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Medical Image Analysis

by: Abe Jones

Medical Image Analysis CS 778

Abe Jones
GPA 3.77


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This 20 page Class Notes was uploaded by Abe Jones on Saturday September 12, 2015. The Class Notes belongs to CS 778 at West Virginia University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/202770/cs-778-west-virginia-university in ComputerScienence at West Virginia University.

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Date Created: 09/12/15
Medical Image Analysis CS 778 578 Computer Science and Electrical Engineering Dept West Virginia University January 14 2009 39 Outline 0 Intro 9 Isotropic Diffusion Formulation 39 Outline 0 Intro D Image smoothing a Isotropic Diffusion Formulation Using sliding windows for image processing Figure Pixel ij in the output image shown in red depends on the pixels in the neighborhood of ij in the input image shown in blue iljl ijl iljl iljl ij l ilj l Coordinates in the 3 x 3 neighborhood of pixel ij CS 778 578 West Virginia Universityquot ImageA alysis I image smoothing Image smoothing We can compute several things within each window a Local averaging mean a Weighted local average weight the center pixel higher 0 Nonlinear ltering median a We can look at larger neighborhoods 5 x 5 7 x 7 and larger 0 Why are these odd dimensions a In 3D 3 x 3 x 3 neighborhoods and larger January 14 2009 5 28 image smoothing Local averaging We associate weights wkm with each voxel in the sliding window The new intensity gij computed from from the original image f and the weights w 1 1 gm Z Z kafi kj m m71k71 is the mean of intensity values within the sliding window When applied to the whole image we write g w X f I January 14 2009 6 HO Local averaging results 1 Ex I I39 a a Edges are destroyed a Smoothmg rs a lowrpass ltenng operatron o ngh frequency components are attenuated low frequency components hermage are pre e t can be shown that rf rmage f rs corrupted by addrtrye Gaussran norse of vananoe a thenthe vananoe of the norse on rmage 3 rs 29mm 72n I imagesmoodjmg Fourier Transform Convolution Theorem The continuous de nition of convolution fg frgxirdr lt1 Wm U fTgtgx7TdT 62 in 2 Using the fact that equot 676 7 00 00 727rik7 727rikciT 7 NM 7 e Mite gltx Wdr lt3 January 141009 8 my image smoothing Substituting 2 x i 739 and dy dx ffgi 727rik7fTdTi 6 2 igydy 4 flfakgl flflflgl 5 o Convolution can be implemented as multiplication in the frequency domain 0 Gaussian convolution is Gaussian multiplication in frequency domain 0 Gaussian convolution is lowpass ltering January 14 2009 a my image smoothing Image Gradient V1067 Y 6 Vector points in the direction of fastest increase in I 67 y o Edges are discontinuities in intensity 0 HVIH is the edge strength sharpness of the edge 0 VI is perpendicular to the edge Edge detection using the gradient OHgma Wage 20 An 60 an mo 20 20 Au so an um 20 Pama dewanve mm vespect m x Pama denvatwe wnh vespecl m y 7 0 so an mom 20 u so no mo 20 Image from quotVisualizing calculus The use of the gradient in image processingquot http ama 1colorado edu 3 image smoode g Divergence For the vectorvalued function VX7 y 1E7 J the divergence is 2 7 de ned as 7 8V1 8V2 d1VVXy 7 a 87y 7 More notation diVVI V VI V21the Laplacian J Imag smoomg Divergence of a vector eld divVX7 y Zia 12 Physical interpretation sources or sinks a velocity eld 8 21 I 39lg zx lI 1 2 K 1 LiL lx J gt 11 I 39 Ill l27 div V gt 0 left div V lt 0 right Image from quotThe idea of divergence and curl httpwwwrnaLhnmnedu cs 778578 WesIVir 39 39 gr 39 January 142009 1320 39 Outline 9 Intro 0 Isotropic Diffusion Formulation 0 Description 0 Solution The Heat Equation Diffusion Equation Let image Ix y z t 91 E d1VVI 9 o The equation describes the way physical systems achieve equilibrium 0 When describing heat transfer I x is temperature 0 When describing diffusion I x is molecular concentration am X X X CS 778 578 West Virginia University Medical Enage Analysis January 1 4 15 l Solution Solving the heat equation In one dimension 8 821 a w 10 Take the Fourier transform of both sides 81 821 f a f Q 11 Let Ix7 t Uw7 t and Recall that iffx lt gt Fw thenf x lt gt inw and f x lt gt 7w2Fw I Januany llng39ODR 16 HO I Solution The Heat Equation Diffusion Equation 3Uw7 t 7 2 8t 7 7w UuJ7 t 12 with initial conditions WW7 0 UOW 13 You can verify that How young 14 Is a solution to the problem Januany izt 09 17 x23 I Solution The Heat Equation Diffusion Equation Taking the inverse Fourier transform f 1Uw7t f liUowe Z i lt15 we see that 2 367 t 0x 27 16 where a 2t This is Gaussian convolution 18 20 39 Solution The Heat Equation Diffusion Equation 0 Evolving I x according to the heat equation is equivalent to convolving with a Gaussian kernel 0 Longer evolution corresponds to convolution with Gaussians of higher variance 0 This gives us rm footing for studying PDEs in the context of image processing Jammy1422009 mm Solution Next Class Variational calculus Numerical methods for solving the heat equation


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