Many image processing problems are naturally expressed as energy minimization or shape optimization problems, in which the free variable is a shape, such as a curve in 2d or a surface in 3d. Examples are image segmentation, multiview stereo reconstruction, geometric interpolation from data point clouds. To obtain the solution of such a problem, one usually resorts to an iterative approach, a gradient descent algorithm, which updates a candidate shape gradually deforming it into the optimal shape. Computing the gradient descent updates requires the knowledge of the first variation of the shape energy, or rather the first shape derivative. In addition to the first shape derivative, one can also utilize the second shape derivative and develop a Newton-type method with faster convergence. Unfortunately, the knowledge of shape derivatives for shape energies in image processing is patchy. The second shape derivatives are known for only two energies in the literature and many results for the first shape derivative are limiting, in the sense that they are either for curves on planes, or developed for a specific representation of the shape or for a very specific functional form in the shape energy. In this work, we overcome these limitations and compute the first and second shape derivatives of large classes of shape energies that we consider to be representative of the energies found in image processing. Many of the formulas we obtain are new and some generalize previous existing results. Our results are valid for general surfaces any number of dimensions. We intend this work to serve as a cookbook for researchers that deal with shape energies for various applications in image processing and need to develop algorithms to compute the shapes minimizing these energies.
Citation: SIAM Journal on Imaging Sciences
Pub Type: Journals
Shape derivatives, shape optimization, energy minimization, gradient descent, image segmentation