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Abstract
An efficient and noise-resistant single-pixel imaging (SPI) technique based on Pseudo-Zernike moments (PZ-SPI) is proposed. In this technique, the illumination light fields are modulated to satisfy the Pseudo-Zernike polynomials. Then the modulated light fields are projected onto the object. And the single-pixel detector is used to measure the reflected light intensities to calculate the Pseudo-Zernike moments. Finally, the object image is reconstructed by iterative summation of the product of the Pseudo-Zernike polynomials and the Pseudo-Zernike moments. Through the numerical simulation and experimental demonstration, PZ-SPI can effectively reconstruct image at low sampling ratios. Besides, comparing with the Fourier-SPI and Zernike-SPI, PZ-SPI has good robustness to background noise in SPI system. These advantages expand the application of PZ-SPI in complex environments.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Traditional imaging tools are cameras based on array detectors such as CCD or CMOS. Since the detectors are made of silicon-based materials, they usually work at the visible light waveband. Single-pixel imaging (SPI) is an emerging imaging technique that uses a single-pixel detector such as photon multiplier tube or avalanche photodiode. The single-pixel detector can be made of silicon, germanium and other materials with low cost, which have the advantage of board working waveband. It makes SPI widely applied in non-visible waveband imaging such as terahertz light [1,2], X-ray [3], infrared light [4], as well as multispectral [5] and hyperspectral imaging [6]. Besides, due to the high detection sensitivity and the high quantum efficiency of the single-pixel detector, SPI can play an important role in weak light detection [7], remote sensing imaging [8], imaging through the scattering medium [9], 3D imaging [10] and so on.
In SPI system, a series of illumination light fields modulated by spatial light modulator (SLM) are sequentially projected onto the object. And the corresponding reflected light intensities are measured by a single-pixel detector. Using the known illumination light fields and the light intensities, the object image can be reconstructed by various algorithms [11–14]. Therefore, the spatial distribution of illumination fields and the reconstructing algorithms are important factors affecting the imaging quality and efficiency.
Initially, the random speckles are used as illumination light fields to project onto the object [15]. Due to the nonorthogonality of the random speckles, the image is usually reconstructed by oversampling which results in poor efficiency. Therefore, the compressive sensing (CS) algorithms are applied to reduce the measurements required for imaging [11]. CS-SPI can break the limitations of Nyquist-Shannon sampling theorem and reconstructs image under sub-Nyquist sampling. However, this method consumes lots of computation time and is susceptible to noise. To further improve the imaging efficiency, the orthogonal basis fields have been used as the illumination light fields for SPI [16–22]. For example, Fourier SPI (F-SPI) [16–18], Hadamard SPI (H-SPI) [18,19] and Discrete Cosine Transform SPI (DCT-SPI) [20] have been proposed. It has been shown that they can reduce computation time and theoretically achieve prefect reconstruction in full sampling. We refer perfect reconstruction means that the reconstructed image is exactly the same as the original image in an ideal case [18]. In addition, there are some other orthogonal moments with special applications are applied to SPI. For example, Krawtchouk moments SPI can achieve region-of-interest feature extraction [21]. And the object classification can be achieved by Zernike moments SPI (Z-SPI) [22].
Pseudo-Zernike moments (PZMs) are orthogonal moments defined on the unit circle. The low-order moments contain most of the information for a good information extraction ability. Besides, they have good robustness to noise due to the loose constraints [23]. Considering the above advantages, PZMs have been widely applied in pattern recognition and image processing, such as image retrieval [24,25], face recognition [26–28], image watermarking [29,30] and so on.
In this paper, PZMs are first applied in SPI system to achieve efficient and noise-resistant imaging. The imaging process of PZ-SPI is verified by numerical simulation and experimental demonstration. In this process, the illumination light fields are generated from the Pseudo-Zernike polynomials. After illuminating the object, the reflected light intensities are measured by the single-pixel detector. And the PZMs are calculated through the intensities. Finally, the image is reconstructed by the q-recursive algorithm which combines the illumination light fields with the PZMs [31]. Comparing with the F-SPI and Z-SPI, PZ-SPI can obtain better image quality at low sampling ratios. Besides, we also prove that PZ-SPI has better robustness to background noise.
2. Principles and methods
2.1 Pseudo-Zernike moments
Pseudo-Zernike moments (PZMs) are defined on the polar coordinate system which are derived from the Zernike moments (ZMs) by releasing from the constraint condition. They are based on the Pseudo-Zernike polynomials that are orthogonal on the unit circle. The two-dimensional PZMs of order m with repetition n of an image are defined as [32]
The Pseudo-Zernike polynomials satisfy the following orthogonality
In numerical simulation and experiment, the discretized Pseudo-Zernike polynomials are usually adopted. Therefore, we convert them to the discrete domain. When the image is represented by $f(x,y)$ in Cartesian coordinate system, the integral is replaced by the following summation formula
Combining the above equations, the reconstructed image $F(x,y)$ is deduced
From Eq. (7), the reconstructed image $F(x,y)$ can be obtained by the Pseudo-Zernike polynomials ${T_{mn}}(x,y)$ and the PZMs ${Z_{mn}}$. In the reconstruction process, as ${m_{\max }}$ tends to infinity, the $F(x,y)$ will approach to $f(x,y)$. It means the reconstructed image quality is better. Due to the factorial function in Eq. (3), the increase of the order affects the accuracy of the computation, causing the reconstructed image degradation. Therefore, some algorithms are applied to optimize the calculation process. The q-recursive algorithm is selected to improve the accuracy and efficiency [31]. The reason is that if using the direct method and the coefficient method to calculate ${R_{mn}}(r)$ in Eq. (3), the time complexity is $O({N^3}{M^2})$ which is considered to be very slow. And when high order moments are calculated, the results become inaccurate and unstable, resulting in degraded image quality. However, the time complexity of q-recursive algorithm is $O({N^2}{M^2})$ better than the former. The detailed mathematical explanation is given in Ref. [31].
2.2 Method of PZ-SPI
According to the principle of the single-pixel imaging and the mathematical form of the PZMs, PZ-SPI can be implemented by the following process.
- (1) Generating the illumination light fields that satisfy the Pseudo-Zernike polynomials. They correspond to ${T_{mn}}(x,y)$ in Eq. (7).
- (2) Loading the illumination light fields onto the SLM and then illuminating the object. The reflected light intensities are measured by a single-pixel detector.
- (3) Calculating the PZMs based on the measured intensities, corresponding to ${Z_{mn}}$ in Eq. (7).
The details of PZ-SPI are explained below. Combining Eq. (1) and Eq. (2), the PZMs can be expressed as follows
And the light intensities measured by the single-pixel detector are defined as
Equation (9) indicates that the illumination fields contain negative values, so they cannot be directly loaded onto the SLM in SPI system. The differential method are applied to solve the problem [21,22]. The illumination fields ${P_{mn}}(x,y)$ are divided into two complementary parts $P_{mn}^ + (x,y)$ and $P_{mn}^ - (x,y)$ which satisfy the following equation
In $P_{mn}^ + (x,y)$, the positive values correspond to the values of ${P_{mn}}(x,y)$ in the same position of the pixels and the negative values of ${P_{mn}}(x,y)$ are replaced by zeros. Similarly, in $P_{mn}^ - (x,y)$, the positive values are corresponding to the absolute values of the negative values in ${P_{mn}}(x,y)$ and the positive values of ${P_{mn}}(x,y)$ are set to zeros. Figure 1 shows the grayscale image of partial orders of Pseudo-Zernike illumination fields on the unit disk with a 128 × 128 pixels rectangle.
According to the Eq. (9), Eq. (10) and Eq. (11), the intensities ${I_{mn}}(x,y)$ are split to ${I^ + }_{mn}(x,y)$ and ${I^ - }_{mn}(x,y)$ corresponding to the $P_{mn}^ + (x,y)$ and $P_{mn}^ - (x,y)$, expressed as
Combining the differential modulation and Eq. (7), the reconstructed images are presented by
3. Numerical simulation
3.1 High efficiency of PZ-SPI
In this section, we use a typical grayscale image which is named ‘Cameraman’ (128 × 128 pixels) as the object. The images are reconstructed by PZ-SPI at different sampling ratios. The sampling ratio is defined as the ratio of the number of illumination light fields used to reconstruct the image to the total number of pixels. The illumination light fields are deduced by Pseudo-Zernike polynomials. And the sampling strategy is equivalent to the illumination sequences which depend on the orders of the PZMs that the low-order one contains most of the image information [23]. Therefore, the field sequence increases from the order $m = 0$. And in the same order m, the sequence of n is from $- m, - m + 1$ to $m - 1,m$ (corresponding to the sequence in Fig. 1, from top to bottom and from left to right). Figure 2 shows the original image and the results of PZ-SPI when the sampling ratios are from 1% to 35%. As the sampling ratio increases from 1% to 10%, the reconstructed images become clearer, especially in the central area of the image. When the sampling ratio is greater than 10%, the edges gradually become blurred. The calculation accuracy of the factorial function in higher order PZMs is affected.
In order to compare the reconstructed image quality, the evaluation parameter root-mean-square error (RMSE) is introduced in this paper for quantitative analysis. The formula for calculating the RMSE is as follows:
M and N indicate that the image has M pixels and N pixels in the X-axis and Y-axis directions. $F(x,y)$ and $f(x,y)$ represent the pixel values corresponding to the reconstructed and original images at the coordinate $(x,y)$. It can be seen that the smaller RMSE value means the better image quality. Figure 3 shows that the RMSE values decrease as the sampling ratio increases from 1% to 10%. And when the sampling ratio exceeds 10%, the RMSE values increase. Corresponding to Fig. 2, the edge of the reconstructed images is gradually blurred. The reason is that when calculating PZMs in the discrete domain, the increase of the order will lose the computation accuracy resulting in the numerical error, especially in the edge of the circle. The above results show that PZ-SPI has an optimal sampling ratio. And the high-quality images can be obtained when the sampling ratio is lower than 10%.
Besides, we test the optimal sampling ratio of four typical grayscale images (Cameraman, Lena, Baboon, Peppers, 128 × 128 pixels). The results are shown in Table 1. The first column represents the image types and each row indicates the RMSE values of different sampling ratios. The value in red on each row is the minimum RMSE value. We can see that for ‘Cameraman’ and ‘Baboon’, 10% is the optimal sampling ratio. And 12% is the optimal sampling ratio for ‘Lena’ and ‘Peppers’. Therefore, there are different optimal sampling ratios for different images.
Table 1. The RMSE values of different images in different sampling ratios
As we know, the information of the natural image is mainly concentrated on the low-frequency waveband of the Fourier spectrum [16,18]. And the low-order Zernike moments contain most of the object information [32]. Therefore, both of them can be used in SPI system. Besides, they can reconstruct the image at low sampling ratio with high efficiency [16,22]. We perform simulations to compare the above three methods using two typical grayscale images. The ‘Cameraman’ and ‘Baboon’ are selected for better comparison, because their optimal sampling ratios are both 10%. Three methods all adopt the differential way, the difference is that the modulated light field of each method is calculated according to their corresponding formula. Besides, the Pseudo-Zernike polynomials are defined on a circle, the values outside the circle are set to zeros. Z-SPI and F-SPI perform the same process to ensure the fairness of comparison. In the reconstruction algorithm, both PZ-SPI and Z-SPI adopt q-recursive algorithm [31]. While F-SPI adopts four-step phase-shifting method combining with the circular sampling path that has high-efficiency imaging [18]. Figure 4 respectively shows the simulation results obtained by two types of images in different SPIs and different sampling ratios. It can be seen that as the sampling ratio increases from 1% to 10%, the reconstructed images have clearer details. And the central area of the PZ-SPI images is sharper. Figure 5 shows the trend of RMSE values of different SPIs at low sampling ratios. The RMSE values of PZ-SPI images are the smallest, indicating the highest quality and efficiency at low sampling ratios. Both types of images have the above conclusions.
Fig. 5. The RMSE values of different SPIs at low sampling ratios by simulation: (a) Cameraman; (b) Baboon
3.2 Noise resistance of PZ-SPI
In the practical application of SPI, it cannot avoid being affected by noise, especially the additive noise. It generally refers to thermal noise and shot noise, which are the background noises of the SPI system. This will cause image degradation. The noise can be suppressed by increasing the number of illumination light fields, but it takes more time to collect the data. Some orthogonal moments are robust to noise due to their properties. In this section, we perform a simulation to demonstrate that PZ-SPI has better robustness to background noise and compare it with F-SPI and Z-SPI. We use the grayscale image (Baboon, 128 × 128 pixels) polluted by Gaussian white noise as the object. The 128 × 128 pixels Gaussian white noise matrix is generated through the built-in function of MATLAB firstly. And then it is superimposed with the object image. The result with many noise points is the polluted image. The noise level can be adjusted by setting the parameters of the function. The larger parameter value indicates the more serious pollution level. In section 3.1, we have verified that PZ-SPI can reconstruct the images effectively when the sampling ratio is lower than 10%. Besides, by comparing with the high-efficiency methods F-SPI [16] and Z-SPI [22], PZ-SPI has better image quality under low sampling ratios. Therefore, the low sampling ratio 8% is set to ensure its high efficiency.
Figure 6 shows the simulation results of three SPIs with different noise levels. It can be seen that as the noise level increases from 0.01 to 0.05, the images are more seriously affected by noise. The edges of PZ-SPI and Z-SPI images become blurred, while there are many noise points in F-SPI images. In the same noise level, the PZ-SPI images have clearer details at the central area, indicating the better image quality. Figure 7 shows the trend of RMSE values of the reconstructed images at different noise levels. It can be seen that the RMSE values of PZ-SPI images are always the smallest at each noise level. As the noise level increases, the RMSE values of PZ-SPI images increase slowly and the images are less affected. While the RMSE values of F-SPI images increase rapidly and the image quality degradation is serious. In summary, we can infer that PZ-SPI has best robustness to noise. The background noise can be effectively suppressed. Therefore, it can be applied to many complex environments. Especially when the thermal noise and shot noise seriously affect the SPI system.
4. Experimental demonstration
The high efficiency and noise resistance of PZ-SPI were demonstrated by experiments. Figure 8 shows the experimental optical path diagram. The light source was a solid-state laser with a wavelength of 532 nm (LSR532NL). The emitted light passing through the collimated beam expanding system (BE) was projected onto the DMD 1 screen. The DMD 1 (Texas Instruments DLP7000 V-7001) had a 1024 × 768 micro-mirror array. It could modulate the light by loading the generated illumination field sequence (as shown in Fig. 1). The modulated light was projected onto the DMD 2 through a projection lens (PL) with a focal length of 300 mm. The DMD 2 had the same performance parameters as the DMD 1. It was used to load the imaging object. In order to obtain a clear image, the DMD 1 and DMD 2 had to be located in the conjugated position of the PL. Therefore, the distances from the DMD 1 to the PL and from the PL to the DMD 2 were both 600 mm. The reflected light from the DMD 2 was then measured by a single-pixel detector (SD, Thorlabs PMT-PMM02) through a collecting lens (CL). The detected light intensities were converted into voltage signals by the data acquisition card (DAC, NI USB-6361). The illumination fields and the corresponding light intensities were recorded simultaneously through the self-developed data synchronization acquisition software.
Fig. 8. Experimental optical path diagram. BE (beam expanding), PL (projection lens), CL (collecting lens), SD (single-pixel detector), DAC (data acquisition card)
4.1 PZ-SPI experiment
In the experiment, the DMD 2 loaded an English letter ‘E’ as the imaging object. To achieve a better modulation effect, 512 × 512 pixels from the center of the DMD 1 were selected to generate a 128 × 128 pixels modulation field by combining each set of 4 × 4 pixels into a single resolution cell. A series of 128 × 128 pixels Pseudo-Zernike illumination fields, Zernike illumination fields [22] and Fourier illumination fields (adopting the four-step phase-shifting method [18]) were loaded onto the DMD 1 to implement SPI. And the qualities of the reconstructed images were compared. In order to be consistent with the simulation, the sampling ratios were also set to 1%, 3%, 5%, 8% and 10%.
Figure 9 shows the original image and the experimental results of imaging at different SPIs. It could be seen that all of them can reconstruct images when the sampling ratio was lower than 10%. As the sampling ratio increased, the reconstructed images details became clearer. The difference was that the center of the PZ-SPI images was affected, and the edges of the Z-SPI images became blurred. While the F-SPI image produced some noise points. Figure 10 shows the quantitative comparison curve of three SPIs. It was concluded that the RMSE values of the PZ-SPI images were always the smallest as the sampling ratio increased from 1% to 10%, indicating the best image quality. Therefore, PZ-SPI had a better ability to extract image information under the low sampling ratio, confirming its high efficiency.
4.2 Noise resistance experiment
In this section, the robustness of PZ-SPI to noise was experimentally demonstrated. The experimental procedure and the optical path were basically the same as in section 4.1. The imaging object was replaced by an English letter ‘E’ polluted by the Gaussian white noise. To be consistent with the simulations, the noise adjustment process was the same as in section 3.2, and the noise level was set from 0.01 to 0.05. The sampling ratio was also set to 8%.
Figure 11 shows the experimental results of different SPIs affected by the Gaussian white noise. It could be seen that as the noise level increased, the degradation of the reconstructed images was becoming more serious. The F-SPI images were covered by many noise points, indicating that the images were affected most seriously. The image difference between PZ-SPI and Z-SPI was small. Both of them generated some points and the edges became blurred. But the image details of PZ-SPI were clearer. Figure 12 shows the trend of RMSE values of the reconstructed images at different noise levels by experiment. The RMSE values of the PZ-SPI images were the smallest at each noise level. It meant that PZ-SPI images were least affected by noise and the image degradation is not obvious. In summary, we concluded that PZ-SPI had good robustness to noise. It could effectively suppress the background noise. This characteristic expanded the application of PZ-SPI in complex environments.
5. Discussion and conclusion
From the above analysis, PZ-SPI has better image quality and robustness to background noise than F-SPI and Z-SPI at low sampling ratios. As for PZ-SPI, when the sampling ratio exceeds above the optimal ratio, the reconstructed image will be degraded. The same situation exists for Z-SPI. However, the image quality of F-SPI will be better as the sampling ratio increases. Therefore, PZ-SPI is suitable for high quality imaging at low sampling ratios. As for the noise-resistance performance, PZ-SPI has good robustness to background noise, but it needs further research and demonstration for more complex noisy environments.
In summary, a novel single-pixel imaging technique based on Pseudo-Zernike moments (PZ-SPI) is proposed in this paper. We verify the high efficiency and noise resistance of this technique through numerical simulation and experimental demonstration. Comparing with the efficient Fourier-SPI and Zernike-SPI, PZ-SPI has a better information extraction ability and it can obtain better image quality at low sampling ratios. Furthermore, we demonstrate that PZ-SPI has good robustness to background noise. According to the simulation and experiment results of different SPIs, the PZ-SPI are least affected by background noise while maintaining its high efficiency. Combining with the high-speed modulation of DMD and the high detection efficiency of single-pixel detector, PZ-SPI can be widely applied in extreme environmental conditions.
Funding
Youth Innovation Promotion Association of the Chinese Academy of Science (2020438).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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