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This 47 page Class Notes was uploaded by Abe Jones on Saturday September 12, 2015. The Class Notes belongs to CS 593B at West Virginia University taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/202766/cs-593b-west-virginia-university in ComputerScienence at West Virginia University.
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Date Created: 09/12/15
Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 20th March 2006 Outline 0 Coordinate Transformations lVlidterm 0 Take home exam 0 Due Tuesday 328 0 Will cover material through 324 Proj ect Some topics 0 Tensor anisotropic diffusion o TVnorm minimization o Segmentation using snakes or some valiant o ChanVese segmentation 0 Hidden Markov Measure Field Model 0 Segmentation using graph cuts 0 Rigid registration 0 Non rigid registration 0 Multimodal image registration 0 Scalar Vector or Tensor Field Visualization Check Supplementary Reading on the website and the bibliographies of the papers Proj ect 0 Submit your team members and topic 0 Reserve presentation time 0 Two per day on 417 19 21 24 2628 Outline 0 Coordinate Transformations 0 Global Transformations 0 Local Transformations Problem De nition Image registration is the process of determining a coordinate transformation between two images that are misaligned mTjndistIlxIzTX o T is a coordinate transformation 0 1x and 2x are 2 images to be aligned o distU 112 is a metric which determines how well the images match 0 distU 112 can be based on image intensities or extracted features Linear transformations 0 Translation 0 Rotation o Scaling o Shear m 1m 2D Linear transformations Translation 2 parameters T x x t Rotation about the origin 1 parameter cos0 isin0 TXsz sin0 cos0 i Nonuniform scaling 2 parameters TXSXSX 0x 0 sy Shear 1 parameter 2D Linear transformations A more general 2D transformation can be obtained by composing several transformations such as TX RzSx 7 c t 0 Translate so that center of the rotation is 00 0 Scale the coordinate systems 9 Rotate about the origin 0 Translate 0 Total of 7 parameters m 1m 3D Linear transformations 0 Translation 3 parameters 0 Scale 3 parameters 0 Shear 2 parameters 0 Rotation 3 Euler angles Using Euler angles the 3D rotation is represented as 3 consecutive rotations about the coordinate axes TX Rnysz where RnyRz l 0 0 cos 1 0 SimJ cos 9 7 sin 9 0 0 cos 1 7 sin 1 0 l 0 sin 9 cos 9 0 0 sin 1 cos j 7 SimJ 0 cos 1 0 0 l Displacement Field Compute a displacement vector for each voxel T x x tx To constrain the displacement eld to represent physically plausible deformations we may impose smoothness constraints Iftx ux7y7 VX7yl will constrain the displacement eld to be smooth Displacement Field 0 Consider the deformation eld to be the velocity eld of some Viscous uid a More suitable for large deformations o Constrain the eld to obey the NavierStokes equation a Smoothness of the eld is controlled by the Viscosity of the simulated uid 0 Computationally expensive Spline based transformations 0 Fewer control points than image pixels 0 The spline may interpolate or approximate the control points 0 Sum of shifted basis functions 0 Basis functions may have local or global support 0 Basis functions are generally low degree 3 polynomials m 11Tx Representing the transformation Let the parametric curve pt xtyt be represented as the sum of basis functions i1 We can form a tensor product parametric surface ps7 t by n m PS7 t Z ZMtASPii i1 j1 The transformationis given by TX psx7 BSpline surface Coordinate Transformations 7000000000000 Local Transformations BSpline basis functions Local Transfonnations Cubic BSpline Piecewise cubic intervals which join together smoothly 1 3 3 1 pk1 1 3 6 3 0 pk r r3 r2 r 1 p i6 3 0 3 0 pk1 1 4 1 0 Pk2 l l I k kl k2 If we want m cubic segments we need m3 control points Coordinate Transformations 000000000000 Local Transformations BSpline curves II 391 0 Local control moving a control point changes 4 nearby intervals 0 Smoothness guaranteed across intervals Wednesday More transformations and matching metrics Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 22nd February 2007 39 Outline 0 Introduction 6 Level Set Methods for Segmentation 9 Shape Recovery Results 0 Conclusions 39 Outline 0 Introduction 0 Geometry of Implicit Curves 0 Implicit Curve Evolution 39 Origins Stanley Osher and James A Sethian quotFronts propagating with curvaturedependent speed algorithms based on HamiltonJacobi formulations J Comput Phys 1988 Tracking dynamic boundaries and interfaces in o Fluid Mechanics 0 Flame Propagation o Crystallography Especially where surfaces may split merge form sharp comers I Z A 3952 Surface Representation o Surfaces are represented as the zero level set of an embedding function 0 This function is evolved implicitly evolving the embedded curve 0 The previous Lagrangian approach was to track points on the interface 22nd Fehwary 200 39 Lagrangian Approach This was the quotsnakequot approach to curve evolution 0 Discretize the curve into individual particles 0 Track these particles as they move through a eld l s 39sz 39 Eulerian Approach o Discretize the embedding function 1Z1x y to represent and evolve the curve 0 Evolve 1Z1x y by updating at xed grid locations 0 For simplicity the function 11x y can be discretized to have the same resolution as the image we are segmenting 752 39 Problems with snakes The Lagrangian approach does not handle a Splitting merging boundaries topological change 0 Selfintersection 9 Sharp corners or other discontinuities 39 Levelset methods The Eulerian approach can handle a Splitting merging boundaries topological change a Selfintersection 0 Sharp comers of other discontinuities quotHamiltonJacobi type equations 19 7 F V 0 at H M have been extensively studied under this framework especially interfaces moving with curvature dependent speed F 6 I L 952 l Parametric VS Implicit Parametric Curve Evaluating the function gives coordinates of points on the curve Implicit Curve Way C Normal and Curvature The gradient of the embedding function is perpendicular to the level curve WWW N x HWJOWW Recall Directional Derivative d Imp 71m hu Vf HWHcow o Duf p has the least magnitude 0 when u is parallel to the curve 0 The directions of least magnitude and greatest magnitude are perpendicular I 1152 Normal and Curvature The curvature of the level curve is the rate of change of the normal vector V111 amp 7 div 7 lWl This can be rewritten as 7 zwxwywxy w 15 5 I msz Example a circle Consider the implicit equation for a circle xiafwibf rz The gradient is W l i Z l HWH 4xia24cyibz 2 xia2 yew ViJ i l x761 HWH xiazyibz yib I 135z 39 Example a circle Computing the curvature FL 7 WWW2y wew m2 yy2 xy0 1 7 8y7b28x7a2 i 7 wwiav4wiwe r I V r Signed Distance Function One possible embedding function for implicit curves 0 1Jxy distance from x7y to the curve 0 11xy 0 if x7y is on the curve 0 11xy lt 0 if x7y is inside the curve 0 11xy gt 0 if x7y is outside the curve WW 1 Almost everywhere I H 39 15752 39 Irnplicit Curve Evolution Suppose we have an evolving curve ct xtyt Let s derive the evolution equation for for 11xy t which has ct as a level set Let Ct be the zero level set of so that zJxtyt7 t 0 for all t This implies that 04ng t 0 By the chain rule amzx way 81 dt EEEEE vwcrf 39 issz 39 Implicit Curve Evolution Decompose c t into components tangent and normal to Ct 0 V1 VNNt VTT1 2 W VNNo 2 since Viz is perpendicular to the tangent to Ct Substituting the level set de nition for the normal to the embedded curve 7 V1 81 0 7 ViJ Van 5 1752 Implicit Curve Evolution We can reWIite this result 7 V1 811 0 7 W 8i 0 7 VNV1JHV1ZJH at HWHZ aw 0 VNHWH 5 811 0 VNllV Jll E i Evolving the embedding function by it 81 CS 593 791 West Virginia University Medical Image Analysis h r icimufv EWlu m Smoothing the curve Evolving the embedding function by Hx7yV 0 re lt 0 Where contour is locally convex 0 H gt 0 Where contour is locally concave 007 207 52 Medical Image Introduction CurvatureB ased Evolution a fide El 5 13 QQC J Medical Image Aim mm id 39 Curvature Based Evolution Letting VN be function of curvature we have evolution equation 2 7meva The segmentation problem is now reduced to nding an appropriate function Frlt We can factor FM kFA Fg where o F A the advection term is usually a constant 0 FG depends on the geometry curvature of the level set a k is a stopping term to slow evolution near boundaries We can use 1 kxy m 2252 Introduction Entropypreserving solution The upwind nite difference scheme we used for TV norm minimization prevents singularities like this swallowtail from developing Medical hm Impnmcmmmm Narrowband update For faster computation x Since We are primarily interested in the zero level set of 1b We may evolve only in a small region smrounding the level set zzganmgznnv 24752 39 Outline 9 Level Set Methods for Segmentation o Discretizing the evolution equation 0 Extending the Speed Function
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