# Adaptive finite-element meshes for progressive contour models

With this scheme, a solution hascontinuous derivatives up toorder atthenodal 2 points.

3.2 Multigrid Algorithm

To achieve good convergence we make use a rate, of themultigrid algorithm have been successwhich fully applied a broad to range problems. the proof In cess correspondence search mentioned 3of inSection 2.1, this multigrid cansignificantly the scheme reduce search space.coarse element fewer Ina mesh, elements are considered each and correspondence search inspects only the points a coarse On the other the in grid. hand, size a search of region be reduced a finer can in ele{ C k } converges the conto the has been improved tour defined the by final correspondence mapping. mentmesh since initial contour This iterations. property due totheseparationthedisplacement by previous is of Another advantageincorporating of the multigrid alcomponent from contour the shape. While the displacegorithm the isadaptation ofregularization parameters. ment of a deformation process issmoothed, smoothmodelsarises when a method ness constraintsnotapplied thecontour are to shape The needforadaptive basedon the regularization fails fit theory to high directly. Therefore, contour smoothly prothe will and curvature featuresthe smoothness constraints. due to gressively approach uniquely a defined contour through A convenient solution to reduce regularization is the these deformations. parameters implicitly an element mesh transwhen is formed into finer a mesh. Let's consider contour a 3 Implementation measured two different scales. In the discretize in formof Euler-Lagrange equation, of normaliza3 set 3.1 Finite-Element Method tion regularization (ci, should fixed parameters p) be in order maintain samelevel accuracy the deto the of in The FEM (Finite-Element Method) a technique rivatives. isalso is This true for the FEM because the of constructing approximation required functions in fitting ability an element depends of on the element a piecewise application variational of a method such length A scale h. invariant relation between the ele(151. Although finite differencesment and smoothness constraints imposed as theRitz methods the canbe provide simpler of discretization, a way they require by specifying fixedas follows: (&, p) the contour represented tobe by relatively dense points; otherwise, estimation and curvatures the ofslopes is inaccurate. representation become problem This may a especially when the modelisextended 3-D contour to t 1 parameter. canbe This where E [0,1isthe unified space. The FEM becomes more appealing than FDM verifiedchanging parameter [0, to E [0,1] by the sE I] t since helps reduce the number it to of variables and in the internal energy as follows: (3) providesstable a solution controllable precision. with 1 ' In thefirst of thefinite-element step analysis, the Ei,t(C) 5 = (.jt)llmt)l12 P(t)llfi"(t)l12 dt (6) 0 contour can be geometrically divided into finite a number of subcontours subdomains) l , . ., (or i l iln, . RM, When an element mesh is subdivided, precision more called elements. Then, displacements the D and their canbe obtained tothereduced element length. due In D' atthe oftwo first derivatives intersections elements order match the to smoothness constraintswe withh , areset the nodal as variables. an interpolation Using should (ci,)again. simplify implementation, fix p To the function, each element be interpolated nodal can by the instead using andmodifyingap)whenever of (3), ( , the variables.variational A formulationEquation 5 ) of ( grid changes, usethe equivalent energy we tan internal canbe constructedeachelement using nodal for these in( 6 )where(&,P) , andt are used. variables asunknowns. Finally, values the the of nodal by elevariables be obtained solving the assembled 3.3 Unconstrained can Optimization ment equation ofthe whole contour. finite-element The Itis typical a situationa numerical solution in of analysisthe contour model the of using Hermite cubic forthe a function as the interpolation is function provided [9]. nonlinear problems userto provide starting in

1

+

127

use of the model where an initial contour can be specifiedarbitrarily. The aforementioned interactive process repeated is several timeswith different selections control points of for each the images in 1. of Fig.

2. e., the Euler-Lagrange equation, required is in a variational approach. Also, the inverse-matrix calculationis not necessary the optimization.This is in especially important whenextending the model3-D to because assembled element the equations are longer no to banded andtheybecome either difficultor expensive solve. Furthermore, quasi-Newton algorithms suchas the model-trust regionapproach [17] for nonlinear optimization have been developedhandlethe situation to in which the starting point not closeto the solution. is As a consequence, the contours not required initial are to be closeto the desired boundaries.

4

ExperimentalResults

The purpose the experiment to evaluate the of is performance the model. We first quantitizevarious of factorsthat may affect the computation and test then the method selected on images controlling measby or uringthese factors. It is obvious thatthe computation is dependent cost on the difference between the initial contourand the desiredcontour. To analyzethis factor,we measure the difference between two contours computing by the areaof the exclusive-or resultof two contour regions,

(c) ( 4 Figure 1. Images and the contours of (a) a tumor, (b) a vertebra, (c) a liver and (d) a ventrical.

whereR(C)denotes the regioncircumscribed C, and by A denotes the of a region. area In our model, priorknowledge contained in the the is initial contour.In orderto quantify this amount, the experiment conducted thefollowing is in way. The user incrementally specifies control points, whicharelocated on thedesired contour, until the program ableto genis erate the desired contour.Thisis an interactive process betweenthe user andthe program. The initial contour is interpolated from the specifiedcontrol points. The numberof the required control pointsthis process in indicates amountof prior knowledge provided the by the user.Note that the restriction of specifying control points in this experiment not necessary for general is

We ran the program a 486 DX 33 MHz personal on computer. Figs. 2(a) and 2(b)plot thecomputation time t in seconds versus the naturallogarithmof the C)) under the attraction of contourdifferenceln(d(C0, all edges undertheattraction the desired and of contour edges, respectively. The resultsshow that the ratio t/ln(d(Co,C))is bounded a similarrange.Thisverifies by thatthe comusing the multigrid putational complexity a program of algorithm O(ln(d)). is

5

Concluding Remarks

We have developed progressive a contour model based on the ideaof deforminga contourfrom an initial shape minimizinga definedcontourenergy by to extracta desired contour from animage. The internal component the contour of energyis usedto impose the smoothness constraintsthe deformation on and to incorporate initial contouras a source the of prior knowledge. external component conThe of the tour energy usedto locate the correspondence for is

128

active contour models,” International Journal Comof vol. 321-331, 1987. puter Vision, 1,pp.

[a]L.D. Cohen,

March1991.

“On contour active models and balloons,”

Computer Vision, Graphics, and Image Processing: vol. no.2, pp.211-218, Image Understanding, 53, [3] A. Davatzikos, J .L.Prince, C. and “Segmentation and

mapping highly-convoluted with applicaof contours Proc.IEEE Conf.Acoustions medical to images,” New tics, Speech, and Signal Processing, York, NY, pp.111.569-111.571, 1992. [4]N. Rougon, and F. Preteux, “Deformable markers: Figure 2. Plots of computation time vs. natfor models mathematical morphology active contour ural logarithm of the contour difference (a) SPIE, and control,” ImageAlgebrea Morphological Imunder attractions of all edges, and (b) under II, 1568, 78-89, vol. pp. 1991. age Processing [5] Samadani, “Adaptive snakes: damping R. of control attractions of contour edges only. SPIE,Geometric Methods and material parameters,” vol. pp. in ComputerVision, 1570, 202-213, 1991. [6] Terzopoulos, D. “Regularization ofinverse visual probthe contour through local correspondence mapping. A lems involving discontinuities,” IEEE Trans. Pattern vol. 413Anal. Machine Intell., PAMI-8,no.4,pp. sequence deformations the contour determined of of is Jul. 1986. 423, by repeatedly deforming updating and the initialcon[7]D. Terxopoulos, “The computation of visible-surface tour. We haveshown thatthecontour deformed by this IEEE Trans. Pattern Anal. Machine representations,” a sequence will smoothly and progressively approach vol. no. 417-438, Jul. 1988. Intell., PAMI-10, 4,pp. uniquely defined contour. [8]A. Blake and A. Zisserman, Visual Reconstruction, The separation the correspondence criterion of from The MIT Press, 1987. the external energy calculation allows us to apply [9] Lin, R. W.-C. Lin, C.-T. and Chen, “Progressive Conthis model more flexibly. The model can be easily tour Models,” ProceedingsSPIE, in of Optical Engincustomized defininga specificcorrespondence criby Chicago, vol. IL., 2622, 824pp. eering Midwest ’95, terion.Moreover, complex forms basiccontours of cre835, May,1995. ate more applications the model. For example, for a [lo] Terzopoulos, “Sampling D. and reconstruction with contour model with an auxiliarynode inside conthe adaptive meshes,” Proc. IEEE Conf.Computer Vistour provides extraconstraints the contour on shape. ionPattern Recognition, 70-75, pp. 1991. Also, multi-contour models maintaingeometric rela[ I l l T.Tanaka and Kishino, H. F. “Adaptive mesh genfor parallel hierarcheration surface reconstruction: tionsbetween contours. ical triangulation discontinuities,” without Proc. IEEE The employment the unconstrained optimization of Conf.ComputerVision Pattern Recognition, 88pp. to in our method seems be the most attractive feature 94,1993. work. This motivates to investigate contour models us [la] Weiss, I. “Shape reconstructionvarying on mesh,” employing constrained optimizationorderto solve in IEEE Trans.Pattern Anal. Machine Intell., vol. problems subjectto certain constraints as area such PAMI-12, 4,pp. no. 345-362, Apr. 1990. preserving deformation and volume preserving deform[13] Terxopoulos, D. J.Platt, Barr, K. Fleischer, A. and ation. Computer Graphics, “Elastically deformable models,” A 2-D contourmodelsuchas the progressive convol. 21, 110.4, pp. 205-214, 1987. Jul. tour modelcan only be used process singleimage to a [14] D. Landau L. andE.M. Lifshitz, of elasticity, Theory at a time. For example, the application 3-D reconin of Pergamon Press, London, 1959. UK, struction fromserialcross-sectional images, desired the [15] N. Reddy, introduction tofinite J. An the element contour every sliceneeds be obtained of to separately. Inc., method,McGraw-Hill 1984. This limitationsuggests extension the 2-D conan of [16] Fletcher, R. PracticaI methods of optimization, Wiley, 1987. tour model a 3-D shellmodelwhichcan process to a [17]J.E. Dennis, and Jr., B. Schnabel, R. Numerical wholeset of 3-D datasimultaneously.

References

[l] M.

Methods forUnconstrained Optimization Nonlinand earEquations, Prentice-Hall, 1983.

Kass, . Witkin, D. Terzopoulos, A and “Snakes:

129