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Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 12th April 2006 Outline o Tensor Field Processing 0 Tensor Field Visualization 9 Beyond Tensor Fields Tensor Field Registration For scalarvalued images 0 Transforming the image involved only changing the coordinate system 0 Image intensities were unaffected Will this work for vector and tensor valued images Tensor Field Visualization Beyond Tensor Fields c Ouu Tensor Reorientation Tensor Field Registration Not when there is a connection between the coordinate system of the image and the vectortensor at each voxel o For example no reorientation is required for color images a For diffusion tensor images reorientation is required for a spatial transformation to be meaningful 0 Tm mm m Tensor Field Registration For rigid rotation of vector image T V RV For rigid rotation of tensor image TD RDRT Consider D X AX T This is equivalent to rotating the eigenvectors TD RXARXT 7 RXAXTRT R XAXTRT RDRT For nonrigid transformations we must factor out the rigid part of the transformation Q a 7 Tensor Slmilaritx o In the problems of segmentation and registration the notion of similarity is important 0 We compared intensity images by looking at the magnitude of intensity differences 0 How can we compare tensors One possibility Matrix norms like the Frobenius norm m m 22 law 2 i1 j1 MW can be used to de ne tensor distance dT17T2HT1T2HF Tensor Slmilarit Frobenius norm is invariant under rigid transformations dT1 T2 dRT1RTRT2RT but is not invariant to af ne transformations n theoretic tensor distance Recall that tensors describe a stochastic diffusion process 1 6X irTD Ir 27rquot 2tD p 4t and that KL divergence is an information theoretic quotdistancequot 7rit7D KLQWI px10gdx Which is not symmetric KLp7 q 7 KLqp Tm l Information theoretic tensor distance A true distance can be obtained by symmetrizing KL 1 JP7 q 50am 11 KLUMD This is called Jdivergence The tensor distance a7T17 T2 Jpr t T1pr t H can be expressed in closedform as 1 dT1 T2 E trTf1T2 T2 1T17 2n where n is the dimension of the tensor This distance is invariant under af ne transformation Tensor Field Processing Tensor Field Visualization Beyond Tensor 06360000 OOQUCC Tensor Similarity Measures Geodesic distance Consider 2 x 2 symmetric positivede nite matrices Da bac b2gt0agt0 b c The parameter space of SPD matrices forms a manifold embedded in R3 speci cally a cone Distances between points on the cone can be interpreted a distances between tensors Tensor Field sualization Q C O Q Q Q Q Q Q Ellipsoids and superquadrics Beyond Tensor Fields Tensor Field Processing Tensor Field Visualization Be ond Tensor Fields 00000000 000000 Tensor Visualization Gl Texture mapped glyphs Tensor Field Proce 11g Tensor Field Visualization Beyond Tensor Fields 000000013 00000 Volume Rendering Direct Volume Rendering A ray casting approach A1 2 n I q A1 A2 A3 3 lt 2W 3 GP A1 l 2 3 2 3A3 CS A1 A2 l A3 3 Fl De ne a coordinate system based on anisotropy Beyond Tensor F Tensor Field V Tensor Field Pro 000000 00000000 V 111116 Rendering DJ Vo me Transfer function over this coordinate system can determine color and opacity Tensor Field Processing Tensor Field Visualization Beyond Tensor Fields 000000 0 0 Vblume Rendering 0 O G Anisotropic lighting Generalized Diffusion Tensor Imaging Recall the relation between images S and tensor D from DT MRI 3 3 S So 6XpibgTDg So 6Xp7b Z ZgigjDij i1 j1 The assumption is the diffusion process is described by a rank2 tensor components are de ned by 2 indices We may generalize to rank 4 tensors odd rank tensors imply negative diffusivity Mu S So 6Xp7bgTDg So 6Xp7b I Z gigjgkngijkl 3 1 11 3 111 w Beyond Tens Fields C ODOOCJ RANK 0 RANK 2 RANK 4 g o y z b I z RANK 6 RANK 3 x39 b b t m Xawmw x e x Diffusivity pro les for tensors of varying ranks High angular resolution diffusion imaging A more general model of diffusion but requires many more images 0 DT MRI Require at least 7 images to t the Gaussian tensor model 0 HARDI We would like as many images as possible but time is a constraint The relation between the acquired images S the diffusion encoding gradients q and the diffusion displacement probability pr is pm fexpltizmqrgtdq High angular resolution diffusion imaging pm fexpltizmqrgtdq o If we have Sq values on a cartesian grid in qspace we can directly compute pr FFTSqSo 9 Given Sq on a sphere we can interpolateextrapolate exponentially to compute a cartesian grid in qspace o It is often suf cient to examine the behavior of the marginal distribution 1709 gt obtained by integrating out the radial component ofp Tensor Field Processing Tensor Field Visualization Beyond Tensor Fields 00000000 000000 D H 44 Beyond Tensor 1d 000 Orientation Distribution Function vaooaooa a 0 orovmi Hoooaoo a quotu Q 0 atIo hhhbblbsu h hu w Q nuh s 3 s stsQm Q s5u Q gym Q m HARDI can capture complex ber geometry 1 oooollrl ul39 I urlIl O o I P J 39 1v Open research areas Solve the problems we have examined in this class for elds of probability distributions 0 Restoration regularization o Segmentation a Registration 0 Visualization o Neuronal ber tracking 0 Atlas construction Monday Final quiz return homework Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 12th January 2007 39 Outline 0 Discretizing the heat diffusion equation 9 Matrix forms of the discretized heat equation 9 Scale space image representation 39 Outline 0 Discretizing the heat diffusion equation 0 Numerical differentiation 0 First derivative approximations 0 Second derivative approximations 9 Matrix forms of the discretized heat equation 9 Scale space image representation I 336 I Introduction The names heat equation and diffusion equation are used interchangeably the same equation describes both phenomena I 1D 7 8t 8x2 8M azu azu IHZD E 7 Q FW I 3D 8M 7 azu azu azu 8t 8x2 ay2 822 3M In general 7 d1VVu at a Heat u is temperature 0 Diffusion u is concentration 0 Images u is image intensity 7 j a i 1th January 2007 A 36 l Introduction 0 We know the heat equation can be solved analytically x2 gt Ix7 I 10x 6 675 o In the future we will consider nonlinear variants of the diffusion equation for which no simple analytical solution eXists a In these cases we approximate a solution numerically 12111 Ia39mfarylom 573 s l Introduction Problem Estimate derivatives of some unknown smooth function f x given only samples Motivation Find approximate solutions to PDEs governing evolution off I f T t 636 l Forward Difference Equation Expand f x in a Taylor series about x0 fx fx0 x xolfxo 00 Then evaluate at x x0 h fxo h fxo hf xo 001 First order forward di erence fxO zfx0 i h fx0 I 736 l Backward Difference Equations Replace h with 7h in the previous derivation fxo 7 h fxo 7 hf xo 000 First order backward di erence fx0fx0h fxo h l Centered Difference Equation Subtract the rst order expansions fxo h fxo h xo fxo 7 h fxo 7 h xo fx0 h fx0 h thxo Divide by 2h to get the centered di erence fx0hfx0h N x 2h I 9236 l Centered Difference Equation The second order terms in the expansion cancel so the remainder is 0h2 hz 2 fx0 h fx0 hfxo if x0 001 mo 7 h fxo 7 h xo Elmo owl For images Use h 1 pixel I 39 1036 39 Second Centered Difference Equation We can approximate the second derivative by adding two second order expansions mo h fxo hf xo ElfKm 00 mo 7 h fxo 7 hf xo Elfx0 0W fx0 h fxo 7 h 2fxo W xo 0012 m W1 1136 I Second Centered Difference Equation Rearrange fxo h fxo 7 h 2fxo hz x0 0h2 to get the second order second centered difference fxO fx0 h 7 2650 fx0 7 h I L 39 12136 39 Outline 0 Discretizing the heat diffusion equation 0 Matrix forms of the discretized heat equation 0 Forward difference method 0 Backward difference method 9 Scale space image representation 39 EXplicit Method Recall the heat equation a 81 81 8t 7 8x2 8y2 Use forward difference in time and second central difference in space 16 z w 7 w 7 It 7 211 It It 7 211 It 7 x1y xy x71y xyl xy x9171 o Superscripts time o Subscripts position I 12th Ianufazy 2007 12 W36 39 Forming the linear system of equations 5 icy 6Lly 7 41y Lily Lg1 1y71 0 We want to simultaneously solve for 6 at all x7 y a The image can be stretched into a vector w by stacking the columns of the image on top of each other gt Matlab reshape command 1 15913 2 261014 3 371115Hw 481216 1 16 The evolution equation will be w 5 Awt I 1th Iandary 2007 15 Ms 39 Reshaping image I into array w 1 15913 2 2 61014 7 3 1 371115 Hw l481216j 1 16 allows us to rewrite the equations 1 6 z z z z z 1 IX Ag 6xly 7 41xy Ixily xyl Ixy71 in terms of w For example 16 1 t t t t 1 W10 W10 6W14 4w10 W6 W11W9 this can be rewritten as a row vector times a column vector 163354 6 6 1746 6 6 w 12th Janufazy 2007 raj3936 39 Forming the linear system of equations In general we have I 6 I I I I I I Wi Wi l 6Wi17 4Wi l Wi71 Win l wiin 0 Collect the coef cients of w into a matrix A o Ifthe image Ixy is size n x n the vector w is n2 x 1 o the matrix of coef cients A is n2 x 112 0 Most elements ofA are 0 ie A is sparse I 12th Ianufazy 2007 17 8 6 39 Linear system of equations 5 7 t t t t t t Wi 139 6Wi1 4Wi wiil Win wiin The matrix of coef cients 1746 6 0 6 0 0 6 1746 6 0 6 0 A 0 6 1746 6 0 6 0 Properties of A o Symmetric o Sparse gt 5 nonzero diagonals for a 2D image gt 7 nonzero diagonals for a 3D image gt Matlab Use sparse spdiags I 12m Janufazy 2007 18393 s 39 EXplicit or forward solution Each iteration is a matrix multiplication w l s Aw Convergence Criterion Steadystate is reached when w Jt S z w Check llw 5 7 w H lt e Problem 0 For large 6 we may overshoot the solution a The iteration will oscillate and never converge 0 Stability is only guaranteed for small 6 and then convergence is slow I 1th January 2007 1936 39 Implicit Method Recall the heat equation 81 7 821 82 8t 7 8x2 8y2 Use backward difference in time and second central difference in space Ly 7 56 t t t t t t 6 xly 7 21xy Ixily Lg1 7 21xy Ixyil I Linear system for backward difference method The generic difference equation 76 Ly Icy 7 6L1y 7 41y Lily Lg1 1y71 has the vector form 6 I I I I I I W Wi 6Wi1 4Wi W121 Win wiin 39 Linear system of equations wti5Bwt The matrix of coef cients 146 76 0 76 0 0 76 146 76 0 76 0 B o 76 146 76 o 76 0 Properties of B o Symmetric o Sparse gt 5 nonzero diagonals for a 2D image gt 7 nonzero diagonals for a 3D image I 1 M 39 Irnplicit or backward solution Each iteration requires solution of a linear system inversion or factorization Bw 7 75 This method is stable however setting 6 too large will result in slow convergence 1 W Mgr 23 3 5 39 MatriX stability analysis If we have some small error e0 in the initial condition w0 w1 Aw0 e0 Aw0 Ae0 and at the neXt iteration we have w2 AAw0 Aeo Azw0 Azeo In general at iteration n we have wquot Aquotw0 Aquote0 Whether HAquote0H HeOH depends on the condition number ofmatriX A Condition numberA z 1 is wellconditioned I 1th Iandary 2007 24 W36 39 MatriX stability analysis Computing the condition number is dif cult but As a general rule Stiictly diagonally dominant matiices are wellconditioned De nition A matIiX A is strictly diagonally dominant if lan l gt Z laijl i for all rows 139 I 1th Ianufazy 2007 1536 39 Stability of the forward difference equation A typical row of matrix A Ai0 6 0 6 1746 6 0 6 0 For what values of 6 is the matrix diagonally dominant 1746lgt4l6l 1746gt1H1746gt46H1gt86 46 01 1746 lt71H1746lt746H1lt0 46 The forward difference method is stable when 6 lt 12m Janufazy 2007 2536 39 Stability of the backward difference equation A typical row of matrix B B0 76 0 76 146 76 0 76 0 For what values of 6 is the matrix diagonally dominant l146lgt4l6l 1646gt1H146gt46H1gt0 01 146 46 The backward difference method is stable for all 6 Unconditionally stable I 1th Janufazy 2007 27 3 6 39 MatriX stability analysis For more details about matrix condition number and spectral radius see Golub and Van Loan quotMatrix Computations or another numerical linear algebra teXt In Matlab use cond condest 1 Lu Mum 39 Implicit or backward solution Each iteration requires solution of a linear system inversion or factorization Bw w s This method is unconditionally stable however setting 6 too large will result in slow convergence Problem Error is of order 06 J I 1th Ianufazy 2007 29 36 39 llixed ExplicitImplicit Method We can get order 062 error by averaging the two difference equations Note that the second order terms in the Taylor series cancel just like they did when we computed central differences 1 iw l s 7AW 2 1sz5 1 z 5W B Iw 5 A Iw 1th Ianufazy 2007 30 8 6 39 Solutions to the linear system B Iw 5 A Iw 0 Don t try to invert the matrix B I 0 Instead use Gaussian elimination LU decomposition 39 Outline 0 Discretizing the heat diffusion equation 9 Matrix forms of the discretized heat equation 9 Scale space image representation 0 The concept 0 A requirement 0 NeXt Class x W What is scale space mage so we wantto represmLan 1mage over a conunuum ofscales coarse w ne llh hmiy an mm x W What is scale space mage so we wantto represmt an 1mage over a conunuum ofscales coarse w ne 039 mvmriymm 3336 r Scalespace requirement new detarls o Images generated by rsotroprc dxffusxon sausfy thrs requrremerrt o Equwalmt to convolutrorr lowrpass lterm o However edges at the coarse scales are blurred e We must track features up to the nest scale to get therrtme locatrerrs mvmriymm auaa New scale space representation Perona and Malik suggest a new technique for generating the scale space of images Which preserves edges in ScaleSpace and Edge Detection Un ng Anisotropic Di usion They propose the use of n Inhomogeneous diffusion rate of diffusion varies spatially Weickert in A Review ofNonlinear Di un39on Filtering proposes a Anisotropic diffusion rate of diffusion at a point varies With direction to generate the scale space of images and perform denoising 12m JammyZElm 35 ms Wednesday IVILK B day is Monday a Quiz about the reading a The physical process of diffusion 0 Discuss the 2 papers Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 2nd April 2007 39 Outline 0 Topological mesh correction a Cortical surface mesh inflation 9 Spherical m esh param eterization 39 Outline Topological mesh correction Topological mesh correction Mesh correction Application B Fischl A Liu A Dale Automated manifold surgery Constructing geometrically accurate and topologically correct models of the human cerebral cortex IEEE Transactions on Medical Imaging 2001 The extracted cortical surface may contain small errors Fixing these errors is necessary for correct Visualization and to simplify texture mapping CS 593 791 West Virginia University Medical Image Analysis Topological mesh correction Topology These surfaces are different from each other in a fundamental way Topology is the study of these properties CS 59 4 irginia Univer 39 y Medical Image An Topological mesh correction Topology o Topology is sometimes called rubber sheet geometry objects which can be continuously deformed into one another are considered topologically equivalent 0 Continuous deformation no cutting or gluing o Topologically equivalent objects have some common properties CS 59 4 irginia Univer 39 y Medical have All a a Topological mesh correction Topology These shapes each have a different genus or number of handles Technically genus is the maximum number of nonintersecting cuts along simple closed curves which do not disconnect the manifold The curves must be topologically distinct CS 59 4 irginia Univer 39 y lVIedical have An a logical mesh correction Topology 0 Differential geometry gt normal vector gt tangent vector gt curvature o Topology gt genus number of handles gt orientability gt connected components lVledical Lnage Anal 39 39 Euler number A mesh is a piecewise planar approximation to a manifold For a closed surface mesh the Euler number x XV7EF272g 0 V number of vertices o E number ofedges o F number of faces 0 g genus If surfaces f1 fz have the same genus then there eXists a mappingM f1 Hfz That mapping M is continuous invertible and M 1 is continuous M is a homeomorphism I an Apnl 2007 9 W33 39 Euler number Examples o Thecubehas V8E12F6sz2andg0 o The subdivided cube has V 12E 20F 10 so X 2 andg 0 o The square has V 4E 4F 1 so X 1 gt Cannot use this version of the Euler characteristic to compute genus gt This shape is not closed gt There is another form of X which accounts for open boundaries 2nd April 2007 10 3933 39 Euler number We know the cortical surface has genus 0 topologically equivalent to a sphere The Euler number x tells us how many topological errors there are in our mesh The Euler number does not tell us where those errors are or how to X them The approach of Fischl et al 0 Find a mapping M from the original surface to a sphere 0 Remove all regions where the inverse mapping is multivalued 9 Retriangulate the holes 2nd April 2007 1133 39 Spherical In ation In ate the mesh into a sphere by updating vertex positions X using x2 x2 F3 WK 0 F3 smoothing force 9 FR radial force moves vertices toward the surface of the enclosing sphere I Zhd April 2007 was 39 Smoothing Term 1 1 V FS ngkj Xk g 2 2niniTXXj Xi 139 16M 0 First term move each vertex toward the centroid of neighbors 0 Second term project outward to correct for shrinkage I Zhd April 2007 13 W33 39 Radial Force FR Rk Xk where Rk is the radial projection of xk onto the sphere of radius R After convergence every vertex is projected onto the sphere ZhdAp39rjl 2007 12 33 39 Detection of surface defects The topological errors will appear as overlapping triangles on the sphere These are regions where the mapping is noninvertible Discard all vertices belonging to overlapping triangles 2ndApn39I 2007 15 3933 Topological mesh con ection Types of surface defects Handles and holes Medical Image Analy 39 39 Surface retriangulation 0 Form a sorted list of the removed edges We will see the sorting criterion later 9 Traverse list adding edges back to the mesh if they don t result in overlap 9 Triangulate the remaining small holes I zhd April 2007 17 33 Topological Inesh correction Surface retriangulation Observe When smoothing the mesh the edges belonging to the topological error will be stretche Retriangulate using an edge list sorted on this edge length CS 593 791 Wes rginia University Medical Image Analysis Topolob Hection Results Si 39 quotWR L xv t x p 5 B lm I39 W W E Small handle successfully removed quottVirginia UI39 7 39 I 2 dAp l32007 1933 39 Outline Mesh in ation Application B Fischl M Serene A Dale Cortical surfacebased analysis Neurolmage 1999 Visualization accumer ponmy distances 1m ApanEIW 2133 39 Distortion minimization Distortion energy V Ed 2di39quot 7 d ny i1n N whereom HXHLH This repiesents the change in local shape between time 0 and time t 39 Smoothness constraint Spring energy between neighboring vertices 1 V Eg zzuxkxw i1 MEN 9 Vertices in concave regions move outward o Vertices in convex regions move inward I Zhd April 2007 23 33 39 Surface in ation minEs ME 0 Ad controls the tradeoff between smoothness and distortion 0 Where curvature is high smooth the mesh 0 Where curvature is low minimize distortion zhd April 2007 24 I33 39 Results This gives us the smoothed shape but not the means to texture map the surface ind Ap 39ljzpm r25 3339 39 Outline 9 Spherical mesh parameterization Spherical mesh parameterization Mesh parameterization Application Texture mapping is a mapping from a mesh into the domain of some image T s gt R2 The 3 vertices of each triangle map to 3 vertices in the plane A triangular region of the image can be mapped to each triangular face of the mesh Man12w 27 CS 593 791 West Virginia University Medical Image twalysis Application In ating the brain surface allows the areas Within the deep folds to be seen This allows the activation map to be more easily Visualize 7 2mman 2233 Spherical mesh parameterization Texture mapping We cannot map our closed genus 0 surface to the plane without introducing cuts so we map the surface to a sphere and compose this mapping with a known spherical texture mapping scheme quot 1 r H 1 quot I gt 39 w39 I 1 39 A 51H r quotFllv Ma aaf nequot a an quot35 vquot 3v 13 Haquot w CS 593 5 791 West nia University l 2nd April 2007 29 33 39 Disto ion minimization Previously We saw how to map a surface to a sphere While minimizing the squared change in edge length O This does not prevent stretching area distonion or angular distortion We may prefer to minimize a combination of angular and stretch dislonion am pnbm an my Distortion minimization i It 211d Ap l ilOO 31 W33 Spheiical mesh 1 unsteiiz ation Distortion minimization W E 391 2 ea 1 any i A AV a i MW 9 a oysgma A Aquot 5 mK v v 1 a V g g in V s V 1 Vs 15 431 W6 lt p t VA lt5 5quot 4 Q yen Minimizing only stretch can lead to angular distortion 52nd April 2007 32 33 CS 593 a 791 West Virginia Univeisityt What about texture mapping higher genus spirfaees We must cut the surface green lines into a disk then map the disk to the image plane The problem the cut path will in uence the distortion of the mapping m Wgrjuawrmmim 2nd Apri12007 3333 Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 18th April 2007 39 Outline 0 Reactiondiffusion 39 Outline Reactiondiffusion 0 Turing Model 0 Other Models and applications 0 Tensor Field Visualization O Generalized ReactionDiffusion I mung Model Introduction 9 Proposed as a model of pattern formation in animals gt zebra stripes gt leopard spots a It models the chemical reaction between two morphogens o Simultaneously those chemicals diffuse Within a developing organism TU39RING A M 1952 The chemical basis of morphogenesis Philosophical Transactions of the Royal Society B 237 3772 ismApni 2007 A 3923 39 Turing Model 39mnng Moaei The reaction model proposed by Turing at 3V 5 duV2uuviuia dVVZV B 7 uv7 0 M7 v concentration of the 2 morphogens o a 7 dv are diffusion rates a a B are constants controlling the reaction process lBLhJApnl 2007 5723 TumgM a4 Generated textures TURK G 1991 39 on arbitrary 39 39 Aquot 39 In Proceedings ofthe 18th annual conference on Computer graphics and interactive techniques 289298 imApmznm 5 23 Generated textures TU39RK G 1991 Generating textures on arbitrary surfaces usmg reactionrdtffusxon 1n meeedmgs ofthe 1m annual conference on Computer graphics and interactive techniques 2897298 m magmas 19mman 72 I 39mnng Moaei Anisotropic reactiondiffusion diVDuVu uV 7 u 7 Oz 8 3 diVDVVv 3 7 W o Du7 DV are diffusion tensors WITKIN A AND KASS M 1991 Reactiondiffusion textures In Proceedings of the 18th annual conference on Computer graphics and interactive techniques 299308 18mApn1 2007 Tunng Model Anisotropic reac onndiffusion 15336211 m aw 39 18m Apnl 2007 9 Z3 Fowler at al Seashells FOWLER D R MEINHARDT H AND PRUSINKIEWICZ P 1992 Modeling seashells In Computer Graphics Proceeding of ACM SIGGRAPH 92 ACM Press New York NY USA 379387 I rsmApdlzom 1023 Kondo and Asai Fish c04 08 12 16 20 24 28 32 36 40 44 48 KONDO 5 AND ASAI R 1995 A reactiondiffusion wave on the skin of the marine angel sh pomacanthus Nature 376 765768 1723 39 Preusser and Rumpf Flow Visualization divAv VP5VP mp a Only a single equation not a coupled system c Tensor1027 Vpe is constructed so that el is parallel to ow direction v 0 Reaction function f p improves the contrast of the resulting image PREUSSER T AND RUMPF M 1999 Anisotropic nonlinear diffusion in ow visualization In IEEE Visualization 325332 1r8mAp ri1 2007 12 23 Preusser and Rumpf The reaction function fltpgt 00M5 10 7 Concentrations image intensity 6 0 l 0 Two stable concentrations O and l 0 One conditionally stable concentration 05 CS 593 791 West Virginia University Medical Image Analysis li8thApri1200 7 13 39 Preusser and Rumpf Constructing the tensor A Construct from the eigenvector decomposition In general 7 MD 0 AiBWV 0 GWVMWWA 3 where a G are scalar functions which determine the eigenvalues of A and p5 is a smoothed version of p The eigenvectors are given by the columns of B l l BhMtwwd where v is the ow velocity and Vf V1 are orthogonal to v 1398LhJApril 2007 123 3923 Results Tznsax mg Visuhmnan Evolution of p 2114 different times mh Am 2m 1523 39 Tschumperle and Deriche DT NIRI Visualization Using the fact that diVDVI trDH V diVD where H is the Hessian matIiX of I and diVD11D12T div D d1VD21DzzT the reactiondiffusion equation was approximated as 3M 5 trDH IBLhJAp l 2007 1523 o O X 39 w r D Tschumperle and R Deiiche Tensor Field Visualization with PDE s and Application to DT MRI Fiber Vlsualization In Proc of Vaiiational and Level Set Methods 2003 18111Ap 1 2007 1 39 Kindlmann et a1 Reaction Diffusion in 3D Gordon Kindlmann David Weinstein David Hart Strategies for Direct Volume Rendering of Diffusion Tensor Fields IEEE Transactions on Visualization and Computer Graphics 62 1247138 AprilJune 2000 I ixmApnl zum 18123 39 NonGaussian Diffusion Einstein s formulation for the concentration C of particles undergoing Brownian motion in 1D is Cxt739 p7rCx7rtdr 1 Expanding C x 7 r7 t in a Taylor series about C x7 t ac r282C r3830 Cx7 r7t Cx7t 7 FE 7 2 Substituting 2 into 1 we see that 8C 00m T 7 cm mmmdr 7 Emrrdr 182C 1 83C EWp7rrzdr p7rr3drm Note that the quantities in square braces are momentsrof the distribution 771quot I l39SLHJAp l 2007 19 23 I Since fp7rdr l and using the fact that lim Cxt 739 7 Cx7t 8g 7H0 739 at then for 739 ltlt 1 ac aclt gt82Cltr2gt 83Cltr3gt 7 7 3x2 2739 8x3 3l739 E 7 75 where angle brackets lt gt denote moments of 771quot 0 First term drift 0 o Truncating this series at the second term yields the familiar isotropic diffusion equation a Extending to 3D yields the KramersMoyal expansion of the diffusion equation 1398LhJAphl 2007 V20 3923 39 KramersMoyal expansion 31 2 3 4 E d1vDS vju D vjku ijk vjw The nabla symbol V00 here is a compact notation denoting a tensor of rank n Whose components are partial derivatives of degree n V531 L quot 8xi18xiz8xn The rankn tensors Dquot are related to the central moments of the molecular displacement probability 0 1st moment mean is a rank 1 tensor vector 0 2nd moment covariance is a rank 2 tensor 0 3rd moment skewness is a rank 3 tensor 9 4th moment kurtosis is a rank 4 tensor I l398LhAphl 2007 21 23 smmmwxw Generalized diffusion propagators av xy 1 Amsolxopxc Gaussian c Noanausslan Rank f i xx 9v w w 3 a Noanausslan Rank 4 a NomGaussm Rank 4 a Nnanzussxan Rank 4 Generalized reactiondi usion textures s 791 mm irginia University t lathApn39l 21101 23 123
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