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.

This paper introduces a clever way of computing inner products between images in order to drastically reduce the computational complexity of fitting appearance models to images. This speedup is possible since computing the hessian matrix for the parameter updates becomes several orders of magnitude faster which in turn has enormous impact on applications. Contrary to previous work within the area, our method has no negative side-effects on accuracy, convergence and robustness of the algorithm to which it is applied. The inner products are computed in a linear space defined by the appearance model which makes the computation time independent of the model image size.

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.

In this paper we consider region-based variational segmentation of two- and three-dimensional images by the minimization of functionals whose fidelity term is the quotient of two integrals. Users often refrain from quotient functionals, even when they seem to be the most natural choice, probably because the corresponding gradient descent PDEs are nonlocal and hence require the computation of global properties. Here it is shown how this problem may be overcome by employing the structure of the Euler-Lagrange equation of the fidelity term to construct a good initialization for the gradient descent PDE, which will then converge rapidly to the desired (local) minimum. The initializer is found by making a one-dimensional search among the level sets of a function related to the fidelity term, picking the level set which minimizes the segmentation functional. This partial extremal initialization is tested on a medical segmentation problem with velocity- and intensity data from MR images. In this particular application, the partial extremal initialization speeds up the segmentation by two orders of magnitude compared to straight forward gradient descent.

The problem of feature-based surface reconstruction is considered in this paper. Our main contribution is the ability to handle visibility constraints, obtained from the projections of points, curves and silhouettes, in the surface fitting process. While traditional methods often ignore such information, we show that visibility constraints not only give better initial surface estimates and faster convergence, but also provide an important cue for determining surface topology. The problem is cast as a variational problem with constraints within the level set framework. It is shown how to evolve the surface without violating the visibility constraints using methods from variational calculus. Applications of the theory are detailed for a number of important cases of geometric primitives: points, curves and visual hulls. Several experiments on real image sequences are given to demonstrate the performance of the approach.

We consider the problem of geometrical surface reconstruction from one or several images using learned shape models. While humans can effort- lessly retrieve 3D shape information, this inverse problem has turned out to be difficult to perform automatically. We introduce a framework based on level set surface reconstruction and shape models for achieving this goal. Through this merging, we obtain an efficient and robust method for reconstructing surfaces of an object category of interest. The shape model includes surface cues such as point, curve and silhou- ette features. Based on ideas from Active Shape Models, we show how both the geometry and the appearance of these features can be modelled consistently in a multi-view context. The complete surface is obtained by evolving a level set driven by a PDE, which tries to fit the surface to the inferred 3D features. In addition, an a priori 3D surface model is used to regularize the solution, in particular, where surface features are sparse. Experiments are demonstrated on a database of real face images.