Non‐iterative blind deconvolution algorithm based on power‐law distribution

IET Image ProcessingVolume 14, Issue 17 p. 4499-4506 Research ArticleFree Access Non-iterative blind deconvolution algorithm based on power-law distribution Weizhe Gao, Weizhe Gao orcid.org/0000-0001-8312-2916 School of Electrical Engineering, Hebei University of Science & Technology, No. 26 Yanxiang Road, Shijiazhuang, 050018 People's Republic of ChinaSearch for more papers by this authorXuebin Xu, Xuebin Xu School of Computer Science & Technology, Xi'an University of Posts & Telecommunications, Chang'an South Road, Xi'an, 710121 People's Republic of ChinaSearch for more papers by this authorYikang Yang, Corresponding Author Yikang Yang yangyk76@mail.xjtu.edu.cn Systems Engineering Institute, Xi'an Jiaotong University, No. 28, Xianning West Road, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this authorZhiguang Zhang, Zhiguang Zhang Systems Engineering Institute, Xi'an Jiaotong University, No. 28, Xianning West Road, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this author Weizhe Gao, Weizhe Gao orcid.org/0000-0001-8312-2916 School of Electrical Engineering, Hebei University of Science & Technology, No. 26 Yanxiang Road, Shijiazhuang, 050018 People's Republic of ChinaSearch for more papers by this authorXuebin Xu, Xuebin Xu School of Computer Science & Technology, Xi'an University of Posts & Telecommunications, Chang'an South Road, Xi'an, 710121 People's Republic of ChinaSearch for more papers by this authorYikang Yang, Corresponding Author Yikang Yang yangyk76@mail.xjtu.edu.cn Systems Engineering Institute, Xi'an Jiaotong University, No. 28, Xianning West Road, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this authorZhiguang Zhang, Zhiguang Zhang Systems Engineering Institute, Xi'an Jiaotong University, No. 28, Xianning West Road, Xi'an, 710049 People's Republic of ChinaSearch for more papers by this author First published: 25 February 2021 https://doi.org/10.1049/iet-ipr.2020.0647AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The spectral amplitude of most natural images is approximately isotropic and follows the power law. In this study, the authors propose a new non-iterative blind image deconvolution algorithm that builds an isosceles curve model to approximate the spectrum amplitude of the real image. In the authors' proposed algorithm, the optical transfer function (OTF) is obtained by comparing the reconstructed and degraded spectra. Then they employ the integrated multidirectional comprehensive estimation to reduce the OTF estimation error. The restored image is then obtained by applying the estimated OTF and the Wiener filter. Experiments on image deconvolution tasks indicate that the proposed algorithm provides a significant performance gain by obtaining an accurate OTF, reducing ringing artefacts compared with existing algorithms, and realising real-time image restoration. 1 Introduction Digital images are inevitably affected by distortions and noise during acquisition and processing. The degradation process can be formulated as a shift-invariant model, g x , y = f x , y ∗ h x , y + n x , y ,(1) where g(x,y) is the degraded image, f(x,y) is the ideal image, h x , y is the point spread function (PSF) that models the blur, n x , y is the noise, and * denotes the 2D linear convolution operation. In the frequency domain, the blur model in (1) can be represented as G ( μ , ν ) = F ( μ , ν ) H ( μ , ν ) + N ( μ , ν ) ,(2) where G ( μ , ν ), F ( μ , ν ), H ( μ , ν ), and N ( μ , ν ) are the discrete Fourier transforms of g x , y , f x , y , h x , y , and n x , y , respectively. Hereafter H ( μ , ν ) is referred to as optical transfer function (OTF). The blur removal from the degraded image is important and is widely used in different applications, e.g. remote sensing, astronomical observation, and medical imaging [1]. Image deconvolution is an inverse problem that mainly aims to restore the real image, f(x,y), from an observed version, g(x,y). The main challenging issue in image deconvolution is to obtain an accurate estimation of the PSF. To address this issue, blind deconvolution [2] is often used in the literature to estimate the original image and the PSF from the blurred image, g(x,y). Several techniques, such as maximum likelihood method [3], simulated annealing [4], Richardson–Lucy algorithm [5], and total variation [6, 7], are proposed to solve this problem. However, many of these algorithms are iterative or recursive thus they are potentially unstable, divergent and time-consuming. A non-iterative algorithm suitable for dealing with image restoration, APEX (the approximate PSF estimating), is proposed in Carasso [8, 9]. This algorithm directly substitutes the spectrum of the original image with a pre-set constant. This, however, may lead to false results. APEX also uses the SECB method for image reconstruction, which is sensitive to noise. This method also based on trial and error which leads to a time-consuming parameter selection process. Although APEX algorithm has been later significantly considerably improved in [10-12], the above issues remained unsolved. To address these issues, in this paper, we propose a non-iterative blind deconvolution algorithm based on power law (BDPL). Based on this algorithm, we further build an isosceles curve model to approximate the spectrum amplitude of the real image. Moreover, we use the multidirectional comprehensive estimation to obtain the OTF. To evaluate the performance of the proposed method, BDPL is then applied on publicly available data sets. The rest of this paper is organised as follows. In Section 2, the BDPL algorithm is briefly described and summarised in the schematic diagram. Then in Section 3, the measures to describe the quality of the restored image are provided. The experimental results are presented in Section 4 followed by conclusions in Section 5. 2 Blind deconvolution algorithm 2.1 Class G In imaging systems, the PSF is generally viewed as a probability density function (PDF) because it is non-negative and can be integrated into unity. The Class G is a universal class of shift-invariant blurring, consisting of the finite convolution products of radially symmetric Lévy stable PDFs. Lévy process is defined in 2D Fourier transform as [8] H ( μ , ν ) = ∫ R 2 h ( x , y ) exp − 2 π i ( μ x + ν y ) d x d y = exp − α ( μ 2 + ν 2 ) β α > 0 , 0 < β ≤ 1 . (3) Lévy process is used to model various imaging systems. In (3), β is determined by the type of blurs, and α is determined by the atmospheric conditions, e.g. β = 5 / 6 describes long-exposure turbulence blurring, and β = 1 corresponds to Gaussian density which is widely used in many imaging systems, and β = 1 / 2 corresponds to Cauchy (or Lorentzian) distribution which can best-fit the empirically obtained PSFs in astronomical 'adaptive optics'. Nevertheless, defocus and motion blurs cannot be expressed in (3). 2.2 Natural image model A relevant characteristic of natural images concerns spectra's gross behaviour. Let A denote the spectrum of the natural images which tends to follow a power-law decay [13, 14] E F μ , ν 2 = E P F = A s / ω γ s .(4) A directional parameter model is introduced due to the difference in various directions of the actual power spectrum. Therefore E F μ , ν 2 = E P F , θ = A s θ / ω γ s θ ,(5) where θ = arctan v / u , ω = μ 2 + ν 2 , A s θ is a scaling factor for each direction revealing the amplitude, and γ s θ is the frequency exponent for each direction. For the spectrum amplitude, (5) is rewritten as E F μ , ν = A θ / ω γ θ .(6) In a seminal study presented in [15], it is shown that the value of γ θ is between 0.8 and 1.5. Fig. 1 indicates a remote sensing image and its spectrum. Fig. 1Open in figure viewerPowerPoint Amplitude spectrum (a) Remote sensing image, (b) Spectral amplitude in the frequency domain and, (c) Spectral amplitude in the frequency domain of horizontal and vertical sections Taking the logarithm of both sides of (6), yields ln F μ , ν = ln ⁡ A θ − γ θ ln μ 2 + ν 2 .(7) For a Class G PSF, (2) is then reformulated as G ( μ , ν ) = F ( μ , ν ) H ( μ , ν ) + N ( μ , ν ) = exp − α ( μ 2 + ν 2 ) β A ( θ ) ( μ 2 + ν 2 ) ( 1 / 2 ) γ 1 ( θ ) + N ( μ , ν ) . (8) By taking the logarithm of both sides and ignoring the noise, we then obtain ln G ( μ , ν ) ≃ ln H μ , ν + ln F μ , ν = − α μ 2 + ν 2 β − 1 2 γ θ ln ⁡ ( μ 2 + ν 2 ) + ln ⁡ A ( θ ) . (9) Since f x , y is real, ln G ( μ , ν ) is conjugate symmetric. Parameters α and β can be then estimated along any ray through the origin in the discrete μ , ν plane. For μ = 0, (7) is reduced to ln G ( 0 , ν ) ≃ ln H 0 , ν + ln F 0 , ν = − α ν 2 β − γ ln ν + ln ⁡ A . (10) Equation (10) is compared with the APEX algorithm ln G ∗ 0 , ν = − α ν 2 β − A(11) In (11), constant A is unable to reflect the decay of the image spectrum. Additionally, the BDPL algorithm does not have a normalised part and determining the appropriate value for A in (9) needs a process of trial-and-error. 2.3 OTF estimation As shown in Fig. 2, the medium and high-frequency parts of the signal are significantly reduced, whereas the low-frequency part is retained. Therefore, in low-frequency regions, ln F ( 0 , ν ) can be set to ln G ( 0 , ν ) . As described in the previous subsection, ln F ( 0 , ν ) can be represented by − γ ln ν + ln ⁡ A and obtained as the following equation: ln F ( 0 , ν ) = ln G ( 0 , ν ) ν < n g 1 + γ × ln n − γ × ln ν n ≤ ν ≤ N − 1 g 2 + γ × ln n − γ × ln ν − N ≤ ν ≤ − n (12) where γ represents the slope of the real spectrum, 0.8 ≤ γ ≤ 1.2, and 6 ≤ n ≤ 10. Since ln G ( 0 , ν ) oscillates, g 1 and g 2 are set as g 1 = smooth ln G 0 , n ,(13) g 2 = smooth ln G 0 , − n .(14) Fig. 3 shows the reconstructed spectrum of the ideal image. The advantage of the prior model in (10) is that it can reflect the spectrum of the natural images. Equation (10) can be amended as − α ν 2 β = ln G 0 , ν − ln F 0 , ν .(15) Fig. 4 shows the result of − α ν 2 β . For a fixed value of β the actual curve − α ν 2 β is bell-shaped and monotonically increases (decreases) for − N ≤ ν ≤ 0 0 ≤ ν ≤ N . Hence, the suitable data where ν ∈ [ − s c a l e , s c a l e ] s c a l e < N is selected to ensure an accurate PSF estimation. Fig. 2Open in figure viewerPowerPoint Spectrum comparison Fig. 3Open in figure viewerPowerPoint Spectrum reconstruction Fig. 4Open in figure viewerPowerPoint Result of − α ν 2 β calculated by (15) The amplitude of the spectrum of the degraded image Ω = { ( μ , ν ) | μ 2 + ν 2 ≤ ω 2 } does not completely satisfy the conjugate symmetry conditions due to random factors (e.g. noise and rounding error). Therefore, it can only approximately satisfy the symmetrical condition. Varying PSF values are then obtained for different orientations. We select four rays including μ = 0, ν = 0, μ = ν and μ = − ν to estimate ( α , β ). In our previous work [16], α takes the arithmetic mean, and β takes the geometric mean. Here we use the data of the four rays together to estimate the OTF to improve accuracy and stability of our estimation. Fig. 5 shows the fitted result of − α ν 2 β . Fig. 5Open in figure viewerPowerPoint Result curve of − α ν 2 β 2.4 Wiener filter Blind deconvolution is a typical inverse problem. In the image reconstruction part, noise is often amplified considerably and even overwhelms the original signal. Wiener filtering can suppress the noise to achieve a linear estimation by minimising the mean squared error. Equation (16) is the transfer function of the Wiener filter [17] F ( μ , ν ) = 1 H ( μ , ν ) H ( μ , ν ) 2 H ( μ , ν ) 2 + S G ( μ , ν )(16) where S is the reciprocal of the signal-to-noise ratio. The block diagram of the proposed algorithm is shown in Fig. 6. The proposed algorithm is summarised in Fig. 7. Fig. 6Open in figure viewerPowerPoint Block diagram of the BDPL algorithm Fig. 7Open in figure viewerPowerPoint Algorithm: BDPL 3 Evaluation criteria To conduct quantitative comparisons that measure the quality of the restored images and evaluate the overall performance. We use different assessment methods in this paper. Let f i , j be the ideal image, and g i , j be the restored image with sizes of M × N. 3.1 Normalised mean square error (NMSE) NMSE describes the normalised mean square error between the restored and original images NMSE = ∑ i = 1 M ∑ j = 1 N f i , j − g i , j 2 M × N . (17) 3.2 Peak signal-to-noise ratio (PSNR) PSNR measures the restoration quality of the degraded image PSNR = 10 lg L 2 × M × N ∑ i = 1 M ∑ i = 1 N f ( i , j ) − g ( i , j ) 2 ,(18) where L denotes the maximum possible pixel value of the image. 3.3 Laplacian sum (LS) The LS uses extended Laplacian operator for each pixel in a 3 × 3 neighbourhood to obtain the absolute value of the differential value in eight neighbourhoods. Then it finds the summation of these values throughout the image range LS = 1 ( M − 2 ) ( N − 2 ) ∑ i = 2 M − 1 ∑ j = 2 N − 1 8 g ( i , j ) − g ( i , j − 1 ) − g ( i − 1 , j ) − g ( i + 1 , j ) − g ( i , j + 1 ) − g ( i − 1 , j − 1 ) − g ( i − 1 , j + 1 ) − g ( i + 1 , j − 1 ) − g ( i + 1 , j + 1 ) . (19) In general, the clearer the image and the sharper the outline, the greater the change in the grey values near each pixel, and the greater the LS value. 3.4 Ringing metric (RM) RM [18] is employed to measure the ringing artefacts in the restored images. RM is calculated as follows: (i) Create the reference edge profile, E ref , by adopting the Canny edge detector in the degraded image. (ii) Use dilate operator r × r to the profile E ref , where r is an approximate width which can cover the ringing artefacts and then obtain a binary edge mask, E M ref . (iii) Compute the edge profile of the restored image, E n , by using the Canny edge detector. (iv) Use the logic operator AND between the mask, E M ref , and the profile, E n , to obtain E m . RM is calculated as follows: R M = ∑ i ∑ j E m − ∑ i ∑ j E ref / ∑ i ∑ j E ref (20) 3.5 Sharpness gain (SG) We use grey mean grads (GMG) to measure the sharpness of the images as the following equation: GMG f = ∑ i = 1 M − 1 ∑ j = 1 N − 1 f i + 1 , j − f i , j 2 + f i , j + 1 − f i , j 2 2 M − 1 N − 1 (21) Then we obtain the SG as SG = GMG f − GMG g / GMG g × 100 % .(22) 4 Experiments This section describes our experiments using Matlab on a desktop PC (Intel Core(TM) i5-3337 at 1.8 GHz × 2 CPU; 4 GB memory). For comparison, we consider Luo's algorithm (mid-frequency based algorithm) [11], Goldstein's algorithm [19], and Perrone's algorithm (TVBD) [7]. Luo's algorithm is non-iterative, and the other two are iterative. 4.1 Remote sensing The remote sensing image [20] is an 8-bit grey-level image with 480 × 480. Figs. 8a and b are obtained from NASA. Fig. 8Open in figure viewerPowerPoint Remote sensing image restoration (a) Original image, (b) Degraded image with fierce atmospheric turbulence, (c) Luo, (d) Goldstein, (e) Perrone and, (f) BDPL algorithms Fig. 8 shows that the BDPL is superior compared to other algorithms in several aspects especially with its significant reduction of ringing artefacts compared with Figs. 8c and d. As shown in Table 1, the pair α , β estimated by BDPL is (0.002631, 5/6), which is relatively close to the true value of (0.0025, 5/6). However, Luo's estimation is (0.003755, 5/6), which is considerably different from the true value. Therefore, the BDPL algorithm can obtain an accurate OTF and the image restoration results are clearer and more similar to the original image. Fig. 8f has higher PSNR and SSIM values and a lower NMSE value compared with Figs. 8c–e. Its NMSE value is decreased by 54.55, 40.30, and 50.31% in Figs. 8c–e, respectively. Nevertheless, the LS of Fig. 8f is lower than that of Fig. 8c. Ringing artefacts produce edges and LS regards ringing artefacts as image details. Thus, an LS may be unable to objectively reflect image restoration quality. Table 1. Image restoration results in Fig. 8 α , β Estimated α , β NMSE PSNR LS RM SSIM Process time blurred (0.0025, 5/6) — 0.0392 21.54 4.008 — 0.4733 — Luo — (0.003755, 5/6) 0.0352 22.01 25.95 0.2736 0.6612 1.024 Goldstein — — 0.0268 23.20 14.01 0.2060 0.5965 1463 Perrone — — 0.0322 22.68 11.13 0.1444 0.5683 1332 BDPL — (0.002631, 5/6) 0.0160 25.43 18.52 0.2003 0.7475 0.739 Goldstein's and Perrone's algorithms use 2D convolution to obtain image deconvolution. The Luo's algorithm and BDPL adopt Wiener filtering to restore the image. In MATLAB, 2D convolution and Wiener filtering are realised through FFT/IFFT. The time complexity of 2D-FFT/IFFT is O n 2 lg ⁡ n . Goldstein's algorithm calls nine FFT/IFFT operations per iteration, and the processing time in one iteration is 1.829 s. Further, Perrone's algorithm calls 12 FFT/IFFT operations per iteration with the processing time 2.446 s. Luo's algorithm and BDPL are non-iterative algorithm and call five FFT/IFFT operations. The processing time of BDPL is 0.739 s, which is relatively short and can be used in real-time restoration tasks. 4.2 Jeep The image is a 950 × 620 8-bit grey level. Fig. 9b is synthetically blurred with Class G parameter (0.04, 0.5) which fits Lorentzian distribution. Fig. 9Open in figure viewerPowerPoint Experiment on image restoration (a) Ground truth, (b) Synthetically blurred image with α = 0.04, β = 0.5, (c) Luo, (d) Goldstein, (e) Perrone and, (f) BDPL algorithms Only the detail texture of tires and tree leaves is shown in Fig. 9f. The ringing artefacts are evident in Figs. 9c and d. As shown in Table 2, Fig. 9f has a higher PSNR and SSIM and a lower NMSE and RM value compared with Figs. 9c–e. Its PSNR value is increased by by10.55, 15.38, and 22.47% in Figs. 9c–e, respectively. The SSIM value in Fig. 9f is increased by 21.48, 21.69, and 61.51% in Figs. 9c–e, respectively. The processing time of BDPL is 1.217 s, which is much shorter than other algorithms. Table 2. Image restoration results of Fig. 9 α , β Estimated α , β NMSE PSNR LS RM SSIM Process time blurred (0.04, 0.5) — 0.0666 17.19 2.431 — 0.3017 — Luo — (0.06163, 0.5) 0.0506 18.39 12.53 0.2176 0.3962 1.502 Goldstein — — 0.0813 17.62 8.107 0.1874 0.3955 1732 Perrone — — 0.0923 16.60 5.423 0.2608 0.2980 2936 BDPL — (0.04251, 0.4953) 0.0318 20.33 11.64 0.1636 0.4813 1.217 4.3 Real image The proposed algorithm works efficiently on real pictures. Fig. 10a is an 8-bit grey-level image with the size of 512 × 512. The BDPL-processed image shown in Fig. 10b reveals the buildings in the distance, Holstein cows grazing in the meadow, and many other fine-scale details that are not readily apparent in Fig. 10a. Processing time is also relatively short and is 0.863 s. Fig. 10c is an 8-bit scanning electron microscopy (SEM) image of a catalyst with the size of 768 × 512. Fig. 10d illustrates the restored image. Using the BDPL algorithm, the visual quality is significantly enhanced and the details of the fabric become more discernible and useful. This experiment demonstrates that the BDPL algorithm can achieve real-time BID. Fig. 10Open in figure viewerPowerPoint Experiment on real pictures (a) Goldhill, (b) Sharpened image of (a), (c) SEM image of a catalyst with size 768 × 512 and, (d) sharpened image of (c) 4.4 Colour image Blind deblurring of the coloured images is still an immature problem (Table 3). It is mainly because the distinct OTF associated with each colour channel is difficult to estimate. Moreover, unbalanced sharpening of the individual channels may cause some issues. With the use of uncertain iteration in blind image deconvolution (BID), the restored coloured image may also display physically false colours. Generally, the coloured image can be described as a 2D manifold in 5D space, namely, x , y , R x , y , G x , y , B x , y , where R, G and B are the red, green, and blue channels of the colour image g x , y , respectively. We estimate the OTF of each colour and then restore the image. As shown in Figs. 8b and d, the image quality has been greatly improved, and the details are highly discernible. This experiment uses real pictures from NASA, which are colour images taken by Hubble Telescope on the Galaxy Tadpole tail and M51 (the spiral galaxy). Table 3. Statistical data of Fig. 10 Original image Size Restored image α β Processing time, s SG, % a 512 × 512 b 0.0813 0.3136 0.863 99.07 c 768 × 512 d 0.0127 0.4326 1.016 91.41 Fig. 11 indicates the efficacy of BDPL restoration on the colour images. Nevertheless, some details in the image are discernible. GMG, LS and total variation (TV) are used as the evaluation indices. Table 4 shows that the GMG value of M51 increases from 3.667 to 10.43 with an increasing rate of 216.8%, the LS value also increases from 16.49 to 44.08 with an increasing rate of 167.3%, and the TV value increases from 83,259 to 236,850. Hence, the quality of the colour image is significantly improved by BDPL. Fig. 11Open in figure viewerPowerPoint Colour image sharpening (a) Tadpole's tail image, (b) Sharpened image of (a), (c) M51 image and (d) Sharpened image of (c) Table 4. Image restoration results of Fig. 11 GMG GMG improvement LS LS improvement TV TV improvement (a) 3.041 — 15.23 — 11,052 — (b) 6.650 118.7% 36.07 136.8% 24,164 118.6% (c) 3.667 — 16.49 — 83,259 — (d) 10.43 216.8% 44.08 167.3% 236,850 184.5% 5 Conclusion To solve the BID problem, this paper proposes an ultrafast non-iterative blind deconvolution algorithm, BDPL, which is easy to implement. Experimental results prove the effectiveness of the proposed algorithm and confirm its following advantages. Firstly, the BDPL [17] algorithm utilises the power-law isosceles model to accurately estimate the PSFs and ensure successful image restoration. Secondly, the proposed algorithm employs integrated multidirectional comprehensive estimation which can reduce the influence of random factors and thereby improve its stability. Thirdly, this algorithm reconstructs the image by applying the Wiener filter which can effectively suppress noise. Finally, BDPL is ultrafast. In our experiments, we noticed that almost all the process time was spent on estimating OTF, Wiener filtering and FFTs, which are avoidable. So it has a greater advantage over other algorithms in speed. However, the BDPL algorithm employs the PSFs of Class G which cannot describe motion and defocus blurring. 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