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Efficient computation of gapped substring kernels on large alphabets
 Journal of Machine Leaning Research
, 2005
"... We present a sparse dynamic programming algorithm that, given two strings s and t, a gap penalty λ, and an integer p, computes the value of the gapweighted lengthp subsequences kernel. The algorithm works in time O(pMlogt), where M = {(i, j)si = t j} is the set of matches of characters in the ..."
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We present a sparse dynamic programming algorithm that, given two strings s and t, a gap penalty λ, and an integer p, computes the value of the gapweighted lengthp subsequences kernel. The algorithm works in time O(pMlogt), where M = {(i, j)si = t j} is the set of matches of characters in the two sequences. The algorithm is easily adapted to handle bounded length subsequences and different gappenalty schemes, including penalizing by the total length of gaps and the number of gaps as well as incorporating characterspecific match/gap penalties. The new algorithm is empirically evaluated against a full dynamic programming approach and a triebased algorithm both on synthetic and newswire article data. Based on the experiments, the full dynamic programming approach is the fastest on short strings, and on long strings if the alphabet is small. On large alphabets, the new sparse dynamic programming algorithm is the most efficient. On mediumsized alphabets the triebased approach is best if the maximum number of allowed gaps is strongly restricted.
Computing rank convolutions with a mask
, 2006
"... Rankconvolutions have important applications in a variety of areas such as signal processing and computer vision. We define a “mask ” as a function taking only values zero and infinity. Rankconvolutions with masks are of special interest to image processing. We show how to compute the rankk convo ..."
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Rankconvolutions have important applications in a variety of areas such as signal processing and computer vision. We define a “mask ” as a function taking only values zero and infinity. Rankconvolutions with masks are of special interest to image processing. We show how to compute the rankk convolution of a function over an interval of length n with an arbitrary mask of length m in O(n √ m log m) time. The result generalizes to the ddimensional case. Previously no algorithm performing significantly better than the brute force O(nm) bound was known. Our algorithm seems to perform well in practice. We describe an implementation, illustrating its application to a problem in image processing. Already on relatively small images, our experiments show a signficant speedup compared to brute force.
The minconvolution of f and g is the function h = f \Omega ming: [2n1]d!R defined by
"... Abstract Minconvolution and more generally, rankconvolutions play an important role in many application areas, including nonlinear signal processing, pattern recognition, computer vision, and combinatorial optimization. In computer vision they have been used for object recognition, depth estimat ..."
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Abstract Minconvolution and more generally, rankconvolutions play an important role in many application areas, including nonlinear signal processing, pattern recognition, computer vision, and combinatorial optimization. In computer vision they have been used for object recognition, depth estimation, and image restoration. Rankconvolutionssuch as the medianconvolution have the advantage of being more robust against noise.We define a &quot;mask &quot; as a function taking only values zero and infinity. Rankconvolutions with masks are of special interest; they correspond to a type of orderstatistic filtering used in image processing. We show how to compute a rankconvolution of any function with amask in O(n3/2(log n)1/2). No improvement better than a logarithmicfactor over the O(n2) brute force bound appears to have been previously known even for the special case of minconvolutions. Our result
On Table Arrangements, Scrabble Freaks, and Jumbled Pattern Matching?
"... Abstract. Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of Parikh vector q (a “jumbled string”) in the text s requires to find a substring t of s with p(t) = q. The corresponding decision problem is to verify whether a ..."
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Abstract. Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of Parikh vector q (a “jumbled string”) in the text s requires to find a substring t of s with p(t) = q. The corresponding decision problem is to verify whether at least one such match exists. So, for example for the alphabet Σ = {a, b, c}, the string s = abaccbabaaa has Parikh vector p(s) = (6, 3, 2), and the Parikh vector q = (2, 1, 1) appears once in s in position (1, 4). Like its more precise counterpart, the renown Exact String Matching, Jumbled Pattern Matching has ubiquitous applications, e.g., string matching with a dyslectic word processor, table rearrangements, anagram checking, Scrabble playing and, allegedly, also analysis of mass spectrometry data. We consider two simple algorithms for Jumbled Pattern Matching and use very complicated data structures and analytic tools to show that they are not worse than the most obvious algorithm. We also show that we can achieve nontrivial efficient average case behavior, but that’s less fun to describe in this abstract so we defer the details to the main part of the article, to be read at the reader’s risk... well, at the reader’s discretion. 1
Master’s Paper [ROUGH DRAFT]: 2 Dimensional MinFilters with Polygons
, 2007
"... Computing minfilters is an important operation in image processing. Minfilters are the basic building blocks for translation invariant morphological operators, which are used to perform operations such as noise suppression, image smoothing, contrast enhancement, and edge detection. Minfilters are ..."
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Computing minfilters is an important operation in image processing. Minfilters are the basic building blocks for translation invariant morphological operators, which are used to perform operations such as noise suppression, image smoothing, contrast enhancement, and edge detection. Minfilters are a special case of minconvolutions, which are used for a variety of applications, including signal processing, and combinatorial optimization. The most widely used approach for computing minfilters decomposes the filtering element as a Minkowsky sum of smaller filters. However this approach can be expensive for large filtering elements[1]. There is an algorithm to perform the minfilter efficiently, but it only applies to axis parallel rectangles [2]. We present an efficient algorithm to compute the minfilter when the filtering element is a polygon for certain polygons. Because the polygon will be given as a non digital geometric object, we relax the conditions by allowing the algorithm to choose a valid digitization of the polygon for each placement. With this relaxation, we present an algorithm to compute the minfilter by arbitrary rectangles and isosceles right triangles in logarithmic time per pixel, in the size of the polygon. We also show how to compute the minfilter by arbitrary triangles in time O per pixel, where A is the area of � log A sinγ the triangle, and γ is the size of the smallest angle in the triangle. 2 1