1. Istituto per le Applicazioni del Calcolo "M. Picone" Multigrid Computation for Variational Image Segmentation Problems: Euler equations and approximation Rosa Maria Spitaleri Istituto per le Applicazioni del Calcolo-CNR Viale del Policlinico 137, 00161 Rome, Italy e-mail: spitaleri@iac.rm.cnr.it Advances in Numerical Algorithms - Graz, September 10-13, 2003

2. Istituto per le Applicazioni del Calcolo "M. Picone" Variational Image Segmentation and Computational Approach minimization of the Mumford-Shah functional definition of a sequence of -convergent functionals solution of associated Euler equations finite difference approximation nonlinear system solution multigrid computation geometric and synthetic images visualization : computed solution (reconstructed image and edge ), convergence histories

3. Istituto per le Applicazioni del Calcolo "M. Picone" Segmentation Problem appropriate decomposition of the domain of a function (computer vision ) is the strength of the light signal striking a plane domain at the point with coordinates the function is called image

4. Istituto per le Applicazioni del Calcolo "M. Picone" Discontinuity Causes light reflected off surfaces of solid objects, seen from, the camera or eye point, will strike the domain (retina or film) in various open subsets which could have common boundaries (edges of the objects in foreground), surfaces with different orientation (edges of a cube), discontinuity in illumination (shadows), textured, partially transparent, highly-reflecting objects,...

5. Istituto per le Applicazioni del Calcolo "M. Picone" the segmentation problem consists in computing a decomposition of such that the image varies smoothly and/or slowly within each the image varies discontinuously and /or rapidly across most of the boundary between different computing optimal approximations of by piece-wise smooth functions (restrictions to the pieces differentiables)

6. Istituto per le Applicazioni del Calcolo "M. Picone" Mumford-Shah Functional Given the image let be a real function defined on a domain and a decomposition of such that, where and the boundary of

7. Istituto per le Applicazioni del Calcolo "M. Picone" MSF Definition the MSF is defined in the following form: where, the Hausdorff measure of, assigned parameters.

8. Istituto per le Applicazioni del Calcolo "M. Picone" MSF Minimization approximation of by smooth on each the boundary as short as possible

9. Istituto per le Applicazioni del Calcolo "M. Picone" Parameters can be calibrated to eliminate false edges, created by noise, and save the actual image; is a scaling parameter, controls the noise effects; defines the threshold to detect the edge

10. Istituto per le Applicazioni del Calcolo "M. Picone" Interest and Expectation is a cartoon of the actual image : sa new image in which the edges are drawn sharply and precisely and the objects are drawn smoothly without texture, sidealization of a complicated image, representing essentially the same scene droping any of the three terms : swithout the first:, swithout the second:, swithout the third:

11. Istituto per le Applicazioni del Calcolo "M. Picone" Variational Convergence the problem of minimizing has been conjectured to be well posed (open problem), these functionals have minimizers in the spaces of Special functions of Bounded Variation (SBV), the minimization problem is difficult for the presence of the set of discontinuity contours as unknown variational convergence to solve minimization of functional depending on discontinuities: –approximation of a variational problem by a sequence of more tractable problems

12. Istituto per le Applicazioni del Calcolo "M. Picone" the sequence of functionals on a metric space is -convergent to the functional if, : (i) sequence converging to (ii) a sequence converging to such that -Convergence

13. Istituto per le Applicazioni del Calcolo "M. Picone" Properties variational property- sequence of functionals on -convergent to -if a sequence of minimizers of converges, then the limit is a minimizer of and converges to the minimal value of k the -convergence is a variational convergence -convergence stability under continuous perturbations- Let be a continuous functional

14. Istituto per le Applicazioni del Calcolo "M. Picone" -Convergent Functionals sequence of -convergent functionals or stability property:

15. Istituto per le Applicazioni del Calcolo "M. Picone" Discontinuity Curves by Control Function the function controls the gradient of and has values ranging between 0 and 1, the minimizer is close to 0 in a neighbourhood of the set, which shrinks as, and close to 1 in the continuity regions the gradient of thus is permitted to become arbitrarily large along (jumps in the solution) the minimizers converge to a function equal to 0 along and 1 everywhere else : the -limit does not depend on

16. Istituto per le Applicazioni del Calcolo "M. Picone" minimizers of are the solutions of the following coupled Euler equations Neumann boundary conditions local minimum of the associated functional more accurate approximation as k increases Euler Equations

17. Istituto per le Applicazioni del Calcolo "M. Picone" the discretization of with grid spacing h andthe finite difference operator ( complete approximation of u and z ) Finite Difference Approximation

18. Istituto per le Applicazioni del Calcolo "M. Picone" Solution Algorithm equation systems on the grid, with mesh size h and covering the domain : onegrid (, l is the grid level ) computation of the solution

19. Istituto per le Applicazioni del Calcolo "M. Picone" initial guess for GS-relaxation applied to each system associated to the functional for a fixed value of the index k observations: the discretization step of the finite difference method should decrease as k increases optimal choice of the parameters and is a delicate problem Gauss-Seidel relaxation rotated lexicographical ordering: a grid point precedes another point if and only if

20. Istituto per le Applicazioni del Calcolo "M. Picone" Image Segmentation a given image, and geometrical and realistic problems: –one or more squares and circles, a vase computed results: smoothed image and control function (discontinuity contours) convergence histories: residuals, norms, logarithm values, iteration numbers result visualization

21. Istituto per le Applicazioni del Calcolo "M. Picone" Experimental Evaluation experimental choice of the parameters:, image resolution: 64x64, 128x128, 256x256, 101x101 brightness measurements : 256 levels initial guess: equal to the input image, equal to 1 everywhere even small values of k can be used

22. Istituto per le Applicazioni del Calcolo "M. Picone"

24. k and h link discontinuity set, we can define in, where is the distance of from and (convergence in ) between 1 and 0 in, : we have mh =10 l = 20/k where m is the node number in this interval for a given h by p = hk we can control the gap approximation

25. Istituto per le Applicazioni del Calcolo "M. Picone" Conclusion We have defined a multigrid finite difference method able to improve numerical solution of Euler equations in variational image segmentation Application to segmentation problems shows the capabilities of the method in computing solutions and providing satisfactory convergence histories Future research deals with improving performances of multigrid computation