Multiscale 3-D Shape Representation and Segmentation Using Spherical Wavelets

Abstract

This paper presents a novel multiscale shape representation and segmentation algorithm based on the spherical wavelet transform. This work is motivated by the need to compactly and accurately encode variations at multiple scales in the shape representation in order to drive the segmentation and shape analysis of deep brain structures, such as the caudate nucleus or the hippocampus. Our proposed shape representation can be optimized to compactly encode shape variations in a population at the needed scale and spatial locations, enabling the construction of more descriptive, nonglobal, nonuniform shape probability priors to be included in the segmentation and shape analysis framework. In particular, this representation addresses the shortcomings of techniques that learn a global shape prior at a single scale of analysis and cannot represent fine, local variations in a population of shapes in the presence of a limited dataset. Specifically, our technique defines a multiscale parametric model of surfaces belonging to the same population using a compact set of spherical wavelets targeted to that population. We further refine the shape representation by separating into groups wavelet coefficients that describe independent global and/or local biological variations in the population, using spectral graph partitioning. We then learn a prior probability distribution induced over each group to explicitly encode these variations at different scales and spatial locations. Based on this representation, we derive a parametric active surface evolution using the multiscale prior coefficients as parameters for our optimization procedure to naturally include the prior for segmentation. Additionally, the optimization method can be applied in a coarse-to-fine manner. We apply our algorithm to two different brain structures, the caudate nucleus and the hippocampus, of interest in the study of schizophrenia. We show: 1) a reconstruction task of a test set to validate the expressiveness of our multiscale prior and 2) a segmentation task. In the reconstruction task, our results show that for a given training set size, our algorithm significantly improves the approximation of shapes in a © 2007 IEEE Correspondence to: Allen Tannenbaum, tannenba@ece.gatech.edu. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. testing set over the Point Distribution Model, which tends to oversmooth data. In the segmentation task, our validation shows our algorithm is computationally efficient and outperforms the Active Shape Model algorithm, by capturing finer shape details. NIH Public Access

Shape-Driven 3D Segmentation Using Spherical Wavelets

“…We use a technique that takes into account biorthogonality and estimates which coefficients can be truncated (set to 0) without significantly affecting the function approximation [9]. In the caudate dataset, 74% of the coefficients were removed resulting in a reconstruction error smaller than 0.1% of the total shape size.…”

“…We use the area formula, and discrete divergence theorem to express the region sum in (5) as a surface sum [9]. Using the notation of (4), the gradient with respect to each pose parameter p k ∈ p is given by:…”

“…We start with an initial shape and iterate between (8) and (9). We update the α parameters in a multiresolution fashion.…”

Left Ventricle Segmentation Using Diffusion Wavelets and Boosting

“…A promising line of research considering wavelets for the representation of shapes was initiated in [3] by build hierarchical active shape models of 2-D anatomical objects using 1-D wavelets, which are then used for shape based image segmentation. A further extension was proposed in [4] where spherical wavelets are used to characterize shape variation in a local fashion in the space and frequency domain.…”

Shape Analysis with Overcomplete Spherical Wavelets

“…The application of these wavelets to closed 2D surfaces demonstrated great utility in both segmentation and shape analysis [11,17]. Unfortunately, the bi-orthogonal wavelet transform on the sphere suffers from the same aliasing problems observed in Euclidean images.…”