The Chan-Vese Algorithm

Transcription

1 The Cha-Vese Algorithm Proect Report Rami Cohe, Itroductio to Medical Imagig, Sprig 010 Techio, Israel Istitute of Techology This report is accompaied by a MATLAB package that ca be requested by mail. 1

2 Table of Cotets 1. Itroductio...3. The model Level Set formulatio Numerical scheme Reiitializatio of Summary of the algorithm Code Results Coclusios Bibliography Appedix Table of equatios Table of figures

3 1. Itroductio Segmetatio is the process of partitioig a digital image ito multiple segmets (sets of pixels). Such commo segmetatio tasks icludig segmetig writte text or segmetig tumors from healthy brai tissue i a MRI image, etc. Cha-Vese model for active cotours [1] is a powerful ad flexible method which is able to segmet may types of images, icludig some that would be quite difficult to segmet i meas of "classical" segmetatio i.e., usig thresholdig or gradiet based methods. This model is based o the Mumford-Shah fuctioal [] for segmetatio, ad is used widely i the medical imagig field, especially for the segmetatio of the brai, heart ad trachea [3]. The model is based o a eergy miimizatio problem, which ca be reformulated i the level set formulatio, leadig to a easier way to solve the problem. I this proect, the model will be preseted (there is a extesio to color (vectorvalued) images [4], but it will ot be cosidered here), ad MATLAB code that implemets it will be itroduced. Figure 1: left: usig Cha-Vese method, right: usig gradiet based method 3

4 . The model Let be a bouded ope set of give image, ad Cs is a piecewise 1 0,1, with its boudary. Let u : 0 be a C parameterized a curve. Let's deote the regio iside C as, ad the regio outside C as \. Moreover, c 1 will deote the average pixels' itesity iside C, ad c will deote the average itesity outside C (i.e., c c C, c c C ). 1 1 Figure : the image is defided o (the big rectagle), the (arbitrary) red curve is C The obect of Cha-Vese algorithm is to miimize the eergy fuctioal,, defied by: 1,, F c c C Legth C Area iside C u x, y c dxdy u x, y c dxdy iside C outside C Equatio 1: The eergy fuctioal F c c C, Where 0, 0, 1, 0 are fixed parameters (should be determied by the user). As suggested by the paper, the preferred settigs are 0, 1 1. It should be oted that the term Legth C could be re-writte more geerally as p Legth C for p 1, but usually p 1. 1 I other words, we are lookig for c1, c, C that will be the solutio to the miimizatio problem: if F c c c C, c, C 1,, 1 Equatio : The miimizatio problem 4

5 3. Level Set formulatio Istead of searchig for the solutio i terms of C, we ca redefie the problem i the level set formalism. I the level set method, C is represeted by the zero level set of some Lipschitz fuctio 1 :, s.t.:, :, 0, :, 0 C x y x y iside C x y x y outside C \ x, y : x, y 0 Equatio 3: Level set formulatio Figure 3: The siged distace fuctio I the followig implemetatio, give a cotour C, xy, siged distace fuctio from C, where outside C the sig of xy, is defied as the is egative. 1 Lipschitz fuctio is a fuctio f such that f x f y C x y for some costat C which is idepedet of x ad y. For example, ay fuctio with a bouded first derivative is Lipschitz. 5

6 Our obect is to evolve xy, zero level set of x, y, t. Figure 4: Example for ad its zero level set, whe the evolved cotour C i each time t is the We ca re-write the fuctioal F c, c, C i terms of xy, 1 oly: 1. Legth C ca be calculated as the legth of the zero level set xy, 0,, Legth C H x y dxdy x y x y dxdy Where. Hz is the Heaviside fuctio: H z 1, if z 0 0, if z 0 Area iside C ca be calculated as the area of the regio i which xy, 0:, Area iside C H x y dxdy 3. u0 x, y c1 dxdy ca be calculated i terms of xy, iside C cosiderig oly the regio i which xy, 0:, 0:, whe 6

7 iside C x, y : x, y 0 u x, y c dxdy u x, y c dxdy u x, y c H x, y dxdy 4. i a similar way: outsidec x, y: x, y0 u x, y c dxdy u x, y c dxdy u x, y c H 1 x, y dxdy 5. The average itesities: c 1 0,, u x y H x y dxdy H x, y dxdy, c 0, 1, u x y H x y dxdy H 1 x, y dxdy The above leads us to the eergy fuctioal i terms of c1, c, (where, ad c c c c 1 1 is Dirac delta fuctio): 0 x 1,, 0,,, F c c x y x y dxdy H x y dxdy,,, 1, u x y c H x y dxdy u x y c H x y dxdy Equatio 4: Eergy fuctioal i terms of Observig the terms i equatio 4, we ca say that the evolutio of the curve is iflueced by two terms ( is usually set to 0, so we will igore it): the curvature regularizes the curve ad makes it smooth durig evolutio; the "regio term" u c u c affects the motio of the curve [5] The term 0 x, y, x y dxdy is the pealty o the total legth of the curve C. For example, if the boudaries of the image are quite smooth, we will give a larger value, to prevet C from beig a complex curve. 1, affect the desired uiformity iside C ad outside C, respectively. For example, It would be advisable to set 1 whe we expect a image with quite uiform backgroud ad varyig grayscale obects i the foregroud. 7

8 Usig Euler-Lagrage equatios ad the gradiet-descet method, it is show i the paper that xy, which miimizes the eergy,, artificial time): F c c satisfies the PDE ( t is a 1 p1 p dxdy div 1 u0 c1 u0 c t p1 p dxdy 1 u0 c1 u0 c x, y,0 o x, y, 0 o Equatio 5: PDE for x, y, t Where is the curvature of the evolvig curve (for some specific height level i ). We saw at class that the curvature ca be calculated usig the spatial derivatives of up to secod order: xx y xy x y yy x 3/ x y Equatio 6: Curvature Figure 5: Curvature at a poit ca tell how fast the curvature is turig there 8

9 4. Numerical scheme First, we defie regularizatios of H x ad d ): dx x (where x H x H x 1 x 1 arcta 1 x x Equatio 7: Regularizatios of the Heaviside ad Dirac delta fuctio For some costat 0. The values used i the simulatios are h 1, where h is the space step (it is reasoable to choose h, sice h is the smallest space step i the problem). These regularizatios achieve good results i simulatios, as described i the paper, i the sese that they usually lead to the global miimum of the eergy. 1 Regularizatio of the Heaviside fuctio Regularizatio of the Dirac delta fuctio Figure 6: Regularizatios 9

10 Let's defie i, t, xi, y where t is the time step. The PDE ca be discretisized by usig the followig otatios for spatial fiite differeces (where h h h ): x y h h /, / h x x i, i, i1, i, i1, i, /, / h y y i, i, i, 1 i, i, 1 i, The liearized, discretized PDE becomes: x x i, x 1 i, / i, 1 i, 1 / i, i, h h h i, t h y y i, y i, / h i1, i 1, 1 / h u c u c i, i, h i, 1 o 1 o Equatio 8: The liearized, discretized PDE Defiig the followig costats (for a give C C ): 1 1, C / 4 / 4 i1, i, i, 1 i, 1 i, i 1, i1, i 1, , C i1, i 1, / 4 i, 1 i, i1, 1 i 1, 1 / 4 i, i, We get the simplified equatio: p1 p L C C C C 1 t i, 1 h i, h t h t h i, 1 u0 c i, 1 u0 c i, p1 p L C C C C i, h i, 1 i 1, i 1, 3 i, 1 4 i, 1 Equatio 9: Simplified PDE 11

11 Where we calculate c1, c as discretized sums, usig the regularized Heaviside fuctio. I additio, L is Legth C as was calculated at sectio 3. As suggested i the paper, i order to solve equatio 10, we ca use a iterative way, as poited out by [6], propositio 6.1, i which it is also proved that there exists a solutio to the equatio. 4.1 Reiitializatio of I each step, we eed to reiitialize xy, to be the siged distace fuctio to its zero level set. This procedure prevets the level set fuctio from becomig too "flat", a effect which is caused due the use of the regularized delta fuctio x, which causes blurrig. The reiitializatio process is made by usig the followig PDE: sig 0 x, y, t x, y, t1 Equatio 10: Reiitializatio The solutio of this equatio,, will have the same zero level set as x, y, t, ad away from this level set, will coverge to 1, as it should be for a distace fuctio. The umerical equatio for equatio 11: Where the "flux" i,,, sig x y t G 1 i, i, i, Equatio 11: Numerical scheme for reiitializatio G is defied usig the otatios a, b, c, d, defied by: 11

12 x i, i, i1, x i, i1, i, y i, i, i, 1 y i, i, 1 i, a / h / h b / h / h c / h / h d / h / h ad a b c d xi y t max, max, 1,,, 0 G max a, b max c, d 1, x, y, t 0 i, i where a max a,0, a mi a,0 0, Equatio 1: defiitio of G ad so o. 4. Summary of the algorithm 1. Iitialize to some Lipschitz fuctio 0 0 i,. Compute c1, c 3. Solve the PDE of equatio 9 4. Reiitialize equatio 11 otherwise by usig 1 1 i, to be the siged distace fuctio to i, 0 5. Check whether the solutio is statioary. If ot, cotiue. Else, stop. I practice, the process should be stopped whe Q t h i, h 1 i, i, (this should be checked at stage o. 5) where M is the umber of grid poits which satisfy i, h, because x, y, t is ot expected to chage aymore (except for maybe some small umerical chages). 1

13 5. Code The umerical scheme above was implemeted usig MATLAB. The attached source files are (more documetatio ca be foud i the source code): 1. CV.m the mai file.. delta.m regularized Dirac delta fuctio (equatio 7). 3. heaviside.m - regularized Heaviside fuctio (equatio 7). 4. reiit.m reiitializatio of by the method proposed at i, 5. cur_diff checkig whether i, reached its steady state accordig to the method proposed at 4. The iput images should be grayscale oly; otherwise they will be coverted ito grayscale. Moreover, there is a optio to select oly a (rectagular) part of the image, ad the segmetatio process will be applied oly to this part. It ca come i hady whe the image is large ad we wat to segmet ust a part of it. As always, larger images eed more computatioal time. It is highly recommeded to use images of a small size (i.e., o more tha 56x56 pixels), ad to ot stop the segmetatio process before it eds (i other words, do't use ctrl+c). I weak computers it ca cause some istability. 13

14 6. Results The first example is similar to the example which appears at the paper a image with (approx.) oly two gray levels. Fial segmetatio Figure 7: left: clear obects, right: segmetatio time elpased: secods, 1 iteratios As ca be see from figure, a good segmetatio was achieved. Let's see how itdraw segmets a MRI image a rectagle usig your(resized): mouse Figure 8: left: MRI image, right: segmetatio Sice Cha-Vese algorithm is ot based o gradiet methods, it achieves good results eve whe the image is blurred: Figure 9: left: blurred obects, right: segmetatio 14

15 Figure 10: left: o oise, middle: modest oise, right: stroger oise I figure 10 we ca see that the algorithm deals quite well with oisy images ( p was set to i this case, to prevet C from surroudig the oisy dots). The algorithm ca also detect quite precisely thi edges: Figure 11: segmetaio of image with thi edges Some optimizatio has bee applied to the code, i order to decrease ruig time (workig o matrices without loops, etc.). For example, Image of size pixels eeds o average about 0.5 miute (whe usig the maximal umber of iteratios) for the segmetatio process. Whe preferrig shorter time o accuracy, a quite fair segmetatio ca be achieved withi ust several secods (3-10 iteratios). The complexity of the algorithm is OMN (i each iteratio) where M N is the size of the image. Improvemet for ruig time ca be achieved by applyig ay efficiet method that ca detect the regios i xy, which udergo the mai chages, so the updatig process will cocetrate o these regios (see later). Of course, may other results ca be obtaied by simply usig the attached code package, with proper selectio of the parameters. Usig the default parameters would also give fair results. 15

16 7. Coclusios Cha-Vese algorithm was implemeted i this proect. From the results above, it ca be see that this algorithm deals quite well eve with images which are quite difficult to segmet i the regular methods, such as gradiet-based methods or thresholdig. This ca be explaied by the fact that CV algorithm relies o global properties (itesities, regio areas), rather tha ust takig ito accout local properties, such as gradiets. Oe of the mai advatages of this approach is better robustess for oise, for example. As metioed before, the algorithm is sometimes quite slow, especially whe dealig with large images. It ca pose a problem for real time applicatios, such as video sequeces, ad a efficiet implemetatio is very importat. There are some papers which suggest refiemets to this algorithm, especially for the time-cosumig computatio of the PDE solutio. These methods use values that were already computed, i order to decrease the computig time of the ext values. Such approaches ca be foud i [5], [7]. CV algorithm is a very powerful algorithm, as we have also see i the results above. This algorithm marks some "moder" approach for image segmetatio, which relies o calculus ad partial differetial equatios. 16

17 8. Bibliography [1] T. Cha ad L. Vese, "Active cotours without edges," i IEEE trasactios o image processig 10(), 001, pp [] D. Mumford ad J. Shah, "Optimal approximatio by piecewise smooth fuctios ad associated variatioal problems," Comm. Pure Appl. Math, vol. 4, pp , [3] Olivier Rousseau ad Yves Bourgault, "Heart segmetatio with a iterative Cha-Vese algorithm," Uiversity of Ottawa, Otario, 009. [4] Cha et. al, "Active Cotours without Edges for Vector-Valued Images," Joural of Visual Commuicatio ad Image Represetatio, vol. 11, pp , 000. [5] Pa et. al, "Efficiet Implemetatio of the Cha-Vese Models Without Solvig PDEs," i Iteratioal Workshop O Multimedia Sigal Processig, 006, pp [6] G. Aubert ad L. Vese, "A variatioal method i image recovery," SIAM J. Num. Aal., vol. 34/5, pp , [7] Zygmut L. szpak ad Jules R. Tapamo, "Further optimizatio for the Cha-Vese active cotour model," i High perfomace computig & simulatio coferece,

18 9. Appedix 9.1 Table of equatios Equatio 1: The eergy fuctioal...4 Equatio : The miimizatio problem...4 Equatio 3: Level set formulatio...5 Equatio 4: Eergy fuctioal i terms of...7 Equatio 5: PDE for x, y, t...8 Equatio 6: Curvature...8 Equatio 7: Regularizatios of the Heaviside ad Dirac delta fuctio...9 Equatio 8: The liearized, discretized PDE Equatio 9: Simplified PDE Equatio 10: Reiitializatio Equatio 11: Numerical scheme for reiitializatio Equatio 1: defiitio of G Table of figures Figure 1: left: usig Cha-Vese method, right: usig gradiet based method...3 Figure : the image is defided o (the big rectagle), the (arbitrary) red curve is C...4 Figure 3: The siged distace fuctio...5 Figure 4: Example for ad its zero level set...6 Figure 5: Curvature at a poit ca tell how fast the curvature is turig there...8 Figure 6: Regularizatios...9 Figure 7: left: clear obects, right: segmetatio Figure 8: left: MRI image, right: segmetatio Figure 9: left: blurred obects, right: segmetatio Figure 10: left: o oise, middle: modest oise, right: stroger oise Figure 11: segmetaio of image with thi edges