7748348
Shape fitting with outliers
Given a set <i>H</i> of <i>n</i> hyperplanes in <i>R<sup>d</sup></i>, we present an algorithm that <i>e</i>-approximates the extent between the top and bottom <i>k</i> levels of the arrangement of <i>H</i> in time <i>O(n + (k/e)<sup>c</sup>)</i>, where <i>c</i> is a constant depending on <i>d</i>. The algorithm relies on computing a subset of <i>H</i> of size.<i>O(k/e<sup>d-1</sup>)</i>, in near linear time, such that the <i>k</i>-level of the arrangement of the subset approximates that of the original arrangement. Using this algorithm, we propose efficient approximation algorithms for shape fitting with outliers for various shapes. This is the first algorithms to handle outliers efficiently for the shape fitting problems considered.
2273352
Near-Linear Time Approximation Algorithms for Curve Simplification
AbstractWe consider the problem of approximating a polygonal
 curve P under a given error criterion by another
 polygonal curve P’ whose vertices are a
 subset of the vertices of P. The goal is to minimize
 the number of vertices of P’ while ensuring that the
 error between P’ and P is below a certain threshold.
 We consider two different error measures: 
 Hausdorff and Frechet. For both error criteria, we present
 near-linear time approximation algorithms that, given a
 parameter ε > 0, compute a simplified polygonal
 curve P’ whose error is less than ε and
 size at most the size of an optimal simplified
 polygonal curve with error ε/2. We consider
 monotone curves in ℝ2 in the case of the Hausdorff error measure
 under the uniform distance metric
 and arbitrary curves in any dimension for the Frechet error measure
 under Lp metrics. 
 We present
 experimental results demonstrating that our algorithms
 are simple and fast, and produce close to optimal
 simplifications in practice.
378133
LFM-Pro: a tool for detecting significant local structural sites in proteins
MOTIVATION
The rapidly growing protein structure repositories have opened up new opportunities for discovery and analysis of functional and evolutionary relationships among proteins. Detecting conserved structural sites that are unique to a protein family is of great value in identification of functionally important atoms and residues. Currently available methods are computationally expensive and fail to detect biologically significant local features.


RESULTS
We propose Local Feature Mining in Proteins (LFM-Pro) as a framework for automatically discovering family-specific local sites and the features associated with these sites. Our method uses the distance field to backbone atoms to detect geometrically significant structural centers of the protein. A feature vector is generated from the geometrical and biochemical environment around these centers. These features are then scored using a statistical measure, for their ability to distinguish a family of proteins from a background set of unrelated proteins, and successful features are combined into a representative set for the protein family. The utility and success of LFM-Pro are demonstrated on trypsin-like serine proteases family of proteins and on a challenging classification dataset via comparison with DALI. The results verify that our method is successful both in identifying the distinctive sites of a given family of proteins, and in classifying proteins using the extracted features.


AVAILABILITY
The software and the datasets are freely available for academic research use at http://bioinfo.ceng.metu.edu.tr/Pub/LFMPro.
312809
Segmenting molecular surfaces
This paper presents a new method for segmentation of molecular surfaces. Topological analysis of a scalar function defined on the surface and its associated gradient field reveals the relationship between the features of interest and critical points of the scalar function. The segmentation is obtained by associating segments with local minima/maxima. Controlled simplification of the function merges segments resulting in a hierarchical segmentation of the molecular surface. This segmentation is used to identify rigid components of protein molecules and to study the role of cavities and protrusions in protein-protein interactions.
37729130
Extreme Elevation on a 2-Manifold
Given a smoothly embedded 2-manifold in ${\Bbb R}^3,$ we define the elevation of a point as the height difference to a canonically defined second point on the same manifold. Our definition is invariant under rigid motions and can be used to define features such as lines of discontinuous or continuous but non-smooth elevation. We give an algorithm for finding points of locally maximum elevation, which we suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.
15443235
Computing the Writhing Number of a Polygonal Knot
AbstractThe writhing number measures the global geometry of
 a closed space curve or knot.
 We show that this measure is related to the average winding number
 of its Gauss map.
 Using this relationship, we give an algorithm for computing
 the writhing number for a polygonal knot with n edges
 in time roughly proportional to n1.6.
 We also implement a different, simple algorithm and provide
 experimental evidence for its practical efficiency.


15617422
R*-Histograms: efficient representation of spatial relations between objects of arbitrary topology
Representation of relative spatial relations between objects is often required in many multimedia database applications because spatial relations between objects in an image convey important information about the image. Quantitative representation of spatial relations taking into account shape, size, orientation and distance is often required. The R-Histogram is such a quantitative representation of spatial relations between two objects. However, this method only considers pixels on the object boundary, assuming that the objects are homeomorphic to a 2-ball. For objects with more complicated topology, we propose in this paper the R*-Histogram, a new extension to the R-Histogram. The R*-Histogram generalizes the R-Histogram by taking into account all the pixels in the objects. We also introduce an efficient O(kN log N) time algorithm to compute the R*-Histogram, which is asymptotically faster than the original O(N<sup>2</sup>) time algorithm for the R-Histogram even when k=O(n). Here, N=n<sup>2</sup> denotes the number of pixels in the processed n x n image and k is the number of different directions considered. The effectiveness of the R*-Histogram is evaluated empirically with a Query By Example (QBE) system on a database of 2000 synthetic images containing objects with complicated shape and topology. Experiments have shown that the similarly search results match human intuition very well.
18478207
Prefix-Free Regular-Expression Matching
We explore the regular-expression matching problem with respect to prefix-freeness of the pattern. We show that the prefix-free regular expression gives only linear number of matching substrings in the size of a given text. Based on this observation, we propose an efficient algorithm for the prefix-free regular-expression matching problem. Furthermore, we suggest an algorithm to determine whether or not a given regular language is prefix-free.
206873031
Lossy compression of images using logic minimization
A technique for lossy compression of images is presented, utilizing ideas of logic minimization. The approach specifically addresses the compression of the binary image data originated in block truncation coding (BTC). The binary vector corresponding to a block of 4/spl times/4 pixels is treated as the output of a Boolean function and prime cubes are generated. The largest prime cube is encoded. Bit rate less than 1.5 bits/pel is attained in BTC without many perceivable errors in the reconstructed grey scale image. Training, pre-stored tables or codebooks, and prior knowledge of the image source are not required by the technique which uses simple logic operations. Computational simplicity of the algorithm makes it suitable for VLSI implementation. Potential of the technique in attaining a rate less than 0.5 bit/pel by applying it on image sequences and extending to blocks of larger size is indicated.
18269009
Maximum Flows and Critical Vertices in AND/OR Graphs
We will study this problem and present an algorithm for finding the minimum-time-cost solution graph in an AND/OR graph. We will also study the following problems which often appear in industry when using AND/OR graphs to model manufacturing processes or to model problem solving processes: finding maximum (additive and nonadditive) flows and critical vertices in an AND/OR graph. Though there are well known polynomial time algorithms for the corresponding problems in the traditional graph theory, we will show that generally it is NP-hard to find a non-additive maximum flow in an AND/OR graph, and it is both NP-hard and coNP-hard to find a set of critical vertices in an AND/OR graph. We will also present a polynomial time algorithm for finding a maximum additive flow in an AND/OR graph, and discuss the relative complexity of these problems.
1746465
Pyramidal search of maximum coherence direction for biomedical image interpolation
A multiresolution technique for biomedical image interpolation is presented. It is an extension of the work on directional coherence interpolation (DCI) - a novel gray-level image interpolation method that interpolates the missing image data along the smoothed maximum coherence directions (MCDs). We propose to apply a pyramidal search strategy for MCD estimation. This coarse-to-fine scheme requires less computation time by starting with the reduced amount of data and propagating searching results to finer resolutions. In addition, it also improves robustness compared with our previous single resolution DCI.