Niels Christian Overgaard

A geometric formulation of gradient descent for variational problems with moving surfaces

Developments within the computer vision community have led to the formulation of many interesting problems in a variational setting. This paper introduces the manifold of admissible surfaces and a scalar product on its tangent spaces. This makes it possible to properly define gradients and gradient descent procedures for variational problems involving m-surfaces. These concepts lead to a geometric understanding of current state of the art evolution methods and steepest descent evolution equations. By geometric reasoning, common procedures within the variational level set framework are explained and justified. Concrete computations for a general class of functionals are presented and applied to common variational problems for curves and surfaces.

Initialization Techniques for Segmentation with the Chan-Vese Model

This paper introduces an effective initialization approach for segmentation using the Chan-Vese model. The initial curve is found by searching among the extremals of the fidelity term, as a form of intelligent thresholding where the regularity of the threshold level is incorporated. The method has a nice connection to the curvature of the optimal initial partition boundary. The method is tested on several examples and gives considerable increase in performance

Automatic segmentation of zona pellucida in HMC images of human embryos

An important prognostic parameter for assessing the success of an in vitro fertilization treatment is the variation in thickness of the zona pellucida. Zona pellucida, the envelope of the human embryo, is usually visualized using Hoffman modulation contrast microscopy (HMC). This paper addresses the problem of segmenting the zona pellucida in HMC images of embryos. We propose a variational method based on an image model for the zona which takes advantage of the characteristic appearance of HMC images. The simple topology of the embryo allows us to focus on parametric models. Our approach is partly inspired by the works of Chan and Vese, to which it has some similarities.

An analysis of variational alignment of curves in images

In this paper a common variational formulation for alignment of curves to vector fields is analyzed. This variational approach is often used to solve the problem of aligning curves to edges in images by choosing the vector field to be the image gradient. The main contribution of this paper is an analysis of the Gateaux derivative and the descent motion of the corresponding alignment functional, improving on earlier research in this area. Several intermediate results are proved and finally a theorem concerning necessary conditions for extremals of the alignment functional is derived. The analysis of the evolution is performed using a level set formulation and results from distribution theory.

A two-step area based method for automatic tight segmentation of zona pellucida in HMC images of human embryos

An important prognostic parameter for assessing the success of an in vitro fertilization treatment is the variation in thickness of the zona pellucida. Zona pellucida, the envelope of the human embryo, is usually visualized using Hoffman modulation contrast microscopy (HMC). This paper considers automatic segmentation of zona pellucida in HMC images of human embryos. There are two subproblems: (a) the embryo should be separated from the background and (b) the zona should be separated from the rest of the embryo. (a) is solved using a robust formulation of a classical area based method and (b) is solved using a probabilistic method. Both solutions are set in a variational framework using a novel image model for the zona. This variational framework is adapted to handle images in which large artefacts are covered with masks. Since the zona has a simple topology we focus on parametric models and a representation by trigonometric sums is considered.

A gradient descent procedure for variational dynamic surface problems with constraints

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 infinite-dimensional level set framework. The method is illustrated with examples and the results are valid for surfaces of any dimension.

Classifying and solving minimal structure and motion problems with missing data

In this paper we investigate the structure and motion problem for calibrated one-dimensional projections of a two-dimensional environment. In a previous paper the structure and motion problem for all cases with non-missing data was classified and solved. Our aim is here to classify all structure and motion problems, even those with missing data, and to solve them. Although our focus here is on one-dimensional retina, the classification part works equally well for ordinary cameras, and we give some results for those as well.

The variational origin of motion by Gaussian curvature

A variational formulation of an image analysis problem has the nice feature that it is often easier to predict the effect of minimizing a certain energy functional than to interpret the corresponding Euler-Lagrange equations. For example, the equations of motion for an active contour usually contains a mean curvature term, which we know will regularizes the contour because mean curvature is the first variation of curve length, and shorter curves are typically smoother than longer ones. In some applications it may be worth considering Gaussian curvature as a regularizing term instead of mean curvature. The present paper provides a variational principle for this:We show that Gaussian curvature of a regular surface in three-dimensional Euclidean space is the first variation of an energy functional defined on the surface. Some properties of the corresponding motion by Gaussian curvature are pointed out, and a simple example is given, where minimization of this functional yields a nontrivial solution.

The Minimal Structure and Motion Problems with Missing Data for 1D Retina Vision

In this paper we investigate the structure and motion problem for calibrated one-dimensional projections of a two-dimensional environment. The theory of one-dimensional cameras are useful in several areas, e.g. within robotics, autonomous guided vehicles, projection of lines in ordinary vision and vision of vehicles undergoing so called planar motion. In a previous paper the structure and motion problem for all cases with non-missing data was classified and solved. Our aim is here to classify all structure and motion problems, even those with missing data, and to solve them. In the classification we introduce the notion of a prime problem. A prime problem is a minimal problem that does not contain a minimal problem as a sub-problem. We further show that there are infinitely many such prime problems. We give solutions to four prime problems, and using the duality of Carlsson these can be extended to solutions of seven prime problems. Finally we give some experimental results based on synthetic data.