This paper addresses the problem of segmenting an image into regions consistent with user-supplied seeds (e.g., a sparse set of broad brush strokes). We view this task as a statistical transductive inference, in which some pixels are already associated with given zones and the remaining ones need to be classified. Our method relies on the Laplacian graph regularizer, a powerful manifold learning tool that is based on the estimation of variants of the Laplace-Beltrami operator and is tightly related to diffusion processes. Segmentation is modeled as the task of finding matting coefficients for unclassified pixels given known matting coefficients for seed pixels. The proposed algorithm essentially relies on a high margin assumption in the space of pixel characteristics. It is simple, fast, and accurate, as demonstrated by qualitative results on natural images and a quantitative comparison with state-of-the-art methods on the Microsoft GrabCut segmentation database.

Kernel PCA has received a lot of attention over the past years and showed usefull for many image processing problems. In this paper we analyse the issue of normalization in Kernel PCA for the pre-image problem. We present a geometric interpretation of the normalization process for the gaussian kernel. As a consequence, we could formulate a correct normalization criterion in centered feature space. Furthermore, we show how the proposed normalization criterion improves previous pre-image methods for the task of image denoising.

We propose a novel method for computing a four-dimensional (4D) representation of the spatio-temporal visual hull of a dynamic scene, based on an extension of a recent provably correct Delaunay meshing algorithm. By considering time as an additional dimension, our approach exploits seamlessly the time coherence between different frames to produce a compact and high-quality 4D mesh representation of the visual hull. The 3D visual hull at a given time instant is easily obtained by intersecting this 4D mesh with a temporal plane, thus enabling interpolation of objects' shape between consecutive frames. In addition, our approach offers easy and extensive control over the size and quality of the output mesh as well as over its associated re- projection error. Our numerical experiments demonstrate the effectiveness and flexibility of our approach for generating compact, high-quality, time-coherent visual hull representations from real silhouette image data.

We introduce a non-linear shape prior for the de- formable model framework that we learn from a set of shape samples using recent manifold learning techniques. We model a category of shapes as a finite dimensional manifold which we approximate using Diffusion maps, that we call the shape prior manifold. Our method computes a Delaunay triangulation of the reduced space, considered as Euclidean, and uses the resulting space partition to identify the closest neighbors of any given shape based on its Nystrom extension. Our contribution lies in three aspects. First, we propose a solution to the pre-image problem and define the projection of a shape onto the manifold. Based on closest neighbors for the Diffusion distance, we then describe a variational framework for manifold denoising. Finally, we introduce a shape prior term for the deformable framework through a non-linear energy term designed to attract a shape towards the manifold at given constant embedding. Results on shapes of cars and ventricule nuclei are presented and demonstrate the potentials of our method.

Image segmentation with shape priors has received a lot of attention over the past years. Most existing work focuses on a linearized shape space with small deformation modes around a mean shape, which is relevant only when considering similar shapes. In this paper, we introduce a new framework that can handle more general shape priors. We model a category of shapes as a finite dimensional manifold, the shape prior manifold, which we approximate from the shape samples using dimensionality reduction techniques suchlike Laplacian eigenmaps. Unfortunately, this model does not provide an explicit projection operator onto the manifold. Our contribution is twofold. First, we calculate the low dimensional representation of any point not in the training set. Second, we properly define a projection operator onto the manifold by interpolating between shape samples using local weighted means. We show results both on synthetic and real shapes and demonstrate the potential of our method for segmentation tasks.

In this paper, we focus on the retrospective topology correction of surfaces. We propose a technique to accurately correct the spherical topology of cortical surfaces. Specifically, we construct a mapping from the original surface onto the sphere to detect topological defects as minimal nonhomeomorphic regions. The topology of each defect is then corrected by opening and sealing the surface along a set of nonseparating loops that are selected in a Bayesian framework. The proposed method is a wholly self-contained topology correction algorithm, which determines geometrically accurate, topologically correct solutions based on the magnetic resonance imaging (MRI) intensity profile and the expected local curvature. Applied to real data, our method provides topological corrections similar to those made by a trained operator.

In vivo quantification of neuroanatomical shape variations is possible due to recent advances in medical imaging and has proven useful in the study of neuropathology and neurodevelopment. In this paper, we apply a spherical wavelet transformation to extract shape features of cortical surfaces reconstructed from magnetic resonance images (MRIs) of a set of subjects. The spherical wavelet transformation can characterize the underlying functions in a local fashion in both space and frequency, in contrast to spherical harmonics that have a global basis set. We perform principal component analysis (PCA) on these wavelet shape features to study patterns of shape variation within normal population from coarse to fine resolution. In addition, we study the development of cortical folding in newborns using the Gompertz model in the wavelet domain, which allows us to characterize the order of development of large-scale and finer folding patterns independently. Given a limited amount of training data, we use a regularization framework to estimate the parameters of the Gompertz model to improve the prediction performance on new data. We develop an efficient method to estimate this regularized Gompertz model based on the Broyden-Fletcher-Goldfarb-Shannon (BFGS) approximation. Promising results are presented using both PCA and the folding development model in the wavelet domain. The cortical folding development model provides quantitative anatomic information regarding macroscopic cortical folding development and may be of potential use as a biomarker for early diagnosis of neurologic deficits in newborns.