Multiscale 3D Shape Analysis using Spherical Wavelets

©2005 Springer. The original publication is available at www.springerlink.com: http://dx.doi.org/10.1007/11566489_57DOI: 10.1007/11566489_57Shape priors attempt to represent biological variations within a population. When variations are global, Principal Component Analysis (PCA) can be used to learn major modes of variation, even from a limited training set. However, when significant local variations exist, PCA typically cannot represent such variations from a small training set. To address this issue, we present a novel algorithm that learns shape variations from data at multiple scales and locations using spherical wavelets and spectral graph partitioning. Our results show that when the training set is small, our algorithm significantly improves the approximation of shapes in a testing set over PCA, which tends to oversmooth data

A Multiscale Approach for Statistical Characterization of Functional Images

Increasingly, scientific studies yield functional image data, in which the observed data consist of sets of curves recorded on the pixels of the image. Examples include temporal brain response intensities measured by fMRI and NMR frequency spectra measured at each pixel. This article presents a new methodology for improving the characterization of pixels in functional imaging, formulated as a spatial curve clustering problem. Our method operates on curves as a unit. It is nonparametric and involves multiple stages: (i) wavelet thresholding, aggregation, and Neyman truncation to effectively reduce dimensionality; (ii) clustering based on an extended EM algorithm; and (iii) multiscale penalized dyadic partitioning to create a spatial segmentation. We motivate the different stages with theoretical considerations and arguments, and illustrate the overall procedure on simulated and real datasets. Our method appears to offer substantial improvements over monoscale pixel-wise methods. An Appendix which gives some theoretical justifications of the methodology, computer code, documentation and dataset are available in the online supplements

Regularity scalable image coding based on wavelet singularity detection

In this paper, we propose an adaptive algorithm for scalable wavelet image coding, which is based on the general feature, the regularity, of images. In pattern recognition or computer vision, regularity of images is estimated from the oriented wavelet coefficients and quantified by the Lipschitz exponents. To estimate the Lipschitz exponents, evaluating the interscale evolution of the wavelet transform modulus sum (WTMS) over the directional cone of influence was proven to be a better approach than tracing the wavelet transform modulus maxima (WTMM). This is because the irregular sampling nature of the WTMM complicates the reconstruction process. Moreover, examples were found to show that the WTMM representation cannot uniquely characterize a signal. It implies that the reconstruction of signal from its WTMM may not be consistently stable. Furthermore, the WTMM approach requires much more computational effort. Therefore, we use the WTMS approach to estimate the regularity of images from the separable wavelet transformed coefficients. Since we do not concern about the localization issue, we allow the decimation to occur when we evaluate the interscale evolution. After the regularity is estimated, this information is utilized in our proposed adaptive regularity scalable wavelet image coding algorithm. This algorithm can be simply embedded into any wavelet image coders, so it is compatible with the existing scalable coding techniques, such as the resolution scalable and signal-to-noise ratio (SNR) scalable coding techniques, without changing the bitstream format, but provides more scalable levels with higher peak signal-to-noise ratios (PSNRs) and lower bit rates. In comparison to the other feature-based wavelet scalable coding algorithms, the proposed algorithm outperforms them in terms of visual perception, computational complexity and coding efficienc

Photonic band-gap engineering for volume plasmon polaritons in multiscale multilayer hyperbolic metamaterials

We theoretically study the propagation of large-wavevector waves (volume plasmon polaritons) in multilayer hyperbolic metamaterials with two levels of structuring. We show that when the parameters of a subwavelength metal-dielectric multilayer ("substructure") are modulated ("superstructured") on a larger, wavelength scale, the propagation of volume plasmon polaritons in the resulting multiscale hyperbolic metamaterials is subject to photonic band gap phenomena. A great degree of control over such plasmons can be exerted by varying the superstructure geometry. When this geometry is periodic, stop bands due to Bragg reflection form within the volume plasmonic band. When a cavity layer is introduced in an otherwise periodic superstructure, resonance peaks of the Fabry-Perot nature are present within the stop bands. More complicated superstructure geometries are also considered. For example, fractal Cantor-like multiscale metamaterials are found to exhibit characteristic self-similar spectral signatures in the volume plasmonic band. Multiscale hyperbolic metamaterials are shown to be a promising platform for large-wavevector bulk plasmonic waves, whether they are considered for use as a new kind of information carrier or for far-field subwavelength imaging.Comment: 12 pages, 10 figures, now includes Appendix

Random Forests and Networks Analysis

D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: 1) a random-walk-based notion of well-distributed points in a graph 2) how to describe metastable dynamics in finite settings by means of Markov intertwining dualities 3) coarse graining schemes for networks and associated processes 4) wavelets-like pyramidal algorithms for graph signals.Comment: Survey pape

From homogeneous to fractal normal and tumorous microvascular networks in the brain

We studied normal and tumorous three-dimensional (3D) microvascular networks in primate and rat brain. Tissues were prepared following a new preparation technique intended for high-resolution synchrotron tomography of microvascular networks. The resulting 3D images with a spatial resolution of less than the minimum capillary diameter permit a complete description of the entire vascular network for volumes as large as tens of cubic millimeters. The structural properties of the vascular networks were investigated by several multiscale methods such as fractal and power- spectrum analysis. These investigations gave a new coherent picture of normal and pathological complex vascular structures. They showed that normal cortical vascular networks have scale- invariant fractal properties on a small scale from 1.4 lm up to 40 to 65 lm. Above this threshold, vascular networks can be considered as homogeneous. Tumor vascular networks show similar characteristics, but the validity range of the fractal regime extend to much larger spatial dimensions. These 3D results shed new light on previous two dimensional analyses giving for the first time a direct measurement of vascular modules associated with vessel-tissue surface exchange