A Gradient Descent Procedure for Variational Dynamic Surface Problems with Constraints

Abstract

Abstract. Many problems in image analysis and computer vision involving boundaries and regions can be cast in a variational formulation. This means that m-surfaces, e.g. curves and surfaces, are determined as minimizers of functionals using e.g. the variational level set method. In this paper we consider such variational problems with constraints given by functionals. We use the geometric interpretation of gradients for functionals to construct gradient descent evolutions for these constrained problems. The result is a generalization of the standard gradient projection method to an innite-dimensional level set framework. The method is illustrated with examples and the results are valid for surfaces of any dimension.

Shape from Specular Reflection and Optical Flow

“…This renders the above approach useless: the Neumann condition would then force n,ô = 0, which will generally be violated by the sought-after surface. We shall therefore extend the proof of the normal speed equation (4.4), as shown in [SO05b], by formally deriving the natural boundary condition.…”

“…We will look for a starting point by unconstrained minimization of J as presented in the following section. For an outline of the basic concepts in abstract form, refer to [SO05b]. The derivation of such gradient descents has also been extensively discussed by the active contours community [ABFJB03, JBHBA06].…”

“…In current vision related level set literature, it is common practice [GM04,SO05b] to postulate that S is closed and fully contained in the computational domain Ω, i.e. S ∩ ∂Ω = ∅.…”

A Gauss-Newton Method for the Integration of Spatial Normal Fields in Shape Space

“…[9,10] and the references therein. Less is known in the three-dimensional case: Tasdizen et al [11], Solem et al [12], Chang et al [13], and finally Goldlücke et al [14] derived a formula for the first-order shape differential of (2), Goldlücke's being the most general one. The main challenge in computing this derivative stems from the fact that, unlike the name implies, shape spaces are inifinite-dimensional nonlinear manifolds, not vector spaces.…”

“…Implicit dynamic surfaces refrain from the use of local bases and thus neatly comply with the shape space paradigm. On that account, they were chosen for representing the gradient flows of (2) presented in [11][12][13]. A similar line of work linearizes the integration problem by continuing the integral (2) to a neighborhood of , cf.…”