Jan Erik Solem

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.

A variational analysis of shape from specularities using sparse data

Looking around in our every day environment, many of the encountered objects are specular to some degree. Actively using this fact when reconstructing objects from image sequences is the scope of the shape from specularities problem. One reason why this problem is important is that standard structure from motion techniques fail when the object surfaces are specular. Here this problem is addressed by estimating surface shape using information from the specular reflections. A specular reflection gives constraints on the surface normal. The approach differs significantly from many earlier shapes from specularities methods since the normal data used is sparse. The main contribution is to give a solid foundation for shape from specularities problems. Estimation of surface shape using reflections is formulated as a variational problem and the surface is represented implicitly using a level set formulation. A functional incorporating all surface constraints is proposed and the corresponding level set motion PDE is explicitly derived. This motion is then proven to minimize the functional. As a part of this functional a variational approach to normal alignment is proposed and analyzed. Also novel methods for implicit surface interpolation to sparse point sets are presented together with a variational analysis. Experiments on both real and synthetic data support the proposed method.

Reconstructing open surfaces from unorganized data points

In this paper a method for fitting open surfaces to an unorganized set of data points is presented using a level set representation of the surface. This is done by tracking a curve, representing the boundary, on the implicitly defined surface. This curve is given as the intersection of the level set describing the surface and an auxiliary level set. These two level sets are propagated using the same motion vector field. Special care has to be taken in order for the surfaces not to intersect at other places than at the desired boundary. Methods for accomplishing this are presented and a novel fast scheme for finding good initial values is proposed. This novel method gives a piecewise linear approximation of the initial surface boundary using a partition of the convex hull. With the described method open surfaces can be fitted to point clouds obtained using structure from motion techniques. This paper solves an important practical problem since in many cases the surfaces in the scene are open or can only be viewed from certain directions. Successful experiments on several data sets support the method.

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

Minimal Solvers for Relative Pose with a Single Unknown Radial Distortion

In this paper, we study the problems of estimating relative pose between two cameras in the presence of radial distortion. Specifically, we consider minimal problems where one of the cameras has no or known radial distortion. There are three useful cases for this setup with a single unknown distortion: (i) fundamental matrix estimation where the two cameras are uncalibrated, (ii) essential matrix estimation for a partially calibrated camera pair, (iii) essential matrix estimation for one calibrated camera and one camera with unknown focal length. We study the parameterization of these three problems and derive fast polynomial solvers based on Gröbner basis methods. We demonstrate the numerical stability of the solvers on synthetic data. The minimal solvers have also been applied to real imagery with convincing results.

Reconstructing Open Surfaces from Image Data

In this paper a method for fitting open surfaces to data obtained from images is presented using a level set representation of the surface. This is done by tracking a curve, representing the boundary, on the implicitly defined surface. This curve is given as the intersection of the level set describing the surface and an auxiliary level set. These two level sets are propagated using the same motion vector field. Special care has to be taken in order for the surfaces not to intersect at other places than at the desired boundary. Methods for accomplishing this are presented and a fast scheme for finding initial values is proposed. This method gives a piecewise linear approximation of the initial surface boundary using a partition of the convex hull of the recovered 3D data. With the approach described in this paper, open surfaces can be fitted to e.g. point clouds obtained using structure from motion techniques. This paper solves an important practical problem since in many cases the surfaces in the scene are open or can only be viewed from certain directions. Experiments on several data sets support the method.

Variational Surface Interpolation from Sparse Point and Normal Data

Many visual cues for surface reconstruction from known views are sparse in nature, e.g., specularities, surface silhouettes, and salient features in an otherwise textureless region. Often, these cues are the only information available to an observer. To allow these constraints to be used either in conjunction with dense constraints such as pixel-wise similarity, or alone, we formulate such constraints in a variational framework. We propose a sparse variational constraint in the level set framework, enforcing a surface to pass through a specific point, and a sparse variational constraint on the surface normal along the observed viewing direction, as is the nature of, e.g., specularities. These constraints are capable of reconstructing surfaces from extremely sparse data. The approach has been applied and validated on the shape from specularities problem

Geodesic Curves for Analysis of Continuous Implicit Shapes

A method is proposed for performing shape analysis of m-surfaces, e.g. planar curves and surfaces, with a geometric interpretation. The analysis uses an implicit surface representation and connects the popular level set approach with shape analysis. The representation is continuous and completely landmark-free. Shapes are represented as points on an infinite-dimensional manifold and the distance between two surfaces is given by the length of a path on this manifold. The analysis is valid in any dimension and examples of applications such as interpolation and clustering are given.

Phase contrast MRI segmentation using velocity and intensity

This paper presents a method for three-dimensional (3D) segmentation of blood vessels, i.e. determining the surface of the vessel wall, using a combination of velocity data and magnitude images obtained using phase contrast MRI. In addition to standard MRI images, phase contrast MRI gives velocity information for blood and tissue in the human body. The proposed method uses a variational formulation of the segmentation problem which combines different cues; velocity and magnitude. The segmentation is performed using the level set method. Experiments on phantom data and clinical data support the proposed method.

Velocity Based Segmentation in Phase Contrast MRI Images

Phase contrast MRI is a relatively new technique that extends standard MRI by obtaining flow information for tissue in the human body. The output data is a velocity vector field in a three-dimensional (3D) volume. This paper presents a method for 3D segmentation of blood vessels and determining the surface of the inner wall using this vector field. The proposed method uses a variational formulation and the segmentation is performed using the level set method. The purpose of this paper is to show that it is possible to perform segmentation using only velocity data which would indicate that velocity is a strong cue in these types of segmentation problems. This is shown in experiments. A novel vector field discontinuity detector is introduced and used in the variational formulation. The performance of this measure is tested with satisfactory results.