Digital Surface Regularization With Guarantees.

Voxel based modeling is a very attractive way to represent complex multi-material objects. Beside artistic choices of pixel/voxel arts, representing objects as voxels allows efficient and dynamic interactions with the scene. For geometry processing purposes, many applications in material sciences, medical imaging or numerical simulation rely on a regular partitioning of the space with labeled voxels. In this article, we consider a variational approach to reconstruct interfaces in multi-labeled digital images. This approach efficiently produces piecewise smooth quadrangulated surfaces with some theoretical stability guarantee. Non-manifold parts at intersecting interfaces are handled naturally by our model. We illustrate the strength of our tool for digital surface regularization, as well as voxel art regularization by transferring colorimetric information to regularized quads and computing isotropic geodesic on digital surfaces.

Optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension.

In binary images, the Distance Transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for d-dimensional images. We also present a d-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape.

Adaptive estimation of normals and surface area for discrete 3-D objects: application to snow binary data from X-ray tomography.

Estimating the normal vector field on the boundary of discrete three-dimensional objects is essential for rendering and image measurement problems. Most of the existing algorithms do not provide an accurate determination of the normal vector field for shapes that present edges. Here, we propose a new and simple computational method in order to obtain accurate results on all types of shapes, whatever their local convexity degree. The presented method is based on the gradient vector field analysis of the object distance map. This vector field is adaptively filtered around each surface voxel using angle and symmetry criteria so that as many relevant contributions as possible are accounted for. This optimizes the smoothing of digitization effects while preserving relevant details of the processed numerical object. Thanks to the precise normal field obtained, a projection method can be proposed to immediately derive the surface area from a raw discrete object. An empirical justification of the validity of such an algorithm in the continuous limit is also provided. Some results on simulated data and snow images from X-ray tomography are presented, compared to the Marching Cubes and Convex Hull results, and discussed.

A comparative evaluation of length estimators of digital curves.

This paper compares previously published length estimators in image analysis having digitized curves as input. The evaluation uses multigrid convergence (theoretical results and measured speed of convergence) and further measures as criteria. This paper also suggests a new gradient-based method for length estimation, and combines a previously proposed length estimator for straight segments with a polygonalization method.

Globally optimal cortical surface matching with exact landmark correspondence.

We present a method for establishing correspondences between human cortical surfaces that exactly matches the positions of given point landmarks, while attaining the global minimum of an objective function that quantifies how far the mapping deviates from conformality. On each surface, a conformal transformation is applied to the Euclidean distance metric, resulting in a hyperbolic metric with isolated cone point singularities at the landmarks. Equivalently, each surface is mapped to a hyperbolic orbifold: a pillow-like surface with each point landmark corresponding to a pillow corner. An initial surface-to-surface mapping exactly aligns the landmarks, and gradient descent is used to find the single, global minimum of the Dirichlet energy of the remainder of the mapping. Using a population of real MRI-based cortical surfaces with manually labeled sulcus endpoints as landmarks, we evaluate the approach by how much it distorts surfaces and by its biological plausibility: how well it aligns previously-unseen anatomical landmarks and by how well it promotes expected associations between cortical thickness and age. We show that, compared to a painstakingly-tuned approach that balances a tradeoff between minimizing landmark mismatch and Dirichlet energy, our method has similar biological plausibility, superior surface distortion, a better theoretical foundation, and fewer arbitrary parameters to tune. We also compare to conformal mapper in the spherical domain to show that sacrificing exact conformality of the mapping does not cause noticeable reductions in biological plausibility.