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78
OneDimensional Quantum Walks
 STOC'01
, 2001
"... We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, ..."
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Cited by 90 (11 self)
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We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range [\Gamma t= p
3D Zernike Descriptors for Content Based Shape Retrieval
 In The 8th ACM Symposium on Solid Modeling and Applications
, 2003
"... Content based 3D shape retrieval for broad domains like the World Wide Web has recently gained considerable attention in Computer Graphics community. One of the main challenges in this context is the mapping of 3D objects into compact canonical representations referred to as descriptors, which serve ..."
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Cited by 59 (1 self)
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Content based 3D shape retrieval for broad domains like the World Wide Web has recently gained considerable attention in Computer Graphics community. One of the main challenges in this context is the mapping of 3D objects into compact canonical representations referred to as descriptors, which serve as search keys during the retrieval process. The descriptors should have certain desirable properties like invariance under scaling, rotation and translation. Very importantly, they should possess descriptive power providing a basis for similarity measure between threedimensional objects which is close to the human notion of resemblance. In this paper we advocate the usage of socalled 3D Zernike invariants as descriptors for content based 3D shape retrieval. The basis polynomials of this representation facilitate computation of invariants under the above transformations. Some theoretical results have already been summarized in the past from the aspect of pattern recognition and shape analysis. We provide practical analysis of these invariants along with algorithms and computational details. Furthermore, we give a detailed discussion on influence of the algorithm parameters like type and resolution of the conversion into a volumetric function, number of utilized coefficients, etc. As is revealed by our study, the 3D Zernike descriptors are natural extensions of spherical harmonics based descriptors, which are reported to be among the most successful representations at present. We conduct a comparison of 3D Zernike descriptors against these regarding computational aspects and shape retrieval performance.
Shape retrieval using 3d zernike descriptors
 Computer Aided Design
, 2004
"... We advocate the usage of 3D Zernike invariants as descriptors for 3D shape retrieval. The basis polynomials of this representation facilitate computation of invariants under rotation, translation and scaling. Some theoretical results have already been summarized in the past from the aspect of patter ..."
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Cited by 27 (2 self)
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We advocate the usage of 3D Zernike invariants as descriptors for 3D shape retrieval. The basis polynomials of this representation facilitate computation of invariants under rotation, translation and scaling. Some theoretical results have already been summarized in the past from the aspect of pattern recognition and shape analysis. We provide practical analysis of these invariants along with algorithms and computational details. Furthermore, we give a detailed discussion on influence of the algorithm parameters like the conversion into a volumetric function, number of utilized coefficients, etc. As is revealed by our study, the 3D Zernike descriptors are natural extensions of recently introduced spherical harmonics based descriptors. We conduct a comparison of 3D Zernike descriptors against these regarding computational aspects and shape retrieval performance using several quality measures and based on experiments on the Princeton Shape Benchmark. 1
A Fingerprint Orientation Model Based on 2D Fourier Expansion (FOMFE) and Its Application to SingularPoint Detection and Fingerprint Indexing
, 2007
"... In this paper, we have proposed a fingerprint orientation model based on 2D Fourier expansions (FOMFE) in the phase plane. The FOMFE does not require prior knowledge of singular points (SPs). It is able to describe the overall ridge topology seamlessly, including the SP regions, even for noisy finge ..."
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Cited by 26 (11 self)
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In this paper, we have proposed a fingerprint orientation model based on 2D Fourier expansions (FOMFE) in the phase plane. The FOMFE does not require prior knowledge of singular points (SPs). It is able to describe the overall ridge topology seamlessly, including the SP regions, even for noisy fingerprints. Our statistical experiments on a public database show that the proposed FOMFE can significantly improve the accuracy of fingerprint feature extraction and thus that of fingerprint matching. Moreover, the FOMFE has a lowcomputational cost and can work very efficiently on large fingerprint databases. The FOMFE provides a comprehensive description for orientation features, which has enabled its beneficial use in featurerelated applications such as fingerprint indexing. Unlike most indexing schemes using raw orientation data, we exploit FOMFE model coefficients to generate the feature vector. Our indexing experiments show remarkable results using different fingerprint databases.
Convergence Of Multidimensional Cascade Algorithm
 Numer. Math
, 1998
"... . A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in L 2 (R d ) of the multidimensional cascade algorithm for any starting function OE 0 whose shifts form a partition of unity. 1. Introduction This paper is a continuation o ..."
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Cited by 26 (10 self)
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. A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in L 2 (R d ) of the multidimensional cascade algorithm for any starting function OE 0 whose shifts form a partition of unity. 1. Introduction This paper is a continuation of [8]. In [8] we obtained a complete characterization of stability and orthonormality of the shifts of a refinable function in terms of its refinenent mask. In this paper we present a complete characterization of the convergence in L 2 (R d ) of the multidimensional cascade algorithm with an arbitrary dilation matrix M in terms of the mask. For fixed integers d 1 and m 2; let M be a d \Theta d dilation matrix with j det(M)j = m: A dilation matrix is an integer matrix with all eigenvalues of modulus ? 1: Let ` 2 (Z d ); where Z d is the set of all multiintegers, be the space of all squaresummable sequences, and L 2 (R d ) the space of all squareintegrable functions. The...
3D Zernike Moments and Zernike Affine Invariants for 3D Image Analysis and Recognition
 In 11th Scandinavian Conf. on Image Analysis
, 1999
"... Guided by the results of much research work done in the past on the performance of 2D image moments and moment invariants in the presence of noise, suggesting that by using orthogonal 2D Zernike rather than regular geometrical moments one gets many advantages regarding noise effects, informati ..."
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Cited by 22 (0 self)
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Guided by the results of much research work done in the past on the performance of 2D image moments and moment invariants in the presence of noise, suggesting that by using orthogonal 2D Zernike rather than regular geometrical moments one gets many advantages regarding noise effects, information suppression at low radii and redundancy, we have worked out and introduce a complete set of 3D polynomials orthonormal within the unit sphere that exhibits a "form invariance" property under 3D rotation like the 2D Zernike polynomials do in the plane. For that reason we call this set 3D Zernike polynomials. The role of the angular exponential function in the 2D Zernike polynomials set is now played by the spherical harmonics on the surface of the unit sphere. Spherical harmonics and spherical moments are introduced in a very succinct, selfcontained and compact way using algebraically powerful tools like 'power substitutions ' and generating functions. Unambiguous affine normalization and unique affine pose determination using 3D image moments of degree not greater than three as well as derivation of complete, uncorrelated affine invariants are naturally accomplished using the concepts we introduce in the present paper.
Hardy's Theorem And The ShortTime Fourier Transform Of Schwartz Functions
, 2001
"... We characterize the Schwartz space of rapidly decaying test functions by the decay of the shorttime Fourier transform or crossWigner distribution. Then we prove a version of Hardy's theorem for the shorttime Fourier transform and for the Wigner distribution. ..."
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Cited by 19 (7 self)
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We characterize the Schwartz space of rapidly decaying test functions by the decay of the shorttime Fourier transform or crossWigner distribution. Then we prove a version of Hardy's theorem for the shorttime Fourier transform and for the Wigner distribution.
DiscreteTime, DiscreteFrequency TimeFrequency Representations
 in Proc. of the IEEE Int. Conf. on Acoust., Speech, and Signal Processing
, 1995
"... A discretetime, discretefrequency Wigner distribution is derived using a grouptheoretic approach. It is based upon a study of the Heisenberg group generated by the integers mod N , which represents the group of discretetime and discretefrequency shifts. The resulting Wigner distribution satisfi ..."
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Cited by 17 (3 self)
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A discretetime, discretefrequency Wigner distribution is derived using a grouptheoretic approach. It is based upon a study of the Heisenberg group generated by the integers mod N , which represents the group of discretetime and discretefrequency shifts. The resulting Wigner distribution satisfies several desired properties. An example demonstrates that it is a fullband timefrequency representation, and, as such, does not require special sampling techniques to suppress aliasing. It also exhibits some interesting and unexpected interference properties. The new distribution is compared with other discretetime, discretefrequency Wigner distributions proposed in the literature. 1. INTRODUCTION The Wigner distribution is an important tool for analyzing signals. Its usefulness arises from the fact that it satisfies many desired mathematical properties. Such properties include the time and frequency marginals, Moyal's formula, the relationship with the ambiguity function, and the rel...
An Uncertainty Principle For Hankel Transforms
"... There exists a generalized Hankel transform of order ff \Gamma1=2 on R, which is based on the eigenfunctions of the Dunkl operator T ff f(x) = f 0 (x) + \Gamma ff + 1 2 \Delta f(x) \Gamma f(\Gammax) x ; f 2 C 1 (R): For ff = \Gamma1=2 this transform coincides with the usual Fourier transf ..."
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Cited by 16 (4 self)
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There exists a generalized Hankel transform of order ff \Gamma1=2 on R, which is based on the eigenfunctions of the Dunkl operator T ff f(x) = f 0 (x) + \Gamma ff + 1 2 \Delta f(x) \Gamma f(\Gammax) x ; f 2 C 1 (R): For ff = \Gamma1=2 this transform coincides with the usual Fourier transform on R. In this paper the operator T ff replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on R. It generalizes the classical WeylHeisenberg uncertainty principle for the position and momentum operators on L 2 (R); moreover, it implies a WeylHeisenberg inequality for the classical Hankel transform of arbitrary order ff \Gamma1=2 on [0; 1[:
Approximating Probability Distributions Using Small Sample Spaces
 Combinatorica
, 1995
"... We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group. The approximate distribution is characterized by a parameter ffl, the quality of the approximation, which is a bound on the difference between corresponding Fourier coefficients of the two d ..."
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Cited by 15 (0 self)
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We formulate the notion of a "good approximation" to a probability distribution over a finite abelian group. The approximate distribution is characterized by a parameter ffl, the quality of the approximation, which is a bound on the difference between corresponding Fourier coefficients of the two distributions. It is also required that the sample space of the approximate distribution be of size polynomial in the representation length of the group elements as well as 1=ffl. Such approximations are useful in reducing or eliminating the use of randomness in randomized algorithms. We demonstrate the existence of such good approximations to arbitrary distributions. In the case of n random variables distributed uniformly and independently over the range f0; : : : ; d \Gamma 1g, we provide an efficient construction of a good approximation. The constructed approximation has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the ...