Singular Value Decomposition Wavelength-Multiplexing Ghost Imaging

Abstract

To enhance imaging quality, singular value decomposition (SVD) has been applied to single-wavelength ghost imaging (GI) or color GI. In this paper, we extend the application of SVD to wavelength-multiplexing ghost imaging (WMGI) for reducing the redundant information in the random measurement matrix corresponding to multi-wavelength modulated speckle fields. The feasibility of this method is demonstrated through numerical simulations and optical experiments. Based on the intensity statistical properties of multi-wavelength speckle fields, we derived an expression for the contrast-to-noise ratio (CNR) to characterize imaging quality and conducted a corresponding analysis. The theoretical results indicate that in SVDWMGI, for the m-wavelength case, the CNR of the reconstructed image is $\sqrt{m}$ times that of single-wavelength GI. Moreover, we carried out an optical experiment with a three-wavelength speckle-modulated light source to verify the method. This approach integrates the advantages of both SVD and wavelength division multiplexing, potentially facilitating the application of GI in long-distance imaging fields such as remote sensing.

1. Introduction

Ghost imaging (GI), also known as correlated imaging, is a nonlocal imaging method that reconstructs the image of an object through intensity correlation measurement. It has been demonstrated that GI can be achieved not only with entangled light [1] but also with classical thermal light [2,3,4,5,6,7,8]. Moreover, computational GI was proposed and demonstrated [9,10,11]. To date, GI has found many potential applications in lidar or remote sensing [12,13,14,15,16,17], 3D imaging [10], atmospheric imaging [18], multi-spectral imaging [19,20], bioimaging [21], optical encryption [22,23,24,25], moving object imaging [26], and X-ray imaging [27], etc. Especially, three-dimensional GI lidar exhibits a benefit in anti-interference against harsh environment and is considered to be one of the best potential applications in GI [14]. To promote the application of GI, many methods, including high-order GI [28,29,30], compressive GI [31], differential [32] or normalized GI [33], logarithmic GI, and exponential GI [34], have been developed to improve the image quality. Liu compares the imaging efficiency and error tolerance of these GI algorithms [35]. The probability theory, including the probability density function, is also employed to analyze GI [36,37,38,39,40], deepening the understanding of the essence of GI and the factors influencing its imaging quality. Additionally, there have been reports on leveraging deep learning to enhance the quality of GI [41]. Instead of using random speckle illumination, structured basis scanning illumination techniques, such as Fourier basis [42,43,44], Hadamard basis [45], or wavelet basis [46], are employed to achieve single-pixel imaging, thereby enhancing imaging quality.

Over the past few years, pseudo-inverse and SVD have been utilized in GI to enhance the image quality. Zhang et al. introduced pseudo-inverse ghost imaging (PGI) as a method for reconstructing images with exceptional quality [47]. Despite the randomness of the measurement matrix, its pseudo-inverse can be achieved through the application of SVD, in accordance with the Moore-Penrose theorem [47]. Gong et al. proposed an improved PGI technique, which exhibits the advantage of substantially improving the spatial transverse resolution of thermal light GI [48]. Subsequently, iterative PGI was proposed, integrating the advantages of iterative denoising and PGI [49]. This method can further eliminate background noise and achieve high-quality imaging with fewer measurements. In 2018, Zhang et al. introduced the concept of SVDGI from the standpoint of matrix reconstruction [50]. Different from conventional GI, SVDGI involves decomposing random matrices into their SVD components, thereby enhancing matrix orthogonality and improving image quality. Chen et al. utilized truncated SVD in GI, and the findings revealed that image reconstruction is feasible by selecting the first few largest singular values and the imaging quality can be further enhanced through additive filtering [51]. Wang et al. proposed a dual-mode adaptive SVDGI, which allows for switching between imaging and edge detection modes [52]. Recently, there are some related studies that have applied the pseudo-inverse or SVD to compressed GI [53], color GI [54,55], computational GI [56], encrypted GI [57], and target recognition [58]. These studies have contributed to a further improvement in the quality of GI. However, most of the aforementioned studies focused on applying SVD to single-wavelength GI or color GI.

In recent years, wavelength-multiplexing ghost imaging (WMGI), alternatively referred to as multi-wavelength GI, has been introduced [59,60]. The cornerstone of this technique lies in enhancing imaging efficiency through the parallel processing of multiple wavelengths. To improve the issue of long data acquisition time in traditional single-wavelength imaging, Zhang et al. implemented WMGI using a tricolor modulated light field as the light source and demonstrated that, with three-wavelength multiplexing, the CNR of the object image obtained is increased by a factor of $\sqrt{3}$ compared to the single-wavelength case [60]. However, multi-wavelength modulation necessitates the generation and storage of a vast number of speckle patterns, resulting in a multiplicative increase in the number of speckles required for a single imaging session. Therefore, in this paper, we extend SVD into WMGI to investigate the impact of the number of singular values within the measurement matrix on the CNR. The rest of this paper is organized as follows. In Section 2, we present a theoretical analysis of the proposed method. Section 3 displays the numerical simulation results. The experimental setup and results are provided in Section 4. Finally, we present the discussion and conclusion in Section 5 and Section 6, respectively.

2. Theory and Methods

In optical communication, wavelength division multiplexing (WDM) is a technology that multiplexes optical signals of different wavelengths into a single fiber for transmission. These optical signals carry different information. At the transmitting end, a multiplexer combines them into a single fiber for transmission. At the receiving end, a demultiplexer separates the various wavelength optical carriers to facilitate further processing and signal recovery. Here, wavelength-multiplexing technology is integrated with GI and SVD to facilitate image reconstruction, that is SVD-WMGI. In this method, by using a multi-wavelength light source combined with SVD, speckle patterns of different wavelengths are modulated, multiplexed and superimposed at the transmitting end, demultiplexed at the receiving end through wavelength-selective detectors, and then image reconstruction is performed using second-order intensity correlation. Unlike color GI, which focuses on restoring colored objects based on intensity correlation, WMGI primarily aims to enhance imaging efficiency through independent and parallel intensity correlation of speckle patterns at different wavelength channels. For the sake of simplicity, we assume that the reflectivity function of a target object is $O(x,y)$ and its reflectivity is wavelength-insensitive. Additionally, the wavelength channels are indexed by $i=1,2,\dots ,m$, where m denotes the total number of wavelength channels. For the i-th wavelength channel, assuming there are M (the number of measurements) independent illumination patterns $\left\{I_j^i(x,y)\right\}(j=1,2,\dots ,M)$ sequentially projected onto the target object. In WMGI, the image reconstruction $\tilde{O}(x,y)$ of the object is achieved by the summing of second-order intensity correlation function for each wavelength channel, which can be expressed as

$$\begin{array}{cc}\hfill \tilde{O}(x,y)=\sum _{i=1}^m{O}_i(x,y)\propto G(x,y)& ={\displaystyle \frac{1}{m}}\sum _{i=1}^m{G}_i(x,y)\hfill \\ & ={\displaystyle \frac{1}{m}}\sum _{i=1}^m\left[{\displaystyle \frac{1}{M}}\sum _{j=1}^M{B}_j^i{I}_j^i(x,y)-{\displaystyle \frac{1}{M^2}}\sum _{j=1}^M{B}_j^i\sum _{j=1}^M{I}_j^i(x,y)\right],\hfill \end{array}$$

(1)

where ${\tilde{O}}_i(x,y)$ and $G_i(x,y)$ denote the image reconstruction and second-order intensity correlation function for the i-th wavelength channel, respectively, and $B_j^i=\int \int I_j^i(x,y)O(x,y)dxdy$ represents bucket detector intensity of the j-th measurement at the i-th wavelength channel.

To facilitate the extension of SVD into WMGI, the ensuing analysis is conducted from the perspective of matrix theory. For the i-th wavelength channel, suppose the illumination pattern $I_j^i(x,y)$ projected onto the target object has a resolution of $a\times b$ pixels. It can be reshaped into a $1\times N$ row vector, where $N=a\times b$. Then, the row vectors corresponding to M illumination patterns (M measurements) can form an $M\times N$ dimensional large matrix, which is referred to as the random measurement matrix ${\phi }_i$. It can be represented as

$${\phi }_i=\left[\begin{array}{cccc}{I}_1^i(1,1)& I_1^i(1,2)& \dots & I_1^i(a,b)\\ I_2^i(1,1)& I_2^i(1,2)& \dots & I_2^i(a,b)\\ ⋮& ⋮& \ddots & ⋮\\ I_M^i(1,1)& I_M^i(1,2)& \dots & I_M^i(a,b)\end{array}\right].$$

(2)

Assuming that the target object $O(x,y)$ also has a resolution of $a\times b$ pixels, it can be reshaped into an N-dimensional column vector. Accordingly, the measurement process of the bucket signal can be described as

$$\left[\begin{array}{c}{B}_1^i\\ B_2^i\\ ⋮\\ B_M^i\end{array}\right]={\phi }_i\left[\begin{array}{c}O(1,1)\\ O(1,2)\\ ⋮\\ O(a,b)\end{array}\right].$$

(3)

Hence, the image reconstruction ${\tilde{O}}_i(x,y)$ at the i-th wavelength channel, characterized by the second-order intensity correlation, can be rewritten in matrix form

$${\tilde{O}}_i(x,y)\propto G_i(x,y)={\displaystyle \frac{1}{M}}\left[\begin{array}{cccc}{I}_1^i(1,1)& I_2^i(1,1)& \dots & I_M^i(1,1)\\ I_1^i(1,2)& I_2^i(1,2)& \dots & I_M^i(1,2)\\ ⋮& ⋮& \ddots & ⋮\\ I_1^i(a,b)& I_2^i(a,b)& \dots & I_M^i(a,b)\end{array}\right]\left[\begin{array}{c}{B}_1^i\\ B_2^i\\ ⋮\\ B_M^i\end{array}\right]-〈B^i〉\left[\begin{array}{c}〈I^i(1,1)〉\\ 〈I^i(1,2)〉\\ ⋮\\ 〈I^i(a,b)〉\end{array}\right],$$

(4)

where $\langle \dots \rangle$ denotes the ensemble average of the input variable, that is, $\langle B^i\rangle ={\displaystyle \frac{1}{M}}\sum _{j=1}^M{B}_j^i$ and $\langle I^i(x,y)\rangle ={\displaystyle \frac{1}{M}}\sum _{j=1}^M{I}_j^i(x,y)$. For simplicity, we temporarily omit the second background term and substitute Equations (2) and (3) into Equation (4), yielding

$$\begin{array}{cc}\hfill {\tilde{O}}_i(x,y)& \propto G_i(x,y)={\displaystyle \frac{1}{M}}{\phi }_i^T{\phi }_i\left[\begin{array}{c}O(1,1)\\ O(1,2)\\ ⋮\\ O(a,b)\end{array}\right],\hfill \end{array}$$

(5)

where ${\phi }_i^T$ denotes the transpose of matrix ${\phi }_i$. Equation (5) indicates that the perfect image reconstruction can be achieved when the condition ${\phi }_i^T{\phi }_i=n\mathbf{I}$ is satisfied, where $\mathbf{I}$ is the identity matrix and n is any real constant. Moreover, the closer the matrix ${\phi }_i^T{\phi }_i$ resembles the identity matrix, the more superior the quality of the resulting image becomes.

As a powerful matrix decomposition tool, SVD has widespread application in image processing and data compression. SVD enables the decomposition of a matrix into the product of three matrices: two orthogonal matrices and one non-negative diagonal matrix of singular values, each containing significant characteristics of the original matrix. Correspondingly, performing SVD on the $M\times N$-dimensional random measurement matrix ${\phi }_i$, one can obtain the left singular vector matrix $U_i$ of size $M\times M$, the $M\times N$-dimensional singular value matrix $S_i$, and the right singular vector matrix $V_i$ of size $N\times N$. Hence, the random measurement matrix ${\phi }_i$ can be decomposed as

$$\begin{array}{c}\hfill {\phi }_i=U_i{S}_i{V}_i^T.\end{array}$$

(6)

To begin with, we discuss the first case where all non-zero singular values are set to 1. That is manipulating the singular value matrix $S_i$ and, for simplicity, uniformly setting all its non-zero singular values to 1, we can artificially generate a measurement matrix, which is specifically represented as

$$\begin{array}{c}{\phi }_i^{\prime }=U_{M\times M}^i{\left[\begin{array}{cc}{\mathrm{\Gamma }}_{M\times M}^i\hfill & 0\hfill \end{array}\right]}_{M\times N}{V}_{N\times N}^{iT},\end{array}$$

(7)

where ${\mathrm{\Gamma }}^i$ is an $M\times M$-dimensional unit matrix. By replacing the random measurement matrix ${\phi }_i$ with the artificial measurement matrix ${\phi }_i^{\prime }$ constructed based on SVD and then substituting Equation (7) into Equation (5) while considering the orthogonality of matrix $U_i$, Equation (5) can be rewritten as

$${\tilde{O}}_i(x,y)\propto G_i(x,y)={\displaystyle \frac{1}{M}}{V}_{N\times N}^i{\left[\begin{array}{cc}{\mathrm{\Gamma }}_{M\times M}^i& 0\\ 0& 0\end{array}\right]}_{N\times N}{V}_{N\times N}^{iT}\left[\begin{array}{c}O(1,1)\\ O(1,2)\\ ⋮\\ O(a,b)\end{array}\right].$$

(8)

By incorporating Equation (8) into Equation (1), the image achieved through WMGI based on SVD can ultimately be described as

$$\tilde{O}(x,y)\propto G(x,y)={\displaystyle \frac{1}{m}}\sum _{i=1}^m\left\{{\displaystyle \frac{1}{M}}{V}^i{\left[\begin{array}{cc}{\mathrm{\Gamma }}_{M\times M}^i& 0\\ 0& 0\end{array}\right]}_{N\times N}{V}^{iT}O(x,y)\right\}.$$

(9)

Taking into account the orthogonality of matrix $V_i$, it can be found from Equation (9) that by assigning a value of 1 to all non-zero singular values to construct an artificial measurement matrix for WMGI, the perfect image reconstruction is attainable when the number of measurements M is equal to the object’s resolution $N=a\times b$.

The aforementioned simplified analysis, which uniformly sets all non-zero singular values to 1, may be more beneficial in practice by either retaining larger singular values or setting a threshold to cut off the smaller ones.

Subsequently, we consider more complex scenarios, where instead of setting all non-zero singular values to 1, we analyze the impact of retaining different numbers of larger singular values on the imaging results. For the sake of simplification, we assume herein that the reflective object under consideration is a binary (black-and-white) one, and its reflective function $O(x,y)$ can be given by

$$O(x,y)=\left\{\begin{array}{cc}{r}_1\hfill & \mathrm{white}\mathrm{region},\hfill \\ r_2\hfill & \mathrm{black}\mathrm{region},\hfill \end{array}\right.$$

(10)

where $r_1$ and $r_2$ represent the reflectivity of object in the white area and black area, respectively.

The imaging quality can be quantified using the contrast-to-noise ratio (CNR), which is defined as [16,36]

$$\mathrm{CNR}={\displaystyle \frac{\langle G_{r_1}\rangle -\langle G_{r_2}\rangle }{\sqrt{{\sigma }_{r_1}^2+{\sigma }_{r_2}^2}}},$$

(11)

where $\langle G_{r_1}\rangle$ and $\langle G_{r_2}\rangle$ denote the ensemble average of second-order correlation function at any pixel where the reflectivity is $r_1$ and $r_2$, respectively, and ${\sigma }^2=\langle G^2\rangle -{\langle G\rangle }^2$ represents the variance of the corresponding second-order correlation function. Considering the statistical independent properties of random speckle patterns [36,61], and based on Equations (1), (10), and (11), the expression for the CNR in WMGI to evaluate the imaging quality of binary reflective objects is derived as

$$\mathrm{CNR}=(r_1-r_2){\left[{\displaystyle \frac{mM-1}{2(M_1{r}_1^2+M_2{r}_2^2)+(r_1^2+r_2^2)[(1-\frac{1}{mM}){({\gamma }_I/{\sigma }_I)}^4-2+\frac{3}{mM}]}}\right]}^{1/2},$$

(12)

where $M_1$ and $M_2$ are defined as the ratios of the white area and black area of the object to the speckle size, ${\gamma }_I^4/{\sigma }_I^4$ is the kurtosis of speckle field, ${\sigma }_I^2=\langle I^2\rangle -{\langle I\rangle }^2$ and ${\gamma }_I^4=\langle {(I-\langle I\rangle )}^4\rangle$ are the variance and fourth moment of the intensity fluctuation for each random speckle pattern, respectively.

We proceed to conduct a detailed analysis of the CNR in WMGI, specifically addressing the scenarios involving random measurement matrix and artificially constructed measurement matrix.

Firstly, we focus on the case where the random measurement matrices exhibits a negative exponential distribution. The intensity statistics of random illumination patterns in each wavelength channel follows a negative exponential distribution, i.e., its probability density function is given by [61]

$$\begin{array}{c}P\left(I\right)={\displaystyle \frac{1}{\langle I\rangle }}exp\left(-{\displaystyle \frac{I}{\langle I\rangle }}\right),\hfill \end{array}$$

(13)

where $\langle I\rangle$ represents the average intensity of the speckle pattern. According to the above equation, the n-th moment of intensity for random speckle pattern with a negative exponential statistical distribution can be calculated as

$$\begin{array}{c}\hfill {\mu }_n=\langle I^n\rangle =\int I^{n}P\left(I\right)dI=\int {\displaystyle \frac{I^n}{\langle I\rangle }}exp\left(-{\displaystyle \frac{I}{\langle I\rangle }}\right)dI=n!{\langle I\rangle }^n.\end{array}$$

(14)

Hence, the variance ${\sigma }_I^2$ and the fourth-order moment ${\gamma }_I^4$ of intensity fluctuations can be obtained as, respectively

$$\begin{array}{c}\hfill {\sigma }_I^2={\mu }_2-{\mu }_1^2={\langle I\rangle }^2,\\ \hfill {\gamma }_I^4={\mu }_4-4{\mu }_3{\mu }_1+6{\mu }_2{\mu }_1^2-3{\mu }_1^4=9{\langle I\rangle }^4.\end{array}$$

(15)

Thus, the kurtosis ${\gamma }_I^4/{\sigma }_I^4$ can be easily calculated to be 9. Substituting this value into Equation (12), one can obtain the expression for the CNR of reflective ghost image under negative exponential intensity statistics and speckle illumination

$$\mathrm{CNR}=(r_1-r_2){\left[{\displaystyle \frac{mM-1}{2(M_1{r}_1^2+M_2{r}_2^2)+(r_1^2+r_2^2)(7-\frac{6}{mM})}}\right]}^{1/2}.$$

(16)

The above expression indicates that, for the case of a random measurement matrix corresponding to random speckle patterns with negative exponential intensity statistics, when the number of measurements M is large enough, the CNR is proportional to $\sqrt{mM}$. Therefore, the CNR of the ghost image corresponding to m-wavelength multiplexing is $\sqrt{m}$ times that of the single-wavelength case.

Now, we delve into the analysis of the CNR in a scenario where the measurement matrix is artificially constructed using SVD. Here, we explore a more practical situation where the measurement matrix is crafted by truncating the larger singular values, rather than setting all non-zero singular values uniformly to 1. From Equation (12), which is the general expression for the CNR, it can be seen that when the ratio of the object size to the speckle size is large enough, that is $M_1{r}_1^2+M_2{r}_2^2\gg 1$, the CNR is almost independent on the kurtosis ${\gamma }_I^4/{\sigma }_I^4$ of speckle patterns. Therefore, although the artificial measurement matrix is different from the original random original measurement matrix, that is, the kurtosis ${\gamma }_I^4/{\sigma }_I^4$ of the speckle patterns corresponding to the artificially constructed measurement matrix deviates from a negative exponential distribution, the CNR is almost the same as that of random measurement matrix case. Even when the singular value truncation ratio is relatively small, the image reconstruction can still be achieved as long as the speckle patterns, corresponding to the artificially constructed measurement matrix, exhibit intensity fluctuations. Additionally, when the number of measurements M is large enough, the CNR of WMGI is proportional to $\sqrt{mM}$. Accordingly, the CNR for m-wavelength-multiplexing GI based on SVD is $\sqrt{m}$ times better than that of the single-wavelength case. This improvement arises because multiplexing multiple wavelengths effectively increases the amount of information available for reconstruction, leading to a better CNR. The use of SVD helps to optimize the measurement matrix, reducing the redundant information.

3. Numerical Simulations

To validate the aforementioned theoretical analysis, we conduct numerical simulations.

The numerical simulations are carried out in a three-wavelength multiplexing GI using the constructed measurement matrix based on truncated singular values. The target object used is a black-and-white binary “FIT” characters of size $128\times 96$ pixels. The random speckle patterns have a resolution of $128\times 96$ pixels and adhere to a negative exponential distribution for their intensity statistics. The random measurement matrix ${\phi }_i$ is achieved through M random speckle patterns according to Equation (2). Then, by performing SVD on the random measure matrix ${\phi }_i$ and retaining the k largest singular values (where $k\le r=rank\left({\phi }_i\right)$) and their corresponding singular vectors, the artificial measurement matrix is constructed. We adopt the singular value truncation ratio to characterize matrix data compression. The singular value truncation ratio $\eta$ refers to the ratio of the number of retained singular values k to the total number of non-zero singular values r of the original random measurement matrix, that is $\eta =k/r$. The artificially constructed measurement matrix, based on the above method, is utilized to perform GI of the reflective object. Figure 1 and Figure 2 present the numerically reconstructed images with $M=1500,3000,4500$ and their CNRs for single-wavelength (red, green, blue light beams) and 3-wavelength multiplexing cases, when the singular value truncation ratio $\eta$ is 0.4 and 0.8, respectively.

The simulation results indicate that, on the one hand, it is feasible to achieve WMGI by truncating singular values to compress redundant data in the original random measurement matrix and constructing an artificial measurement matrix. On the other hand, when SVD is applied to three-wavelength multiplexing GI, the CNR for imaging binary reflective object is $\sqrt{3}$ times that of single-wavelength GI, which is consistent with traditional three-wavelength multiplexing. In addition, for SVD-based two-wavelength (RG, GB, or RB) multiplexing GI, the CNR for imaging binary reflective object is $\sqrt{2}$ times higher than that of single-wavelength GI. Unlike the SVD-R, SVD-G, and SVD-B cases that rely solely on single-color patterns, the SVD-WMGI integrates information from all the multiplexed patterns at the same time for image reconstruction, resulting in better performance.

4. Optical Experiments

To further validate the SVD-WMGI, we carry out an optical experiment. The schematic diagram of the experimental setup is shown in Figure 3, which consists of spatial light modulation system, three color-selective (red, green, and blue) photodetectors acting as bucket detectors, a reflective object, and a computer. A 3LCD projector (Sony, VPL-EW578, Sony (China) Co., Ltd., Shanghai, China) is employed as spatial light modulation system which includes a UHP (ultra high pressure) lamp, two dichroic mirrors $D_1$ and $D_2$ used for generating RGB light, three mirrors $M_1$, $M_2$, and $M_3$, three liquid-crystal-display panels $L_1$, $L_2$, and $L_3$, and a beam combiner prism B. The detector is a silicon photodetector with a spectral range of approximately 400–1000 nm, a bandwidth of about 10 MHz, a sensitivity of around $0.4$ mA/mV, a noise power of approximately 10 mV, and a detection signal-to-noise ratio of about 15 dB. Red, green, and blue filters with center wavelengths of approximately 630 nm, 550 nm, and 450 nm, respectively, are placed in front of their corresponding detectors. The incoherent white light from the UHP lamp in the projector is separated into red, green, and blue beams by dichroic mirrors $D_1$, $D_2$. Note that the power and spectral range of the UHP are approximately 200 W and 400–680 nm, respectively. Based on 3LCD modulation technology, we are able to independently modulate three light beams of different wavelengths. These color-modulated light beams are subsequently combined to form a superposed RGB random pattern, which illuminates a binary reflective object located approximately 0.85 m away from the light source. The object used in the experiment is a black-and-white reflective object with a size of approximately 21.0 cm × 15.7 cm, bearing the letters “TEL” on its surface. The surface roughness of the object is approximately 1.5 μm, and the reflectivity of its white and black areas is about 0.86 and 0.11, respectively. A typical example of an illuminated random pattern is provided in Figure 4. The resolution of projected pattern illuminating the object is 128 × 96. The reflected light signals from the object are captured by three color-selective (red, green, blue) photodetectors. It should be noted that the three signals detected in a single measurement are mutually independent, and each matches the modulated light of its respective color.

We perform SVD on the original random measurement matrix that satisfies a negative exponential intensity distribution, and then construct different artificial measurement matrices for experiment by varying the singular value truncation ratio and their corresponding singular vector matrices. The image reconstruction of reflective object is achieved through the second-order intensity correlation between the detected bucket values and the man-made measurement matrix. The experimental reconstructed images with $M=1500,3000,4500$ and their corresponding CNRs for both single-wavelength (red, green, blue light beams) and three-wavelength multiplexing cases are presented in Figure 5 and Figure 6, respectively, when the singular value truncation ratio $\eta$ is 0.4 and 0.8.

The experimental results presented in Figure 5 and Figure 6 further demonstrate that, by leveraging SVD to compress random measurement matrix data through singular value truncation and constructing an artificial measurement matrix, WMGI can be realized. The CNR of three-wavelength multiplexing GI based on the artificial measurement matrix is approximately $1.6$ to $1.7$ times better than that of single-wavelength GI, which is basically consistent with theoretical analysis. Moreover, based on the three-wavelength experiment, we can perform any two wavelengths (RG, GB, or RB) for multiplexing, resulting in a two-wavelength multiplexed ghost imaging CNR approximately $1.32$ to $1.4$ times that of single-wavelength ghost imaging. These results are generally consistent with theoretical simulations.

To comprehensively analyze the impact of random speckle data compression on imaging quality, we further varied the truncation ratio to remove redundant information and constructed corresponding artificial measurement matrices for image reconstruction. Figure 7 presents the CNRs of the SVD-WMGI reconstructed images vary with the singular value truncation ratio. It can be seen from Figure 7 that the imaging reconstruction can still be performed even when the singular value truncation ratio is as low as 0.1. When the truncation ratio exceeds 0.5, the change in the CNR curve decelerates. This result can be understood through the higher-order kurtosis of speckle intensity fluctuations that determines the CNR, as obtained in the theoretical analysis.

5. Discussion

Based on the above experimental and numerical simulation results, on the one hand, it can be found that the experimental results are generally consistent with the theoretical results in terms of CNR improvement. On the other hand, the overall CNR of the experimental results is lower than that of the numerical simulation results. The specific discussion is as follows. In SVDWMGI, beams of different wavelengths are independently modulated and detected. Intensity correlation exists between beams of the same wavelength, but not between beams of different wavelengths. Therefore, SVDWMGI can be considered as a parallel operation of single-wavelength SVDGI. Hence, the improvement in CNR is attributed to the fact that in SVD-based WMGI, the multiplexing of multi-wavelength speckle fields effectively increases the number of measurement frames indirectly. Although the improvement factor of the measured CNR closely matches the theoretical simulation result, the experimentally measured CNR values are lower than obtained from the simulations. This discrepancy could be attributed to two key factors: (1) the non-ideal binary object used in the experiment, whose surface roughness may introduce additional optical effects degrading imaging quality, and (2) the presence of experimental background noise.

It should be noted that the effectiveness of the proposed methodology is demonstrated through simulation and experiments with simple binary objects. In practical applications, the reflectance maybe vary across different wavelengths. The proposed method also has limitations. On one hand, it only employs binary objects and the assumption of equal reflectance primarily to simplify the analysis complexity, and more complex objects such as multi-spectral, grayscale and dynamic object remain to be further studied. Furthermore, the method proposed in this paper, when integrated with a strategically designed arrangement of bucket detectors and reconstruction algorithms from prior 3D ghost imaging research [10], holds promise for 3D object image reconstruction. On the other hand, the UHP-based projector used in the experiment, while having the advantage of high brightness, has a limited spectral range, and an LED-based projector may be more suitable.

6. Conclusions

In conclusion, we have extended SVD to WMGI and the validity of this approach has been confirmed through both numerical simulations and optical experiments. We utilized the SVD method to construct a wavelength-multiplexed modulation matrix for ghost imaging to improve the sampling ratio and imaging quality. We investigated the influence of the singular value truncation ratio of the measurement matrix on the imaging CNR. The theoretical results reveal that, similar to conventional WMGI, in SVD-based WMGI, for an m-wavelength multiplexing, the CNR of the reconstructed object image is $\sqrt{m}$ times better than that of single-wavelength GI. In the experiment, we validated the theoretical results using a three-color light source as an example. This method integrates the advantages of SVD and wavelength division multiplexing, thereby holding potential for enhancing the application of GI in long-distance imaging domain.