Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.Comment: 14 page

Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling

In this thesis, we focus on the image labeling problem which is the task of performing unique pixel-wise label decisions to simplify the image while reducing its redundant information. We build upon a recently introduced geometric approach for data labeling by assignment flows [ APSS17 ] that comprises a smooth dynamical system for data processing on weighted graphs. Hereby we pursue two lines of research that give new application and theoretically-oriented insights on the underlying segmentation task. We demonstrate using the example of Optical Coherence Tomography (OCT), which is the mostly used non-invasive acquisition method of large volumetric scans of human retinal tis- sues, how incorporation of constraints on the geometry of statistical manifold results in a novel purely data driven geometric approach for order-constrained segmentation of volumetric data in any metric space. In particular, making diagnostic analysis for human eye diseases requires decisive information in form of exact measurement of retinal layer thicknesses that has be done for each patient separately resulting in an demanding and time consuming task. To ease the clinical diagnosis we will introduce a fully automated segmentation algorithm that comes up with a high segmentation accuracy and a high level of built-in-parallelism. As opposed to many established retinal layer segmentation methods, we use only local information as input without incorporation of additional global shape priors. Instead, we achieve physiological order of reti- nal cell layers and membranes including a new formulation of ordered pair of distributions in an smoothed energy term. This systematically avoids bias pertaining to global shape and is hence suited for the detection of anatomical changes of retinal tissue structure. To access the perfor- mance of our approach we compare two different choices of features on a data set of manually annotated 3 D OCT volumes of healthy human retina and evaluate our method against state of the art in automatic retinal layer segmentation as well as to manually annotated ground truth data using different metrics. We generalize the recent work [ SS21 ] on a variational perspective on assignment flows and introduce a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs. The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was introduced in J. Math. Imaging & Vision 58(2), 2017. Due to this parameterization, solving the G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with re- spect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions (DC) decomposition of this potential and show that the basic geometric Euler scheme for inte- grating the assignment flow is equivalent to solving the G-PDE by an established DC program- ming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit higher-order information of the vector field that drives the assignment flow, in order to devise a novel accelerated DC programming scheme. A detailed convergence analysis of both numerical schemes is provided and illustrated by numerical experiments

Fast parallel algorithms for a broad class of nonlinear variational diffusion approaches

Variational segmentation and nonlinear diffusion approaches have been very active research areas in the fields of image processing and computer vision during the last years. In the present paper, we review recent advances in the development of efficient numerical algorithms for these approaches. The performance of parallel implement at ions of these algorithms on general-purpose hardware is assessed. A mathematically clear connection between variational models and nonlinear diffusion filters is presented that allows to interpret one approach as an approximation of the other, and vice versa. Numerical results confirm that, depending on the parametrization, this approximation can be made quite accurate. Our results provide a perspective for uniform implement at ions of both nonlinear variational models and diffusion filters on parallel architectures

Active skeleton for bacteria modeling

The investigation of spatio-temporal dynamics of bacterial cells and their molecular components requires automated image analysis tools to track cell shape properties and molecular component locations inside the cells. In the study of bacteria aging, the molecular components of interest are protein aggregates accumulated near bacteria boundaries. This particular location makes very ambiguous the correspondence between aggregates and cells, since computing accurately bacteria boundaries in phase-contrast time-lapse imaging is a challenging task. This paper proposes an active skeleton formulation for bacteria modeling which provides several advantages: an easy computation of shape properties (perimeter, length, thickness, orientation), an improved boundary accuracy in noisy images, and a natural bacteria-centered coordinate system that permits the intrinsic location of molecular components inside the cell. Starting from an initial skeleton estimate, the medial axis of the bacterium is obtained by minimizing an energy function which incorporates bacteria shape constraints. Experimental results on biological images and comparative evaluation of the performances validate the proposed approach for modeling cigar-shaped bacteria like Escherichia coli. The Image-J plugin of the proposed method can be found online at http://fluobactracker.inrialpes.fr.Comment: Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging and Visualizationto appear i