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An Image Processing Approach to Computing Distances Between RNA Secondary Structures Dot Plots
, 2007
"... Background: Computing the distance between two RNA secondary structures can contribute in understanding the functional relationship between them. When used repeatedly, such a procedure may lead to finding a query RNA structure of interest in a database of structures. Several methods are available fo ..."
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Background: Computing the distance between two RNA secondary structures can contribute in understanding the functional relationship between them. When used repeatedly, such a procedure may lead to finding a query RNA structure of interest in a database of structures. Several methods are available for computing distances between RNAs represented as strings or graphs, but none utilize the RNA representation with dot plots. Since dot plots are essentially digital images, there is a clear motivation to devise an algorithm for computing the distance between dot plots based on image processing methods. Results: We have developed a new metric dubbed ’DoPloCompare’, which compares two RNA structures. The method is based on comparing dot plot diagrams that represent the secondary structures. When analyzing two diagrams and motivated by image processing, the distance is based on a combination of histogram correlations and a geometrical distance measure. We illustrate the procedure by an application that utilizes this metric on RNA sequences in order to locate peculiar point mutations that induce significant structural alternations relative to the wild type predicted secondary structure. The method was tested on several RNA sequences with known secondary structures to affirm their prediction, as well as on a data set of ribosomal pieces. These pieces were computationally cut from a ribosome for which an experimentally derived secondary structure is available, and on
Lossless Representation of Graphs using Distributions
, 2007
"... We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their subtriangles. In the second part, we propose graph represent ..."
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We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their subtriangles. In the second part, we propose graph representations in terms of onedimensional distributions (e.g., distribution of the node weights, sum of adjacent weights, etc.). For the case when the weights of the graph are realvalued vectors, we show that all graphs, except for a set of measure zero, are uniquely determined, up to isomorphism, from these distributions. The motivating application for this paper is the problem of browsing through large sets of graphs.
Constructing Uniquely Realizable Graphs
"... In the Graph Realization Problem (GRP), one is given a graph G, a set of nonnegative edgeweights, and an integer d. The goal is to find, if possible, a realization of G in the Euclidian space Rd, such that the distance between any two vertices is the assigned edge weight. The problem has many appl ..."
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In the Graph Realization Problem (GRP), one is given a graph G, a set of nonnegative edgeweights, and an integer d. The goal is to find, if possible, a realization of G in the Euclidian space Rd, such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NPhard when the dimension d is fixed. Characterizing tractable instances of GRP is a classical problem, first studied by Menger in 1931. We construct two new infinite families of GRP instances whose solution can be approximated up to an arbitrary precision in polynomial time. Both constructions are based on the blowup of fixed small graphs with large expanders. Our main tool is the Connelly’s condition in Rigidity Theory, combined with an explicit construction and algebraic calculations of the rigidity (stress) matrix. As an application of our results, we give a deterministic construction of uniquely kcolorable vertextransitive expanders.
Embedding and Similarity Search for Point Sets under Translation
, 2008
"... Pattern matching in point sets is a well studied problem with numerous applications. We assume that the point sets may contain outliers (missing or spurious points) and are subject to an unknown translation. We define the distance between any two point sets to be the minimum size of their symmetric ..."
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Pattern matching in point sets is a well studied problem with numerous applications. We assume that the point sets may contain outliers (missing or spurious points) and are subject to an unknown translation. We define the distance between any two point sets to be the minimum size of their symmetric difference over all translations of one set relative to the other. We consider the problem in the context of similarity search. We assume that a large database of point sets is to be preprocessed so that given any query point set, the closest matches in the database can be computed efficiently. Our approach is based on showing that there is a randomized algorithm that computes a translationinvariant embedding of any point set of size at most n into the L1 metric, so that with high probability, distances are subject to a distortion that is O(log² n).
Invariant Histograms
"... We introduce the concept of the Euclideaninvariant distance histogram function for curves. For sufficiently regular plane curves, we prove convergence of the cumulative distance histograms based on discretizing the curve by either uniformly spaced or randomly chosen sample points. Robustness of the ..."
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We introduce the concept of the Euclideaninvariant distance histogram function for curves. For sufficiently regular plane curves, we prove convergence of the cumulative distance histograms based on discretizing the curve by either uniformly spaced or randomly chosen sample points. Robustness of the curve histogram function under noise and pixelization of the curve is also established. We argue that the histogram function serves as a simple, noiseresistant shape classifier for regular curves under the Euclidean group of rigid motions. Extensions of the underlying ideas to higher dimensional submanifolds, as well as to area histogram functions invariant under the equiafffine group of areapreserving transformations are discussed. 1 Introduction. Given a finite set of points contained in R n, equipped with the usual Euclidean metric, consider the histogram formed by the mutual distances between all distinct pairs of points. An interesting question, first studied in depth by Boutin and Kemper, [2, 3], is to what extent the distance histogram uniquely determines the point set. Clearly, if the point set is
Algorithms for Molecular Biology BioMed Central
, 2009
"... An image processing approach to computing distances between RNA secondary structures dot plots ..."
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An image processing approach to computing distances between RNA secondary structures dot plots
FAITHFUL SHAPE REPRESENTATION FOR 2D GAUSSIAN MIXTURES
"... It has been recently discovered that a faithful representation for the shape of some simple distributions can be constructed using invariant statistics [1, 2]. In this paper, we consider the more general case of a Gaussian mixture model. We show that the shape of generic Gaussian mixtures can be rep ..."
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It has been recently discovered that a faithful representation for the shape of some simple distributions can be constructed using invariant statistics [1, 2]. In this paper, we consider the more general case of a Gaussian mixture model. We show that the shape of generic Gaussian mixtures can be represented without any loss by the distribution of the distance between two points independently drawn from this mixture. In other words, we show that if their respective distributions of distances are the same, then there exists a rigid transformation mapping one Gaussian mixture onto the other. Our main motivation is the problem of recognizing the shape of an object represented by points given noisy measurements of these points which can be modeled as a Gaussian mixture. Index Terms — Object recognition, shape, invariant statistics. 1.
Point Sets Up to Rigid Transformations are Determined by the Distribution of their Pairwise Distances
, 2008
"... This report is a summary of [BK06], which gives a simpler, albeit less general proof for the result of [BK04]. 1 ..."
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This report is a summary of [BK06], which gives a simpler, albeit less general proof for the result of [BK04]. 1