ADTPString Algorithms CS 791B
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Date Created: 09/12/15
Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 26th January 2007 39 Outline 0 Bilateral ltering 9 Properties 9 Relation to anisotropic diffusion 0 Review of image denoising 39 Outline Bilateral ltering 0 Domain ltering 0 Range ltering 0 Domain and range ltering 9 Properties 9 Relation to anisotropic diffusion 9 Review of image denoising 39 Bilateral ltering 9 Convolutionbased can be implemented as discrete convolution 0 Adaptive ltering kernel varies depending on local intensities The main idea Replace image intensity values with the average of nearby and similar pixels I 4h 1 my m M21 39 Bilateral ltering h x If H CE7xsf 7fxd5 hx output image 0 f x input image a c x closeness function gt Nearness of pixel locations 5 and x o sf fx similarity function gt Nearness of pixel intensities f 5 and f x o Denominator normalizes the kernel function I rm 1 my m 5721 Domain39hlta ing Domain ltering Gaussian Convolution Continuous W fl fl News Discrete 7 fix i 7 39 t 39739 1 IX7y M if j w oyj 7 W07 em 2 0D Choose one parameter 0D I y T 67quot2 1 Range ltmng Range ltering Range lter is independent of spatial distribution of intensities fie ffg esm x xnds Lo Lo Sf 7fxd Mean intensity if 14 is the frequency distribution of gray levels in f E Lf f wow We can also rewrite other integrals depending on intensity in terms of the histogram ffgff d5 0 g gtv gtd gt hx I 26th January 2007 7 21 Range ltmng Range ltering The range lter can be interpreted as a histogram operation Example Mean intensity this is a range lter with s gt7 f constant we gtV gtd gt In general the result is a weighted mean 00 W gtT gt7fd gt 0 where S gt7fV gt f5 S gt7fV gtd gt The range lter causes the image histogram to become more peaked TQM I 26th January 2007 821 l Domain and range ltering Combining domain and range ltering results yields a bilateral lter Common domain and range lter kernels Gaussian Closeness function 7 1 My 0E7x e 2 D 7 x HE xH Similarity function 1 505JXz sex WM l gt fl 0 How does the lter behave as TR becomes very large a How does the lter behave as a D becomes very large I 26th January 2007 9 21 Domainmdmgemtermg The lter kernel The kernel adapts to the shape of edges in the image t 1 41 333 3 W 3 Original imageleft kernel at edge middle Filtered imageright 26thIanuary 2007 10 3921 39 Outline Bilateral ltering 9 Properties 9 Relation to anisotropic diffusion 0 Review of image denoising Edge preserving Ongma mage Ew ateva Hm mage Gausswan Fmad Wage The results ofsmoothmg a symmeuc mage mymmyznm 1221 Ejtsjg mw Small scale texture and wise are smoothed Filtered images rm ttmvhxgm ll m i m 26th JanuaIy 2007 13 21 Original one iteration ve iterations Looks like a scalespace image representation but is it 39 Bilateral Mesh Smoothing T R Jones F Durand and M Desbrun Noniterative feature preserving mesh smoothing ACM Transactions on Graphics special issue on Proc of ACM SIGGRAPH 2003 I 1 Original noisy restored mesh 0 Domain distance on surface a Range distance from tangent plane Trilateral Filter P Choudhury J Tumblin The trilateral lter for high contrast images and meshes EGRW 03 Proceedings of the 14th Eurographics workshop on Rendering 2003 l X WWW Unilateral Filter Bilateral Filter Trilateral Filter a C o Preserves gentle slopes and gradients CS 593 791 West Virginia University lele39clicall Image Analysis 26th January 72007 16 3231 39 Outline Bilateral ltering 9 Properties 9 Relation to anisotropic diffusion 0 Review of image denoising 39 lD discrete Gaussian convolution It1 x 01 x 71 0211x C3I x 1 Since we know 01 02 03 1 then 02 17 017 03 It1x71 x 01I x 717I x 03I x 1 7I x For a symmetric kernel cl 03 c 1 1x 7 1 x c1 x 7 1 7 21 x 1 x 1 which is the discretized heat equation 8 2 i I at CV 25th Janufazy 2007 ref3921 39 lD adaptive ltering Itlx c x711 x71 c x x c x 1I x 1 Since we know c x 71 c x dx 1 1 thenc x 1 ic x7 1 761x 1 It1x 61x71 x711ictx717c x1I xc x1ltx1 1 7 I m 7 c x 71I x 7171 x c x 1I x 171 x which is an implementation of PeronaMalik smoothing 81 7 8t 7 W V 25th Iandary 2007 191211 V2 000106 39 Outline Bilateral ltering 9 Properties 9 Relation to anisotropic diffusion 0 Review of image denoising 39 Review of image denoising o Isotropic diffusion equation 8m Vzu gt Equivalent to Lowpass ltering signal processing gt Equivalent to Gaussian convolution gt Analogous to physical diffusion heat transfer process gt lLinimizes membrane spline energy minu f9 HVul lzdx 0 Inhomogeneous diffusion equation 8m diVcVu gt Slows diffusion at edges gt Can perform edge enhancement a Anisotropic diffusion equation 8m diVDVu gt Directs diffusion along edges 0 TVnorm minimization 8m diV Vu HVuH gt lLinimizes TV norm mim f9 HVqux o Bilateral ltering gt Convolution with a spatially adaptive lter gt Combines domain and range ltering gt Connection with inhomogeneous diffusion I 26th January 2007 21 21