chen, K

Harding, S

Williams, BM

zheng, Y

University of Liverpool Repository

WILLIAMS, CHEN, ZHENG, HARDING: ADVANCED BLUR REMOVAL METHODS 1Advanced Blur Removal Methods withApplications to Retinal Imaging forOphthalmologyBryan Williams1bryan@liv.ac.ukKe Chen1www.liv.ac.uk/cmitYalin Zheng2cmit@liv.ac.ukSimon Harding2cmit@liv.ac.uk1 Department of Mathematical SciencesThe University of LiverpoolLiverpool, UK2 Department of Eye and Vision ScienceThe University of LiverpoolLiverpool, UKAbstractBlurring of images occurs in many fields, causing significant problems in retinalimaging. In any diabetic retinopathy screening programme, up to 10% of the images areungradable due to inadequate clarity or poor field definition. It is important to obtain asmuch information as possible of retinal vessels and other structures.Deblurring is a major technique that may be developed to restore the lost true image.We present non-blind and blind approaches to tackle this problem, including an approachfor multi-channel images. We propose a new solution algorithm for removing unknownblur (blind case) and show results for retinal images.1 IntroductionImage processing techniques, such as image reconstruction which includes removing im-age noise from a given image (denoising) [12], reconstructing an image from an givenblurred image (deblurring) [6], reconstructing the missing or damaged portion of an im-age (inpainting) [3], emphasizing the boundaries of an image by different filters or seg-menting a image into subregions (segmentation) [5], have been widely used in manyareas. Despite significant development in photographic techniques and technology, bluris still a major cause for image quality degradation in clinical settings. While new seg-mentation models can cope with noise, they become inflective for blurred images. This isdue to many factors such as motion of the camera or more commonly in the case of retinalimages the target scene, defocusing of the lens system, imperfections in the electronic,photographic, transmission medium, or obstructions.An observed blurred image can be written as a convolution of the true image with aLinear Shift-Invariant (LSI) blur, known as the Point Spread Function (PSF) or spatialinvariant/variant unknown kernel K [9].There are three main deconvolution problems: (1) blind deconvolution, which in-cludes the cases when both kernel and image are unknown, (2) semi-blind restoration,c© 2012. The copyright of this document resides with its authors.It may be distributed unchanged freely in print or electronic forms.2 WILLIAMS, CHEN, ZHENG, HARDING: ADVANCED BLUR REMOVAL METHODSin which the kernel is assumed to belong to a class of parametric functions, or (3) non-blind deconvolution where only the image is unknown. All three types are important notonly in many scientific applications such as astronomical imaging, medical imaging, andremote sensing, but also for consumer photography.Deconvolution in the case of known blur, assuming the linear degradation model,has been investigated widely in the last few decades giving rise to a variety of solutions[2, 10, 13]. In non-blind deconvolution, the point spread function is assumed known eventhough this information is not available in most of the real applications. In cases whenthe blur is not known the problem becomes harder and much more challenging. Blinddeconvolution, which is our main concern below, was first introduced by [6] and a lot ofwork has been carried out so far to improve the model [1, 7, 11] and an excellent tutorialhas been provided by [9].This paper is organised as follows: in Section 2, we present the formulation for non-blind deblurring and the splitting idea with some results. In Section 3 we show theformulation of the blind method with results, including applications in retinal imaging.In the Section 4, we give the conclusion.2 Non-Blind DeblurringThe idea of minimising an energy functional of the formminu{||k ∗u− z||22 +α∫ΩL(u)dΩ}(1)where the first term is the least squares term which aims to keep the restored image asclose to the true image as possible and the second term is a regularisation term which aimsto restore edges lost in the reconstruction, was proposed by [12]. A number of functionsare commonly selected for L(u), such as L(u) = u2 (Tikhonov [14]) or L(u) = |∇u|2. TheTotal Variation (TV) regularisation term given by L(u) = |∇u| has been widely used dueto its effectiveness with preserving edges. Minimising equation (1) we derive the EulerLagrange equationKT Ku−KT z−α∇ ·( ∇u|∇u|)= 0,where K is an n2×n2 dense matrix of structured blocks. To cope with ∇u= 0, we modify|∇u| by |∇u|β =√(∇u)2 +β for β > 0. We therefore aim to solveKT Ku−KT z−α∇ ·(∇u|∇u|β)= 0. (2)To solve this non-linear partial differential equation (PDE) we use iterative methods.Time marching is effective but typically proves to be slow to obtain good results. Instead,we use a Conjugate Gradient method aided by preconditioners which aim to increase thestability and speed of the system, such as the Product Preconditioner [15] or the CosineTransform Preconditioner [4].2.1 Splitting Deblurring and DenoisingWe modify the functional by replacing the restored image variable in the least squaresterm with a new variable, distinct from that used in the regularisation term, and add afurther term to minimise the difference between them. Our modified functional is givenbyf (u,v) = 12||k ∗ v− z||22 +α∫Ω|∇u|dΩ+ γ2||u− v||22. (3)WILLIAMS, CHEN, ZHENG, HARDING: ADVANCED BLUR REMOVAL METHODS 3Minimisation of equation (3) with respect to u and v yields the Euler Lagrange equationsKT Kv−KT z− γ(u− v) = 0 (4)γ(u− v)−α∇ ·(∇u|∇u˜|β)= 0 (5)which we solve for v and u respectively [8]. We may use fixed point and the 2-dimensionalfast Fourier transform (fft) to solve equation (4) for v. In order to solve equation (5) foru we use Time Marching or Conjugate Gradient. We make an initial estimate u of thetrue image utrue, which we typically take to be the received image z and repeatedly solveequations (4) and (5) for u and v respectively until u and v are sufficiently close. Ouralgorithm is given asAlgorithm 1 Non-blind Deblurring1: procedure NONBLINDSPLITTING(z,k,α,γ , tol)2: u← z3: repeat4: Solve (KT K + γI)v = KT z+ γu for v5: Solve γ(u− v)−α∇ ·(∇u|∇u˜|β)= 0 for u6: until ||u− v||22 ≤ tol7: end procedure2.2 Experimental ResultsWe present experimental results of this method in the non-blind case using the satelliteand retinal image examples corrupted by motion and Gaussian blur in figures 1—3.Figure 1: Retina image with motion blur. The PSNR increases from 22.847 in the receivedimage to 27.145 in the restored image. The CPU time to obtain the restored image is 1.48.Figure 2: Retina image with strong Gaussian blur. The PSNR increases from 17.862 in thereceived image to 27.003 in the restored image. The CPU time to obtain the restored imageis 1.63.4 WILLIAMS, CHEN, ZHENG, HARDING: ADVANCED BLUR REMOVAL METHODSFigure 3: Satellite image with motion blur. The PSNR increases from 142.688 in the receivedimage to 147.510 in the restored image.3 Blind DeblurringA model for Blind Deblurring, given by minimising the functionalf (u,k) = 12||k ∗u− z||22 +α1∫Ω|∇u|dΩ+α2∫Ω|∇k|dΩ. (6)with respect to u and k and solving via an alternate minimisation scheme was proposedby [6]. The model was shown to give excellent results for model problems, but notfor testing general images where convergence is an issue. We modify this functional,splitting the restored image and kernel terms as follows:f (u,v,k,h)= 12||h∗v−z||22+α1∫Ω1|∇u|dΩ1+α2∫Ω2|∇k|dΩ2+γ12||u−v||22+γ22||k−h||22.Minimising with respect to u, v, k and h respectively, we obtain the following EulerLagrange equationsγ1(v(x,y)−u(x,y))+h(−x,−y)∗ (h(x,y)∗ v(x,y)− z(x,y)) = 0 (7)γ1(u(x,y)− v(x,y))−α1∇ ·(∇u(x,y)|∇u(x,y)|β)= 0 (8)γ2(h(x,y)− k(x,y))+ v(−x,−y)∗ (v(x,y)∗h(x,y)− z(x,y)) = 0 (9)γ2(k(x,y)−h(x,y))−α2∇ ·(∇k(x,y)|∇k(x,y)|β)= 0 (10)An overall algorithm is given in Algorithm 2.3.1 Experimental ResultsWe present experimental results of the blind restoration using this model of the satelliteand retina images corrupted by motion blur in figures (4) and (5).4 ConclusionsA new splitting method algorithm is proposed for blind deconvolution restoration. Testresults in retinal images are encouraging and show that it is potentially useful in medicalimaging.WILLIAMS, CHEN, ZHENG, HARDING: ADVANCED BLUR REMOVAL METHODS 5Algorithm 2 Blind Deblurringfunction BLINDSPLITTING(z,kinitial,α1,α2,γ1,γ2, tol,maxit)u← zk← kinitialfor i← 1 to maxit dorepeatSolve u(−x,−y)uk−u(−x,−y)z+ γ(k−h) = 0 for kSolve γ(h− k)−α∇ ·(∇h|∇˜h|)= 0 for huntil ||h− k||22 ≤ tolImpose: h(x,y)← 0 if h(x,y)< 0Impose: h(x,y)← (h(x,y)+h(−x,−y))/2 ∀x,y ∈ΩImpose: h← h/∫Ω h(x,y)dxdyrepeatSolve k(−x,−y)kv− k(−x,−y)z+ γ(v−u) = 0 for vSolve γ(u− v)−α∇ ·(∇u|∇u˜|)= 0 for uuntil ||u− v||22 ≤ tolImpose: u(x,y)← 0 if u(x,y)< 0end forend functionReferences[1] Mariana S. C. Almeida and Luís B. Almeida. Blind and semi-blind deblurring ofnatural images. IEEE Transactions on Image Processing.[2] L. Bar, N. Sochen, and N. Kiryati. Semi-blind image restoration via Mumford-Shah regularization. IEEE Transactions on Image Processing, 15(2):483—493,Feb 2006.[3] C. Brito-Loeza and K. Chen. Multigrid method for a modified curvature drivendiffusion model for image inpainting. Journal of Computational Mathematics, 26(6):856—875, 2008.[4] Raymond H. Chan, Tony F. Chan, and Chiu Kwong Wong. Cosine transform basedpreconditioners for total variation deblurring. 1995.[5] Tony F. Chan and Luminita A. Vese. Active contours without edges, 1998.[6] Tony F. Chan and C. K. Wong. Total variation blind deconvolution. IEEE Transac-tions on Image Processing, 7(3):259—285, March 1998.[7] Tony F. Chan and C. K. Wong. Convergence of the alternating minimization algo-rithm for blind deconvolution. Linear Algebra Appl, 316:259—285, 2000.[8] Yumei Huang, Michael K. Ng, and You-Wei Wen. A fast total variation minimiza-tion method for image restoration. SIAM, 2008.[9] D. Kundur and D. Hatzinakos. Blind image deconvolution. IEEE Signal ProcessingMagazine, 13(3):43—64, may 1996.6 WILLIAMS, CHEN, ZHENG, HARDING: ADVANCED BLUR REMOVAL METHODSFigure 4: Satellite image corrupted with motion blur. The CPU time to obtain the restoredimage is 16.71. On the top left we show the received image, on the top-right is the restoredimage, bottom left is the initial estimate of the kernel and bottom right is the restored kernel.Figure 5: Retina image corrupted with motion blur. The CPU time to obtain the restoredimage is 18.77. On the top left we show the received image, on the top-right is the restoredimage, bottom left is the initial estimate of the kernel and bottom right is the restored kernel.[10] R. L. Lagendijk, I. Biemond, and D. E. Boekee. Regularized iterative image restora-tion with ringing reduction. IEEE Trans. Acoust. Speech Signal Processing, 36(12):1874—1888, 1988.[11] R. Molina, J. Mateos, and A.K. Katsaggelos. Blind deconvolution using a varia-tional approach to parameter, image, and blur estimation. IEEE Transactions onImage Processing, 15(12):3715—3727, Dec 2006.[12] L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removalalgorithms. 1992.[13] M.I. Sezan and H.J. Trussell. Prototype image constraints for set-theoretic im-age restoration. IEEE Transactions on Signal Processing, 39(10):2275—2285, oct1991.[14] Curtis R. Vogel. Computational Methods for Inverse Problems. SIAM, 2002.[15] Curtis R. Vogel and Mary E. Oman. Fast, robust total variation-based reconstructionof noisy, blurred images. 1998.

Advanced blur removal methods with applications to retinal imaging for ophthalmology

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