High-Quality and Enhanced-Resolution Single-Pixel Imaging Based on Spiral Line Array Laser Source

Single-pixel imaging (SPI) is a novel computational imaging technique which combines illumination light fields and single-pixel detection values to reconstruct the image. Therefore, the generation method of the illumination light fields affects the imaging efficiency and quality. We propose a spiral line array laser source which can generate random illumination light fields without periodicity in the normalized second-order correlation function g(2). It also has a lower full width at half maxima value (FWHM). In numerical simulations and experiments, the compressed sensing based on total variation algorithm is adopted to reconstruct the image. We demonstrate that the novel array is capable of obtaining images of superior quality and resolution compared to existing array laser sources, including hexagonal and Fermat spiral arrays. Combined with the fiber lasers and electro-optical phase modulators, it is expected to achieve high-speed modulation for light fields and high emitting power. Therefore, this method has significant potential for application in remote target detection and recognition.


I. INTRODUCTION
T HE single-pixel imaging (SPI) technique utilizes struc- tured light fields to illuminate the object.A single-pixel detector (SPD) is used to collect light intensity from the object.Subsequently, the image is obtained through the implementation of reconstruction algorithms [1], [2].Compared with traditional array detection imaging, it has significant superiorities in many fields.For example, SPD has the advantages of wide response band and low cost, so it can be applied to non-visible wavelength imaging, including X-ray [3], infrared light [4], [5], terahertz [6], [7].In addition, it is also applied in low light detection [8], remote sensing [9], [10], underwater imaging [11], 3D imaging [12], [13] due to its high detection efficiency and sensitivity.However, since SPD has no spatial resolution, there should be more efforts on other parts of SPI to improve the imaging quality and efficiency.
Based on the basic principle of SPI, the generation method of the illumination light fields is an important factor affecting the quality and efficiency of imaging.At present, the commonly used methods depend on spatial light modulation components, including rotating ground glass (RGG) [14], [15], digital micromirror device (DMD) [16], [17], [18], liquid-crystal spatial light modulator (LC-SLM) [19], [20], [21], LED array light source [13], [22] and silicon-based optical phased array (OPA) [23], [24].Among them, the light field refresh frequency of RGG and LC-SLM is several hundred Hertz level, which seriously restricts the speed of sampling.The refresh frequency of DMD can reach up to 22.4 kHz, but its diffraction loss is large and the applications are limited.The refresh rate of LED array light source can reach MHz level, but the divergence angle is large and cannot be transmitted over long distance.To further improve the light field refresh frequency, the researchers propose the use of OPA integrated chip for modulation with the speed of 100 MHz.A multimode fiber (MMF) is used to project the illumination light fields onto the object.However, it cannot be applied to remote detection due to the complex manufacturing process and limited output power.
To solve the above problems simultaneously, researchers have proposed to use laser arrays as illumination light sources, such as fiber laser arrays [25], [26], [27].Specifically, the random speckle light field is formed by the far-field interference of the sub-light sources of the laser array.Therefore, the light field refresh rate is expected to be increased to GHz by equipping the electro-optical phase modulators.The emitting power can be also increased by improving the power of the sub-light source [28].The drawback of the method is that the periodic array will cause the periodicity of the normalized second-order intensity correlation function g (2) of the light fields.It will introduce significant periodicity to the image and damage the image quality.In 2018, Wu proposed the spatial arrangement of sparse structured light source optimized by genetic algorithm to effectively suppress periodicity [29].Meanwhile, Liu proposed a method using a square array to illuminate and a low-pixel APD array to detect [25].Through theoretical analysis and simulations, it was showed that the method can reduce the sampling amount and eliminate the periodicity of the image.In 2023, Lai proposed a novel Fermat spiral array without spatial periodicity, which can eliminate image periodicity and improve image quality [26].It was also demonstrated that the more sub-light In this paper, we propose an array laser source based on spiral line.Different from the existing hexagonal arrays, the sub-light source arrangement meets the aperiodic spiral line function.The g (2) of the generated light field is not periodic.It can effectively eliminate the periodicity of reconstructed images with better image quality.Compared with the Fermat spiral array, the g (2) of the light fields generated by the spiral line array has a lower full width at half maxima (FWHM).It indicates a higher resolution of the reconstructed image.Through the simulations and experiments, we prove the high-quality imaging and resolution enhancement performance of the proposed array.Combined with the fiber lasers and electro-optical phase modulators, it is expected to be applied in remote target detection and recognition.

II. THE MODEL OF FAR-FIELD INTERFERENCE
The random speckle illumination light fields are obtained by far-field interference of the array laser source.The far-field interference physical model of a fiber laser array with M sub-light sources is illustrated in Fig. 1.The seed light source is a fiber laser.The power is enhanced by an amplifier.It is divided into multiple unmodulated lasers by a beam splitter.Then, through the phase modulators, the output lasers are modulated by random voltages.The modulated lasers are connected to a still spiral line collimator to generate a spiral line array light source.It is worth mentioning that the arrangement of the array light source is related to the design of the mask, but not to the arrangement of the modulator.Therefore, the distribution of the phase modulators is not restricted.The outgoing beam passes through a convergent lens with a focal length F in the emission plane (x, y, z).Finally, the random illumination light field is obtained by interference of multiple lasers in the focal plane (u, v, z) which is viewed as the far-field distribution of a laser array.
In the emission plane, the light field U em of M sub-light can be represented as (1).
where m denotes the m-th sub-light source.A m is the amplitude and (x m , y m ) is the central coordinate of corresponding sub-light source.ϕ m refers to the random phase.The emitted light can be viewed as Gaussian beam with a waist radius of w 0 .r represents the radius of the truncated circular aperture.And the δ(r) stands for the truncation function as shown in (2).
After passing through the lens, the array laser propagates the distance f in free space to the focal plane, where interference occurs.The light field U F is expressed as (3) according to the Fraunhofer diffraction theory.
λ is the wavelength and k = 2π / λ represents the wavenumber of the light.-ik (x 2 +y 2 ) / 2F is the phase factor introduced by the curvature of the lens.Furthermore, the light field intensity distribution I F of the focal plane is derived as shown in (4).

III. THE MODEL OF NORMALIZED SECOND-ORDER CORRELATION FUNCTION
According to the existing theory, the spatial intensity distribution of illumination light field will affect the imaging quality of SPI [25], [29].The corresponding evaluation parameter is the normalized second-order correlation function g (2) (x, y; x 0 , y 0 ), which is expressed as (5).
where <•> denotes the ensemble average.I (x, y) and I (x 0 , y 0 ) express the light intensity at the position (x, y) and (x 0 , y 0 ) of the focal plane.
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As we know, the point spread function (PSF) represents the response of an imaging system to a point light source or a point object.It can be used to measure the resolution of the image.The lower full width at half maxima (FWHM) indicates the higher resolution of the image.In SPI system, g (2) represents the spatial cross-correlation coefficient of (x, y) and (x 0 , y 0 ) based on the above analysis.Therefore, g (2) can theoretically be equivalent to the PSF of SPI.The lower FWHM of g (2) represents the higher resolution of image.In the following, we analyze the g (2) of illumination light fields generated by different array laser sources.Further, the quality and resolution of the different reconstructed images are compared.

IV. THE MODEL OF SPIRAL LINE ARRAY
As we analyzed, the g (2) of illumination light fields affects the quality of the reconstructed image.The researchers have discussed the effect of hexagonal array and Fermat spiral array on the image quality of SPI [26].The hexagonal array has been widely applied in coherent beam combining (CBC) system due to the compact structure [28].It is conductive to the output of high-power laser.However, when it is applied to SPI, the periodicity of its spatial arrangement will lead to the periodicity of the g (2) of illumination light fields.This will cause the reconstructed image overlap and damage the image quality.To eliminate the periodicity, an aperiodic array based on Fermat spiral function is introduced [26], [30].It is demonstrated that the g (2) of the illumination light fields generated by Fermat spiral array has only one main peak.The reconstructed image has no periodicity.However, the resolution of the reconstructed image requires to be improved.
To eliminate the periodicity and enhance the resolution of the image simultaneously, we propose a novel laser array based on the spiral line function [31] as shown in (6).
ρ 0 is the minimum radius of the spiral line.s denotes the number of lines in spiral line and z = F is the position of the observed plane.M represents the total number of sub-light sources and N is the number of sub-light sources on each line.(ρ n , θ n ) denotes the central position coordinate of the n-th sub-light source from inside to outside on the laser array plane in the polar coordinate.

V. SIMULATION ANALYSIS OF ILLUMINATION LIGHT FIELDS
In this section, we firstly analyze the g (2) of illumination light fields generated by hexagonal array, Fermat spiral array and spiral line array.Subsequently, the quality of reconstructed images under different arrays illumination is compared.The wavelength of laser λ is 1064 nm.The focal length F of the convergent lens is 1 m.The number of sub-light sources M of the hexagonal array is 37, and that of the other two arrays is 36.The waist radius w 0 of the Gaussian beam and the radius R of the truncated circular aperture are set to w 0 = R = 3 mm.Therefore, the aperture D of the sub-light source is D = 2R = 6 mm.For the hexagonal array, the distance L from the center of the two sub-light sources is set to 8 mm and the duty cycle is 0.75.The other two arrays have a similar spacing of sub-light sources.
In order to more accurately analyze the characteristics of the illumination light fields, the sampling rate is set to 25%.As mentioned above, g (2) is equivalent to the PSF of a SPI system.And the lower FWHM value indicates the higher resolution of the reconstructed image.Therefore, the far-field g (2) distribution of different arrays are analyzed as shown in Fig. 2. Fig. 2(a1)-(c1) represents the hexagonal array, Fermat spiral array and spiral line array arrangement respectively.Fig. 2(a2)-(c2) represents the two-dimensional distribution of g (2) corresponding to the illumination light fields.Fig. 2(a3)-( c3) and (a4)-(c4) are the g (2)  distribution of the illumination light fields along the x section and y section, respectively.Fig. 2(a5)-(c5) expresses the 2D intensity distributions of the different illumination light fields.It can be seen that the g (2) of the hexagonal array illumination light fields has the same sidelobes of similar size to the main peak value along the y section, which has obvious periodicity.Therefore, the reconstructed image of hexagonal array illumination has periodicity theoretically.On the contrary, the g (2)  of the illumination light fields generated by Fermat spiral and spiral line array has only one main peak value, which is not affected by the sidelobes.In addition, intuitively, the spiral line array generates the finest speckle of the illumination light field.Therefore, in theory, the reconstructed image of spiral line array illumination has the higher resolution and no periodicity.
Furthermore, we compare the g (2) of the illumination light fields generated by the Fermat spiral and spiral line arrays in details, as shown in Fig. 3.The blue line represents the Fermat spiral array, and the red line corresponds to the spiral line array.Fig. 3(a) shows the overall distribution of g (2) .They both have no periodicity.Fig. 3(b) is an amplification of the main peak of g (2) .Intuitively, the width of the red line is significantly narrower than the width of the blue line.The FWHM value of the red line is 0.061, and the FWHM value of the blue line is 0.083.Therefore, in theory, the image obtained by the spiral line array illumination has a higher resolution.

VI. SIMULATION ANALYSIS OF IMAGE PERIODICITY
In order to compare the quality of SPI reconstructed images, we carry out the simulation of SPI.The basic parameters are the same as above.And the resolution of illumination light fields is 64 × 64.For a more objective comparison, we introduce the parameter peak signal to noise ratio (PSNR) [32] to evaluate the overall image quality.The higher PSNR value indicates the better image quality.We select a typical algorithm to reconstruct the image.It is compressed sensing based on total variation prior (CS-TV) [33].A binary image (transverse three slits) and a grayscale image (drone) are chosen as the imaging objects.The simulation results are shown in Fig. 4. Fig. 4(a) represents the reconstructed images of 3 slits and Fig. 4(b) shows the results of drone.The corresponding PSNR value is placed below the image.Intuitively, it is obvious that the reconstructed image of hexagonal array illumination has periodicity and image overlap, which seriously damages the image quality.On the contrary, Fig. 2. Simulation g (2) analysis of illumination light fields corresponding to different arrays.(a 1 -c 1 ): Different arrays; (a 2 -c 2 ): The two-dimensional distribution of g (2) ; (a 3 -c 3 ): The distribution of g (2) along the X-axis cross section; (a 4 -c 4 ): The distribution of g (2) along the Y-axis cross section.(a 5 -c 5 ): The 2D intensity distributions of the different illumination light fields.Fig. 3.The g (2) of the illumination light fields generated by the Fermat spiral and spiral line arrays.(a) The overall distribution of g (2) ; (b) the amplified main peak of g (2) .the reconstructed images illuminated by Fermat spiral and spiral line array have no periodicity.There is only partial noise.Through the quantitative comparison, it can be seen that the PSNR value (marked red) of the reconstructed image illuminated by spiral line array is highest, indicating the best image quality.
Furthermore, in order to analyze the effect of array design on the quality of reconstructed images more comprehensively.Except the number of sub-light sources, the other parameters are the same as the above simulations.A complex grayscale image 'Peppers' is selected as the imaging object with a resolution of 64 × 64.The reconstruction algorithm is also CS-TV.We compare the quality of reconstructed images with different number of sub-light sources.Fig. 5 shows the reconstructed images of different number of sub-light sources.M is the number of sub-light sources, increasing from left to right.It can be seen that as the M increases, the images have better quality.Below each image is its corresponding PSNR value.The data highlighted in red represents the highest PSNR value in the different conditions.When the M is 44, the reconstructed image has the highest PSNR value.The results indicate that as the number of sub-light sources increases, the image quality will be better.
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VII. SIMULATION ANALYSIS OF IMAGE RESOLUTION
According to the published studies [26] and the above simulations, both Fermat spiral and spiral line array illumination light fields have obvious advantages in eliminating image periodicity.Furthermore, we compare the resolution of the reconstructed images of the two arrays illumination.The simulation parameters are consistent with the above sections.In order to make a more objective comparison, a parameter R is selected as the evaluation criterion of the image resolution.It is defined as the ratio of the intensity at the image center to the maximum intensity [34], [35].The mathematical expression is (7).
where Num is the pixels number in the Y-axis.(x, y) expresses the position of the pixel.H(x, y) represents the correlation function in SPI theory [35].It is expressed as (8).It is also worth noting that the average pixel value on the Y-axis is calculated as the value of y = 0 to reduce the effect of noise.

H(x, y)
where Q is the total measurements.I q and S q represent the qth illumination light field intensity distribution and single-pixel detection value, respectively.Therefore, the lower value of R indicates the higher resolution of the reconstructed image.The line 3 pair in the USAF 1951 resolution board is selected as the imaging object.The reconstructed images are compared as shown in Fig. 6 and below the image are the corresponding PSNR and R values.illumination and spiral line array illumination, respectively.For a more intuitive comparison, we use a colormap to represent the image.As we can see, the lines in (c) are clearer and the gaps are more obvious.They indicate the higher image resolution.On the contrary, the lines in (b) are fuzzy.The lines and gaps cannot be clearly distinguished, indicating a poor image resolution.In addition, it can be seen that the reconstructed image obtained by spiral line array illumination has a higher PSNR value and a lower R value (marked red) than the other one.This proves that it has better image quality and clearer resolution.Therefore, the spiral line array has great application value in SPI.
In addition, we analyze the effect of each hole size on the image resolution.In fact, the increase of each hole size means that the array arrangement is more compact.The simulation parameters are consistent with those in Sections III and VI.The diameter D of each hole is set to 3 mm, 4 mm, 5 mm, 6 mm, 7 mm.Fig. 7 shows the reconstructed images of different size of each hole.D is the diameter of the hole, increasing from left to right.Intuitively, the details of image are clearer with the increasing of hole diameter.Further, we calculate the R value of each image, which is placed below the image.The data highlighted in red represents the lowest R value in the different conditions.It can be seen that as the diameter increases, the R value gradually decreases.When the diameter is 7 mm, the image has a lowest R value of 0.33, indicating the best image resolution.

VIII. EXPERIMENTS
In order to further demonstrate the superiority of spiral line array, we built an SPI experiment system with the LC-SLM, as shown in Fig. 8.A fiber laser with a collimator emitted the laser with a wavelength of 1064 nm.The size of the outgoing laser beam increased ten times after passing through the beam expander (BE).Then, the expanded laser passed through a mask with the specific array configuration to generate an array illumination laser source, including hexagonal array, Fermat spiral array and spiral line array.To be more convincing, the mask parameters were consistent with the numerical simulation.The array laser illuminated onto the SLM loading the random phase for modulation with a maximum modulation frequency of 60 Hz.Therefore, each sub-light source had a different phase.The modulated light was divided into two beams by beam splitter (BS) after passing through a projection lens (PL) with a focal length of 1 m.The first one was the reference light and the illumination light field distribution was recorded by a charge coupled device (CCD) camera with a maximum fps of 30 at the focal plane position of the lens.The other one was the signal light, which illuminated the object at the focal plane.The transmitted light intensity was collected by an SPD with a response speed more than 3 kHz.The limiting factor of the acquisition rate in this experiment is the camera.Therefore, the actual time in the experiment is 0.3 s for a sampling.The Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.   (analysis of illumination light fields corresponding to different arrays.(a 1 -c 1 ): Different arrays; (a 2 -c 2 ): The two-dimensional distribution of g (2) ; (a 3 -c 3 ): The distribution of g (2) along the X-axis cross section; (a 4 -c 4 ): The distribution of g (2) along the Y-axis cross section; (a 5 -c 5 ): the 2D intensity distributions of the different illumination light fields.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 10.The FWHM of g (2) of the illumination light fields generated by the Fermat spiral and spiral line arrays.intensity was converted into a voltage signal by a data acquisition card (DAC).A controller was used to synchronously load random phases, record the illumination light fields and collect the detection signals.Subsequently, we use fiber laser arrays, lithium niobate electro-optical phase modulators and high-speed cameras to improve the sampling speed.
Firstly, we analyzed the g (2) distribution of the illumination light fields generated by the different arrays, as shown in Fig. 9.The number of illumination light fields was 1024.Similarly, Fig. 9(a1)-(c1) represents hexagonal array, Fermat spiral array and spiral line array arrangement respectively.Fig. 9(a2)-(c2) represents the two-dimensional distribution of g (2) corresponding to the illumination light fields.Fig. 9(a3)-( c3) and (a4)-(c4) are the g (2) distribution of the illumination light fields along the x and y section.Fig. 9(a5)-(c5) expresses the 2D intensity distributions of the different illumination light fields.The g (2) distribution of the hexagonal array light fields had periodicity, while the other two arrays had no periodicity.They only had one main peak.Similar to the simulation results, the light fields generated by spiral line array have the finest speckle.
Further, we compared g (2) distribution of the illumination light fields generated by Fermat spiral and spiral line array in detail.They were shown in Fig. 10.The blue line represented the Fermat spiral array, and the red line represented the spiral line array.It can be intuitively seen that the red line had a narrower FWHM with a value of 0.057, while the blue line had a wider FWHM with a value of 0.068.Therefore, the reconstructed images of spiral line array illumination theoretically had a higher resolution.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.In experiments, we chose the letter 'w' and the number '3' as objects.The sampling rate was set to 6.25 % and the reconstruction algorithm was CS-TV.Fig. 11 shows the reconstructed images illuminated by different arrays.Below the image was the corresponding PSNR value.It can be seen that the image obtained by the hexagonal array illumination had multiple object contours with the obvious periodicity.In contrast, there was no periodicity in the other two images, but there was a small amount of noise.In addition, the image obtained by the spiral line array illumination had a highest PSNR (marked red), indicating the best image quality.
Finally, the line 3 pair in the USAF 1951 resolution target was experimentally imaged as an object for the comparison of resolution.The sampling rate was also set to 6.25 %.The experimental results were shown in Fig. 12.It was obvious that the image quality of (c) was better than that of (b).The three horizontal slits in (c) allowed the basic outline to be seen, while the outline in (b) was blurred and connected.The three longitudinal slits in (c) could be clearly separate the lines and intervals, while those in (b) cannot be distinguished.For a more convincing comparison, the PSNR and R values were calculated and placed below the image.The images obtained by spiral line array illumination had a higher PSNR value and a lower R value, indicating the better image quality and superior resolution.We demonstrated that spiral line array had more obvious advantages in SPI.

IX. DISCUSSION AND CONCLUSION
Discussion: Based on the above results, we will further improve the performance of the system from the following aspects: 1) In order to obtain better image quality and resolution, the g (2) of illumination light fields can be used as a guide function to further optimizing the array arrangement.
2) The deterministic and repeatable random illumination light fields can be obtained by locking each laser phase.It can optimize the system structure and reduce the difficulty of sampling.3) Currently, in SPI, structured light fields such as Hadamard basis fields and Fourier basis fields have been shown by many researchers to have more efficient sampling efficiency than random light fields.Therefore, under the framework of this scheme, it is expected that the structured light field can be generated by accurately calculating and strictly controlling the phase value of each laser.4) More efficient reconstruction algorithms (e.g., deep learning) can be applied to achieve higher image quality.It is necessary to choose the appropriate algorithm according to the actual application.Conclusion: In this paper, we propose an array laser source based on spiral line function applied to single-pixel imaging (SPI).It generates random illumination light fields by far-field interference of multiple lasers.Through numerical simulations and experiments, we analyze the characteristics of the illumination light fields and the quality of the reconstructed images.The reconstruction algorithm is compressed sensing based on total variation.Compared with the common hexagonal array, the normalized second-order correlation function g (2) of the illumination light fields generated by spiral line array is aperiodic.Moreover, the full width at half maxima (FWHM) of g (2) is narrower and has a lower value when compared with the existing Fermat spiral array.Intuitively, the reconstructed image of spiral line array illumination has no periodicity, fewer noise points and clearer details.It also has a higher PSNR value and a lower R value, indicating superior image quality and higher resolution.In addition, we demonstrate that the image quality will be better with the increase of sub-light sources number.And as each hole size increases, the reconstructed image resolution will be higher.In practical application, it is necessary to design a suitable array arrangement.Furthermore, when the array light sources are combined with fiber lasers and lithium niobate electro-optical phase modulators, it is expected to achieve high power output (up to kW) and high refresh frequency (up to MHz) of light fields.It provides guidance for the SPI technique illuminated by array laser sources.Particularly, it has important application values in the field of remote target detection and recognition.

Fig. 5 .
Fig. 5. Reconstructed images of different number (M) of sub-light sources.The sampling rate is set to 25%.
Fig. 6(a) is the original image.Fig. 6(b) and (c) represent the reconstructed images of Fermat spiral array

Fig. 6 .
Fig. 6.Comparison of the resolution.(a) Original image; (b) reconstructed image of Fermat spiral illumination; (c) reconstructed image of spiral line illumination.

Fig. 12 .
Fig. 12. Experimental comparison of the resolution.(a) Original image; (b) reconstructed image of Fermat spiral illumination; (c) reconstructed image of spiral line illumination.