_{TH}$ iteration. Finally, after $k$ iterations, we can obtain the $k$-sparse representation (approximate solution $\overrightarrow{\widehat{\theta }}$), which comprises k non-zero values such as $c^{(1)},c^{(2)}\cdots \cdots c^{(k)}$. This is an optimization problem for the smallest norm of $l_1$, which can be mathematically expressed as follows:

$$\underset{\overrightarrow{\theta }}{\min }{\left|\left|\overrightarrow{\theta }\right|\right|}_{l_1}s.t.\overrightarrow{y}=B\psi \overrightarrow{\theta }$$

(7)

Algorithm 1 presents the execution steps of the OMP algorithm. As shown in Figure 2, we multiply the approximate solution $\overrightarrow{\widehat{\theta }}$ by the sparse basis matrix $\psi$; then, the original signal recovered is $\overrightarrow{\widehat{x}}=\Psi \times \overrightarrow{\widehat{\theta }}$. The final reconstructed image is obtained as follows:

$$\left[\overrightarrow{{\widehat{x}}_1},\overrightarrow{{\widehat{x}}_2},\cdots \cdots ,\overrightarrow{{\widehat{x}}_n}\right]=\Psi \times \left[\overrightarrow{{\widehat{\theta }}_1},\overrightarrow{{\widehat{\theta }}_2},\cdots \cdots ,\overrightarrow{{\widehat{\theta }}_n}\right]$$

(8)

Algorithm 1: Orthogonal Matching Pursuit

Input: Sensor matrix B, Sparseness k

Output: $\mathrm{Sparse} \mathrm{representation} \overrightarrow{\theta }$

Initialize: $\mathrm{Residual}{\overrightarrow{r}}_0=\overrightarrow{y}$, $\mathrm{Index} \mathrm{set}{\Lambda }_0=\varnothing$, t = 1

Loop performs the following five steps: (1) $\mathrm{Find} \mathrm{out}:q:q_t=\arg {\max }_{j=1\cdots \cdots N}\left|〈{\overrightarrow{r}}_{t-1},{\alpha }_j〉\right|$; (2) Update the index set: ${\Lambda }_t={\Lambda }_{t-1}\cup \left\{q_t\right\}$; Reconstruction of atomic collection: $B_t=[B_{t-1},{\alpha }_q]$; (3) Least-squares method:${\overrightarrow{\theta }}_t=\arg \min {\left|\left|\overrightarrow{y}-B_t\overrightarrow{x}\right|\right|}_2$; (4) Update the residual: ${\overrightarrow{r}}_t=\overrightarrow{y}-B_t{\overrightarrow{\theta }}_t,t=t+1$; (5) Judgment: If t > k, stop the iteration, or go to step (1).

3.2. The Reconstruction Principle of the TL-GOMP Algorithm Based on Compressed Sensing

In this section, we introduce an improved TL-GOMP algorithm based on the traditional OMP algorithm. We selected unknown images $D\in R^{N\times N}$. To illustrate the principle of the improved TL-GOMP algorithm, we took the unknown target $D\in R^{N\times N}$ and converted it into the form of a column vector $D=\left[{\overrightarrow{x}}_1,{\overrightarrow{x}}_2\cdots \cdots {\overrightarrow{x}}_n\right]$, as expressed by the equation below:

$$\left[{\overrightarrow{y}}_1,{\overrightarrow{y}}_2\cdots \cdots {\overrightarrow{y}}_n\right]={\left(\begin{array}{ccc}{b}_{11}& \dots & b_{1n}\\ ⋮& \ddots & ⋮\\ b_{m1}& \cdots & b_{mn}\end{array}\right)}_{m\times n}\times \left[{\overrightarrow{x}}_1,{\overrightarrow{x}}_2\cdots \cdots {\overrightarrow{x}}_n\right]$$

(9)

Here, $\Psi ={\left(\begin{array}{ccc}{b}_{11}& \dots & b_{1n}\\ ⋮& \ddots & ⋮\\ b_{m1}& \cdots & b_{mn}\end{array}\right)}_{m\times n}$ is the measurement matrix. Assuming that the residual value after k iterations is ${\overrightarrow{r}}^{(k)}$, we arbitrarily extracted a column $\overrightarrow{y}\in \left[{\overrightarrow{y}}_1,{\overrightarrow{y}}_2\cdots \cdots {\overrightarrow{y}}_n\right]$ and assigned it to the initial value ${\overrightarrow{r}}^{(0)}=\overrightarrow{y}$ of the residual value. By using the transpose form ${[{\overrightarrow{a}}_1,{\overrightarrow{a}}_2\cdots \cdots {\overrightarrow{a}}_n]}^T$ of the sensor matrix $B={\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right)}_{m\times n}=[{\overrightarrow{a}}_1,{\overrightarrow{a}}_2\cdots \cdots {\overrightarrow{a}}_n]$, and multiplying the residual value ${\overrightarrow{r}}^{(k-1)}\in R^{m\times 1}$, we could obtain vector $\overrightarrow{d}\in R^{n\times 1}$ as follows:

$$\overrightarrow{d}=\left[\begin{array}{l}{d}_1\\ ⋮\\ d_n\end{array}\right]=\left(\begin{array}{ccc}{a}_{11}& \dots & a_{1m}\\ ⋮& \ddots & ⋮\\ a_{n1}& \cdots & a_{nm}\end{array}\right)\times {\overrightarrow{r}}^{(k-1)}$$

(10)

Here, we define one parameter $q_s=\left|\left|{\overrightarrow{r}}^{(k-1)}\right|\right|/\sqrt{M}$ and the other parameter $m_s=1$; the parameter $m_s$ can be understood as a variable that controls or adjusts the threshold, which is a range value. The principle of its selection is to constantly change the threshold value and form the corresponding column vector of the first $S$ inner product values in the sensor matrix into a matrix, with the purpose of solving the optimal $S$ least-squares solutions to form a sparse representation. After $k$ cycles, the sparsity of the sparse representation is $kS$. Parameter $M$ represents the number of rows of perception matrix and measurement matrix in compressed sensing theory, or the number of measurements of observation matrix. The threshold $Th$ is then denoted as

$$Th=m_s{q}_s=\left|\left|{\overrightarrow{r}}^{(k-1)}\right|\right|/\sqrt{M}$$

(11)

Subsequently, we took the absolute value of each of the elements in the vector $\overrightarrow{d}$ and placed them in descending order to obtain the vector ${\overrightarrow{d}}^T=[d_{11},d_{22}\cdots \cdots d_{nn}]$. We stored the sequence numbers of the elements satisfying the inequality relation in Equation (12).

$${\overrightarrow{d}}^T=[d_{11},d_{22}\cdots \cdots d_{nn}]\ge \left|\left|{\overrightarrow{r}}^{(k-1)}\right|\right|/\sqrt{M}$$

(12)

The algorithm cycles $k$ times in total, where $k$ refers to the number of non-zero-valued elements in the sparse representation. Each cycle will store the maximum number of elements $S$ that satisfy the threshold conditions. After $k$ cycles, there is a $kS$ value. Thereafter, the column vector of the sensor matrix corresponding to the value of $kS$ is stored in the matrix $A_t$, where $A_t\in R^{M\times kS}$. We then use the least-squares method to obtain the approximate solution $\overrightarrow{\widehat{\theta }}$ for the sparse representation, as expressed by the equation below:

$$\overrightarrow{\widehat{\theta }}={(A_t^T\times A_t)}^{-1}\times A_t^T\times {\overrightarrow{r}}^{(k-1)}$$

(13)

After each iteration, the updated residual value ${\overrightarrow{r}}^{(k)}$ is expressed as:

$${\overrightarrow{r}}^{(k)}={\overrightarrow{r}}^{(k-1)}-A_t\times \overrightarrow{\widehat{\theta }}$$

(14)

Finally, the reconstructed image $\widehat{D}=\left[{\overrightarrow{\widehat{x}}}_1,{\overrightarrow{\widehat{x}}}_2\cdots \cdots {\overrightarrow{\widehat{x}}}_n\right]$ is obtained as follows:

$$\widehat{D}=\left[{\overrightarrow{\widehat{x}}}_1,{\overrightarrow{\widehat{x}}}_2\cdots \cdots {\overrightarrow{\widehat{x}}}_n\right]=\left(\begin{array}{ccc}{\psi }_{11}& \dots & {\psi }_{1n}\\ ⋮& \ddots & ⋮\\ {\psi }_{m1}& \cdots & {\psi }_{mn}\end{array}\right)\times \left[{\overrightarrow{\widehat{\theta }}}_1,{\overrightarrow{\widehat{\theta }}}_2\cdots \cdots {\overrightarrow{\widehat{\theta }}}_n\right]$$

(15)

As listed in Algorithm 2, the core function of the algorithm is to effectively select the maximum number $S$ using the additional threshold value. After the algorithm iterates $k$ times, the sparse representation vector has $kS$ sparsity. The threshold value used was $Th=m_s{q}_s$. In the subsequent simulations, we selected $m_s=1$ and $q_s=\left|\left|{\overrightarrow{r}}^{(k-1)}\right|\right|/\sqrt{M}$.

Algorithm 2: Threshold Limited–Generalized Orthogonal Matching Pursuit

Input: Sensor matrix B, Sparseness k

Output:$\mathrm{Sparse} \mathrm{representation} \overrightarrow{\theta }$$, \mathrm{Residual} {\overrightarrow{r}}_k$

Initialize:$\mathrm{Residual} {\overrightarrow{r}}_0=\overrightarrow{y}$$, \mathrm{Index} \mathrm{set} {\Lambda }_0=\varnothing$$, A_0=\varnothing$, t = 1

Loop performs the following five steps: $(1) \mathrm{Find} \mathrm{out}q:q_t=\arg {\max }_{j=1\cdots \cdots N}\left|〈{\overrightarrow{r}}_{t-1},{\alpha }_j〉\right|, \mathrm{selecting} \mathrm{the} \mathrm{maximum} \mathrm{number} S \mathrm{of} \mathrm{values} \mathrm{that} \mathrm{are} \mathrm{greater} \mathrm{than} \mathrm{the} \mathrm{threshold} \mathrm{value} Th=m_s{q}_s$; $(2) \mathrm{Update} \mathrm{the} \mathrm{index} \mathrm{set}: {\Lambda }_t={\Lambda }_{t-1}\cup \left\{q_t\right\}; \mathrm{Reconstruction} \mathrm{of} \mathrm{atomic} \mathrm{collection}{B}_t=[B_{t-1},{\alpha }_q]$; (3) Least-squares method: ${\overrightarrow{\theta }}_t=\arg \min {\left|\left|\overrightarrow{y}-B_t\overrightarrow{x}\right|\right|}_2$; $(4) \mathrm{Update} \mathrm{the} \mathrm{residual}: {\overrightarrow{r}}_t=\overrightarrow{y}-B_t{\overrightarrow{\theta }}_t,t=t+1$; (5) Judgment: If t > k, stop the iteration, or go to step (1).

The advantage of this algorithm is that the inner product values meeting the threshold conditions can be quickly screened out in time by setting the limiting threshold value $Th=m_s{q}_s$, and corresponding column vectors can be directly found in the sensor matrix according to the serial number of the first $S$ inner product values. These inner product values are represented by logical value 1 in the code, while other inner product values are represented by logical value 0. On the other hand, by setting the threshold coefficient $m_s$ to adjust the limiting threshold, we constantly combine the serial numbers of the first $S$ inner product values into the corresponding column vectors in the sensor matrix to form a matrix, aiming at solving the optimal $S$ least-squares solutions with good universality and flexibility. After $k$ iterations, we can reconstruct the image information of the target through sparse representation $\overrightarrow{\theta }$ with a sparsity of $kS$.

4. Experiments

To demonstrate the performance of the TL-GOMP algorithm (refer to Algorithm 2) in reconstructing the target images, we present some simulation results in this section. First, we used the TL-GOMP algorithm and the same series of OMP, STOMP, and GOMP algorithms to conduct a comparative simulation of the target test images, as shown in Figure 3. In this case, the simulation results show that the image quality reconstructed using the TL-GOMP algorithm is better than that reconstructed using the same series of OMP algorithms. Subsequently, in order to rigorously prove the advantages of the TL-GOMP algorithm, we selected algorithms other than the OMP series to conduct a comparative simulation of the targets shown in Figure 3, and the simulation results once again showed that the image reconstructed by the TL-GOMP algorithm was better than that reconstructed by the other algorithms. We then applied the TL-GOMP algorithm in the field of photonic integrated interference image reconstruction and used this algorithm to reconstruct sparse spatial frequency information collected by the PIC at different distances. The simulation image results show that this algorithm can reconstruct the content information of the detected target well. Finally, we explored the measurement number $M$ and sparsity $k$ in the TL-GOMP algorithm and their influence on the quality of the reconstructed images.

In all the experiments below, we used the Gaussian random matrix as the measurement matrix, which is established by the randn function in the code, and the values of each element in this matrix satisfy the standard normal distribution. Meanwhile, we used the discrete cosine transform matrix as the sparse matrix, whose function is the sparse representation or compression of the original signal. In the experiments, the measurement matrix was updated with the operation of the code every time, which reflects the randomness. Therefore, we conducted several simulation experiments in each research part and verified the reliability of the conclusion through the data results.

4.1. Comparison of Simulation Results of the TL-GOMP and OMP Series Algorithms

For this section, we selected test images with pixel values of 350 × 350, 500 × 500, 650 × 650, and 800 × 800 as the target scenes; four image reconstruction algorithms (OMP, STOMP, GOMP, and TL-GOMP) to perform image reconstruction simulation; and used peak signal-to-noise ratio and mean square error to evaluate the image quality. Figure 4a–d show the simulation results of the 800 × 800 image reconstruction. From an intuitive point of view, the improved TL-GOMP tracing algorithm can be used to further improve the quality of the reconstructed images.

Table 1. P_SNR, M_SE, and time of the 350 × 350 image reconstructed by OMP series algorithms.

	OMP	STOMP	GOMP	TL-GOMP
PSNR (dB)	18.1671	18.4884	18.2944	21.1882
MSE	991.6670	920.9654	963.0243	494.6024
Running time (s)	3.3577	1.7663	2.7278	2.5031

presents the quality evaluation data and the code runtime for the 350 × 350 reconstructed images. From the perspective of quantitative data, we can also see that the P_SNR values of the images obtained by the TL-GOMP algorithm increased by 15.82% (compared with the results of the GOMP algorithm), 14.60% (compared with the STOMP algorithm), and 16.63% (compared with the OMP algorithm). The M_SE values of the images decreased by 48.64%, 46.30%, and 50.12%, respectively, and the code running time was relatively fast. Therefore, from the above simulation data, we conclude that the TL-GOMP algorithm based on compressed sensing can rely on sparse data collected by the PIC to restore the content information of the detected target.

Table 2. P_SNR, M_SE, and time of the 500 × 500 image reconstructed by OMP series algorithms.

	OMP	STOMP	GOMP	TL-GOMP
PSNR (dB)	18.5193	18.5506	18.5136	21.856
MSE	914.4386	907.8610	915.6401	424.1134
Running time (s)	6.7739	3.2574	5.5594	5.6994

shows simulation results, with the test image at a resolution of 500 × 500 pixels selected as the target. The data results show that the P_SNR values of the reconstructed images obtained by the TL-GOMP algorithm increased by 18.05% (compared with the GOMP algorithm), 17.82% (compared with the STOMP algorithm), and 18.02% (compared with the OMP algorithm). The M_SE values of the images decreased by 53.68%, 53.28%, and 53.62%, respectively.

Table 3. P_SNR, M_SE, and time of the 650 × 650 image reconstructed by OMP series algorithms.

	OMP	STOMP	GOMP	TL-GOMP
PSNR (dB)	18.9683	18.9774	18.6246	21.8993
MSE	824.6088	822.8859	892.5204	419.9057
Running time (s)	12.1603	5.5557	12.0085	11.5920

shows the simulation results, with the test image at a resolution of 650 × 650 pixels selected as the target. The data results show that the P_SNR values of the reconstructed images obtained by the TL-GOMP algorithm increased by 17.58% (compared with the GOMP algorithm), 15.40% (compared with the STOMP algorithm), and 15.45% (compared with the OMP algorithm). The M_SE values of the images decreased by 52.95%, 48.97%, and 49.08%, respectively.

Table 4. P_SNR, M_SE, and time of the 800 × 800 image reconstructed by OMP series algorithms.

	OMP	STOMP	GOMP	TL-GOMP
PSNR (dB)	19.1433	19.3284	18.7903	21.4954
MSE	792.0467	759.0033	859.1182	460.8311
Running time (s)	18.8039	8.7040	22.8444	23.6628

shows the simulation results, for which a test image with a resolution of 800 $\times$ 800 pixels was selected as the target. The resulting data show that the P_SNR values of the reconstructed image obtained by the TL-GOMP algorithm increased by 14.40% (compared with the result of the GOMP algorithm), 11.21% (compared with the result of the STOMP algorithm), and 12.29% (compared with the result of the OMP algorithm). The M_SE values of the images decreased by 46.36%, 39.28%, and 41.82%, respectively. In the image quality evaluation, the higher the peak signal-to-noise ratio, the better the image quality; in contrast, the smaller the mean square error value, the better the image quality. The red curve shown in Figure 5 is the simulation result of the image reconstruction with different resolutions using the TL-GOMP algorithm. We can observe that image quality improves with an increase in resolution, and thus this algorithm has the advantage of improving the quality of the reconstructed image.

Table 5. P_SNR, M_SE, and time of the different pixel image reconstructed by TL-GOMP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
21.3302	478.6986	3.0450	21.9785	412.3144	5.7769	21.7634	433.2510	12.1419	21.4110	469.8728	24.8091
21.3607	475.3485	2.4762	21.8199	427.6476	5.7062	21.8976	420.0683	12.2621	21.1822	495.2878	24.8688
21.2495	487.6785	2.4810	21.6895	440.6868	5.7678	21.7275	436.8474	16.2619	21.3847	472.7259	24.6147
21.3407	477.5455	2.4567	21.9302	416.9279	5.6955	21.9430	415.7013	12.2062	21.2746	484.8612	24.4856
21.1693	496.7621	2.4414	21.6719	442.4760	5.7021	21.9532	414.7298	12.2592	21.2543	487.1404	24.2110
21.2787	484.4030	2.4960	21.7925	430.3612	5.8279	21.8741	422.3505	15.7850	21.2940	482.7085	24.5258
21.2071	492.4605	2.4652	21.8286	426.7995	5.6639	21.7166	437.9442	15.4912	21.3495	476.5711	24.5294
21.3974	471.3456	2.4356	21.8625	423.4784	5.6723	21.6169	448.1164	15.5665	21.1921	494.1608	24.6656
21.2362	489.1702	2.4730	21.8421	425.4695	5.6400	21.7768	431.9201	14.7710	21.2277	490.1291	23.7997
21.1882	494.6024	2.5031	21.8560	424.1134	5.6994	21.8993	419.9057	11.5920	21.4954	460.8311	23.6628
PSNR Mean: 21.2758 MSE Mean: 484.8015			PSNR Mean: 21.8272 MSE Mean: 427.0275			PSNR Mean: 21.8168 MSE Mean: 428.0835			PSNR Mean: 21.3066 MSE Mean: 481.4289

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by TL-GOMP algorithms.

Table 6. P_SNR, M_SE, and time of the different pixel image reconstructed by OMP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
18.1671	991.6670	3.9888	18.4080	938.1606	7.5859	18.9861	821.2361	12.3038	19.0968	800.5658	20.3478
18.2682	968.8629	3.2198	18.5454	908.9571	7.7925	18.9347	831.0171	12.5650	19.1620	788.6510	21.0002
18.3101	959.5554	3.4737	18.4466	929.8707	7.2411	18.9399	830.0296	12.5509	19.2571	771.5568	20.4160
18.3829	943.6068	3.6291	18.5152	915.3030	7.1395	19.1091	798.3158	12.4553	19.0280	813.3486	20.9783
18.0393	1021.3	3.6883	18.4265	934.1821	7.7281	18.9553	827.0812	12.3023	19.2065	780.6037	22.2521
18.2515	972.5983	3.2354	18.5735	903.0985	7.4901	18.9398	830.0348	12.4164	19.1368	793.2272	20.6466
18.3869	942.7279	3.2961	18.6313	891.1402	7.5999	19.0351	812.0306	12.4421	19.0966	800.6090	20.4755
18.2823	965.7220	3.2445	18.6419	888.9868	6.7575	18.8986	837.9640	12.5608	19.1766	785.9948	20.5140
18.2712	968.1929	3.2312	18.6113	895.2566	6.7414	19.0885	802.0964	12.4971	19.1135	797.4937	19.9576
18.1671	991.6670	3.3577	18.5193	914.4386	6.7739	18.9683	824.6088	12.1603	19.1433	792.0467	18.8039
PSNR Mean: 18.2523 MSE Mean: 972.5902			PSNR Mean: 18.5319 MSE Mean: 911.9394			PSNR Mean: 18.9855 MSE Mean: 821.4414			PSNR Mean: 19.1417 MSE Mean: 792.4097

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by OMP algorithms.

Table 7. P_SNR, M_SE, and time of the different pixel image reconstructed by STOMP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
18.4565	927.7439	1.9871	18.8762	842.2853	3.2992	19.1958	782.5286	5.3546	19.4015	746.3251	8.9414
18.6146	894.5757	1.7530	18.7574	865.6460	3.0897	19.0214	814.5855	5.5177	19.1294	794.5858	8.4978
18.3691	946.6053	1.7672	18.9131	835.1674	3.2591	19.1162	797.0074	5.3404	19.3628	753.0119	8.3268
18.7747	862.2063	1.6509	18.6830	880.5934	3.1428	19.0212	814.6351	5.4728	19.3334	758.1162	8.3945
18.5523	907.5075	1.6583	18.7680	863.5269	3.1527	19.1821	784.9930	5.8004	19.2737	768.6233	8.8009
18.5249	913.2449	1.7565	18.9101	835.7386	3.1413	19.1222	795.9122	5.2623	19.2195	778.2766	9.1070
18.4711	924.6457	1.8162	18.9665	824.9626	3.0769	19.1388	792.8615	5.3666	19.1329	793.9458	8.9904
18.5027	917.9382	1.7327	18.8543	846.5476	3.2085	19.0365	811.7637	5.2851	19.2444	773.8263	9.3969
18.5617	905.5508	1.6634	18.6145	894.5940	3.1701	19.1874	784.0488	6.3562	19.0907	801.6980	9.0372
18.4884	920.9654	1.7663	18.5506	907.8610	3.2574	18.9774	822.8859	5.5557	19.3284	759.0033	8.7040
PSNR Mean: 18.5316 MSE Mean: 912.0984			PSNR Mean: 18.7894 MSE Mean: 859.6923			PSNR Mean: 19.0999 MSE Mean: 800.1222			PSNR Mean: 19.2517 MSE Mean: 772.7412

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by STOMP algorithms.

Table 8. P_SNR, M_SE, and time of the different pixel image reconstructed by GOMP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
18.5342	911.3028	2.5342	18.5828	901.1544	5.5202	18.6002	897.5575	12.2352	18.6504	887.2426	25.3709
18.5119	915.9863	2.3974	18.5438	909.2880	5.2957	18.7124	874.6603	12.7123	18.7940	858.3842	24.2956
18.4565	927.7495	2.4619	18.5660	904.6536	5.3261	18.7070	875.7537	11.9305	18.6203	893.4049	25.9419
18.4604	926.9053	2.3857	18.5033	917.7947	5.3010	18.6821	880.7937	11.6473	18.7787	861.4088	24.9687
18.4557	927.9302	2.4236	18.5870	900.2903	5.2821	18.8511	847.1601	11.7459	18.6419	888.9819	25.1073
18.5121	915.9399	2.4229	18.4860	921.4620	5.4175	18.6688	883.4962	11.5865	18.7756	862.0250	25.7629
18.6042	896.7213	2.4183	18.4826	922.1833	5.3205	18.6093	895.6809	11.4574	18.7594	865.2522	28.2525
18.4933	919.9185	2.4489	18.5674	904.3666	5.2700	18.5587	906.1624	11.5960	18.5693	903.9659	25.3977
18.3236	956.5777	2.4210	18.5627	905.3299	5.3125	18.6640	884.4572	11.6323	18.6851	880.1824	24.9104
18.2944	963.0243	2.7278	18.5136	915.6401	5.5594	18.6246	892.5204	12.0085	18.7903	859.1182	22.8444
PSNR Mean: 18.4646 MSE Mean: 926.2056			PSNR Mean: 18.5395 MSE Mean: 910.2163			PSNR Mean: 18.6680 MSE Mean: 883.8242			PSNR Mean: 18.7063 MSE Mean: 875.9966

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by GOMP algorithms. The data results show the advantages of the TL-GOMP image reconstruction algorithm once again.

4.2. Comparison of Simulation Results of the TL-GOMP and Other Algorithms

In this section, to verify the excellent performance of the TL-GOMP algorithm in image reconstruction, we adopt the research method of simulating the same target and comparing the results with those of other algorithms. In the following section, we simulate the test images with resolutions of 350 $\times$ 350, 500 $\times$ 500, 650 $\times$ 650, and 800 $\times$ 800. Figure 6 shows the simulation results for the test image with a resolution of 800 $\times$ 800 pixels using different types of algorithms. The figure shows that the image reconstructed by the TL-GOMP algorithm reflects the details or content information of the results. We present the results of the simulation and conduct a quantitative analysis below. In conclusion, the TL-GOMP algorithm has great potential for applications in the field of photonic integrated interference imaging.

In this section, we adopted a test image with a resolution of 350 $\times$ 350 as the scene target and simulated it using a series of six different image reconstruction algorithms: Compressive Sampling Matching Pursuit (CoSaMP), Generalized Back Propagation (GBP), Iterative Hard Thresholding (IHT), Iteration Reweighted Least Square (IRLS), Subspace Pursuit (SP), and TL-GOMP. The image quality was evaluated using the peak signal-to-noise ratio and mean square error.

Table 9. P_SNR, M_SE, and running time of the 350 × 350 image reconstructed by different algorithms.

	CoSaMP	GBP	IHT	IRLS	SP	TL-GOMP
PSNR (dB)	16.8484	19.2067	15.2984	20.1498	18.0819	21.1882
MSE	1343.5	780.5690	1919.8	628.2036	1011.3	494.6024
Running time (s)	8.2561	15.1944	0.9567	10.8665	6.9127	2.5031

lists the quality evaluation data and code running times of the reconstructed images. From the quantitative data, it can be seen that the P_SNR values of the images obtained by the TL-GOMP algorithm are increased by 25.76% (compared with CoSaMP), 10.32% (compared with GBP), 38.50% (compared with IHT), 5.15% (compared with IRLS), and 17.18% (compared with SP). The M_SE values of the images decreased by 63.19%, 36.64%, 74.23%, 21.27%, and 51.09%, respectively, and the code running time was also relatively fast.

Table 10. P_SNR, M_SE, and running time of the 500 × 500 image reconstructed by different algorithms.

	CoSaMP	GBP	IHT	IRLS	SP	TL-GOMP
PSNR (dB)	17.1099	19.7444	15.3828	21.0134	18.2516	21.8560
MSE	1265	689.6748	1882.8	514.9214	972.5734	424.1134
Running time (s)	20.4365	60.4386	2.5600	64.1737	16.4263	5.6994

shows the simulation results for the use of a test image with a resolution of 500 $\times$ 500 as the target. The results show that the P_SNR values of the reconstructed images obtained by the TL-GOMP algorithm improved by 27.74% (compared with CoSaMP), 10.69% (compared with GBP), 42.08% (compared with IHT), 4.01% (compared with IRLS), and 19.75% (compared with SP). The M_SE values of the images decreased by 66.47%, 38.51%, 77.47%, 17.64%, and 56.39%, respectively.

Table 11. P_SNR, M_SE, and running time of the 650 × 650 image reconstructed by different algorithms.

	CoSaMP	GBP	IHT	IRLS	SP	TL-GOMP
PSNR (dB)	17.3410	20.1147	15.3888	21.2962	18.6761	21.8993
MSE	1199.4	633.3042	1880.2	482.4597	881.9959	419.9057
Running time (s)	47.6477	151.2679	5.5886	260.7229	34.7940	11.5920

shows the simulation results for the use of a test image with a resolution of 650 $\times$ 650 as the target. The data in the table show that the P_SNR value of the reconstructed image obtained by the TL-GOMP algorithm improved by 26.29% (compared with CoSaMP), 8.87% (compared with GBP), 42.31% (compared with IHT), 2.83% (compared with IRLS), and 17.26% (compared with SP). The M_SE values of the images decreased by 64.99%, 33.70%, 77.67%, 12.97%, and 52.39%, respectively.

Table 12. P_SNR, M_SE, and running time of the 800 × 800 image reconstructed by different algorithms.

	CoSaMP	GBP	IHT	IRLS	SP	TL-GOMP
PSNR (dB)	17.4872	20.4032	15.5785	21.4114	18.7344	21.4954
MSE	1159.7	592.5921	1799.8	469.8253	870.2398	460.8311
Running time (s)	96.3305	308.2801	10.4591	650.7766	66.1669	23.6628

lists the simulation results for the use of the test image with a resolution of 800 $\times$ 800 as the target. The data in the table show that the P_SNR value of the image reconstructed by the TL-GOMP algorithm increased by 22.92% (compared with CoSaMP), 5.35% (compared with GBP), 37.98% (compared with IHT), 0.40% (compared with IRLS), and 14.74% (compared with SP). The M_SE values of the images decreased by 60.26%, 22.23%, 74.40%, 1.91%, and 47.05%, respectively. In Figure 7, the red curve represents the image data as reconstructed by the TL-GOMP algorithm. It can be observed that the peak signal-to-noise ratio and mean square error of the image reconstructed by this algorithm are higher than those reconstructed by the other algorithms. Its mean square error value is much lower than that of the image reconstructed by the other algorithms, and it is also faster in terms of the code running time. In summary, the TL-GOMP algorithm has better image reconstruction performance.

Table 13. P_SNR, M_SE, and time of the different pixel image reconstructed by CoSaMP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
16.8092	1355.7	8.8949	16.9569	1310.4	21.0424	17.4505	1169.6	53.1953	17.4279	1175.7	104.1132
16.4940	1457.7	8.5638	17.1272	1260	20.5154	17.3648	1192.9	51.8239	17.5628	1139.7	100.1575
16.4971	1456.7	8.6117	17.0410	1285.2	20.8961	17.4114	1180.2	47.8983	17.4618	1166.6	99.2829
16.8151	1353.8	8.6880	17.0706	1276.5	20.2516	17.3835	1187.8	48.2978	17.6096	1127.5	102.0728
16.7148	1385.5	8.6742	17.1166	1263	20.0926	17.4745	1163.1	46.9857	17.4945	1157.8	102.7467
16.5159	1450.4	8.2333	16.9466	1313.5	20.1809	17.3226	1204.5	48.5044	17.6168	1125.6	106.1725
16.7907	1361.5	8.0242	17.1552	1251.9	20.0922	17.1436	1255.2	47.4073	17.5932	1131.8	103.6256
16.6924	1392.7	8.1107	17.1143	1263.7	20.1702	17.3318	1202	47.1307	17.5843	1134.1	102.4037
16.6452	1407.9	8.0281	16.9100	1324.6	20.1803	17.3889	1186.3	46.7126	17.6081	1127.9	102.3405
16.8484	1343.5	8.2561	17.1099	1265	20.4365	17.3410	1199.4	47.6477	17.4872	1159.7	96.3305
PSNR Mean: 16.6823 MSE Mean: 1396.54			PSNR Mean: 17.0548 MSE Mean: 1281.38			PSNR Mean: 17.3613 MSE Mean: 1194.1			PSNR Mean: 17.5446 MSE Mean: 1144.64

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by CoSaMP algorithms.

Table 14. P_SNR, M_SE, and time of the different pixel image reconstructed by GBP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
19.4978	729.9666	16.7171	19.9097	663.9063	60.8671	20.2651	611.7477	152.6819	20.4798	582.2355	313.1546
19.1973	782.2584	15.4194	19.8177	678.1193	58.8448	20.2567	612.9301	151.4318	20.4731	583.1371	307.3742
19.4500	738.0428	15.7188	19.9173	662.7476	60.3878	20.0630	640.8796	152.0517	20.3483	600.1356	306.4753
19.3991	746.7401	15.4124	19.8520	672.7897	60.0065	20.1227	632.1369	152.8503	20.4549	585.5920	315.1839
19.3752	750.8673	15.4889	19.7395	690.4477	59.8597	20.2543	613.2665	151.6609	20.4367	588.0450	307.7829
19.2693	769.3932	15.3593	19.9330	660.3565	62.0048	20.3432	600.8463	151.6051	20.4022	592.7377	311.2913
19.4267	742.0074	15.5287	19.7074	695.5632	60.5518	20.2234	617.6454	150.1376	20.5054	578.8216	306.9459
19.3039	763.2860	15.4373	19.7071	695.6140	60.3061	20.2587	612.6538	150.4258	20.3715	596.9458	308.9359
19.4260	742.1325	15.4490	19.8390	674.8012	60.0366	20.2990	606.9950	149.5369	20.4816	581.9978	308.8351
19.2067	780.5690	15.1944	19.7444	689.6748	60.4386	20.1147	633.3042	151.2679	20.4032	592.5921	308.2801
PSNR Mean: 19.3552 MSE Mean: 754.5263			PSNR Mean: 19.8167 MSE Mean: 678.4020			PSNR Mean: 20.2201 MSE Mean: 618.2406			PSNR Mean: 20.4357 MSE Mean: 588.2240

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by GBP algorithms.

Table 15. P_SNR, M_SE, and time of the different pixel image reconstructed by IHT algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
15.3230	1908.9	1.0292	15.5289	1820.5	2.7308	15.5357	1817.6	5.8068	15.1495	1986.7	10.6558
15.2696	1932.5	0.9657	15.3572	1893.9	2.5608	15.5149	1826.4	5.7885	15.6001	1790.9	10.7756
15.4349	1860.3	0.9223	15.3438	1899.8	2.5887	15.3108	1914.3	5.5971	15.4884	1837.6	10.4584
15.6466	1771.8	0.9237	15.5348	1818	2.5473	15.2720	1931.4	5.6787	15.3543	1895.2	10.7111
15.6298	1778.7	0.9336	15.3886	1880.3	2.5745	15.3944	1877.8	5.5852	15.1589	1982.4	10.4426
15.4950	1834.7	0.9343	15.6286	1779.2	2.5784	15.4219	1865.9	5.6056	15.2874	1924.6	10.5495
15.3671	1889.6	0.9265	15.3710	1887.9	2.5645	15.4774	1842.2	5.6063	15.4666	1846.8	10.5276
15.4187	1867.3	0.9262	15.3187	1910.8	2.5596	15.2540	1939.5	5.6121	15.4023	1874.4	10.4417
15.6738	1760.8	0.9315	15.6133	1785.5	2.5755	15.3534	1895.6	5.5838	15.4003	1875.2	10.4518
15.2984	1919.8	0.9567	15.3828	1882.8	2.5600	15.3888	1880.2	5.5886	15.5785	1799.8	10.4591
PSNR Mean: 15.4557 MSE Mean: 1852.44			PSNR Mean: 15.4468 MSE Mean: 1855.87			PSNR Mean: 15.3923 MSE Mean: 1879.09			PSNR Mean: 15.3886 MSE Mean: 1881.36

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by IHT algorithms.

Table 16. P_SNR, M_SE, and time of the different pixel image reconstructed by IRLS algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
20.6086	565.2282	11.7302	21.2215	490.8249	65.6006	21.3258	479.1809	270.9207	21.3758	473.7010	664.0105
20.6022	566.0515	10.5527	21.0093	515.4109	74.9643	21.4131	469.6444	262.2276	21.5207	458.1548	693.0186
20.6176	564.0528	11.4481	20.8764	531.4202	66.3486	21.4715	463.3728	260.0932	21.3714	474.1812	668.7674
20.4171	590.7103	10.6213	21.0181	514.3672	64.8940	21.6391	445.8292	261.3721	21.5438	455.7273	662.6378
20.7530	546.7416	11.0238	21.2248	490.4610	65.6442	21.2211	490.8747	258.2043	21.6174	448.0666	666.4153
21.0103	515.2937	11.2650	20.8733	531.7981	64.7945	21.2544	487.1234	261.8442	21.5099	459.2951	668.0591
21.1637	497.4021	11.4127	20.9998	516.5319	64.5561	21.2597	486.5327	255.4833	21.3478	476.7588	677.1501
20.4757	582.7821	10.9141	21.1686	496.8444	64.5493	21.2293	489.9456	255.1096	21.3538	476.1018	666.8833
20.6876	555.0313	10.6173	21.1205	502.3765	64.5562	21.3770	473.5656	264.4215	21.3173	480.1209	663.1420
20.1498	628.2036	10.8665	21.0134	514.9214	64.1737	21.2962	482.4597	260.7229	21.4114	469.8253	650.7766
PSNR Mean: 18.6069 MSE Mean: 561.1497			PSNR Mean: 21.0526 MSE Mean: 510.4957			PSNR Mean: 21.3487 MSE Mean: 476.8529			PSNR Mean: 21.4369 MSE Mean: 467.1933

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by IRLS algorithms.

Table 17. P_SNR, M_SE, and time of the different pixel image reconstructed by SP algorithms.

350 × 350 Pixel Values			500 × 500 Pixel Values			650 × 650 Pixel Values			800 × 800 Pixel Values
PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time	PSNR	MSE	Time
17.9046	1053.5	7.1466	18.4409	931.0924	17.3122	18.5753	902.7211	36.5388	18.8447	848.4259	70.6439
17.8879	1057.5	6.9380	18.1981	984.6217	16.6595	18.6085	895.8334	35.2757	18.8404	849.2576	68.9989
17.8405	1069.1	7.0076	18.1930	985.7762	16.7420	18.5203	914.2157	36.1913	18.8246	852.3637	74.357
17.8168	1075	6.8259	18.1758	989.6957	16.6609	18.4910	920.4035	34.9478	18.5796	901.8120	70.4628
17.7943	1080.6	6.7711	18.2247	978.6191	16.6698	18.5779	902.1786	35.1908	18.7513	866.8650	74.8542
17.7769	1084.9	6.739	18.2543	971.9713	16.7688	18.7444	868.2313	35.4837	18.7457	867.9828	70.8790
17.7737	1085.7	6.6753	18.3440	952.0871	17.0695	18.2689	968.6950	35.2054	18.7504	867.0365	71.0472
18.0686	1014.4	6.8211	18.1780	989.1917	17.3466	18.7020	876.7578	35.1299	18.7856	860.0449	69.8509
17.8676	1062.5	6.8499	18.2797	966.3036	16.6775	18.5849	900.7179	35.1921	18.7113	874.8790	71.3209
18.0819	1011.3	6.9127	18.2516	972.5734	16.4263	18.6761	881.9959	34.7940	18.7344	870.2398	66.1669
PSNR Mean: 17.8813 MSE Mean: 1059.45			PSNR Mean: 18.2540 MSE Mean: 972.1932			PSNR Mean: 18.5800 MSE Mean: 903.1750			PSNR Mean: 18.7568 MSE Mean: 865.8907

shows P_SNR values and M_SE values of the different pixel image repeatedly reconstructed by SP algorithms. The data results show the advantages of the TL-GOMP image reconstruction algorithm once again.

4.3. Simulation Results of Single-Column Signal Reconstruction by the CS TL-GOMP Algorithm

Figure 8 shows the 256 $\times$ 256 target test images. Figure 9 shows the simulation results from the use of the TL-GOMP algorithm to reconstruct a single column of the original signals with different running times. We first selected an image with a resolution of 256 $\times$ 256 for testing, arbitrarily selected a 256 $\times$ 1 column vector as the original signal, and then performed signal reconstruction 1, 50, 100, and 200 times. The simulation results of the signal reconstruction are shown in Figure 9. It can be observed that the reconstructed signal swings around the original signal and gradually approaches the original signal with an increase in the reconstruction time.

Table 18. Signal reconstruction times and residual values.

Times of Signal Reconstruction	1	50	100	200
Value of residual	168.5664	161.6117	150.3473	136.5506

shows that the residual values of the TL-GOMP algorithm after different runs are 168.5664, 161.6117, 150.3473, and 136.5506 upon comparison of the reconstructed signal with the original signal. Among these, the residual value is an important index for measuring the size of the error or the degree of deviation. The simulation results show that with an increase in the number of original signal reconstructions, the residual value demonstrates a decreasing trend; that is, the accuracy of the reconstructed signal gradually approaches that of the original signal. The TL-GOMP algorithm exhibits good stability in the reconstruction of the original signal.

Table 19. Multiple simulation data with different signal reconstruction times.

	Residual Values
1 time	168.5664	168.6020	162.6642	165.0314	169.6857	165.8414	158.1089	155.4569	155.4202	149.4011
50 times	161.6117	159.0616	160.1910	155.4877	155.4719	150.6663	142.2037	149.5014	148.7272	155.5769
100 times	150.3473	150.5078	150.1486	147.8456	148.4673	153.6050	153.2938	147.4182	155.6388	149.4355
200 times	136.5506	143.0227	139.9187	143.3128	146.7156	147.7237	146.9315	141.7615	144.6260	143.4051

shows multiple simulation data with different signal reconstruction times.

4.4. Simulation Results of the CS TL-GOMP Algorithm in Image Reconstruction at Different Distances

We used the Photonic Integrated Circuit to collect the spatial frequency information emitted by the target to form the restoration image. Figure 10 shows the imaging results for the frequency information collected by the microlens array on the PIC at different distances d; the resolution of the restored images is 256 $\times$ 256. The “Original image” in Figure 10 represents the restoration image of the microlens array on PIC as the target test image. Because the signal acquisition of the PIC is an under-sampling process, it is necessary to use a sparse signal image reconstruction algorithm in order to recover the content information of the detected target.

In the experiment, the Gaussian random matrix was selected as the measurement matrix, and the discrete cosine transform matrix was used as the sparse matrix. In this part, we conducted several simulation experiments and displayed the experimental data and reconstructed images of one of them, as shown in Figure 11 and

Table 20. P_SNR and M_SE of reconstructed images at different distances.

d (m)	75	125	175	225
PSNR (dB)	13.5770	10.4228	11.2921	12.2664
MSE	2.8525 × 103	5.8992 × 103	4.8292 × 103	3.8587 × 103

.

Figure 11 shows the simulation results after the reconstruction of restored images at d = 75, 125, 175, and 225 m using the compressed sensing TL-GOMP algorithm. We used two image quality evaluation indices, the peak signal-to-noise ratio, and the mean square error to measure the image quality. That is, the higher the peak signal-to-noise ratio, the better the image quality. Conversely, the lower the mean square error, the better the image quality. The simulation data in Table 20 show that the compressed sensing TL-GOMP image reconstruction algorithm is suitable for the content restoration of detected targets at different distances. “1d rec img” in Figure 11 represents the result of target reconstruction by TL-GOMP algorithm. We used the image quality evaluation function for evaluation of the reconstructed image; 13.5770 dB, 10.4228 dB, 11.2921 dB, and 12.2664 dB show the peak signal-to-noise ratio of the reconstructed image.

4.5. Influence of Measurement Number M in the CS TL-GOMP Algorithm

Figure 8 shows the 256 $\times$ 256 target test images. The observation matrix ($M\times N$) $\Phi$ is an important parameter in the CS TL-GOMP algorithm, which can collect the original signal $\overrightarrow{x}$, obtain the sparse representation $\overrightarrow{\theta }$ by combining with the algorithm, and finally reconstruct the desired signal $\overrightarrow{\widehat{\theta }}$. Figure 12a shows the relationship curve of different measurement numbers $M$ on the sparsity $k$ and quality of the reconstructed image. The value of $M$ ranged from 56 to 256, with a step size of 5. The simulation data show that the sparsity $k$ (the number of non-zero values) in the sparse representation $\overrightarrow{\theta }$ is 18. Meanwhile, with an increase in $M$ in the observation vector $\overrightarrow{y}$, the P_SNR value of the image also increases. In addition, the M_SE value of the image exhibits a decreasing trend. Therefore, we can conclude that the quality of the reconstructed image also improves with an increase in $M$ in the observation vector. In the experiment, the Gaussian random matrix was selected as the measurement matrix, and the discrete cosine transform matrix was used as the sparse matrix; Figure 12 shows the multiple simulation results.

4.6. Influence of Measurement Matrix M × N and Sparsity k in the CS TL-GOMP Algorithm on the Quality of the Reconstructed Image

Figure 8 shows the 256 $\times$ 256 target test images. Figure 13 shows the results from the simulation of the relationship between different sparsity $k$ values and the quality of the reconstructed image. For the sparsity $k$, we selected the values of 9, 10, 11, 12, and 13 for the simulation of the test image with a resolution of 256 $\times$ 256. The results show that the P_SNR increases with an increase in $k$, whereas the M_SE decreases with an increase in $k$. Figure 14 shows the simulation results for the relationship between the different measurement matrix sizes and the quality of the reconstructed image. We selected measurement matrices with dimensions of 36 $\times$ 256, 42 $\times$ 256, 64 $\times$ 256, and 85 $\times$ 256 to simulate the same test image. The simulation results show that P_SNR increases with an increase in the size of the size of the measurement matrix. The M_SE decreases with an increase in the size of the measurement matrix.

Table 21. P_SNR and M_SE of the reconstructed images with different sparsity k.

$\mathit{k}$	9	10	11	12	13
PSNR	24.4658	25.9451	26.2747	26.4402	26.5216
MSE	232.5423	165.4131	153.3254	147.5921	144.8496

and

Table 22. P_SNR and M_SE of the reconstructed images with different matrix sizes M × N.

$\mathit{M}$	85	64	51	42	36
PSNR	24.5055	23.9166	23.3133	22.6527	21.3493
MSE	230.4277	263.8865	303.2150	353.0308	476.5985

list one of the multiple simulation results. Therefore, we can conclude that the quality of the reconstructed image improves with an increase in $k$ and the size of the measurement matrix.

We studied the influence of different sparsity and different measurement matrix sizes on reconstruction quality. The measurement matrix selected by us was Gaussian random matrix, and the sparse matrix was discrete cosine transform matrix. For the sparsity $k$, we selected the values of 9, 10, 11, 12, and 13. In each case of sparsity, we conducted several experiments, and the results of the experiments are shown in

Table 23. P_SNR and M_SE of the multiple simulation data with different sparsity k.

k = 9	PSNR	24.4658	24.4578	24.5642	24.6083	24.5429	24.3835	24.4639	24.4566	24.5734	24.5788
	MSE	232.5423	232.9702	227.3323	225.0338	228.4496	236.9926	232.6450	233.0364	226.8511	226.5680
k = 10	PSNR	25.9451	25.9281	25.9644	25.9163	26.1031	26.0670	26.1165	25.9435	26.2481	25.9240
	MSE	165.4131	166.0630	164.6806	166.5131	159.5034	160.8350	159.0111	165.4743	154.2653	166.2188
k = 11	PSNR	26.2747	26.2508	26.3373	26.1912	26.3492	26.2861	26.2062	26.2744	26.2622	26.1795
	MSE	153.3254	154.1708	151.1292	156.3016	150.7153	152.9223	155.7619	153.3364	153.7671	156.7223
k = 12	PSNR	26.4402	26.4753	26.3413	26.2574	26.5338	26.4707	26.3457	26.4020	26.3657	26.5155
	MSE	147.5921	146.4039	150.9896	153.9345	144.4438	146.5585	150.8391	148.8964	150.1456	145.0545
k = 13	PSNR	26.5216	26.4978	26.5798	26.6370	26.6094	26.4142	26.6716	26.4771	26.4774	26.6881
	MSE	144.8496	145.6461	142.9226	141.0527	141.9519	148.4760	139.9344	146.3412	146.3327	139.4024

. Similarly, we selected measurement matrices with dimensions of 36 $\times$ 256, 42 $\times$ 256, 64 $\times$ 256, and 85 $\times$ 256 to simulate the same test image. In the case of the size of each measurement matrix, we also used the same measurement matrix and the same research method to carry out many experiments. The experimental data results are shown in

Table 24. P_SNR and M_SE of the multiple simulation data with different matrix size M × N.

M = 85	PSNR	24.5055	24.6425	24.5528	24.5056	24.5522	24.5659	24.5421	24.5434	24.4919	24.4003
	MSE	230.4277	223.2689	227.9270	230.4216	227.9595	227.2443	228.4929	228.4230	231.1488	236.0733
M = 64	PSNR	23.9166	24.1006	23.9725	24.0803	24.1310	24.0046	24.0576	24.1294	23.9754	24.0435
	MSE	263.8865	252.9398	260.5121	254.1288	251.1774	258.5935	255.4592	251.2689	260.3421	256.2885
M = 51	PSNR	23.3133	23.5388	23.5970	23.4893	23.4474	23.2988	23.2958	23.3413	23.4279	23.3476
	MSE	303.2150	287.8705	284.0408	291.1752	293.9946	304.2291	304.4373	301.2679	295.3188	300.8316
M = 42	PSNR	22.6527	22.7587	23.1023	22.7050	22.5485	22.8871	23.2799	22.9354	22.9467	22.6748
	MSE	353.0308	344.5174	318.3073	348.8021	361.6032	334.4826	305.5548	330.7809	329.9219	351.2371
M = 36	PSNR	21.3493	21.9878	21.9421	21.8499	22.1946	21.9828	21.8089	21.8007	22.1391	21.0789
	MSE	476.5985	411.4304	415.7896	424.7125	392.2991	411.9064	428.7317	429.5465	397.3452	507.2072

. According to the data results for many experiments, we conclude that “the quality of the reconstructed image improves with an increase in $k$ and the size of the measurement matrix”. However, in order to achieve high-quality image reconstruction, the measurement matrix and sparse matrix of compressed sensing theory are constantly being studied. In future research, we will use an updated measurement matrix to verify the above conclusions.

5. Conclusions

In this study, we improved the traditional image reconstruction algorithm and proposed a TL-GOMP tracing algorithm for compressed sensing. In the simulation, we used the TL-GOMP algorithm and the same series of traditional OMP, STOMP, and GOMP algorithms to perform simulations using the same test targets. The results of the simulation show that the quality of the image reconstructed by the TL-GOMP algorithm was better than that reconstructed by the other traditional algorithms in the same series. To illustrate the advantages of this algorithm more rigorously, we also conducted a comparison simulation between the TL-GOMP algorithm and other image reconstruction algorithms. The results also showed that the quality of the image reconstructed by the TL-GOMP algorithm was better than that reconstructed by the other algorithms, which has potential application value. To verify the stability of the algorithm, we arbitrarily extracted a column of the original signal column vectors and performed signal reconstruction several times. The simulation results showed that the accuracy of the reconstructed signal gradually approached that of the original signal with an increase in the number of reconstruction runs. The TL-GOMP algorithm was also used to reconstruct the restored images at different detection distances, and the simulation results showed that the algorithm could reproduce the content information of the target. Therefore, the TL-GOMP algorithm is advantageous for applications in photonic integrated interference imaging. It can reconstruct sparse spatial frequency information collected by the PIC and recover the content information of the detected target. In summary, the TL-GOMP algorithm can reconstruct the sparse and unknown information collected as well as recover the content information of unknown targets. This could benefit scientific and technological exploration and production, and it also has good potential for application in the field of photonic integrated interference detection technology.