alexandre cunha :: shape optimization

shape optimization

shape match poisson stokes navier-stokes

Read an extended abstract about this work.

Find below some of the results for model problems we have tested in one and two spatial dimensions.

shape matching problem

In the shape matching problem we try to match a target shape starting from an arbitrary initial configuration. We formulate this problem as an unconstrained minimization problem where the objective functional is the misfit F between the current and target shapes. X and X* are their corresponding continuous Heavisides/characteristic functionals shown in F. A regularization term R is needed to render the solution unique. Tikhonov and Total Variation are options we consider. We also regularize against a signed distance like function but this is primarily used on PDE constrained problems.

This model somehow resembles the Mumford-Shah model used in image segmentation. We tried it using Tikhonov regularization and segmented MRI brain images, our first attempt at variational image processing. Our results are promising. We need to add noisy though to make it more realistic and apply TV.

Test cases of shape matching follow below. Click on image to play an animation in Quicktime format.

1D examples

quicktime animation 253K (128 elements)

The shape match on a line is the problem of equalizing segment lengths. The animation shows the level set function evolution during the iterations of the nonlinear solver. We use boxes to visualize segments and when they align the matching is absolute. The orange box on the animation is the non-zero part of the binary Heaviside when applied to the evolving level set function which is plotted as an orange curve. The target segment is shown as a blue box. For visualisation purposes only, its height is twice the orange one. An iteration counter is shown on the top right corner. In this example, we started with a parabola and it took the solver 12 iterations to find an optimal solution. Observe the topology change from one to two segments. Used Tikhonov regularization.

2D examples

quicktime movie 632K (256 x 256 mesh). short quicktime 291K (128 x 128 mesh)

We are matching against two kidney beans in the plane. Continuous Heavisides, as opposed to binary Heavisides in the previous example, are plotted and projected onto the zero plane. This is an old, outdated example where we used a pure Newton step and did not apply continuation on the regularization parameter. We like to keep this example for reference reasons (and because it is a nice movie to show on presentations!). The final result is still a very good match but obtained at a much higher price than, for example, the example shown below for the same geometry. In this case, we started the level set function as a sphere. The solver took around 30 iterations (we could have stopped at 12 or so since there was no further improvement on the misfit after that point but we imposed the stop criterion to be the Lagrangian gradient). Used Tikhonov regularization.

quicktime movie 787K (64 x 64 mesh)

We have in this example the same kidney beans but the solver has been considerably improved. It took only 12 iterations for our Gauss-Newton-Krylov solver to obtain a perfectly matching result. Note that the shape is very close to the target after 4 or 5 iterations. The remaining iterations serve to capture the very fine details not seen by the naked eye (!) and to stabilize the regularization. The level set curves are plotted on the plane together with the evolving interface and target beans. We started with a signed distance function and it deforms to something else (observe the non equidistant level curves at the end). Check the 3D animation below. Employed Tikhonov regularization.

quicktime movie 511K (64 x 64 mesh)

A 3D view of the evolution of the level sets for the example shown above.