A Note on the Unsolvability of the Weighted Region Shortest Path Problem

Let S be a subdivision of the plane into polygonal regions, where each region has an associated positive weight. The weighted region shortest path problem is to determine a shortest path in S between two points s, t in R^2, where the distances are measured according to the weighted Euclidean metric-the length of a path is defined to be the weighted sum of (Euclidean) lengths of the sub-paths within each region. We show that this problem cannot be solved in the Algebraic Computation Model over the Rational Numbers (ACMQ). In the ACMQ, one can compute exactly any number that can be obtained from the rationals Q by applying a finite number of operations from +, -, \times, \div, \sqrt[k]{}, for any integer k >= 2. Our proof uses Galois theory and is based on Bajaj's technique.Comment: 6 pages, 1 figur

Applied Similarity Problems Using Frechet Distance

In the first part of this thesis, we consider an instance of Frechet distance problem in which the speed of traversal along each segment of the curves is restricted to be within a specfied range. This setting is more realistic than the classical Frechet distance setting, specially in GIS applications. We also study this problem in the setting where the polygonal curves are inside a simple polygon. In the second part of this thesis, we present a data structure, called the free-space map, that enables us to solve several variants of the Frechet distance problem efficiently. Our data structure encapsulates all the information available in the free-space diagram, yet it is capable of answering more general type of queries efficiently. Given that the free-space map has the same size and construction time as the standard free-space diagram, it can be viewed as a powerful alternative to it. As part of the results in Part II of the thesis, we exploit the free-space map to improve the long-standing bound for computing the partial Frechet distance and obtain improved algorithms for computing the Frechet distance between two closed curves, and the so-called minimum/maximum walk problem. We also improve the map matching algorithm for the case when the map is a directed acyclic graph. As the last part of this thesis, given a point set S and a polygonal curve P in R^d, we study the problem of finding a polygonal curve Q through S, which has a minimum Frechet distance to P. Furthermore, if the problem requires that curve Q visits every point in S, we show it is NP-complete.Comment: arXiv admin note: text overlap with arXiv:1003.0460 by other author

Automated Segmentation of Cells with IHC Membrane Staining

This study presents a fully automated membrane segmentation technique for immunohistochemical tissue images with membrane staining, which is a critical task in computerized immunohistochemistry (IHC). Membrane segmentation is particularly tricky in immunohistochemical tissue images because the cellular membranes are visible only in the stained tracts of the cell, while the unstained tracts are not visible. Our automated method provides accurate segmentation of the cellular membranes in the stained tracts and reconstructs the approximate location of the unstained tracts using nuclear membranes as a spatial reference. Accurate cell-by-cell membrane segmentation allows per cell morphological analysis and quantification of the target membrane proteins that is fundamental in several medical applications such as cancer characterization and classification, personalized therapy design, and for any other applications requiring cell morphology characterization. Experimental results on real datasets from different anatomical locations demonstrate the wide applicability and high accuracy of our approach in the context of IHC analysi

Path Similarity Analysis: a Method for Quantifying Macromolecular Pathways

Diverse classes of proteins function through large-scale conformational changes; sophisticated enhanced sampling methods have been proposed to generate these macromolecular transition paths. As such paths are curves in a high-dimensional space, they have been difficult to compare quantitatively, a prerequisite to, for instance, assess the quality of different sampling algorithms. The Path Similarity Analysis (PSA) approach alleviates these difficulties by utilizing the full information in 3N-dimensional trajectories in configuration space. PSA employs the Hausdorff or Fr\'echet path metrics---adopted from computational geometry---enabling us to quantify path (dis)similarity, while the new concept of a Hausdorff-pair map permits the extraction of atomic-scale determinants responsible for path differences. Combined with clustering techniques, PSA facilitates the comparison of many paths, including collections of transition ensembles. We use the closed-to-open transition of the enzyme adenylate kinase (AdK)---a commonly used testbed for the assessment enhanced sampling algorithms---to examine multiple microsecond equilibrium molecular dynamics (MD) transitions of AdK in its substrate-free form alongside transition ensembles from the MD-based dynamic importance sampling (DIMS-MD) and targeted MD (TMD) methods, and a geometrical targeting algorithm (FRODA). A Hausdorff pairs analysis of these ensembles revealed, for instance, that differences in DIMS-MD and FRODA paths were mediated by a set of conserved salt bridges whose charge-charge interactions are fully modeled in DIMS-MD but not in FRODA. We also demonstrate how existing trajectory analysis methods relying on pre-defined collective variables, such as native contacts or geometric quantities, can be used synergistically with PSA, as well as the application of PSA to more complex systems such as membrane transporter proteins.Comment: 9 figures, 3 tables in the main manuscript; supplementary information includes 7 texts (S1 Text - S7 Text) and 11 figures (S1 Fig - S11 Fig) (also available from journal site

Partial shape matching using CCP map and weighted graph transformation matching

La détection de la similarité ou de la différence entre les images et leur mise en correspondance sont des problèmes fondamentaux dans le traitement de l'image. Pour résoudre ces problèmes, on utilise, dans la littérature, différents algorithmes d'appariement. Malgré leur nouveauté, ces algorithmes sont pour la plupart inefficaces et ne peuvent pas fonctionner correctement dans les situations d’images bruitées. Dans ce mémoire, nous résolvons la plupart des problèmes de ces méthodes en utilisant un algorithme fiable pour segmenter la carte des contours image, appelée carte des CCPs, et une nouvelle méthode d'appariement. Dans notre algorithme, nous utilisons un descripteur local qui est rapide à calculer, est invariant aux transformations affines et est fiable pour des objets non rigides et des situations d’occultation. Après avoir trouvé le meilleur appariement pour chaque contour, nous devons vérifier si ces derniers sont correctement appariés. Pour ce faire, nous utilisons l'approche « Weighted Graph Transformation Matching » (WGTM), qui est capable d'éliminer les appariements aberrants en fonction de leur proximité et de leurs relations géométriques. WGTM fonctionne correctement pour les objets à la fois rigides et non rigides et est robuste aux distorsions importantes. Pour évaluer notre méthode, le jeu de données ETHZ comportant cinq classes différentes d'objets (bouteilles, cygnes, tasses, girafes, logos Apple) est utilisé. Enfin, notre méthode est comparée à plusieurs méthodes célèbres proposées par d'autres chercheurs dans la littérature. Bien que notre méthode donne un résultat comparable à celui des méthodes de référence en termes du rappel et de la précision de localisation des frontières, elle améliore significativement la précision moyenne pour toutes les catégories du jeu de données ETHZ.Matching and detecting similarity or dissimilarity between images is a fundamental problem in image processing. Different matching algorithms are used in literature to solve this fundamental problem. Despite their novelty, these algorithms are mostly inefficient and cannot perform properly in noisy situations. In this thesis, we solve most of the problems of previous methods by using a reliable algorithm for segmenting image contour map, called CCP Map, and a new matching method. In our algorithm, we use a local shape descriptor that is very fast, invariant to affine transform, and robust for dealing with non-rigid objects and occlusion. After finding the best match for the contours, we need to verify if they are correctly matched. For this matter, we use the Weighted Graph Transformation Matching (WGTM) approach, which is capable of removing outliers based on their adjacency and geometrical relationships. WGTM works properly for both rigid and non-rigid objects and is robust to high order distortions. For evaluating our method, the ETHZ dataset including five diverse classes of objects (bottles, swans, mugs, giraffes, apple-logos) is used. Finally, our method is compared to several famous methods proposed by other researchers in the literature. While our method shows a comparable result to other benchmarks in terms of recall and the precision of boundary localization, it significantly improves the average precision for all of the categories in the ETHZ dataset