Graph matching—aligning a pair of graphs to minimize their edge disagreements—has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.

Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching problem is challenging from a computational point of view, and therefore different relaxations are commonly used. Although common relaxations techniques tend to work well for matching perfectly isomorphic graphs, it is not yet fully understood under which conditions the relaxed problem is guaranteed to obtain the correct answer.In this paper we prove that the graph matching problem and its most common convex relaxation, where the matching domain of permutation matrices is substituted with its convex hull of doubly-stochastic matrices, are equivalent for a certain class of graphs, such equivalence being based on spectral properties of the corresponding adjacency matrices. We also derive results about the automorphism group of a graph, and provide fundamental spectral properties of the adjacency matrix.

The use of alternative measures to evaluate classifier performance is gaining attention, specially for imbalanced problems. However, the use of these measures in the classifier design process is still unsolved. In this work we propose a classifier designed specifically to optimize one of these alternative measures, namely, the so-called F-measure. Nevertheless, the technique is general, and it can be used to optimize other evaluation measures. An algorithm to train the novel classifier is proposed, and the numerical scheme is tested with several databases, showing the optimality and robustness of the presented classifier.

We present a computer-aided detection pipeline for polyp detection in Computer tomographic colonography. The¯rst stage of the pipeline consists of a simple colon segmentation technique that enhances polyps, which is followed by an adaptive-scale candidate polyp delineation, in order to capture the appropriate polyp size. In the last step, candidates are classi¯ed based on new texture and geometric features that consider both the information in the candidate polyp location and its immediate surrounding area. The system is tested with ground truth data, including°at and small polyps which are hard to detect even with optical colonoscopy. We achieve 100% sensitivity for polyps larger than 6 mm in size with just 0.9 false positives per case, and 93% sensitivity with 2.8 false positives per case for polyps larger than 3 mm in size.

Computer Tomographic Colonography, combined with computer-aided detection (CAD), is a promising emerging technique for colonic polyp analysis. We present a CAD scheme for polyp flagging based on new texture and geometric features that consider both the information in the candidate polyp location and its immediate surrounding area, testing multiple sizes. The proposed algorithm is tested with ground truth data, including flat and small polyps, with very promising results.