Optimization of Convex Shapes: An Approach to Crystal Shape Identification

We consider a shape identification problem of growing crystals. The shape of the crystal is to be constructed from a single interferometer measurement. This is an ill-posed inverse problem. The forward problem of interferogram from shape is injective if we restrict the problem to convex shapes with known boundary. The problem is formulated as a shape optimization problem. Our aim is to solve this numerically using the gradient descent method. In the numerical computations of this paper we study the behavior of the approach in simplified cases. Using H 1-gradients (inner products) acts as a regularization method. Methods for enforcing the convexity of shapes are discussed.

[1]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[2]  Bertrand Maury,et al.  H1-projection into the set of convex functions : a saddle-point formulation , 2001 .

[3]  L. Vese.  A method to convexify functions via curve evolution , 1999 .

[4]  L. Armijo.  Minimization of functions having Lipschitz continuous first partial derivatives. , 1966 .

[5]  James A. Sethian,et al.  Image Processing: Flows under Min/Max Curvature and Mean Curvature , 1996, CVGIP Graph. Model. Image Process..

[6]  Convex Bodies of Optimal Shape , 1997 .

[7]  Pedro Morin,et al.  Approximating optimization problems over convex functions , 2008, Numerische Mathematik.

[8]  Jan Erik Solem.  Variational Problems and Level Set Methods in Computer Vision - Theory and Applications , 2006 .

[9]  M. Delfour,et al.  Shapes and Geometries: Analysis, Differential Calculus, and Optimization , 1987 .

[10]  Otmar Scherzer,et al.  Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising , 2005, Computing.

[11]  Édouard Oudet,et al.  Minimizing within Convex Bodies Using a Convex Hull Method , 2005, SIAM J. Optim..

[12]  Bertrand Maury,et al.  A numerical approach to variational problems subject to convexity constraint , 2001, Numerische Mathematik.

[13]  T. O’Neil.  Geometric Measure Theory , 2002 .

[14]  A. Chambolle,et al.  Crystalline Mean Curvature Flow of Convex Sets , 2006 .

[15]  R. Jochemsen,et al.  Morphology and Growth Kinetics of 3He Crystals Below 1 mK , 2002 .

[16]  M. Burger.  A framework for the construction of level set methods for shape optimization and reconstruction , 2003 .

[17]  Paolo Guasoni,et al.  Shape optimization problems over classes of convex domains. , 1997 .

[18]  Stanley Osher,et al.  A survey on level set methods for inverse problems and optimal design , 2005, European Journal of Applied Mathematics.

[19]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[20]  John S. Wettlaufer,et al.  A geometric model for anisotropic crystal growth , 1994 .

[21]  Ian M. Mitchell.  The Flexible, Extensible and Efficient Toolbox of Level Set Methods , 2008, J. Sci. Comput..