RESUMEN

This paper introduces assignment flows for density matrices as state spaces for representation and analysis of data associated with vertices of an underlying weighted graph. Determining an assignment flow by geometric integration of the defining dynamical system causes an interaction of the non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Adopting the Riemannian-Bogoliubov-Kubo-Mori metric from information geometry leads to closed-form local expressions that can be computed efficiently and implemented in a fine-grained parallel manner. Restriction to the submanifold of commuting density matrices recovers the assignment flows for categorical probability distributions, which merely assign labels from a finite set to each data point. As shown for these flows in our prior work, the novel class of quantum state assignment flows can also be characterized as Riemannian gradient flows with respect to a non-local, non-convex potential after proper reparameterization and under mild conditions on the underlying weight function. This weight function generates the parameters of the layers of a neural network corresponding to and generated by each step of the geometric integration scheme. Numerical results indicate and illustrate the potential of the novel approach for data representation and analysis, including the representation of correlations of data across the graph by entanglement and tensorization.

RESUMEN

The Turing mechanism describes the emergence of spatial patterns due to spontaneous symmetry breaking in reaction-diffusion processes and underlies many developmental processes. Identifying Turing mechanisms in biological systems defines a challenging problem. This paper introduces an approach to the prediction of Turing parameter values from observed Turing patterns. The parameter values correspond to a parametrized system of reaction-diffusion equations that generate Turing patterns as steady state. The Gierer-Meinhardt model with four parameters is chosen as a case study. A novel invariant pattern representation based on resistance distance histograms is employed, along with Wasserstein kernels, in order to cope with the highly variable arrangement of local pattern structure that depends on the initial conditions which are assumed to be unknown. This enables us to compute physically plausible distances between patterns, to compute clusters of patterns and, above all, model parameter prediction based on training data that can be generated by numerical model evaluation with random initial data: for small training sets, classical state-of-the-art methods including operator-valued kernels outperform neural networks that are applied to raw pattern data, whereas for large training sets the latter are more accurate. A prominent property of our approach is that only a single pattern is required as input data for model parameter predicion. Excellent predictions are obtained for single parameter values and reasonably accurate results for jointly predicting all four parameter values.

RESUMEN

We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.

RESUMEN

With the introduction of spectral-domain optical coherence tomography (OCT), resulting in a significant increase in acquisition speed, the fast and accurate segmentation of 3-D OCT scans has become evermore important. This paper presents a novel probabilistic approach, that models the appearance of retinal layers as well as the global shape variations of layer boundaries. Given an OCT scan, the full posterior distribution over segmentations is approximately inferred using a variational method enabling efficient probabilistic inference in terms of computationally tractable model components: Segmenting a full 3-D volume takes around a minute. Accurate segmentations demonstrate the benefit of using global shape regularization: We segmented 35 fovea-centered 3-D volumes with an average unsigned error of 2.46 ± 0.22 µm as well as 80 normal and 66 glaucomatous 2-D circular scans with errors of 2.92 ± 0.5 µm and 4.09 ± 0.98 µm respectively. Furthermore, we utilized the inferred posterior distribution to rate the quality of the segmentation, point out potentially erroneous regions and discriminate normal from pathological scans. No pre- or postprocessing was required and we used the same set of parameters for all data sets, underlining the robustness and out-of-the-box nature of our approach.

Asunto(s)

RESUMEN

A variational approach is presented to the estimation of turbulent fluid flow from particle image sequences in experimental fluid mechanics. The approach comprises two coupled optimizations for adapting size and shape of a Gaussian correlation window at each location and for estimating the flow, respectively. The method copes with a wide range of particle densities and image noise levels without any data-specific parameter tuning. Based on a careful implementation of a multiscale nonlinear optimization technique, we demonstrate robustness of the solution over typical experimental scenarios and highest estimation accuracy for an international benchmark data set (PIV Challenge).

RESUMEN

We present a probabilistic approach to the segmentation of OCT scans of retinal tissue. By combining discrete exact inference and a global shape prior, accurate segmentations are computed that preserve the physiological order of intra-retinal layers. A major part of the computations can be performed in parallel. The evaluation reveals robustness against speckle noise, shadowing caused by blood vessels, and other scan artifacts.

Asunto(s)

RESUMEN

Imaging plays an important role in experimental fluid dynamics. It is equally important both for scientific research and a range of industrial applications. It is known, however, that estimated velocity fields of fluids often suffer from various types of corruptions like missing data, for instance, that make their physical interpretation questionable. We present an algorithm that accepts a wide variety of corrupted 2-D vector fields as input data and allows to recover missing data fragments and to remove noise in a physically plausible way. Our approach essentially exploits the physical properties of incompressible fluid flows and does not rely upon any particular model of noise. As a result, the developed algorithm performs well and robust for different types of noise and estimation errors. The computational algorithm is sufficiently simple to scale up to large 3-D problems.

RESUMEN

The detection and extraction of complex anatomical structures usually involves a trade-off between the complexity of local feature extraction and classification, and the complexity and performance of the subsequent structural inference from the viewpoint of combinatorial optimization. Concerning the latter, computationally efficient methods are of particular interest that return the globally-optimal structure. We present an efficient method for part-based localization of anatomical structures which embeds contextual shape knowledge in a probabilistic graphical model. It allows for robust detection even when some of the part detections are missing. The application scenario for our statistical evaluation is spine detection and labeling in magnetic resonance images.

Asunto(s)

RESUMEN

We present an approach to parallel variational optical-flow computation by using an arbitrary partition of the image plane and iteratively solving related local variational problems associated with each subdomain. The approach is particularly suited for implementations on PC clusters because interprocess communication is minimized by restricting the exchange of data to a lower dimensional interface. Our mathematical formulation supports various generalizations to linear/nonlinear convex variational approaches, three-dimensional image sequences, spatiotemporal regularization, and unstructured geometries and triangulations. Results concerning the effects of interface preconditioning, as well as runtime and communication volume measurements on a PC cluster, are presented. Our approach provides a major step toward real-time two-dimensional image processing using off-the-shelf PC hardware and facilitates the efficient application of variational approaches to large-scale image processing problems.

Asunto(s)

RESUMEN

This paper investigates the usefulness of bidirectional multigrid methods for variational optical flow computations. Although these numerical schemes are among the fastest methods for solving equation systems, they are rarely applied in the field of computer vision. We demonstrate how to employ those numerical methods for the treatment of variational optical flow formulations and show that the efficiency of this approach even allows for real-time performance on standard PCs. As a representative for variational optic flow methods, we consider the recently introduced combined local-global method. It can be considered as a noise-robust generalization of the Horn and Schunck technique. We present a decoupled, as well as a coupled, version of the classical Gauss-Seidel solver, and we develop several multgrid implementations based on a discretization coarse grid approximation. In contrast, with standard bidirectional multigrid algorithms, we take advantage of intergrid transfer operators that allow for nondyadic grid hierarchies. As a consequence, no restrictions concerning the image size or the number of traversed levels have to be imposed. In the experimental section, we juxtapose the developed multigrid schemes and demonstrate their superior performance when compared to unidirectional multgrid methods and nonhierachical solvers. For the well-known 316 x 252 Yosemite sequence, we succeeded in computing the complete set of dense flow fields in three quarters of a second on a 3.06-GHz Pentium4 PC. This corresponds to a frame rate of 18 flow fields per second which outperforms the widely-used Gauss-Seidel method by almost three orders of magnitude.

Asunto(s)