A quadtree algorithm for template matching on a pyramid computer

Senoussi, Houcine; Saoudi, A.

doi:10.1016/0304-3975(94)00020-j

Cited by 12 publications

(10 citation statements)

References 8 publications

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“…We refer the reader to Bell (1940), Boyer (1968) and Pan (1997) on this fascinating development. The areas influenced by the polynomial root-finding problem in the 19th and 20th centuries included meromorphic functions, algebraic curves, and structured matrices (Householder, 1970;Gauss, 1973;Henrici, 1974); furthermore, Weyl's quadtree root-finder alone (Weyl, 1924) has made an impact on computational geometry, image processing, template matching, and the n-body particle simulation (Samet, 1984;Greengard, 1988;Senoussi, 1994). Presently, polynomial root-finding is still a major research topic with highly important applications to computer algebra, in particular to the solution of polynomial systems of equations (Kapur and Lakshman, 1992;Blum et al, 1997;Pan, 1997;Pan, 1998, 2000) (cf.…”

Section: Some History Of Polynomial Root-findingmentioning

confidence: 99%

Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

Pan

2002

Journal of Symbolic Computation

159

175

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To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new root-finder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and root-finding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be ill-conditioned, forming clusters. (The worst case complexity bounds for root-finding are supported by our previous algorithms as well.) All algorithms allow processor efficient acceleration to achieve solution in polylogarithmic parallel time.

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Section: Some History Of Polynomial Root-findingmentioning

confidence: 99%

Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

Pan

2002

Journal of Symbolic Computation

159

175

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“…The algorithm can be viewed as a 2-dimensional version of the customary bisection of a line interval. Under the name of the quadtree algorithm, this algorithm has been successfully applied to various areas of practical importance, such as image processing, n-particle simulation, template matching, and computational geometry [Sa84,Gre88,Se94].…”

Section: The Functional Iteration Approachmentioning

confidence: 99%

Approximating Complex Polynomial Zeros: Modified Weyl's Quadtree Construction and Improved Newton's Iteration

Pan¹

2000

Journal of Complexity

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We propose a new algorithm for the classical and still practically important problem of approximating zeros z j of an nth degree polynomial p(x) within error bound 2 &b max j |z j |. The algorithm uses O((n 2 log n) log(bn)) arithmetic operations and comparisons for approximating all the n zeros and O((kn log n) log(bn)) for approximating the k zeros lying in a fixed domain (disc or square) and isolated from the other zeros. Unlike the previous fast algorithms of this kind, the new algorithm has its simple elementary description, is convenient for practical implementation, and allows the users to adapt the computational precision to the current level of approximation achieved in the process of computing and ultimately to the requirements to the output precision for each zero of p(x). The algorithm relies on our novel versions of Weyl's quadtree construction and Newton's iteration. Academic Press

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“…Due to serial data input, their architecture requires 22.6 ms for processing a 512 ϫ 512 image and 128 ϫ 128 template. A quadtree based algorithm for template matching on a N ϫ N image with M ϫ M template using a pyramid of log N ϩ 1 levels is described in [42]. A 3D architecture made of a 2D array of linear arrays is described in [43].…”

Section: Scene Matchingmentioning

confidence: 99%