Lee Seng Luan
Email Address
matleesl@nus.edu.sg
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Publication Construction of schauder decomposition on banach spaces of periodic functions(1998) Goh, S.S.; Lee, S.L.; Shen, Z.; Tang, W.S.; MATHEMATICSThis paper deals with Schauder decompositions of Banach spaces X2x of 2π-periodic functions by projection operators Pk onto the subspaces Vk, k = 0,1,..., which form a multiresolution of X2x. The results unify the study of wavelet decompositions by orthogonal projections in the Hilbert space L2x 2 on one hand and by interpolatory projections in the Banach space C2x on the other. The approach, using "orthogonal splines", is constructive and leads to the construction of a Schauder decomposition of X2x and a biorthogonal system for X2x and its dual X2x*. Decomposition and reconstruction algorithms are derived from the construction.Publication Scale-space derived from B-splines(1998) Wang, Y.-P.; Lee, S.L.; MATHEMATICSIt is well-known that the linear scale-space theory in computer vision is mainly based on the Gaussian kernel. The purpose of the paper is to propose a scale-space theory based on S-spline kernels. Our aim is twofold. On one hand, we present a general framework and show how S-splines provide a flexible tool to design various scale-space representations: continuous scalespace, dyadic scale-space frame, and compact scale-space representation. In particular, we focus on the design of continuous scale-space and dyadic scale-space frame representation. A general algorithm is presented for fast implementation of continuous scale-space at rational scales. In the dyadic case, efficient frame algorithms are derived using S-spline techniques to analyze the geometry of an image. Moreover, the image can be synthesized from its multiscale local partial derivatives. Also, the relationship between several scale-space approaches is explored. In particular, the evolution of wavelet theory from traditional scale-space filtering can be well understood in terms of S-splines. On the other hand, the behavior of edge models, the properties of completeness, causality, and other properties in such a scale-space representation are examined in the framework of S-splines. It is shown that, besides the good properties inherited from the Gaussian kernel, the S-spline derived scale-space exhibits many advantages for modeling visual mechanism with regard to the efficiency, compactness, orientation feature, and parallel structure. © 1998 IEEE.Publication Wavelet bases for a set of commuting unitary operators(1993-02) Goodman, T.N.T.; Lee, S.L.; Tang, W.S.; MATHEMATICSLet (U=U1, ..., Ud) be an ordered d-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space ℋ, and let X:={x1, ..., xr} ⊂ ℋ such that {Mathematical expression} is a Riesz basis of the closed linear span V0 of {Mathematical expression}. Suppose there is unitary operator D on ℋ such that V0 ⊂DV0 =:V1 and UnD=DUAn for all n ∈ ℤd, where A is a d ×d matrix with integer entries and Δ := det(A) ≠ 0. Then there is a subset Λ in V1, with r(Δ - 1) vectors, such that {Mathematical expression} is a Riesz basis of W0, the orthogonal complement of V0 in V1. The resulting multiscale and decomposition relations can be expressed in a Fourier representation by one single equation, in terms of which the duality principle follows easily. These results are a consequence of an extension, to a set of commuting unitary operators, of Robertson's Theorems on wandering subspace for a single unitary operator [24]. Conditions are given in order that {Mathematical expression} is a Riesz basis of W0. They are used in the construction of a class of linear spline wavelets on a four-direction mesh. © 1993 J.C. Baltzer AG, Science Publishers.Publication New multifilter design property for multiwavelet image compression(1999) Tham, Jo Yew; Shen, Lixin; Lee, Seng Luan; Tan, Hwee Huat; MATHEMATICSApproximation order, linear phase symmetry, time-frequency localization, regularity, and stopband attenuation are some criteria that are widely used in wavelet filter design. In this paper, we propose a new criterion called good multifilter properties (GMPs) for the design and construction of multiwavelet filters targeting image compression applications. We first provide the definition of GMPs, followed by a necessary and sufficient condition for an orthonormal multiwavelet system to have a GMP order of at least 1. We then present an algorithm to construct orthogonal multiwavelets possessing GMPs, starting from any length-2N scalar CQFs. Image compression experiments are performed to evaluate the importance of GMPs for image compression, as compared to other common filter design criteria. Our results confirmed that multiwavelets that possess GMPs not only yield superior PSNR performances, but also require much lower computations in their transforms.Publication On the spectral radius of convolution dilation operators(2002) Didenko, V.D.; Korenovskyy, A.A.; Lee, S.L.; MATHEMATICSConvolution dilation operators with non-compactly supported kernels are considered and effective formulae for their spectral radii are found. The formulae depend on the behaviour of the eigenvalues of the dilation matrix.Publication Characterization of compactly supported refinable splines(1995-01) Lawton, W.; Lee, S.L.; Shen, Z.; MATHEMATICS; INSTITUTE OF SYSTEMS SCIENCEWe prove that a compactly supported spline function φ of degree k satisfies the scaling equation {Mathematical expression} for some integer m ≥ 2, if and only if {Mathematical expression} where p(n) are the coefficients of a polynomial P(z) such that the roots of P(z)(z - 1)k+1 TM are mapped into themselves by the mapping z →zm, and Bk is the uniform B-spline of degree k. Furthermore, the shifts of φ form a Riesz basis if and only if P is a monomial. © 1995 J.C. Baltzer AG, Science Publishers.Publication Approximate foveated images and reconstruction of their uniform pre-images(2007-07) Didenko, V.; Lee, S.L.; Roch, S.; Silbermann, B.; MATHEMATICSApproximate foveated images can be obtained from uniform images via the approximation of some integral operators. In this paper it is shown that these operators belong to a well-studied operator algebra, and the problem of restoration of the approximate uniform pre-images is considered. Under common assumptions on smoothness of the integral operator kernels, necessary and sufficient conditions are established for such procedure to be feasible. © 2007 Elsevier Inc. All rights reserved.Publication Foveated splines and wavelets(2008-11) GAO XIAOJIE; Goodman, T.N.T.; Lee, S.L.; MATHEMATICS; RISK MANAGEMENT INSTITUTESpline wavelets on a hybrid of uniform and geometric meshes that admits a natural dyadic multiresolution structure are constructed. The construction is extended to other scaling functions. The hybrid splines and wavelets provide good approximation of functions near singularities and efficient representation of images with high resolution around regions of interest. © 2008 Elsevier Inc. All rights reserved.Publication Periodic Orthogonal Splines and Wavelets(1995-07) Koh, Y.W.; Lee, S.L.; Tan, H.H.; MATHEMATICSPeriodic scaling functions and wavelets are constructed directly from non-stationary multiresolutions of L2([0, 2π)), the space of square-integrable 2π-periodic functions. For a multiresolution {Vk : k ≥ 0}, necessary and sufficient conditions for ∪k≥0Vk to be dense in L2([0, 2π)) and characterizations of a function φk for which φk(· - 2πj/2k), j = 0, 1, . . ., 2k - 1, form a basis of Vk are given. The construction of scaling functions and wavelets are done via orthogonal bases of functions, called orthogonal splines . Sufficient conditions are given for a sequence of scaling functions to generate a multiresolution. These conditions are also sufficient for the convergence of convolution operators with the scaling functions as kernels. Sufficient conditions are also given for the wavelets to generate a stable basis of L2([0, 2π)). The orthogonal spline bases give rise to algorithms in which the equations are the finite Fourier transforms of the classical wavelet decomposition and reconstruction equations. Each equation in the orthogonal spline algorithms involves only two terms and its complexity does not depend on the length of the filter coefficients. The general construction given here includes periodic versions of known wavelets. Examples on periodic polynomial spline wavelets and an extension of Chui and Mhaskar′s trigonometric wavelets are given to illustrate the construction. These trigonometric wavelets, in particular Chui and Mhaskar′s wavelets, form a stable basis of L2([0, 2π)). © 1995 Academic Press. All rights reserved.Publication Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets(2002-05) Jia, R.-Q.; Jiang, Q.; Lee, S.L.; MATHEMATICSThe cascade algorithm with mask a and dilation M generates a sequence φn, n = 1, 2,..., by the iterative process φn(x) = ∑αεℤs a(α)φn-1(Mx - α) x ε ℝs, from a starting function φ0, where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets.