Showing papers by "Michael K. Ng published in 1999"
TL;DR: A new approach is developed, which allows the use of the k-means paradigm to efficiently cluster large categorical data sets and a fuzzy k-modes algorithm is presented and the effectiveness of the algorithm is demonstrated with experimental results.
Abstract: This correspondence describes extensions to the fuzzy k-means algorithm for clustering categorical data. By using a simple matching dissimilarity measure for categorical objects and modes instead of means for clusters, a new approach is developed, which allows the use of the k-means paradigm to efficiently cluster large categorical data sets. A fuzzy k-modes algorithm is presented and the effectiveness of the algorithm is demonstrated with experimental results.
486 citations
TL;DR: It is shown that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices, and the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions.
Abstract: Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.
437 citations
TL;DR: A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a previous linear system and numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.
Abstract: In this paper, we consider using conjugate gradient (CG) methods for solving multiple linear systems $A^{(i)} x^{(i)} = b^{(i)},$ for $1 \le i \le s,$ where the coefficient matrices $A^{(i)}$ and the right-hand sides $b^{(i)}$ are different in general.\ In particular, we focus on the seed projection method which generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems onto the generated Krylov subspace to get the approximate solutions.\ The whole process is repeated until all the systems are solved.\ Most papers in the literature [T.\ F.\ Chan and W.\ L.\ Wan, {\it SIAM J.\ Sci.\ Comput.}, 18 (1997), pp.\ 1698--1721; B.\ Parlett {\it Linear Algebra Appl.}, 29 (1980), pp.\ 323--346; Y.\ Saad, {\it Math.\ Comp.}, 48 (1987), pp.\ 651--662; V.\ Simoncini and E.\ Gallopoulos, {\it SIAM J.\ Sci.\ Comput.}, 16 (1995), pp.\ 917--933; C.\ Smith, A.\ Peterson, and R.\ Mittra, {\it IEEE Trans.\ Antennas and Propagation}, 37 (1989), pp. 1490--1493] considered only the case where the coefficient matrices $A^{(i)}$ are the same but the right-hand sides are different.\ We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right-hand sides. A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a previous linear system. Finally, numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.
58 citations
TL;DR: It is shown that the solution of an n-by-n discrete symmetric sinc-Galerkin system can be obtained in O(n log n) operations and it is proved that the condition number of the preconditioned matrix is uniformly bounded by a constant independent of the size of the matrix.
Abstract: The symmetric sinc-Galerkin method developed by Lund, when applied to the second- order self-adjoint boundary value problem, gives rise to a symmetric coefficient matrix. The coefficient matrix has a special structure so that it can be advantageously used in solving the discrete system. In this paper, we employ the preconditioned conjugate gradient method with banded matrices as preconditioners. We prove that the condition number of the preconditioned matrix is uniformly bounded by a constant independent of the size of the matrix. In particular, we show that the solution of an n-by-n discrete symmetric sinc-Galerkin system can be obtained in O(n log n) operations. We also extend our method to the self-adjoint elliptic partial differential equation. Numerical results are given to illustrate the effectiveness of our fast iterative solvers.
36 citations
TL;DR: The major characteristics of the time series astronomical data, data preprocessing techniques to process these time series, and some domain-specific techniques to separate candidate variable stars from the nonvariant ones are presented.
Abstract: In this paper we present some initial results of a project which uses data-mining techniques to search for evidence of massive compact halo objects (MACHOs) from very large time series database. MACHOs are the proposed materials that probably make the “dark matter” surrounding our own and other galaxies. It was suggested that MACHOs may be detected through the gravitational microlensing effect which can be identified from the light curves of background stars. The objective of this project is two-fold, namely, (i) identification of new classes of variable stars and (ii) detection of microlensing events. In this paper, we present the major characteristics of the time series astronomical data, data preprocessing techniques to process these time series, and some domain-specific techniques to separate candidate variable stars from the nonvariant ones. We discuss the use of the Fourier model to represent the time series and the k -means based clustering method to classify variable stars.
27 citations
TL;DR: It is shown that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast and hence can be computed efficiently by fast transform algorithms.
Abstract: In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.
11 citations
TL;DR: The evaluation phase in the policy iteration algorithm for the infinite horizon discounted Markov decision problem can be done in O(mN^2) operations, where N is the number of states of theMarkov decision process and m is thenumber of states in which the decision changes during the policy improvement phase.
6 citations
01 Jan 1999
4 citations
24 Oct 1999
TL;DR: An alternating minimization iterative scheme is devised to recover the image and simultaneously identify the phase in a phase diversity blind deconvolution algorithm based on the minimization method.
Abstract: In this paper, we deal with the restoration of images degraded by the atmospheric turbulence. Atmospheric turbulence imposes a strong limit for observation on long propagation paths. We present a phase diversity blind deconvolution algorithm based on the minimization method. An alternating minimization iterative scheme is devised to recover the image and simultaneously identify the phase.
2 citations
24 Oct 1999
TL;DR: It is found that when there is no subpixel displacement error, the blurring matrix can be diagonalized by the two-dimensional discrete cosine transform matrix, and the BTTB-like linear system can be solved by thecosine transform based PCG method very efficiently.
Abstract: We study the problem of reconstructing high resolution images from multiple undersampled, shifted, degraded frames with subpixel displacement errors. This leads to a formulation involving a spatially-variant imaging system model. The MAP estimation scheme is used subject to the assumption that the original high-resolution image is modeled by a stationary Markov-Gaussian random field. The resulting MAP formulation is expressed as a large linear system, where the coefficient matrix involves block-Toeplitz-Toeplitz-block-like (BTTB-like) blurring matrix and banded BTTB inverse covariance matrix associated with the original image. We find that when there is no subpixel displacement error, the blurring matrix can be diagonalized by the two-dimensional discrete cosine transform matrix. Thus we apply the preconditioned conjugate gradient (PCG) method with cosine transform preconditioners to solve the BTTB-like linear system. Experimental results show that the system can be solved by the cosine transform based PCG method very efficiently.
1 citations