Edge preserving smoothing by Euclidean shortening flow

Transcription

1 Proceeding International Mathematica Sympoium Augut 5, Perth, Autralia UR: Edge preerving moothing by Euclidean hortening flow Deign of Algorithm with Mathematica Bart M. ter Haar Romeny, Marku A. van Almick Eindhoven Univerity of Technology Eindhoven, the Netherland B.M.terHaarRomeny@tue.nl, M.A.v.Almick@tue.nl Initialization Differentiation of dicrete data i done by convolution of the data with the derivative of a Gauian function (ee for a careful explanation [1]). Thi i implemented for D dicrete image with the function gd: gau::uage "gau x, i the normalized Gauian kernel function. For higher dimenion take product, e.g. gau x,subcript,x gau y,subcript,y.\n \t cale in pixel."; gau x_, _ ; : N 1 E x ; gd::uage "gd im,nx,ny,,option calculate the Gauian derivative of a D image by patial convolution. It i optimized for peed by 1D convolution per dimenion. The image i conidered cyclic in each direction. \n im D input image it.\n nx order of differentiation to x Integer.\n ny order of differentiation to y Integer.\n cale in pixel "; Option gd kernelsamplerange 6, 6 ; gd im_it, nx_, ny_, _, opt?optionq : Module x, y, kpleft, kpright, kx, ky, mid, tmp, kpleft, kpright kernelsamplerange. opt. Option gd ; kx N Table Evaluate D gau x,, x, nx, x, Floor kpleft 1, Ceiling kpright 1 ; ky If nx ny, kx, N Table Evaluate D gau y,, y, ny, y, Floor kpleft 1, Ceiling kpright 1 ;

2 36 mid Ceiling ength #1 &; tmp Tranpoe itconvolve kx, im, 1, mid kx, 1, mid kx ; Tranpoe itconvolve ky, tmp, 1, mid ky, 1, mid ky ; SetOption itdenityplot, Meh Fale, ApectRatio Automatic ; SetOption itplot, PlotJoined True ; Off General::"pell1" ; Graphic`; The function GetUR from the Jink example read data from the internet. Need["Jink`"]; GetUR[url_String, opt?optionq] := JavaBlock[ Module[{u, tream, numread, outfile, buf}, IntallJava[]; u = JavaNew["java.net.UR", url]; tream = u@openstream[]; If[tream === $Failed, Return[$Failed]]; buf = JavaNew["[B", ]; (* i an arbitrary buffer ize *) outfile = OpenTemporary[DOSTextFormat->Fale, CharacterEncoding->{}]; While[(numRead = tream@read[buf]) >, WriteString[outFile, FromCharacterCode[If[# <, # + 56, #]& /@ Take[JavaObjectToExpreion[buf], numread]]] ]; tream@cloe[]; Cloe[outFile] (* Cloe return the filename *)]]; Unprotect[Get]; Get[_String] := Module[{tempFile, re}, tempfile = GetUR[]; If[tempFile =!= $Failed, re = Get[tempFile]; DeleteFile[tempFile]; re, (* ele *) $Failed ] ] /; StringMatchQ[, " Protect[Get]; NotebookOpenUR[url_String] := NotebookOpen[GetUR[url]] Introduction Edge preerving moothing i a locally adaptive proce, where the ize of the blurkernel, applied to uppre the noie, i a local function of the edge trength [, 3]. Blurring i decribed by a partial differential equation, the diffuion equation. At trong edge, the conductivity of the diffuion i reduced. Thi wa firt introduced into the realm of computer viion by Perona and Malik [4] in It wa a huge ucce, a it alo enhanced the remaining edge. But at trong edge, the noie remained, and there wa a parameter k that had to be et.

3 Alvarez [5] came up with an elegant olution for both iue, by propoing a new nonlinear image evolution cheme, where 363 the local edge direction wa taken into account. Thi olution i known a Euclidean Shortening Flow. We firt give the theory and implementation of Perona & Malik nonlinear diffuion, then we focu on Euclidean hortening flow. The lat ection dicue an implementation on a noie ultraound image. Thi paper i baed on Chapter 1 of [1]. Nonlinear Diffuion Equation The diffuion equation i given by, where t i the diffuion time, i the gradient of the luminance, t and i the aplacian. Perona and Malik introduced a conductivity coefficient (c) i introduced in the diffuion equation: c, where c i a function of the local (differential) tructure, e.g. edge, c c, x, x,. Conider the analogon of diffuion of heat in a metal plate with a local iolator. Edge in image become the local iolator. Mathematica i ideal to explore the nonlinear diffuion equation firt, find a numerical (forward Euler) approximation, and implement the dicrete derivative by a convolution with Gauian derivative operator by pattern matching. The Perona & Malik Equation Perona and Malik [] propoed to make c a function of the gradient magnitude in order to reduce the diffuion at the location of edge: c with c k. So, the Perona and Malik (P&M) equation become k. Expanding the differential operator for the right hand ide, we get in D: Calculu`VectorAnalyi`; SetCoordinate Carteian x, y, z ; Grad x, y, ; Div Exp. Simplify k 1,1, x,y, 1,, x,y, k k k,1, x, y,,, x, y, 4,1, x, y, 1,, x, y, 1,1, x, y, k 1,, x, y,,, x, y, The mot traightforward numerical approximation of c i the forward-euler approximation c. The function gd[im,nx,ny, ] implement the derivative of the dicrete image im, nx time with repect to x, ny time with repect to y, at cale. It i defined in the initialization package. c E x x,y y x,y k ; pm im_, _, k_ x c x x, y y c y x, y. Derivative n_, m_ x_, y_ gd im, n, m, Simplify; We calculate the variable conductance diffuion firt on a imple mall (64, maximum: 55): 64) noiy tet image of a black dik (minimum: im Table If x 3 y 3 3,, 55 1 Random, y, 64, x, 64 ; Forward-Euler approximation cheme:

4 364 peronamalik im_, _, _, k_, n_ : Module, evolved im; Do evolved pm evolved,, k, n ; evolved ; where im i the input image, i the time tep, i the cale of the differential operator, k i the conductivity control parameter and n i the number of iteration. Here i an example of it performance: line Red, ine, 3, 64, 3 ; DiplayTogetherArray itdenityplot #, Epilog line & im, imp peronamalik im,.1,.7, 5,, itplot im 3, imp 3, ImageSize 38 ; Figure. Example of Perona & Malik edge-preerving moothing. The profile along the red line i depicted below. Note that the noie i removed, the edge have remained their teepne. Euclidean hortening flow Alvarez, Guichard, ion and Morel [5] realized that the P&M variable conductance diffuion wa complicated by the choice of the parameter k. They reaoned that the principal influence on the local conductivity hould be to direct the flow in the direction of the gradient only: thi lead to much diffuion along the edge, but virtually no diffuion acro the edge. Thi led to the propoal to make the intenity flow a function of the unit gradient vector co, in, o one obtain the (nonlinear) diffuion equation. Thi i implemented a follow:

5 Clear ; Calculu`VectorAnalyi`; SetCoordinate Carteian x, y, z ; Grad x, y, ; 365. Div. Simplify,, x, y, 1,, x, y,,1, x, y, 1,, x, y, 1,1, x, y,,1, x, y,,, x, y,,1, x, y, 1,, x, y, The enthuiam of Alvarez et al. about thi equation led them to the name fundamental equation of image proceing (!). Thi PDE i better known a the Euclidean hortening flow equation. The numerical iterative approximation to the nonlinear PDE become: euclideanhortening im_, nrtep_, _, evolutionrange_ : Module, imt, evolutionrange nrtep; imt im; Do imt gd imt,,, gd imt, 1,, gd imt,, 1, gd imt, 1,, gd imt, 1, 1, gd imt,, 1, gd imt,,, gd imt,, 1, gd imt, 1,,, nrtep ; imt ; Thi i an example for an ultraound image with it particular peckle pattern: imagefile = GetUR[" BRomeny/FEV/Image/ultraound-kidney.gif"]; im = Import[imageFile,"GIF"][[1,1]]; DeleteFile[imageFile]; Dimenion im 6, 345 DiplayTogetherArray itdenityplot im, Plotabel "Original", itdenityplot euclideanhortening im, 6,.8, 9, Plotabel "cale " ToString, ImageSize ; Original cale Figure. Euclidean hortening flow on a pixel ultraound image (ource: Note that the peckle i markedly reduced, and the edge remain teep.

6 366 Dicuion Thi edge-preerving moothing cheme i an adaptive blurring proce. It i implemented through an evolution equation, i.e. a nonlinear partial differential equation, the nonlinear diffuion equation. The performance i good, and can be extended to higher dimenional image, uch a 3D and 3D-time. Note that the enhanced image are calculated for improved contour and tructure analyi, not to be viewed for final diagnoi. A doctor will alway prefer the original data, while the enhanced image are good input for conequent tepd in an image analyi routine, e.g. a 3D egmentation for 3D viualization. Thi i a rich field, with many development [] and peed improvement by efficient numerical approximation [6]. See for a detailed overview alo [7]. Reference [1] B. M. ter Haar Romeny, Front-end viion and multicale image analyi. Dordrecht: Kluwer Academic Publiher, 3. Written in Mathematica. [] B. M. ter Haar Romeny (ed.), Geometry-driven diffuion in computer viion. Dordrecht: Kluwer Academic Publiher, [3] W. J. Nieen, B. M. ter Haar Romeny,. M. J. Florack, and M. A. Viergever, A general framework for geometrydriven evolution equation, Intern. Journal of Computer Viion, vol. 1, no. 3, pp , [4] P. Perona and J. Malik, Scale-pace and edge detection uing aniotropic diffuion, in IEEE Computer Society Workhop on Computer Viion, (Miami, F), pp. 16-, [5]. Alvarez, F. Guichard, P.. ion, and J. M. Morel, Axiom and fundamental equation of image proceing, Archive for Rational Mechanic, vol. 13, pp , September [6] J. Weickert, B. M. ter Haar Romeny, and M. A. Viergever, Efficient and reliable cheme for nonlinear diffuion filtering, IEEE Tr. on Image Proceing, Vol. 7, , [7] J. Weickert, Aniotropic diffuion in image proceing, ECMI Serie, Teubner Verlag, Stuttgart, ISBN About the Author Bart M. ter Haar Romeny received an MSc degree in Applied Phyic at Delft Univerity, and a PhD degree (1983) at Utrecht Univerity, the Netherland. He erved from 1983 till 1 a principal phyicit and aociate profeor at the Dept. of Radiology and the Image Science Intitute at Utrecht Univerity. He i now full profeor at Eindhoven Univerity of Technology, Dept. of Biomedical Engineering. Hi interet are medical image analyi and multi-cale computer viion, it mathematical foundation and clinical application. A focu on the human viual ytem give biological inpiration for the mathematic. He i the author of a popular interactive tutorial book on perceptually inpired multi-cale image analyi, written in Mathematica. Bart M. ter Haar Romeny Eindhoven Univerity of Technology Biomedical Engineering Image Analyi and Interpretation PO Box 513, WH MB Eindhoven The Netherland B.M.terHaarRomeny@tue.nl Marku van Almick Eindhoven Univerity of Technology Biomedical Engineering

7 Image Analyi and Interpretation PO Box 513, WH MB Eindhoven The Netherland 367