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Abstract
When insufficient samples in the spatial frequency domain could be effectively compensated by the modified CLEAN algorithm, a novel aperture-synthetic scheme of ghost imaging takes advantage of a superior speed of modulation and an enhancement on the spatial resolution. However, there still exist some imperfections in the modified CLEAN reconstructions, such as the fact that some omitted scatter noise still remains or the object contour may be incomplete. Therefore, we optimize the modified CLEAN algorithm by proposing a density clustering algorithm to overcome these drawbacks and improve the visual quality.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ghost imaging (GI), which used to be considered as a counter-intuitive optical phenomenon, reconstructs the image of objects using the second-order intensity or fluctuation correlations. The first GI realization was demonstrated using the entangled photon pairs in 1995 [1], which used to be considered as a part of quantum imaging. With the increasing attention of the optics community, GI was also demonstrated to be realized using the (pseudo-) thermal light source [2], X-rays [3], or even the sunlight [4,5]. Recently, GI is widely applied in many research fields ranging from the remote imaging to microscopy [6–9]. Being different from the conventional imaging, the repeated signals from object and the idler speckles are recorded separately, and the image is obtained by calculating the second-order correlation of the two recorded sequences, or using various algorithms to achieve the higher performance [10–12]. The unique setup of GI brings it the advantages of high-resolution [13] and high-sensitivity [14], but the repeated sampling feature also makes the imaging speed restricted by the sampling rate. By reducing the number of sampling elements, GI can be effectively speed up at the cost of handling the degradation of image quality. Thus, the concept of aperture synthesis is introduced to GI to solve this problem. The aperture synthesis techniques use a few detectors with small aperture to simulate the responses of a single detector with a large aperture, which takes the advantage of reducing the number of elements [15,16].
However, in aperture synthesis techniques, the number of samples in the spatial-frequency domain is far below the Nyquist limit, which would lead to the repetitive visual artifacts in the reconstructed image and significantly reduces the quality of the reconstructed image. In [17], a Modified CLEAN algorithm based on point spread function (PSF) estimation method was proposed to realize high-quality and high-speed GI under the sparse spatial-frequency sampling conditions. However, during the Modified CLEAN iterations, some of the clean object points may be extracted and substracted repeatedly, which would not only cause the information loss and incomplete object contour, but also introduce the extra noise, which still remains a common obstacle in the application of the aperture-synthetic GI.
In this paper, we revisit the schematic of aperture-synthetic GI with the Modified CLEAN algorithm and analyze the underlying reason of the incomplete object in the reconstructed image. Furthermore, a density clustering-based algorithm is introduced to overcome the scatter noise around the object points in the background. Validations of the proposed algorithm with different parameters and conditions are also presented. The experimental results shows that the missing part of the object in the reconstructed image could be effectively compensated, and the background noise can be filtered out, which leads to a better perceptual result of the reconstructed image.
2. Methods
The experimental setup of the aperture-synthetic GI is shown in Fig. 1. The 532 nm laser beam is expanded by a beam expander and then be randomly modulated by a phase-only spatial light modulator (SLM, Holoeye Pluto VIS-010-A) . Each random phase pattern is recorded to generate the diffraction pattern on the object plane. In order to emulate the aperture-synthetic source in GI, we covered an etched mask with 24 pinholes distributed as a circle on the SLM. After the random phase modulation, the light was through an object and then is collected by a bucket detector (Thorlabs PDA100/A). The size of the random matrix is 100$\times$100, and the repeated measurement number ranges from 100 to 10000.
In order to use the clustering-based CLEAN algorithm to reduce the repetitive visual artifacts, we need first to compute the reconstructed image and PSF [17] by second-order fluctuation correlation (SFC). Usually, we need a point source to get the PSF, which means we need an extra experiment to measure it, but here we use the self-correlation function of the speckle matrix as the substitute for PSF [18].
In GI, the ideal PSF is a sombrero function without any obvious sidelobes, and the ideal reconstructed image can be regarded as a convolution of the PSF and a matrix with the object points [19]. In the experiment, while the sparse spatial frequencies are insufficient, the PSF estimation suffers from the surrounding sidelobes, shown in Fig. 2(a), leading to repetitive visual artifacts in the reconstructed image, shown in Fig. 2(b). The previous Modified CLEAN algorithm extracts the object points distribution with a spatial-domain deconvolution procedure, and the repetitive visual artifacts can be reduced by convolving the object points with an ideal PSF [20]. However, when imaging the ’truck’ object, as shown in Fig. 2(c), the result of Modified CLEAN processing shows that the cab of the truck is already missed as shown in Fig. 2(d), even the repeated measurement number is over 10000.
Fig. 2. (a)PSF; (b)Reconstructed image by SFC; (c)Object; (d)Reconstructed image with Modified CLEAN algorithm.
A brief description of the Modified CLEAN algorithm is as follows [17]:
- 1 Set parameters and get reconstructed image as "Dirty Map" and PSF;
- 2 Find the object points with PSF and reconstructed image and renew the Dirty Map in iterations.
- 3 Get the clean components matrix which contains only object points, and a new reconstructed image called "clean Map" without repetitive visual artifacts.
The key step of the Modified CLEAN algorithm is to find the maximum intensity point in the reconstructed image and subtract the convolution of PSF and this point in iterations. Therefore, the intensity of some object points around the found point may be reduced multiple times and less than other object points or even some noise points after multiple iterations, making them seen as noise points in the algorithm. A brief explanation is given in Fig. 3.
Fig. 3. Explanation of the cause of incomplete object. A, B, C, D, E: Object points; F: Noise points; Solid Lines: intensity of point regarded as object; Broken Lines: intensity of points regarded as noise.
Generally, the spatial resolution of the object in the reconstructed image is not accurate to a single pixel, which means that the points around each object point may also be object points. Therefore, we strengthened the points around the current object point during each iteration to improve the possibility of the points being seen as a target point. In this way, the missing part caused by the algorithm could be compensated. A brief explanation is given in Fig. 4.
Fig. 4. Explanation of the proposed algorithm. A, B, C, D, E: Object points; F: Noise points; Solid Lines: intensity of point regarded as object; Broken Lines: intensity of points regarded as noise.
The details of the algorithm are as follows:
- a) Prepare a Gaussian distribution called enhancement function before iterations.
- b) In each iteration, we calculate the convolution of the enhancement function and the object point found in step 2, assign the intensity of the point found in the previous convolution result to 0, and add the product of the result to the Dirty Map. Set the result as the new Dirty Map.
Moreover, after the repetitive visual artifacts are filtered out by the Modified CLEAN algorithm, the noise remained can be divided into two parts: noise points densely distributed with low intensity introduced by the algorithm and noise points scattered distribution with high intensity that already existed. Therefore, after getting the Clean Map, we added extra modules based on the DB-Scan clustering algorithm [21] to filter the noise:
- 1) Initialize an empty matrix as the target matrix with the same size of the object image. Set parameters such as gain of intensity and gain of clustering.
- 2) Begin iterative loops: Find the first point in clean components, calculate the number of points within the clustering radius and the intensity of the first point.
- 3) Calculate the Evaluation with the number of points and intensity obtained in step 2), and delete it from the clean components. If the Evaluation is bigger than 1, record the point in the target matrix.
- 4) Repeat step 2) and step 3) until there is no point unprocessed in clean components.
- 5) Matching a Gaussian distribution and convolve it with the target matrix to form the clean image. The Evaluation in Step 3) is calculated as
3. Results
The results of the experiment are shown in Fig. 5. Compared with the reconstructed image by SFC in Fig. 5(a), the repetitive visual artifacts is removed successfully after processed by Modified CLEAN algorithm in the image shown in Fig. 5(b), but the cab of the truck is nearly disappeared. Meanwhile, as shown in Fig. 5(c) after processed by clustering-CLEAN algorithm, the cab of the truck is compensated, and the intensity of the body part is also enhanced.
Fig. 5. Images of (a) Object; Reconstructed image by (b) SFC; (c) Modified CLEAN; (d) Clustering-based CLEAN.
In order to make a quantitative analysis of the algorithms, we introduced the signal-to-noise ratio (SNR) to evaluate the quality of reconstructed images. Besides, we designed multiple masks to imitate the light sources with different distributions [22] to verify the effectiveness of the algorithm under different conditions, and the results are shown in Fig. 6 and Fig. 7.
Fig. 6. Reconstructed image by different algorithms under different distribution of pinholes; (a) distribution of the pinholes; (b) SFC; (c) Modified CLEAN; (d) Clustering-enhanced CLEAN;(I) Two circles;(II) Hexagon;(III) Random; $G_r=3$, $G_I=3$.
Fig. 7. SNR of the reconstructed image by SFC, Modified CLEAN algorithm and clustering-based CLEAN algorith under different distribution of pinholes;(a)One cricle;(b)Two circles;(c)Hexagon;(d)Random
The reconstructed images by different algorithms under different distribution of pinholes are shown in Fig. 6. The coverage of sampling with sparse spatial frequencies depends on the distribution of the pinholes. Therefore, the distribution affects the reconstructed images a lot. As shown in Fig. 6(III(b)), the object is hardly to be identified. And as shown in Fig. 6(c), the Modified CLEAN algorithm works well in removing repetitive visual artifacts whatever the distribution of the pinholes is, but the problem of incomplete of the object still exists. The reconstructed images after processed by Clustering-based CLEAN are shown in Fig. 6(d). It can be seen that with the object becoming intact, noise was introduced in the reconstructed images.
As shown in Fig. 7, due to the repetitive visual artifacts, the SFC’s SNR of the reconstructed image keeps less than 1. After being processed by the Modified CLEAN algorithm, it has been dramatically improved because of the artifacts being removed, and when processed by Clustering-based CLEAN algorithm, the complete structure of the object makes the SNR improved about 3dB.
Several parameters mentioned were playing important parts in the algorithm, and the influence of the loop gain, stopping criterion have been studied in earlier work [17]. Thus we discussed the influence of the enhancement function mentioned in step a), the gain of intensity ($G_r$) and gain of intensity ($G_I$) used in step 3). In the experiment, the results with different standard deviation values and the Modified CLEAN algorithm are shown in Fig. 8. Parameters as $G_r$ and $G_I$ affect the reconstructed image differently, so we set $G_r=3$, $G_I=3$ while discussing the influence of the value of standard deviation enhancement function. As shown in Fig. 8, different standard deviation values have little effect on the object in the reconstructed images if we neglect the introduced noise. Thus, noise filtering is necessary.
Fig. 8. Reconstructed image with clustering-based CLEAN algorithm with $G_r=3$, $G_I=3$, and standard deviation of (a) 1; (b) 10; (c) 20; (d)50; (e)100.
While doing noise filtering, the influence of $G_r$ and $G_I$ on reconstructed images is shown in Fig. 9. Fig. 9(I(a)) is the reconstructed image without noise filtered, and Fig. 9(II(a)) is the reconstructed image with $G_r=3$, $G_I=3$. In Fig. 9(I(b))-(I(d)), $G_r$ is set to be 7,5,3 separately while $G_I$ is 0. $G_r$ amplified the density of the points, so when $G_r$ is small, the points with sparse distribution will be wiped out as noise points. It can also be seen that with $G_r$ reducing, the noise points filtered varies from scattered points away from the object to points around the object. In Fig. 9(II(b))-(II(d)). $G_I$ is set to be 6,4,2 separately while $G_r$ is 0. $G_I$ amplified the intensity of the points, so when $G_I$ is small, the points with low intensity will be wiped out as points, which is also can be seen from Fig. 9(II(b))-(II(d)).
Fig. 9. Reconstructed image with different values of parameters:(I)(a)without noise filtered; (I)(b)-(d):$G_I=0$, (b) $G_r=7$; (c) $G_r=5$; (d) $G_r=3$;(II)(a) $G_r=3$, $G_I=3$; (II)(b)-(d):$G_r=0$,(b) $G_I=6$; (c) $G_I=4$;(d) $G_I=2$.
4. Conclusions
In this paper, we make the further developments to the Modified CLEAN algorithm proposed in earlier work [17] to improve the quality of reconstructed images in GI with aperture-synthesis. The results show that our proposed algorithm can not only overcome the repetitive visual artifacts, but also compensate the missing part in the reconstructed images, even capable of removing background scatter noise by the clustering operation. However, there still remains two problems: one is the results of all the CLEAN-based algorithm relies on the quality of the reconstructed image by SFC, if the SFC reconstruction is severely corrupted, these algorithms cannot reach a satisfactory reconstruction. The other one is that the parameters in the clustering algorithm influence the result, whose values were set by experience now. In further study, we hope to find a resolution to solve the problems.
Our method would contribute to GI with sparse spatial frequencies in several ways. The scheme of GI with sparse spatial frequencies, in other words, synthetic aperture brings it potential to improve imaging resolution, and the characteristic of sparse spatial sampling also reduces the demand for transmission bandwidth and storage space in data acquisition. Therefore, it has a broad application prospect in a variety of challenging scenarios. However, the unique scheme also makes repetitive visual artifacts inevitable, in which case our method would be beneficial. Meanwhile, our method is the post-processing method after obtaining reconstructed images. It can also be used as a post-processing method in other scenes or combined with other algorithms to improve the imaging quality.
Funding
Youth Research and Innovation Program of BUPT (2015RC12); National Natural Science Foundation of China (61401036, 61471051, 61531003, 61631014, 61801042).
Disclosures
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
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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