Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging

Mumford-Shah and Potts functionals are powerful variational models for regularization which are widely used in signal and image processing; typical applications are edge-preserving denoising and segmentation. Being both non-smooth and non-convex, they are computationally challenging even for scalar data. For manifold-valued data, the problem becomes even more involved since typical features of vector spaces are not available. In this paper, we propose algorithms for Mumford-Shah and for Potts regularization of manifold-valued signals and images. For the univariate problems, we derive solvers based on dynamic programming combined with (convex) optimization techniques for manifold-valued data. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging), we show that our algorithms compute global minimizers for any starting point. For the multivariate Mumford-Shah and Potts problems (for image regularization) we propose a splitting into suitable subproblems which we can solve exactly using the techniques developed for the corresponding univariate problems. Our method does not require any a priori restrictions on the edge set and we do not have to discretize the data space. We apply our method to diffusion tensor imaging (DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation of the corpus callosum

Colour morphological sieves for scale-space image processing

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Medical image enhancement

Each image acquired from a medical imaging system is often part of a two-dimensional (2-D) image set whose total presents a three-dimensional (3-D) object for diagnosis. Unfortunately, sometimes these images are of poor quality. These distortions cause an inadequate object-of-interest presentation, which can result in inaccurate image analysis. Blurring is considered a serious problem. Therefore, “deblurring” an image to obtain better quality is an important issue in medical image processing. In our research, the image is initially decomposed. Contrast improvement is achieved by modifying the coefficients obtained from the decomposed image. Small coefficient values represent subtle details and are amplified to improve the visibility of the corresponding details. The stronger image density variations make a major contribution to the overall dynamic range, and have large coefficient values. These values can be reduced without much information loss

Image Segmentation by Energy and Related Functional Minimization Methods

Effective and efficient methods for partitioning a digital image into image segments, called ¿image segmentation,¿ have a wide range of applications that include pattern recognition, classification, editing, rendering, and compressed data for image search. In general, image segments are described by their geometry and similarity measures that identify them. For example, the well-known optimization model proposed and studied in depth by David Mumford and Jayant Shah is based on an L2 total energy functional that consists of three terms that govern the geometry of the image segments, the image fidelity (or closeness to the observed image), and the prior (or image smoothness). Recent work in the field of image restoration suggests that a more suitable choice for the fidelity measure is, perhaps, the l1 norm. This thesis explores that idea applied to the study of image segmentation along the line of the Mumford and Shah optimization model, but eliminating the need of variational calculus and regularization schemes to derive the approximating Euler-Lagrange equations. The main contribution of this thesis is a formulation of the problem that avoids the need for the calculus of variation. The energy functional represents a global property of an image. It turns out to be possible, however, to predict how localized changes to the segmentation will affect its value. This has been shown previously in the case of the l2 norm, but no similar method is available for other norms. The method described here solves the problem for the l1 norm, and suggests how it would apply to other forms of the fidelity measure. Existing methods rely on a fixed initial condition. This can lead to an algorithm finding local instead of global optimizations. The solution given here shows how to specify the initial condition based on the content of the image and avoid finding local minima