Diffusion Geometry Derived Keypoints and Local Descriptors for 3D Deformable Shape Analysis

Geometric analysis of three-dimensional (3D) surfaces with local deformations is a challenging task, required by mobile devices. In this paper, we propose a new local feature-based method derived from diffusion geometry, including a keypoint detector named persistence-based Heat Kernel Signature (pHKS), and a feature descriptor named Heat Propagation Strips (HeaPS). The pHKS detector first constructs a scalar field using the heat kernel signature function. The scalar field is generated at a small scale to capture fine geometric information of the local surface. Persistent homology is then computed to extract all the local maxima from the scalar field, and to provide a measure of persistence. Points with a high persistence are selected as pHKS keypoints. In order to describe a keypoint, an intrinsic support region is generated by the diffusion area. This support region is more robust than its geodesic distance counterpart, and provides a local surface with adaptive scale for subsequent feature description. The HeaPS descriptor is then developed by encoding the information contained in both the spatial and temporal domains of the heat kernel. We conducted several experiments to evaluate the effectiveness of the proposed method. On the TOSCA Dataset, the HeaPS descriptor achieved a high performance in terms of descriptiveness. The feature detector and descriptor were then tested on the SHREC 2010 Feature Detection and Description Dataset, and produced results that were better than the state-of-the-art methods. Finally, their application to shape retrieval was evaluated. The proposed pHKS detector and HeaPS descriptor achieved a notable improvement on the SHREC 2014 Human Dataset.

Direction-Averaged Diffusion-Weighted MRI Signal using different Axisymmetric B-tensor Encoding Schemes “Submitted to Magnetic Resonance in Medicine”

PurposeIt has been shown previously that for the conventional Stejskal-Tanner pulsed gradient, or linear tensor encoding (LTE), as well as planar tensor encoding (PTE) and in tissue in which diffusion exhibits a ‘stick-like’ geometry, the diffusion-weighted MRI signal at extremely high b-values follows a power-law. Specifically, the signal decays as a in LTE and 1/b in PTE. Here, the direction-averaged signal for arbitrary diffusion encoding waveforms is considered to establish whether power-law behaviors occur with other encoding wave-forms and for other (non-stick-like) diffusion geometries.MethodsWe consider the signal decay for high b-values for encoding geometries ranging from 2-dimensional planar tensor encoding (PTE), through isotropic or spherical tensor encoding (STE) to linear tensor encoding. When a power-law behavior was suggested, this was tested using in-silico simulations and in-vivo using an ultra-strong gradient (300 mT/m) Connectom scanner.ResultsThe results show that using an axisymmetric b-tensor a power-law only exists for two scenarios: For stick-like geometries, (i) the already-discovered LTE case; and (ii) for pure planar encoding. In this latter case, to first order, the signal decays as 1/b. Our in-silico and in-vivo experiments confirm this 1/b relationship.ConclusionA complete analysis of the power-law dependencies of the diffusion-weighted signal at high b-values has been performed. Only two forms of encoding result in a power-law dependency, pure linear and pure planar tensor encoding and when the diffusion geometry is ‘stick-like’. The different exponents of these encodings could be used to provide independent validation of the presence of stick-like geometries in-vivo.