Showing papers by "Michael K. Ng published in 2001"

A Fast MAP Algorithm for High-Resolution Image Reconstruction with Multisensors

Abstract: In many applications, it is required to reconstruct a high-resolution image from multiple, undersampled and shifted noisy images. Using the regularization techniques such as the classical Tikhonov regularization and maximum a posteriori (MAP) procedure, a high-resolution image reconstruction algorithm is developed. Because of the blurring process, the boundary values of the low-resolution image are not completely determined by the original image inside the scene. This paper addresses how to use (i) the Neumann boundary condition on the image, i.e., we assume that the scene immediately outside is a reflection of the original scene at the boundary, and (ii) the preconditioned conjugate gradient method with cosine transform preconditioners to solve linear systems arising from the high-resolution image reconstruction with multisensors. The usefulness of the algorithm is demonstrated through simulated examples.

Strang‐type preconditioners for systems of LMF‐based ODE codes

Abstract: We consider the solution of ordinary differential equations (ODEs) using boundary value methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. The GMRES method with the Strang-type block-circulant preconditioner is proposed for solving these linear systems. We show that if an A k1,k2 -stable boundary value method is used for an m-by-m system of ODEs, then our preconditioners are invertible and all the eigenvalues of the preconditioned systems are 1 except for at most 2m(k 1 + k 2 ) outliers. It follows that when the GMRES method is applied to solving the preconditioned systems, the method will converge in at most 2m(k 1 +k 2 ) + 1 iterations. Numerical results are given to illustrate the effectiveness of our methods.

Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning

Abstract: We focus on image deconvolution and image reconstruction problems where a sought image is recovered from degraded observed data. The solution is defined to be the minimizer of an objective function combining a data-fidelity term and an edge-preserving, convex regularization term. Our objective is to speed up the calculation of the solution in a wide range of situations. To this end, we propose a method applying pertinent preconditioning to an adapted half-quadratic equivalent form of the objective function. The optimal solution is then found using an alternating minimization (AM) scheme. We focus specifically on Huber regularization. We exhibit the possibility of getting very fast calculations while preserving the edges in the solution. Preliminary numerical results are reported to illustrate the effectiveness of our method.

A Cube Model and Cluster Analysis for Web Access Sessions

Abstract: Identification of the navigational patterns of casual visitors is an important step in online recommendation to convert casual visitors to customers in e-commerce. Clustering and sequential analysis are two primary techniques for mining navigational patterns from Web and application server logs. The characteristics of the log data and mining tasks require new data representation methods and analysis algorithms to be tested in the e-commerce environment. In this paper we present a cube model to represent Web access sessions for data mining. The cube model organizes session data into three dimensions. The COMPONENT dimension represents a session as a set of ordered components {c1, c2, ..., cP}, in which each component ci indexes the ith visited page in the session. Each component is associated with a set of attributes describing the page indexed by it, such as the page ID, category and view time spent at the page. The attributes associated with each component are defined in the ATTRIBUTE dimension. The SESSION dimension indexes individual sessions. In the model, irregular sessions are converted to a regular data structure to which existing data mining algorithms can be applied while the order of the page sequences is maintained. A rich set of page attributes is embedded in the model for different analysis purposes. We also present some experimental results of using the partitional clustering algorithm to cluster sessions. Because the sessions are essentially sequences of categories, the k-modes algorithm designed for clustering categorical data and the clustering method using the Markov transition frequency (or probability) matrix, are used to cluster categorical sequences.

An empirical study on the visual cluster validation method with Fastmap

Abstract: This paper presents an empirical study on the visual method for cluster validation based on the Fastmap projection. The visual cluster validation method attempts to tackle two clustering problems in data mining: to verify partitions of data created by a clustering algorithm; and to identify genuine clusters from data partitions. They are achieved through projecting objects and clusters by Fastmap to the 2D space and visually examining the results by humans. A Monte Carlo evaluation of the visual method was conducted. The validation results of the visual method were compared with the results of two internal statistical cluster validation indices, which shows that the visual method is in consistence with the statistical validation methods. This indicates that the visual cluster validation method is indeed effective and applicable to data mining applications.

Patterns Discovery Based on Time-Series Decomposition

Abstract: Complete or partial periodicity search in time-series databases is an interesting data mining problem. Most previous studies on finding periodic or partial periodic patterns focused on data structures and computing issues. Analysis of long-term or short-term trends over different time windows is a great interest. This paper presents a new approach to discovery of periodic patterns from time-series with trends based on time-series decomposition. First, we decompose time series into three components, seasonal, trend and noise. Second, with an existing partial periodicity search algorithm, we search either partial periodic patterns from trends without seasonal component or partial periodic patterns for seasonal components. Different patterns from any combination of the three decomposed time-series can be found using this approach. Examples show that our approach is more flexible and suitable to mine periodic patterns from time-series with trends than the previous reported methods.

The convergence rate of block preconditioned systems arising from lmf-based ode codes ∗

Abstract: The solution of ordinary an partial differential equations using implicit linear multi-step formulas (LMF)is considered. More precisely, boundary value methods (BVMs), a class of methods based on implicit formulas will be taken into account in this paper. These methods require the solution of large and sparse linear systems \(\hat M\)x = b. Block-circulant preconditioners have been propose to solve these linear systems. By investigating the spectral condition number of \(\hat M\), we show that the conjugate gradient method, when applied to solving the normalize preconditioned system, converges in at most O(log s) steps, where the integration step size is O(1/s). Numerical results are given to illustrate the effectiveness of the analysis.

Comments on "Least squares restoration of multichannel images"

Abstract: For original article see Galatsanos et al. (IEEE Trans. Signal Processing, vol.39, p.2222-36, Oct. 1991). In this correspondence, we give the correct matrix formulation arising from the constrained optimization of the least squares restoration of multichannel images in Galatsanos et al.

High-Resolution Color Image Reconstruction with Neumann Boundary Conditions ∗

Abstract: This paper studies the application of preconditioned conjugate gradient methods in high-resolution color image reconstruction problems. The high-resolution color images are reconstructed from multiple undersampled, shifted, degraded color frames with subpixel displacements. The resulting degradation matrices are spatially variant. To capture the changes of reflectivity across color channels, the weighted H1 regularization functional is used in the Tikhonov regularization. The Neumann boundary condition is also employed to reduce the boundary artifacts. The preconditioners are derived by taking the cosine transform approximation of the degradation matrices. Numerical examples are given to illustrate the fast convergence of the preconditioned conjugate gradient method.

Circulant preconditioners for indefinite Toeplitz systems

Abstract: In recent papers circulant preconditioners were proposed for ill-conditioned Hermitian Toeplitz matrices generated by 2π-periodic continuous functions with zeros of even order. It was shown that the spectra of the preconditioned matrices are uniformly bounded except for a finite number of outliers and therefore the conjugate gradient method, when applied to solving these circulant preconditioned systems, converges very quickly. In this paper, we consider indefinite Toeplitz matrices generated by 2π-periodic continuous functions with zeros of odd order. In particular, we show that the singular values of the preconditioned matrices are essentially bounded. Numerical results are presented to illustrate the fast convergence of CGNE, MINRES and QMR methods.

Half-quadratic regularization, preconditioning and applications

Abstract: The article addresses a wide class of image deconvolution or reconstruction situations where a sought image is recovered from degraded observed image. The sought solution is defined to be the minimizer of an objective function combining a data-fidelity term and an edge-preserving, convex regularization term. Our objective is to speed up the calculation of the solution in a wide range of situations. We propose a method applying pertinent preconditioning to an adapted half-quadratic equivalent form of the objective function. The optimal solution is then found using an alternating minimization (AM) scheme. We focus specifically on Huber regularization. We exhibit the possibility of getting very fast calculations while preserving the edges in the solution. Preliminary numerical results are reported to illustrate the effectiveness of our method.