David Tolliver

Graph Partitioning by Spectral Rounding: Applications in Image Segmentation and Clustering

We introduce a family of spectral partitioning methods. Edge separators of a graph are produced by iteratively reweighting the edges until the graph disconnects into the prescribed number of components. At each iteration a small number of eigenvectors with small eigenvalue are computed and used to determine the reweighting. In this way spectral rounding directly produces discrete solutions where as current spectral algorithms must map the continuous eigenvectors to discrete solutions by employing a heuristic geometric separator (e.g. k-means). We show that spectral rounding compares favorably to current spectral approximations on the Normalized Cut criterion (NCut). Results are given for natural image segmentation, medical image segmentation, and clustering. A practical version is shown to converge.

Gait shape estimation for identification

A method is presented for identifying individuals by shape, given a sequence of noisy silhouettes segmented from video. A spectral partitioning framework is used to cluster similar poses and automatically extract gait shapes. The method uses a variance-weighted similarity metric to induce clusters that cover disparate stages in the gait cycle. This technique is applied to the HumanID Gait Challenge dataset to measure the quality of the shape model, and the efficacy of shape statistics in human identification.

Combinatorial Preconditioners and Multilevel Solvers for Problems in Computer Vision and Image Processing

Linear systems and eigen-calculations on symmetric diagonally dominant matrices (SDDs) occur ubiquitously in computer vision, computer graphics, and machine learning. In the past decade a multitude of specialized solvers have been developed to tackle restricted instances of SDD systems for a diverse collection of problems including segmentation, gradient inpainting and total variation. In this paper we explain and apply the support theory of graphs, a set of of techniques developed by the computer science theory community, to construct SDD solvers with provable properties. To demonstrate the power of these techniques, we describe an efficient multigrid-like solver which is based on support theory principles. The solver tackles problems in fairly general and arbitrarily weighted topologies not supported by prior solvers. It achieves state of the art empirical results while providing robust guarantees on the speed of convergence. The method is evaluated on a variety of vision applications.

Clinical optical coherence tomography of early articular cartilage degeneration in patients with degenerative meniscal tears

OBJECTIVE Quantitative and nondestructive methods for clinical diagnosis and staging of articular cartilage degeneration are important to the evaluation of potential disease-modifying treatments in osteoarthritis (OA). Optical coherence tomography (OCT) is a novel imaging technology that can generate microscopic-resolution cross-sectional images of articular cartilage in near real-time. This study tested the hypotheses that OCT can be used clinically to identify early cartilage degeneration and that OCT findings correlate with magnetic resonance imaging (MRI) T2 values and arthroscopy results. METHODS Patients undergoing arthroscopy for degenerative meniscal tears were recruited under Institutional Review Board-approved protocols. Thirty consecutive subjects completing preoperative 3.0T MRI, arthroscopy, and intraoperative OCT comprised the study group. Qualitative and quantitative OCT results and MRI T2 values were compared with modified Outerbridge cartilage degeneration scores (0-4 scale) assigned at arthroscopy. RESULTS Arthroscopic grades showed cartilage abnormality in 23 of the 30 patients. OCT grades were abnormal in 28 of the 30 patients. Both qualitative and quantitative OCT strongly correlated with the arthroscopy results (P = 0.004 and P = 0.0002, respectively, by Kruskal-Wallis test). Neither the superficial nor the deep cartilage T2 values correlated with the arthroscopy results. The quantitative OCT results correlated with the T2 values in the superficial cartilage (Pearsons r = 0.39, P = 0.03). CONCLUSION These data show that OCT can be used clinically to provide qualitative and quantitative assessments of early articular cartilage degeneration that strongly correlate with arthroscopy results. The correlation between the quantitative OCT values and T2 values for the superficial cartilage further supports the utility of OCT as a clinical research tool, providing quantifiable microscopic resolution data on the articular cartilage structure. New technologies for nondestructive quantitative assessment of human articular cartilage degeneration may facilitate the development of strategies to delay or prevent the onset of OA.

Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing

Several algorithms for problems including image segmentation, gradient inpainting and total variation are based on solving symmetric diagonally dominant (SDD) linear systems. These algorithms generally produce results of high quality. However, existing solvers are not always efficient, and in many cases they operate only on restricted topologies. The unavailability of reliably efficient solvers has arguably hindered the adoptability of approaches and algorithms based on SDD systems, especially in applications involving very large systems. A central claim of this paper is that SDD-based approaches can now be considered practical and reliable. To support our claim we present Combinatorial Multigrid (CMG), the first reliably efficient SDD solver that tackles problems in general and arbitrary weighted topologies. The solver borrows the structure and operators of multigrid algorithms, but embeds into them powerful and algebraically sound combinatorial preconditioners, based on novel tools from support graph theory. In order to present the derivation of CMG, we review and exemplify key notions of support graph theory that can also guide the future development of specialized solvers. We validate our claims on very large systems derived from imaging applications. Finally, we outline two new reductions of non-linear filtering problems to SDD systems and review the integration of SDD systems into selected algorithms.

Robust unmixing of tumor states in array comparative genomic hybridization data

Motivation: Tumorigenesis is an evolutionary process by which tumor cells acquire sequences of mutations leading to increased growth, invasiveness and eventually metastasis. It is hoped that by identifying the common patterns of mutations underlying major cancer sub-types, we can better understand the molecular basis of tumor development and identify new diagnostics and therapeutic targets. This goal has motivated several attempts to apply evolutionary tree reconstruction methods to assays of tumor state. Inference of tumor evolution is in principle aided by the fact that tumors are heterogeneous, retaining remnant populations of different stages along their development along with contaminating healthy cell populations. In practice, though, this heterogeneity complicates interpretation of tumor data because distinct cell types are conflated by common methods for assaying the tumor state. We previously proposed a method to computationally infer cell populations from measures of tumor-wide gene expression through a geometric interpretation of mixture type separation, but this approach deals poorly with noisy and outlier data. Results: In the present work, we propose a new method to perform tumor mixture separation efficiently and robustly to an experimental error. The method builds on the prior geometric approach but uses a novel objective function allowing for robust fits that greatly reduces the sensitivity to noise and outliers. We further develop an efficient gradient optimization method to optimize this ‘soft geometric unmixing’ objective for measurements of tumor DNA copy numbers assessed by array comparative genomic hybridization (aCGH) data. We show, on a combination of semi-synthetic and real data, that the method yields fast and accurate separation of tumor states. Conclusions: We have shown a novel objective function and optimization method for the robust separation of tumor sub-types from aCGH data and have shown that the method provides fast, accurate reconstruction of tumor states from mixed samples. Better solutions to this problem can be expected to improve our ability to accurately identify genetic abnormalities in primary tumor samples and to infer patterns of tumor evolution. Contact: tolliver@cs.cmu.edu Supplementary information:Supplementary data are available at Bioinformatics online.

3D OCT retinal vessel segmentation based on boosting learning

Blood vessel on retina is generally used for medical image registration. Three dimensional (3D) OCT is the new technique capable of providing the detailed 3D structure of retina. Most algorithms of 3D OCT vessel segmentation need to use the result of retinal layer segmentation to enhance vessel pattern. The proposed 3D boosting learning algorithm is an independent pixel (A-scan projection on OCT fundus image) classification algorithm, which does not rely on any processing result. Both 2D features from OCT fundus image and the third dimensional Haar-feature generated from each A-scan are used in the boosting learning. A matched template, second-order Gaussian filter is used to post-process the generated binary vessel image to clean up the false classifications and smooth the vessels. Eleven images were tested and compared with the manually marked reference. The average sensitivity and specificity were 85% and 88% respectively. The proposed algorithm is an efficient way to automatically identify the blood vessel on 3D OCT image without the need of pre-segmentation.

Multilevel Spectral Partitioning for Efficient Image Segmentation and Tracking

An efficient multilevel method for solving normalized cut image segmentation problems is presented. The method uses the lattice geometry of images to define a set of coarsened graph partitioning problems. This problem hierarchy provides a framework for rapidly estimating the eigenvectors of normalized graph Laplacians. Within this framework, a coarse solution obtained with a standard eigensolver is propagated to increasingly fine problem instances and refined using subspace iterations. Results are presented for image segmentation and tracking problems. The computational cost of the multilevel method is an order of magnitude lower than current sampling techniques and results in more stable image segmentations

Corrected Laplacians: closer cuts and segmentation with shape priors

We optimize over the set of corrected Laplacians (CL) associated with a weighted graph to improve the average case normalized cut (NCut) of a graph. Unlike edge-relaxation SDPs, optimizing over the set CL naturally exploits the matrix sparsity by operating solely on the diagonal. This structure is critical to image segmentation applications because the number of vertices is generally proportional to the number of pixels in the image. CL optimization provides a guiding principle for improving the combinatorial solution over the spectral relaxation, which is important because small improvements in the cut cost often result in significant improvements in the perceptual relevance of the segmentation. We develop an optimization procedure to accommodate prior information in the form of statistical shape models, resulting in a segmentation method that produces foreground regions which are consistent with a parameterized family of shapes. We validate our technique with ground truth on MRI medical images, providing a quantitative comparison against results produced by current spectral relaxation approaches to graph partitioning.