Monotone Pieces Analysis for Qualitative Modeling

It is a crucial task to build qualitative models of industrial applications for model-based diagnosis. A Model Abstraction procedure is designed to automatically transform a quantitative model into qualitative model. If the data is monotone, the behavior can be easily abstracted using the corners of the bounding rectangle. Hence, many existing model abstraction approaches rely on monotonicity. But it is not a trivial problem to robustly detect monotone pieces from scattered data obtained by numerical simulation or experiments. This paper introduces an approach based on scale-dependent monotonicity: the notion that monotonicity can be defined relative to a scale. Real-valued functions defined on a finite set of reals e.g. simulation results, can be partitioned into quasi-monotone segments. The end points for the monotone segments are used as the initial set of landmarks for qualitative model abstraction. The qualitative model abstraction works as an iteratively refining process starting from the initial landmarks. The monotonicity analysis presented here can be used in constructing many other kinds of qualitative models; it is robust and computationally efficient

Combinatorial Continuous Maximal Flows

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit metrication artefacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow CCMF problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.Comment: 26 page

TAPER: query-aware, partition-enhancement for large, heterogenous, graphs

Graph partitioning has long been seen as a viable approach to address Graph DBMS scalability. A partitioning, however, may introduce extra query processing latency unless it is sensitive to a specific query workload, and optimised to minimise inter-partition traversals for that workload. Additionally, it should also be possible to incrementally adjust the partitioning in reaction to changes in the graph topology, the query workload, or both. Because of their complexity, current partitioning algorithms fall short of one or both of these requirements, as they are designed for offline use and as one-off operations. The TAPER system aims to address both requirements, whilst leveraging existing partitioning algorithms. TAPER takes any given initial partitioning as a starting point, and iteratively adjusts it by swapping chosen vertices across partitions, heuristically reducing the probability of inter-partition traversals for a given pattern matching queries workload. Iterations are inexpensive thanks to time and space optimisations in the underlying support data structures. We evaluate TAPER on two different large test graphs and over realistic query workloads. Our results indicate that, given a hash-based partitioning, TAPER reduces the number of inter-partition traversals by around 80%; given an unweighted METIS partitioning, by around 30%. These reductions are achieved within 8 iterations and with the additional advantage of being workload-aware and usable online.Comment: 12 pages, 11 figures, unpublishe

Geometric Multi-Model Fitting with a Convex Relaxation Algorithm

We propose a novel method to fit and segment multi-structural data via convex relaxation. Unlike greedy methods --which maximise the number of inliers-- this approach efficiently searches for a soft assignment of points to models by minimising the energy of the overall classification. Our approach is similar to state-of-the-art energy minimisation techniques which use a global energy. However, we deal with the scaling factor (as the number of models increases) of the original combinatorial problem by relaxing the solution. This relaxation brings two advantages: first, by operating in the continuous domain we can parallelize the calculations. Second, it allows for the use of different metrics which results in a more general formulation. We demonstrate the versatility of our technique on two different problems of estimating structure from images: plane extraction from RGB-D data and homography estimation from pairs of images. In both cases, we report accurate results on publicly available datasets, in most of the cases outperforming the state-of-the-art

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

A Posteriori Error Control for the Binary Mumford-Shah Model

The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non properly segmented region. To this end, a suitable strictly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin's functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the $L^2$-norm. In combination with a suitable cut out argument, a fully practical estimate for the area mismatch is derived. This estimate is incorporated in an adaptive meshing strategy. Two different adaptive primal-dual finite element schemes, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl