Regularization of positive definite matrix fields based on multiplicative calculus

Multiplicative calculus provides a natural framework in problems involving positive images and positivity preserving operators. In increasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standard calculus in a nontrivial way. The purpose of this article is to illustrate the basics of multiplicative calculus and its application to the regularization of positive definite matrix fields

Matching two-dimensional articulated shapes using generalized multidimensional scaling

generalized multidimensional scalin

Numerical schemes for linear and non-linear enhancement of DW-MRI

We consider left-invariant di??usion processes on DTI data by embedding the data into the space R3 o S2 of 3D positions and orientations. We then define and solve the diffusion equation in a moving frame of reference defined using left-invariant derivatives. The diffusion process is made adaptive to the data in order to do Perona-Malik-like edge preserving smoothing, which is necessary to handle fiber structures near regions of large isotropic diffusion such as the ventricles of the brain. The corresponding partial differential systems are solved using finite difference stencils. We include experiments both on synthetic data and on DTI-images of the brain

Fiber enhancement in diffusion-weighted MRI

Diffusion-Weighted MRI (DW-MRI) measures local water diffusion in biological tissue, which reflects the underlying fiber structure. In order to enhance the fiber structure in the DW-MRI data we consider both (convection-)diffusions and Hamilton-Jacobi equations (erosions) on the space \mathbbR3 \rtimes S2Unknown control sequence '\rtimes' of 3D-positions and orientations, embedded as a quotient in the group SE(3) of 3D-rigid body movements. These left-invariant evolutions are expressed in the frame of left-invariant vector fields on SE(3), which serves as a moving frame of reference attached to fiber fragments. The linear (convection-)diffusions are solved by a convolution with the corresponding Green’s function, whereas the Hamilton-Jacobi equations are solved by a morphological convolution with the corresponding Green’s function. Furthermore, we combine dilation and diffusion in pseudo-linear scale spaces on \mathbbR3\rtimes S2Unknown control sequence '\rtimes'. All methods are tested on DTI-images of the brain. These experiments indicate that our techniques are useful to deal with both the problem of limited angular resolution of DTI and the problem of spurious, non-aligned crossings in HARDI

Intrinsic Regularity Detection in 3D Geometry

Abstract. Automatic detection of symmetries, regularity, and repetitive structures in 3D geometry is a fundamental problem in shape analysis and pattern recognition with applications in computer vision and graphics. Especially challenging is to detect intrinsic regularity, where the repetitions are on an intrinsic grid, without any apparent Euclidean pattern to describe the shape, but rising out of (near) isometric deformation of the underlying surface. In this paper, we employ multidimensional scaling to reduce the problem of intrinsic structure detection to a simpler problem of 2D grid detection. Potential 2D grids are then identified using an autocorrelation analysis, refined using local fitting, validated, and finally projected back to the spatial domain. We test the detection algorithm on a variety of scanned plaster models in presence of imperfections like missing data, noise and outliers. We also present a range of applications including scan completion, shape editing, super-resolution, and structural correspondence.