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Convergence of vector subdivision schemes in Sobolev spaces

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2002年第1期12卷 128-149页

作者： Chen, DR Jia, RQ Riemenschneider, SD Beijing Univ Aeronaut & Astronaut Dept Appl Math Beijing 100083 Peoples R China Univ Alberta Dept Math Sci Edmonton AB T6G 2G1 Canada W Virginia Univ Dept Math Morgantown WV 26506 USA

In this paper we consider functional equations of the form phi = Sigma(alphais an element ofZs) a(alpha)phi(M-.-alpha). where phi = (phi(1), . . . . phi(r)()T) is an r x 1 vector of functions on the s-dimensional Euclidean space, a(alpha), alpha is an element of Z(S), is a finitely supported sequence of r x r complex matrices, and M is an s x s isotropic integer matrix such that lim(n-->infinity)M(-n) = 0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function phi(j), j = 1,..., r, belonging to the Sobolev space W-p(k)(R-S)? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Q(a) associated with the sequence a is defined by QaF := Sigma(alphais an element ofZs) a(alpha)F(M-.-alpha). F is an element of (W-p(k)(R-s))(r). Let phi(0) be a nontrivial r x 1 vector of compactly supported functions in W-p(k)(R-S). The iteration scheme phi(n) = Q(a)phi(n-1), n = 1, 2, ..., is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0) = m(-1) Sigma(alphais an element ofZs) a(alpha), m = \det M\, and are related to polynomial reproducibility and the classical Strang-Fix conditions. The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution phi). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang-Fix type conditions that determine the feasible set of initial vectors. (C) 2002 Elsev

关键词： refinement equations multiple refinable functions vector subdivision schemes cascade algorithm joint spectral radii transition operators

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Subdivision schemes and refinement equations with nonnegative masks

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JOURNAL OF APPROXIMATION THEORY 2001年第2期113卷 207-220页

作者： Wang, Y Georgia Inst Technol Sch Math Atlanta GA 30332 USA

We consider the two-scale refinement equation f(x) = Sigma(n=0)(N) c(n)f(2x - n) with Sigma(n) c(2n) = Sigma(n)c(2n+1) = 1 where c(0), c(N) not equal 0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all c(n) greater than or equal to 0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 0 < c(0), c(N) < 1 and the greatest common divisor of S = {n: c(n) > 0} is 1. We prove the conjecture for a large class of refinement equations. (C) 2001 Academic Press.

关键词： nonnegative mask cascade algorithm subdivision scheme refinement equation refutable function

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Multiscale edge detection of noisy images using wavelets

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Journal of Harbin Institute of Technology(New Series) 2007年第5期14卷 737-740页

作者： FAROOQ M College of Automation Engineering Nanjing University of Aeronautics & Astronautics

The classical edge detectors work fine with the high quality pictures, but often are not good enough for noisy images because they cannot distinguish edges of different significance. The paper presented a novel approach to multiscale edge detection for noisy images using wavelet transforms based on Lipschitz regularity coefficients and a cascade algorithm. The relationship between wavelet transform and Lipschitz regularity was established. The proposed wavelet based edge detection algorithm combined the coefficients of wavelet transforms along with a cascade algorithm which significantly improves the result. The comparison between the proposed method and the classical edge detectors was carried out. The algorithm was applied to various images and its performance was discussed. The results of edge detection of contaminated images using the proposed algorithm show that it works better than the classical edge detectors.

关键词： edge detection wavelet lipschitz regularity cascade algorithm

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On shape detection in noisy images with particular reference to ultrasonography

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STATISTICS AND COMPUTING 1998年第4期8卷 377-389页

作者： Luan, JA Stander, J Wright, D Univ Plymouth Sch Math & Stat Plymouth PL4 8AA Devon England

We discuss the detection of a connected shape in a noisy image. Two types of image are considered: in the first a degraded outline of the shape is visible, while in the second the data are a corrupted version of the shape itself. In the first type the shape is defined by a thin outline of pixels with records that are different from those at pixels inside and outside the shape, while in the second type the shape is defined by its edge and pixels inside and outside the shape have different records. Our motivation is the identification of cross-sectional head shapes in ultrasound images of human fetuses. We describe and discuss a new approach to detecting shapes in images of the first type that uses a specially designed filter function that iteratively identifies the outline pixels of the head. We then suggest a way based on the cascade algorithm introduced by Jubb and Jennison (1991) of improving and considerably increasing the speed of a method proposed by Storvik (1994) for detecting edges in images of the second type.

关键词： cascade algorithm edge detection ICM simulated annealing

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The collocation method for Hammerstein equations by Daubechies wavelets

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期172卷 846-864页

作者： Maleknejad, K Derili, H Iran Univ Sci & Technol Sch Math Tehran 16844 Iran Islamic Azad Univ Fac Sci Karaj Unit Dept Math Karaj *** Iran

The numerical solutions to the nonlinear integral equations of Hammerstein-type y(t) = f(t)+ integral(1)(0) k(t, s)g(s, y(s))ds, t is an element of [0, 1] with using Daubechies wavelets are investigated. A general kernel scheme basing on Daubechies wavelets combined with a collocation method is presented. The approach of creating Daubechies interval wavelets and their main properties are briefly mentioned. Also we present an algorithm for computing of Daubechies wavelets in collocation points. The rate of approximation solution converging to the exact solution is given. Finally we also give some numerical examples for showing efficiency of the method. (c) 2005 Published by Elsevier Inc.

关键词： cascade algorithm Daubechies wavelets Bezout polynomials collocation method nonlinear integral equations

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CHARACTERIZATION OF L(P)-SOLUTIONS FOR THE 2-SCALE DILATION EQUATIONS

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 1995年第4期26卷 1018-1046页

作者： LAU, KS WANG, JR UNIV PITTSBURGH DEPT MATH & STAT PITTSBURGH PA 15260 USA

We give a characterization of the existence of compactly supported L(p)-solutions, 1 less than or equal to p < infinity, for the two-scale dilation equations. For the L(2)-case, the condition reduces to the determination of the spectral radius of a certain matrix in terms of the coefficients, which can be calculated through a finite step algorithm. For the other cases, we implement the characterization by the four-coefficient dilation equation and obtain some simple sufficient conditions for the existence of the solutions. The results are compared with known ones.

关键词： cascade algorithm COMPACTLY SUPPORTED L(P)-SOLUTIONS DILATION EQUATION FOURIER TRANSFORMATION ITERATION SPECTRAL RADIUS WAVELET

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Superresolution Imaging Using Constrained Iterative Deconvolution 11

Superresolution Imaging Using Constrained Iterative Deconvol...

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IEEE 11th Conference on Industrial Electronics and Applications (ICIEA)

作者： Fu, Xiongjun Peng, Shuilian Qian, Shengqi Zhang, Chengyan Xie, Ming Li, Shuguang Beijing Inst Technol Sch Informat & Elect Beijing 100081 Peoples R China Shanghai Aerosp Elect Technol Inst Shanghai 201109 Peoples R China

ISBN: (纸本)9781467386449

Constrained iterative deconvolution (CID) superresolution algorithm and its fast version, i.e., fast constrained-iterative deconvolution (FCID) are studied. These approaches are able to break through the aperture limiting to angular resolutions and can be applicable to two-dimensional imaging of non-coherent radars. However, FCID algorithm suffers fromd ivergence, which limits the improvement of angular resolutions. A cascade algorithm that FCID iterations alternate with CID iterations is developed in this paper. This algorithm significantly increases permissible FCID iterations, thus higher angular resolution of two-dimensional imaging is achieved with less computational burden.

关键词： CID FCID cascade algorithm higher angular resolution less computational burden

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Constructing Irislet: a New Wavelet Type which Matched for Iris Image Characteristics 2

Constructing Irislet: a New Wavelet Type which Matched for I...

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2nd International Conference on Information and Communication Technology (ICoICT)

作者： Isnanto, R. Rizal Satoto, Kodrat Iman Windasari, Ike Pertiwi Computer Engineering Department Diponegoro University Semarang Indonesia

ISBN: (纸本)9781479935802

Iris has a unique pattern that can be used in biometric recognition. To extract the features of the iris, it can be done based on the textural characteristics of the iris pattern. One method is a texture-based feature extraction using wavelet. To construct a wavelet type which matched for a signal, in this case two-dimensional signal from the iris image, the necessary steps are quite complex. In this research, all stages of wavelet design are carried out, beginning from iris image data acquisition up to the finding of the new wavelet, which will then be referred to as irislet. There are 19 (nineteen) steps in the design of this wavelet. To do all the stages, several basic concepts are required: convolution, circular Hough transform, conversion into unwrapped polar image form, determining the profile of the 1-D line images, signal averaging, concept of Daubechies wavelet basis, calculating signal energy, least squares method, how to construct scaling and wavelet functions, as well as the cascade algorithm. The test results showed that the recognition implementation irislet shows recognition rate is 100% correct.

关键词： irislet scaling function wavelet function least squares method cascade algorithm

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BLOCK TOEPLITZ-LIKE OPERATORS AND MULTIWAVELETS 2

BLOCK TOEPLITZ-LIKE OPERATORS AND MULTIWAVELETS

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2nd Conference on Wavelet Applications

作者： TURCAJOVA, R KAUTSKY, J FLINDERS UNIV S AUSTRALIA DEPT MATH & STATADELAIDESA 5001AUSTRALIA

ISBN: (纸本)0819418447

Because multiresolution analyses and wavelet bases are generated by translating and dilating scaling and wavelet functions, these functions must satisfy some special equations involving Toeplitz and Toeplitz-like operations. These relations can be exploited when such functions are to be constructed or their properties are studied. In the case of multiwavelets, where more than one scaling functions generate the multiresolution analysis, the corresponding operators are block Toeplitz-like. We discuss here some basic properties of these operators and compare them to the classical case of one scaling function. Using these observations we study the convergence of the cascade algorithm and derive sufficient conditions for existence of solution to a two-scale dilation equation.

关键词： BLOCK TOEPLITZ-LIKE OPERATORS cascade algorithm MULTIWAVELETS 2-SCALE EQUATIONS

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Evenly cascaded Convolutional Networks

Evenly Cascaded Convolutional Networks

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IEEE International Conference on Big Data (Big Data)

作者： Ye, Chengxi Devaraj, Chinmaya Maynord, Michael Fermueller, Cornelia Aloimonos, Yiannis Univ Maryland Inst Adv Comp Studies College Pk MD 20740 USA

ISBN: (纸本)9781538650356

We introduce Evenly cascaded convolutional Network (ECN), a neural network taking inspiration from the cascade algorithm of wavelet analysis. ECN employs two feature streams - a low-level and high-level steam. At each layer these streams interact, such that low-level features are modulated using advanced perspectives from the high-level stream. ECN is evenly structured through resizing feature map dimensions by a consistent ratio, which removes the burden of ad-hoc specification of feature map dimensions. ECN produces easily interpretable features maps, a result whose intuition can be understood in the context of scale-space theory. We demonstrate that ECN's design facilitates the training process through providing easily trainable shortcuts. We report new state-of-the-art results for small networks, without the need for additional treatment such as pruning or compression - a consequence of ECN's simple structure and direct training. A 6-layered ECN design with under 500k parameters achieves 95.24% and 78.99% accuracy on CIFAR-10 and CIFAR-100 datasets, respectively, outperforming the current state-of-the-art on small parameter networks, and a 3 million parameter ECN produces results competitive to the state-of-the-art.

关键词： cascade algorithm multi-level feature representation network efficiency recurrent convolution recursive convolution

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