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Outline

Title
Abstract
Figures
Introduction
Proximal Gradient Methods
Convergence Analysis
Convergence Results from Convex Optimization
Simulation Results
Solvability Analysis
Noise Analysis
Experimental Results
Experimental Data
Image Processing and Results
Conclusion
References
All Topics
Mathematics
Applied Mathematics

Nonuniform Defocus Removal for Image Classification

Profile image of Nguyên Hoàng thảoNguyên Hoàng thảo

2021, ArXiv

Abstract

We propose and study the single-frame anisoplanatic deconvolution problem associated with image classification using machine learning algorithms, named the nonuniform defocus removal (NDR) problem. Mathematical analysis of the NDR problem is done and the so-called defocus removal (DR) algorithm for solving it is proposed. Global convergence of the DR algorithm is established without imposing any unverifiable assumption. Numerical results on simulation data show significant features of DR including solvability, noise robustness, convergence, model insensitivity and computational efficiency. Physical relevance of the NDR problem and practicability of the DR algorithm are tested on experimental data. Back to the application that originally motivated the investigation of the NDR problem, we show that the DR algorithm can improve the accuracy of classifying distorted images using convolutional neural networks. The key difference of this paper compared to most existing works on single-fra...

Figures (11)
Fig. 1 Synthetic data depicts (a) an object depth profile for which the data image can b« approximately separable for individual defocus zones of the object versus (b) an inseparabli case where the constant defocus zones are too small. Solution approaches partitioning the data into isoplanatic zones are applicable to (a) but not (b) for which global algorithm: handling the entire data are required.
Fig. 1 Synthetic data depicts (a) an object depth profile for which the data image can b« approximately separable for individual defocus zones of the object versus (b) an inseparabli case where the constant defocus zones are too small. Solution approaches partitioning the data into isoplanatic zones are applicable to (a) but not (b) for which global algorithm: handling the entire data are required.
Fig. 2 Solvability analysis of the DR algorithm with respect to the degradation level of the data image measured by the blur coefficient: (a) the ground true object; (b) the restoration errors (the blue curve) are compared to errors of the input images (the dark curve) for blu coefficients ranging from 0.1 to 0.45. The restoration error is directly proportional to the blur coefficient as depicted by the upward blue curve. (c) The input image with error 18.5% versus (d) the corrected image with visible details (error 3.2%). expected the restoration error is directly proportional to the blur coefficient as depicted by the upward blue curve. For example, the data image with error 18.5% and its restoration with error 3.2% are respectively shown in figure (c) & (d) for visualization. We observe that most details of the restored image shown in (d) are visually recognizable.
Fig. 2 Solvability analysis of the DR algorithm with respect to the degradation level of the data image measured by the blur coefficient: (a) the ground true object; (b) the restoration errors (the blue curve) are compared to errors of the input images (the dark curve) for blu coefficients ranging from 0.1 to 0.45. The restoration error is directly proportional to the blur coefficient as depicted by the upward blue curve. (c) The input image with error 18.5% versus (d) the corrected image with visible details (error 3.2%). expected the restoration error is directly proportional to the blur coefficient as depicted by the upward blue curve. For example, the data image with error 18.5% and its restoration with error 3.2% are respectively shown in figure (c) & (d) for visualization. We observe that most details of the restored image shown in (d) are visually recognizable.
Fig. 3 Noise analysis of the DR algorithm with a comparison to the projected gradient (PG) method. (a) The correction errors for different levels of Gaussian noise with SNR ranging from 30 to 60dB are plotted. The error of DR (the blue curve) gradually reduces from 6.4% to 0.9% which is lower than that of PG (the red curve). (b) A section of the data (the left column) and the corresponding ones corrected by PG (the middle column) and DR (the last columns) for six particular noise levels are shown for visual comparison. For SNR from 36dB the images corrected by both methods are visually recognizable and the one by DR is clearer. conducted. The restoration is terminated either after 150 iterations or when- ever the estimate quality in terms of relative RMS drops by a threshold (107+ is taken). In this section we also compare the DR algorithm with the projected gradient (PG) method. The results are summarized in Fig. 3(a) where the error of the corrected image is plotted versus the SNR. For SNR increasing from 30 to 60dB, the restoration error of the DR algorithm (the blue curve) gradually reduces from 6.4% to 0.9%. It is worth mentioning that the higher error of PG the red curve) compared to DR does not mean that the former is less robust than the latter. This is rather due to the fact that PG would require much more iterations to reach the corresponding accuracy level of DR, see Sect. 5.3 for more details about their convergence speed. Fig. 3(b) shows a section of the data? and the corresponding ones corrected by PG (the middle column) and DR (the last column) for six SNRs. For SNR from 36dB the restorations by both methods are visually recognizable and the one by DR is obviously more visible than the one by PG.
Fig. 3 Noise analysis of the DR algorithm with a comparison to the projected gradient (PG) method. (a) The correction errors for different levels of Gaussian noise with SNR ranging from 30 to 60dB are plotted. The error of DR (the blue curve) gradually reduces from 6.4% to 0.9% which is lower than that of PG (the red curve). (b) A section of the data (the left column) and the corresponding ones corrected by PG (the middle column) and DR (the last columns) for six particular noise levels are shown for visual comparison. For SNR from 36dB the images corrected by both methods are visually recognizable and the one by DR is clearer. conducted. The restoration is terminated either after 150 iterations or when- ever the estimate quality in terms of relative RMS drops by a threshold (107+ is taken). In this section we also compare the DR algorithm with the projected gradient (PG) method. The results are summarized in Fig. 3(a) where the error of the corrected image is plotted versus the SNR. For SNR increasing from 30 to 60dB, the restoration error of the DR algorithm (the blue curve) gradually reduces from 6.4% to 0.9%. It is worth mentioning that the higher error of PG the red curve) compared to DR does not mean that the former is less robust than the latter. This is rather due to the fact that PG would require much more iterations to reach the corresponding accuracy level of DR, see Sect. 5.3 for more details about their convergence speed. Fig. 3(b) shows a section of the data? and the corresponding ones corrected by PG (the middle column) and DR (the last column) for six SNRs. For SNR from 36dB the restorations by both methods are visually recognizable and the one by DR is obviously more visible than the one by PG.
Fig. 4 The change (a) and the error (b) of the temporal estimate by the DR algorithm (the blue curves) are plotted in comparison with the PG method (the red curves). Figure (a) shows that DR converges faster than PG and the convergence of both methods is linear as indicated by the slopes of the tailing parts of the curves. Steady decrease of the restoratior error in figure (b) shows that convergence of the algorithms is towards stable and meaningfu solutions. DR is faster and more accurate than PG, for example, to obtain an estimate with error < 1%, 2500 iterations of PG is needed while that of DR is about 100. summarized in Fig. 4 in which the y-axis quantity is shown in the logarithmic scale for clarity. The iterative change of the temporal estimate is shown in Fig. 4(a) where the slopes of the tailing parts of the curves show linear convergence of the two algorithms. The DR algorithm (the convergence of both algorithms is towards stab DR is also more accurate than PG, for example, < 1%, 2500 iterations of PG is needed while th blue curves) converges much faster than the PG method (the red curves) as expected. The restoration error in iteration is plotted in Fig. 4(b) where its steady decrease shows that the e and meaningful solutions. to achieve a restoration error at of DR is only about 100. The restoration error of 2500 DR iterations is a the one of PG. For the sake of brevity, we chose bout 130 times smaller than to skip the numerical results in the noiseless setting which consistently show linear convergence of the DR algorithm to the correct solution.
Fig. 4 The change (a) and the error (b) of the temporal estimate by the DR algorithm (the blue curves) are plotted in comparison with the PG method (the red curves). Figure (a) shows that DR converges faster than PG and the convergence of both methods is linear as indicated by the slopes of the tailing parts of the curves. Steady decrease of the restoratior error in figure (b) shows that convergence of the algorithms is towards stable and meaningfu solutions. DR is faster and more accurate than PG, for example, to obtain an estimate with error < 1%, 2500 iterations of PG is needed while that of DR is about 100. summarized in Fig. 4 in which the y-axis quantity is shown in the logarithmic scale for clarity. The iterative change of the temporal estimate is shown in Fig. 4(a) where the slopes of the tailing parts of the curves show linear convergence of the two algorithms. The DR algorithm (the convergence of both algorithms is towards stab DR is also more accurate than PG, for example, < 1%, 2500 iterations of PG is needed while th blue curves) converges much faster than the PG method (the red curves) as expected. The restoration error in iteration is plotted in Fig. 4(b) where its steady decrease shows that the e and meaningful solutions. to achieve a restoration error at of DR is only about 100. The restoration error of 2500 DR iterations is a the one of PG. For the sake of brevity, we chose bout 130 times smaller than to skip the numerical results in the noiseless setting which consistently show linear convergence of the DR algorithm to the correct solution.
Fig. 5 (a) Sensitivity analysis of the DR algorithm with respect to the focal position: the restoration error is plotted versus the focal offset ranging from —2 to 2 (DoF). The error is approximately linear to the offset level in both directions. (b) Sensitivity analysis of the DR algorithm with respect to the blur coefficient: the restoration error is plotted versus the blur coefficient offset ranging from —9.6% to 9.6%. The restoration quality deteriorates proportionally relative to the offset level in both directions. to +2 (DoF). The numerical results are summarized in Fig. 5(a) where the restoration error is plotted versus the offset level. The error is approximately linear to the offset level in both directions as depicted by the two almost straight branches of the curve. The smallest error 0.6% corresponds to the case without offset while the largest residual about 4.45% occurs to the one with the most offset.
Fig. 5 (a) Sensitivity analysis of the DR algorithm with respect to the focal position: the restoration error is plotted versus the focal offset ranging from —2 to 2 (DoF). The error is approximately linear to the offset level in both directions. (b) Sensitivity analysis of the DR algorithm with respect to the blur coefficient: the restoration error is plotted versus the blur coefficient offset ranging from —9.6% to 9.6%. The restoration quality deteriorates proportionally relative to the offset level in both directions. to +2 (DoF). The numerical results are summarized in Fig. 5(a) where the restoration error is plotted versus the offset level. The error is approximately linear to the offset level in both directions as depicted by the two almost straight branches of the curve. The smallest error 0.6% corresponds to the case without offset while the largest residual about 4.45% occurs to the one with the most offset.
Fig. 6 The experimental data consists of the image of the ROI in the central part marked by the rectangle and the guide-stars marked by the red circles which are the images of two parallel straight sharp edges. We first describe the imaging setup of our experiment which has been done as simple and independent of the camera specifications as possible. A text object on a piece of paper is imaged by a camera with optical axis not perpendicular to the paper plane. To make it convenient for partitioning the pictorial object into isoplanatic zones, before taking the image we manually rotated the camera around its axis so that each column of the object (approximately) corresponds to asingle defocus value. This adjustment is equivalent to rotating the obtained image around its centre at the same angle but with opposite direction. In the current experiment the rotation angle can be numerically estimated as explained in Sect. 6.2 below, however, adjusting the camera in advance helps avoid possible inexactness of numerically rotating the image posteriorly.
Fig. 6 The experimental data consists of the image of the ROI in the central part marked by the rectangle and the guide-stars marked by the red circles which are the images of two parallel straight sharp edges. We first describe the imaging setup of our experiment which has been done as simple and independent of the camera specifications as possible. A text object on a piece of paper is imaged by a camera with optical axis not perpendicular to the paper plane. To make it convenient for partitioning the pictorial object into isoplanatic zones, before taking the image we manually rotated the camera around its axis so that each column of the object (approximately) corresponds to asingle defocus value. This adjustment is equivalent to rotating the obtained image around its centre at the same angle but with opposite direction. In the current experiment the rotation angle can be numerically estimated as explained in Sect. 6.2 below, however, adjusting the camera in advance helps avoid possible inexactness of numerically rotating the image posteriorly.
Fig. 7 (a) The in-focal point along the lower guide-star is estimated from its sharpness. The cyan curve shows the highest sharpness of each column of the guide-star. The guide-star as the image of a straight homogeneous sharp edge would exhibit its highest sharpness at the in-focal point (around column 40) marked by the blue rectangle. The blur coefficient of the imaging setup is estimated from the sharpness variation of the guide-star measured by the average slope of the sharpness curve (in cyan) via linear approximation (the red line). (b) The plot summarizes the ad-hoc grid search over 17 different values of blur coefficient versus the sharpness variation levels: the larger the blur coefficient, the greater the variation. The sharpness variation of the guide-star marked in red provides an estimate of the blur coefficient about 0.0296. ferent values of blur coefficient ranging from 0.01 to 0.05 uniformly separated by 0.0025. The DoF size s = 5 (pixels) is empirically chosen. The sharpness variations of the obtained 17 images are then computed in the same way as for the one of the guide-star. The plot in Fig. 7(b) confirms that the larger the blur coefficient, the greater the variation. The sharpness variation of the guide-star, marked in red, provides an ad-hoc estimate of the blur coefficient about 0.0296. It is worth mentioning that for this experiment the DR algo- rithm is highly robust with respect to both the focal position and the blur coefficient.
Fig. 7 (a) The in-focal point along the lower guide-star is estimated from its sharpness. The cyan curve shows the highest sharpness of each column of the guide-star. The guide-star as the image of a straight homogeneous sharp edge would exhibit its highest sharpness at the in-focal point (around column 40) marked by the blue rectangle. The blur coefficient of the imaging setup is estimated from the sharpness variation of the guide-star measured by the average slope of the sharpness curve (in cyan) via linear approximation (the red line). (b) The plot summarizes the ad-hoc grid search over 17 different values of blur coefficient versus the sharpness variation levels: the larger the blur coefficient, the greater the variation. The sharpness variation of the guide-star marked in red provides an estimate of the blur coefficient about 0.0296. ferent values of blur coefficient ranging from 0.01 to 0.05 uniformly separated by 0.0025. The DoF size s = 5 (pixels) is empirically chosen. The sharpness variations of the obtained 17 images are then computed in the same way as for the one of the guide-star. The plot in Fig. 7(b) confirms that the larger the blur coefficient, the greater the variation. The sharpness variation of the guide-star, marked in red, provides an ad-hoc estimate of the blur coefficient about 0.0296. It is worth mentioning that for this experiment the DR algo- rithm is highly robust with respect to both the focal position and the blur coefficient.
Fig. 8 (a) To intuitively illustrate the blur coefficient in the experiment data, the PSF for the rightmost zone of the ROI generated using the physical parameters given by Sect. 6.2 is shown. (b) The histogram of the image of the ROI reveals the information about the background intensities of the ROI (most of its pixels are almost dark). This knowledge is helpful for preprocessing the experimental data, e.g., thresholding the data at 0.11 results in better image restoration (without artifacts) as shown in Fig. 9(c) compared to Fig. 9(b). in Fig. 6 which is 90 x 580. The depth of field s = 5 is taken. That is, the ROI consists of 116 equal vertical zones each consisting of 5 adjacent columns. Following the analysis in Sect. 6.2, we set the required physical parameters as follows: the 8th zone is in focal and the blur coefficient is 0.0296. It is more intuitive to conceive the blur coefficient via the corresponding generated PSFs, for example, the one with most defocus for the rightmost zone of the ROI is shown in Fig. 8(a). The ROI is restored by 10 iterations of the DR algorithm with stepsize 1 and the initial acceleration parameter 1. The parameters used in this experiment are also summarized in Table 3.
Fig. 8 (a) To intuitively illustrate the blur coefficient in the experiment data, the PSF for the rightmost zone of the ROI generated using the physical parameters given by Sect. 6.2 is shown. (b) The histogram of the image of the ROI reveals the information about the background intensities of the ROI (most of its pixels are almost dark). This knowledge is helpful for preprocessing the experimental data, e.g., thresholding the data at 0.11 results in better image restoration (without artifacts) as shown in Fig. 9(c) compared to Fig. 9(b). in Fig. 6 which is 90 x 580. The depth of field s = 5 is taken. That is, the ROI consists of 116 equal vertical zones each consisting of 5 adjacent columns. Following the analysis in Sect. 6.2, we set the required physical parameters as follows: the 8th zone is in focal and the blur coefficient is 0.0296. It is more intuitive to conceive the blur coefficient via the corresponding generated PSFs, for example, the one with most defocus for the rightmost zone of the ROI is shown in Fig. 8(a). The ROI is restored by 10 iterations of the DR algorithm with stepsize 1 and the initial acceleration parameter 1. The parameters used in this experiment are also summarized in Table 3.
Fig. 9 (a) The image of the ROI from the experimental data. (b) The restored ROI with unprocessed data suffers artifacts which are slightly visible in its rightmost area. (c) The restored ROI with the data threshold at 0.11 no longer exhibits artifacts compared to (b). The improvement of the correction (c) compared to the data (a) proves both the physical relevance of the NDR problem (4) and the practicability of the DR algorithm.
Fig. 9 (a) The image of the ROI from the experimental data. (b) The restored ROI with unprocessed data suffers artifacts which are slightly visible in its rightmost area. (c) The restored ROI with the data threshold at 0.11 no longer exhibits artifacts compared to (b). The improvement of the correction (c) compared to the data (a) proves both the physical relevance of the NDR problem (4) and the practicability of the DR algorithm.
As the last estimate is independent of individual PSF's and the pixels of r play symmetric roles with respect to n and coordinates (£,7), we get Let rn := Un © (pn * 7) for each n = 1,2,...,N. Then its entries satisfy
As the last estimate is independent of individual PSF's and the pixels of r play symmetric roles with respect to n and coordinates (£,7), we get Let rn := Un © (pn * 7) for each n = 1,2,...,N. Then its entries satisfy

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