Spectral Laplace-Beltrami Wavelets With Applications in Medical Images


The spectral graph wavelet rework (SGWT) has recently been developed to compute wavelet transforms of functions defined on non-Euclidean spaces like graphs. By capitalizing on the established framework of the SGWT, we tend to adopt a fast and efficient computation of a discretized Laplace-Beltrami (LB) operator that allows its extension from arbitrary graphs to differentiable and closed 2-D manifolds (smooth surfaces embedded in the 3-D Euclidean space). This explicit class of manifolds are widely employed in bioimaging to characterize the morphology of cells, tissues, and organs. They're typically discretized into triangular meshes, providing additional geometric information other than straightforward nodes and weighted connections in graphs. In comparison with the SGWT, the wavelet bases made with the LB operator are spatially localized with a a lot of uniform “unfold” with respect to underlying curvature of the surface. In our experiments, we first use synthetic knowledge to point out that ancient applications of wavelets in smoothing and edge detectio can be done using the wavelet bases created with the LB operator. Second, we tend to show that multi-resolutional capabilities of the proposed framework are applicable in the classification of Alzheimer's patients with traditional subjects using hippocampal shapes. Wavelet transforms of the hippocampal form deformations at finer resolutions registered higher sensitivity (96percent) and specificity (90%) than the classification results obtained from the direct usage of hippocampal form deformations. Additionally, the Laplace-Beltrami methodology requires consistently a smaller number of principal elements (to retain a mounted variance) at higher resolution as compared to the binary and weighted graph Laplacians, demonstrating the potential of the wavelet bases in adapting to the geometry of the underlying manifold.

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