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Abstract
Point spread function (PSF) is important for evaluating an optical system and image deblurring. In this paper, we proposed a method to measure space-variant PSF at any depth based on single-pixel imaging (SPI), and we initiated a depth-variant PSF interpolation model. In our method, we regarded space-variant PSF as light transport coefficients from object points to image pixels. By applying SPI to each image pixel to obtain these light transport coefficients at different depths, the PSF of each object point can be extracted. The depth calculation of PSF is based on multi-frequency heterodyne phase-shifting principles and perspective-n-point (PnP) algorithm. In our PSF interpolation model, we interpolated the light transport coefficients from different object points to an image pixel first. We then obtained the interpolated PSF indirectly from the interpolated coefficients. With simple experimental facilities containing a digital camera and a liquid crystal display (LCD) screen to display and capture specific patterns, which relative distance is changed, the proposed method accurately obtained the space-variant PSF at any depth. Without complicated calculation, PSF at a certain depth can be interpolated from the PSF measured data at another depth with our PSF interpolation method. Significant similarities exist between the interpolated PSF and directly measured PSF. Our work is a successful attempt in using SPI to solve traditional optical problems.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Point spread function (PSF) is an impulsive response of an optical system, which can represent the quality of optical system. PSF is a widely used concept in Fourier optics [1,2], astronomic imaging [3–5], medical imaging [6,7], electron microscopy [8], lithography [9], and so on. It can be regarded as liner and position-invariant in simple situations [10]. However, realistic spatially-variant situations cannot be set aside. An accurate PSF measurement and interpolation result has practical and scientific importance.
Traditional PSF acquiring methods can be divided into three categories [11], namely, blind PSF estimation, non-blind PSF estimation, and PSF direct measurement methods.
Blind PSF estimation using natural scene images means estimating PSF from acquired images without any prior information. This method generally relies on sharp edges in images [12] or two photographs at different distances [13]. Through these methods, the accuracy of PSF is affected by the features that images contain, which are unusually unstable.
The second category of methods obtain PSF by taking images of certain patterns, such as checkerboards [12] and disks [14]. These patterns always contain sharp edges. The second method uses prior information, such as the shape of patterns. Thus, this method is more stable and accurate. The first two categories of methods usually regard PSF as Gaussian model. These two methods are used to estimate the parameters of Gaussian model, which is an ideal model.
The last category of methods obtain PSF by taking images of point-like sources [15,16]. These methods are simple and reliable, which directly measure the ground-truth PSF. But there are some requirements. On the one hand, the point-like sources must be very small. On the other hand, the intensity of the source mush be strong enough. If these two requirements cannot be well be satisfied, the SNR (signal-to-noise ratio) of the measured PSF will be low.
Depth-variant PSF (DV-PSF) is essential for DV image restoration algorithm. However, measuring DV-PSF constantly is not feasible. Thus, DV-PSF is interpolated using few measured PSF data. Multiple DV imaging models are presented to solve the interpolation problem of DV-PSF, such as strata-based and PCA-based DV imaging models [17], which require the PSF data of at least two different depths to interpolate another unknown PSF.
Single-pixel imaging (SPI) is an innovative technique which uses correlation measurements to obtain images from non-pixelated detection, which has been widely researched and applied in some areas [18,19]. Fourier single-pixel imaging (FSI) is proposed and demonstrated [20,21]. FSI can acquire images with high SNR based on Fourier spectrum acquisition method. FSI-based PSF acquirement method, which can measure PSF with high SNR, is presented in our previous study [11].
The previous method was further developed. In this work, we have two main contributions. First, the space-variant PSF (SV-PSF) is measured based on SPI, and the accurate depth of each PSF is also obtained. Second, a new PSF interpolation model is presented. Our method has advantages over previous approaches. First, an accurate depth information of each PSF can be extracted. Second, PSF measurement result at a certain depth along with the focus distance are sufficient to interpolate unknown PSF.
This paper is organized as follows. Section 2 presents the principles of our method, including 3D image formation model, our PSF model and interpolation method, SPI-based PSF measurement, and depth measurement method of PSF. Section 3 shows our experiment results, whereas Section 4 discusses our conclusions.
2. Principles
A method to measure the SV-PSF and corresponding depths is presented, and a DV-PSF interpolation model is initiated. In contrast to traditional methods, a digital camera with lens is used for testing, while a liquid crystal display (LCD) screen is used to show specific patterns, as shown in Fig. 1.
Fig. 1. Experimental setup. An LCD screen is used to display the patterns. A digital camera with the lens to be tested captures the images. Each pixel of the image sensor can be regarded as a single pixel detector. The distance between the camera and the screen is changed in our experiment.
The framework for PSF measurement and interpolation comprises the following steps:
Step 1: Measuring SV-PSF at different depths. By displaying and capturing certain sinusoid patterns and applying SPI-tech, the Fourier spectrum of the light transport coefficients from LCD screen points to image sensor pixels is acquired, which can be converted into SV-PSF.
Step 2: Measuring the depth of PSF. By applying multi-frequency heterodyne phase-shifting principles and Perspective-n-Point (PnP) algorithm, the relative pose between the screen and camera is calculated, from which the depth of each PSF is obtained.
Step 3: Modeling and interpolating of PSF. An initial PSF model is established, from which DV-PSF interpolation can be performed using light transport coefficients measured result at one depth along with the focus distance.
This paper gives a 3D image formation mode. Section 2.1 explains how PSF at different depths result in blurred images when camera captures a 3D scene. Section 2.2 is our PSF model and interpolation method. Section 2.3 introduces how to measure SV-PSF by applying SPI. Section 2.4 shows how the depth of each PSF is calculated by multi-frequency heterodyne phase-shifting principles and PnP algorithm.
2.1 3D image formation model using SV-PSF
In traditional image formation model, the degradation of 2D image [10] is shown as follows:
where (x, y) represents the coordinates of the 2D object plane; (u, v) represents the coordinates of image plane; h(x, y, u, v) represents light transport coefficients between object points (x, y) to image pixels (u, v), which also represents SV-PSF; O(x, y) represents the input scene; I(x, y) is the captured image; and Ω is the area where the optical system can observe.In Eq. (1), PSF changes with different positions of object points. However, PSF varies with the depths of object points. If we consider depth, then image formation model can be extended as follows:
where Σ is the 3D surface in the field of view; (x, y, z) represents the points on Σ; O(x, y, z) is the reflectivity or the light intensity of object surface; z is the depth of the object points; and h(x, y, z; u, v) is SV-PSF which contains depth information. According to the first type surface integral, Eq. (2) can be transformed into:To simplify the problem, this paper assumes that the LCD screen is vertical to optical axis of camera. Thus, the depth of each pixel on LCD screen is equal. Then Eq. (3) can be written as follows:
where Ωxy represents the object area at equal depth involved in imaging. Equations (1) and (4) have the same form. The main work in this paper is to measure h(x, y, z; u, v), SV-PSF on different object planes, and measure the different depths z. PSF h(x, y, z1; u, v) at depth z1 and focus distance zf is used to interpolate PSF h(x, y, z2; u, v) at depth z2.2.2 Depth-variant PSF model by interpolation
A fundamental hypothesis is presented in our model, that is, the light from one object point on focus plane will finally be collected by one pixel of image sensor. In other words, the defocus effect and other spread effect at the focus plane can be set aside.
Based on the above hypothesis, a significant conclusion can be made. The light collected by one image sensor pixel, all passes through a certain point on the focus plane.
To express our model better, the actual situation was simplified to a 1-D interpolating situation, as shown in Fig. 2. The response of image sensor pixel to the ray of light at certain direction is definite and expressed as follows:
where y1 and y2 are the ordinates on plane at different depths z1 and z2, and u0 correspond to the image pixel of focus point yf on focus plane zf. The relationship between these coordinates can be expressed as follows:Fig. 2. A schematic diagram of our PSF model. The light collected by one image sensor pixel passes through a certain point on the focus plane. The response of image sensor pixel to the ray of light at certain direction is definite.
Based on the analysis above, PSF on plane z2 is represented using measured light transport coefficients at z1 and the focus distance zf. The PSF at depth z2 is interpolated indirectly. On opposite side of the focus plane, interpolation of PSF on plane z3 follows the same rule. Figure 3 shows that PSF interpolation method is divided into two steps.
Fig. 3. A schematic diagram of our PSF model and interpolation method. The first step is doing light transport coefficients interpolation, the second step is recombine the object PSF.
Step 1: Interpolating the light transport coefficients h(x, y, z2; u0, v0) using h(x, y, z1; u0, v0). Considering Eqs. (5) and (6) in real DV-PSF interpolation, the interpolation process can be expressed as follows:
Figure 4 shows an example when α = 0.6 is given, and the interpolated h(x, y, z2; u0, v0) is a subpixel result, which is supposed to be resampled to obtain result having the same pixel scale as h(x, y, z1; u0, v0).
Fig. 4. Where α = 0.6, the process of interpolation and resampling the light transport coefficients. After interpolation, the size of a 5×5 light transport coefficients array is shrink into a 3×3 area.
Step 2: Adapting step 1 to all image sensor pixels, from which we can obtain h(x, y, z2; u, v). Thus, the PSF h(x0, y0, z2; u, v) of each point (x0, y0, z2) on plane at depth z2 can be obtained.
2.3 SPI-based PSF measurement
In our SPI-based SV-PSF measurement method, a digital camera is used for recording the Fourier coefficients by capturing sinusoidal-structured patterns of different frequency displayed by an LCD screen [11], the LCD screen is vertical to optical axis of camera. By doing inverse Fourier transform (IFT), an image from each single pixel of image sensor can be obtained, which represents the light transport coefficients from screen points to one image sensor pixel. The coefficients from one screen point to different image sensor pixels are then combined. Thereafter, PSF is reconstructed.
The FSI sinusoidal-structured patterns, which are shown by LCD screen, can be written as [20]:
where (x, y, z) represents the points on the LCD screen; z is the distance between the LCD screen and the camera; (fx, fy) represents the spatial frequency; φ is the initial phase; a is the average intensity of the display image; and b is the lightness amplitude. If the resolution of LCD screen is M×N, then the patterns must satisfy fx=i/M and fy=j/N to get the light transport coefficients from all the screen points to one image sensor pixels, where i=0, 1, 2, …, M-1 and j=0, 1, 2, …, N-1. Each combination of (fx, fy) correspond to a pattern image.According to Eq. (4), the response of the digital camera to the display patterns is as follows:
For each combination of spatial frequency (fx, fy), the four-step phase measurement method is used. The screen displays patterns with an initial phase φ (0, π/2, π, 3π/2). For one pixel (u0, v0), the corresponding Fourier coefficients obtained are as follows:
In images having a resolution of M×N using SPI-tech, the number of patterns should be M×N×4/2 if the conjugate symmetry of Fourier domain is considered. This process requires a long period of time.
In fact, most values in the M×N matrix are 0. If the most effective values in matrix h(x, y, z; u0, v0) are concentrated in a m×n small area, then the resolution of pattern needed is m×n, which is lesser than the image resolution M×N. The number of patterns needed is m×n×4/2. The frequency of the patterns in a small area is fx = i/m, fy = j/n, where i=0, 1, 2, …, m-1 and j=0, 1, 2, …, n-1. The M×N patterns shown on LCD is generated by doing periodic extension to small m×n patterns. By applying SPI, we can calculate h(x, y, z; u0, v0) with a resolution of m×n.
2.4 Depth calculation for each PSF
Depth information is necessary when conducting proposed PSF interpolation method. It is the distance between the LCD screen and the optical center of camera. By matching object points with image pixels and solving the PnP problem, the extrinsic camera parameters can be calculated, from which the depth of each PSF can be obtained. The matching principle is multi-frequency heterodyne phase-shifting [22]. An LCD screen is used to display the sinusoidal fringes. The fringes are expressed as follows:
The camera intrinsic parameters can be solved by Zhang’s calibration method with calibration board, and distortion coefficients can be solved at the same time [24]. As object points are matched with image pixels, the following extrinsic parameters can be obtained by applying PnP algorithm: R, t, which devotes the transformation from screen coordinates to camera coordinates.
A point on screen (x, y, 0), there is a transformation from screen coordinates to camera coordinates:
Fig. 5. PSF measurement at different depth and depth calculation. Where zf is the focus distance. The distance between the camera and the screen is changed by moving camera.
3. Experiment
The experimental setup is shown in Fig. 6. An LCD screen is used to show patterns, and a digital camera with the lens to be tested is used to capture images. A workstation is used to control LCD screen, camera, and perform data processing. The camera is fixed on a magnetic base, the screen and the camera along with the magnetic base are both placed on an optical table. The model of the LCD screen is Philips 242P6VPJKEB. Its resolution is 3840×2160, and its pixel size is 0.2715 mm×0.2715 mm. The model of the camera is Basler acA1600-20gm, which is a grayscale camera with a resolution of 1600×1200 and a pixel size of 4.5µm × 4.5µm. The focus length of lens is 8 mm, and F-number is set to 4.0. We move the magnetic base to change the distance between the camera and the screen.
To align the optical axis of the camera to be perpendicular to the LCD screen, a rectangle is displayed on the LCD screen. Then we adjust the pose of the camera until the rectangle is in the center of the captured image and the perspective effect cannot be observed. If the perpendicular between the optical axis of the camera and the LCD screen is not too well, it will lead a low quality of the interpolated PSFs.
Considering the potential impact of the intensity nonlinearity of the LCD screen, gamma correction [25] of the LCD screen is required before displaying patterns. At any depth, the resolution of the screen area corresponding to the image sensor must be higher than the resolution of the image sensor, and the view of camera must be covered by the screen.
3.1 Solving the depth of PSF
Zhang’s calibration method was used to solve the intrinsic parameter of the camera. By displaying and capturing triple-frequency four step phase-shifted sinusoidal fringe, the screen points corresponding to image points are then calculated. The depth of each PSF is also calculated. The total number of fringes is 24.
Following the principle of phase synthesis [26], the width of synthesis phase must cover the screen area, where the lateral resolution of the camera is M = 3840. The values of λ1, λ2, λ3 are 19, 20, and 21 in our experiment, respectively. The requirements are met.
Figure 7 displays part of the fringes shown on screen, captured by the camera, and part of the unwrapped phase image, which is used finally for matching and PnP solving.
Fig. 7. Phase unwrap process of captured image. (a) The fringe (part) shown on screen. (b) The fringe (part) captured by the camera. (c) The unwrapped phase image.
Following the principles in Section 2.4, the screen points can be matched with the image pixels. The extrinsic parameters can be solved, and the distance between the screen and the camera can be measured. R is converted into Euler angles (‘xcyczc’ order) to represent the perpendicularity between the optical axis of the camera and the LCD screen. RMSE (re-projection error) can represent the accuracy of PnP. The results are given in Table 1.
Table 1. Different depths of camera to measure PSF.
3.2 PSF measurement result at any depth
In our experiment, PSF is measured at five different depths. The camera is focused at 360 mm. Fourier sinusoidal-structured patterns are used, with frequencies fx and fy in range [0, 23/24]. The total number of patterns is 1152. 5×7 uniform distribution positions on the image sensor plane are selected for the computation of light transport coefficients and PSF, which start at (200,200). One position is selected every 200 pixels. It takes about 251s to acquire the datasets at a single depth, and it takes about 9s to calculate 35 PSFs, and 16s to read 2.06GB captured patterns from the hard disk.
The normalized light transport coefficients are shown in Fig. 8. Figure 9 shows the normalized SV-PSF at different measured depths, from which a general recognition for the SV-PSF of the camera can be had.
Fig. 8. Applying SPI to different image sensor pixels at five different depths. Resolution of single matrix is 21×21, which represents light transport coefficients from screen points to single image sensor pixel.
The PSF results shown in Fig. 9 mean that PSF does not only vary in the field of view at certain depth but also varies with the change of depth. The shape of PSF is irregular. Traditional Gaussian model cannot fit the PSF well at certain positions.
In a fixed position in the view, if the measurement plane is farther to the focus plane, then the physic size of the PSF is larger. Because, when the camera is defocused, generally, one pixel can receive light from a wide area, which causes a larger size of PSF.
Figures 10 and 11 show the detailed SPI result of single pixel and the PSF result at the center in the image, where (u, v) = (800,600).
Fig. 10. The light transport coefficients from different screen points to image sensor pixel (800, 600) at five different depths.
Figures 10 and 11 also show that the shape of PSF varies when the object plane is at the opposite sides of focus plane. This is because the PSF is not an even function about the distance to the focus plane [27]. This occurrence leads to the difference of PSF at opposite sides of focus plane, even when absolute distance from the focus plane is equal. Light transport coefficients from LCD screen points to camera pixel are similar to PSF result in shape because PSF is nearly invariant in a small area.
The experiment verifies that our method can measure SV-PSF at any depth.
3.3 PSF interpolation experiment
PSF is interpolated using our proposed method in 2.2. We implement our method to obtain the interpolated PSF at depth z2 = 312 mm using measured light transport coefficients at depth z1 = 258 mm and to obtain the interpolated PSF at depth z3 = 409 mm using measured light transport coefficients at depth z4 = 464 mm, and zf = 359 mm. The interpolated results are shown in Figs. 12 and 13.
Fig. 12. Interpolated PSF at depth z2 = 312 mm using measured PSF at depth z1 = 258 mm. (a) Measured PSF at depth = 312 mm. (b) Interpolated PSF at depth = 312 mm using measured light transport coefficients at depth = 258 mm. (c) Error of measured PSF and interpolated PSF.
Fig. 13. Interpolated PSF at depth z3=409 mm using measured PSF at depth z4 = 464 mm. (a) Measured PSF at depth = 409 mm. (b) Interpolated PSF at depth = 312 mm using measured light transport coefficients at depth = 464 mm. (c) Error of measured PSF and interpolated PSF.
Figures 12 and 13 show that the PSF interpolation model can well represent the PSFs not directly measured at some depths. The experiment shows that PSF at a certain depth can be interpolated from the light transport coefficients at another depth and focus distance with our PSF interpolation method. Which could not be achieved by using traditional PSF interpolation methods.
And the PSF interpolation error varies because our interpolation method has some fundamental hypothesis, which is given at Section 2.2. While in a real optical system, such condition cannot be well satisfied for the whole field of view because of different kinds of aberrations. But our interpolation method works well in most situations. When highly irregular PSFs exist, the depth sampling interval should be small to reduce the interpolation error.
To evaluate the PSF model and interpolation method, the captured images, the images deblurred with measured PSF, the images deblurred with interpolated PSF, and the original sharp images are used for calculating the correlation coefficient (CC) [11,28] and the structural similarity index (SSIM) [28]. When the values of them are closer to one, two corresponding images are more similar. We deconvolve the input images using the Lucy-Richardson algorithm.
When the LCD is place at z2 and z3, a sharp image is displayed on the screen and captured by the camera. And the measured PSF and the interpolated PSF are both used to deblur the captured image. Figures 14 and 15 show the four images at depths z2 and z3 respectively.
Fig. 14. Comparison of the original sharp image, captured image, the images deblurred with measured PSF and interpolated PSF at depth z2 = 312 mm. (a) Original sharp image displayed on the LCD screen. (b) Image captured by the camera. (c) Image deblurred using measured PSF. (d) Image deblurred using interpolated PSF.
Fig. 15. Comparison of the original sharp image, captured image, images deblurred with measured PSF and interpolated PSF at depth z3 = 409 mm. (a) Original sharp image displayed on the LCD screen. (b) Image captured by the camera. (c) Image deblurred using measured PSF. (d) Image deblurred using interpolated PSF.
The CC and SSIM are calculated among the sharp image, the captured image, image deblurred using measured PSF and image deblurred using interpolated PSF. The results at different depths z2 and z3 are shown in Tables 2 and 3, and Tables 4 and 5 respectively.
Table 2. CC among images at depth z2 = 312 mm.
Table 3. SSIM among images at depth z2 = 312 mm.
Table 4. CC among images at depth z3 = 409 mm.
Table 5. SSIM among images at depth z3 = 409 mm.
From the comparison among the sharp image and other images at the first column in Tables 2, 3, 4 and 5, we can find that the .image quality can be improved by both the measured and the interpolated PSF to deblur the captured image since the SSIM and CC is larger. CC and SSIM between the sharp image and the image deblurred using measured PSF is larger than CC and SSIM between the sharp image and the Image deblurred using interpolated PSF, which indicates that the performance of the measured PSF is better than the interpolated PSF. And we can also find that image similarity between the image deblurred using measured PSF and interpolated PSF is very high from the CC and SSIM.
4. Conclusion
Detailed steps were followed to measure SV-PSF based on SPI method at any depth with a digital camera and an LCD screen. We measured the depth by applying multi-frequency heterodyne phase-shifting principles and PnP algorithm. Furthermore, we initiated a DV-PSF interpolation model. Experiments showed that our method can measure SV-PSF at any depth. Significant similarities exist between the interpolated PSF and the directly measured PSF. Our work is a successful application of using SPI to solve traditional optical problems.
Funding
National Natural Science Foundation of China (61735003, 61875007); Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R02); Leading Talents Program for Enterpriser and Innovator of Qingdao (18-1-2-22-zhc).
Disclosures
The authors declare no conflicts of interest.
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