Deflation Algorithm Research Articles

An adaptive mesh model for 3D reconstruction from unorganized data points

Reconstructing exact topology mesh from data points is one of the most important tasks in the fields of industrial CAD/CAE/CAM, computational vision and reverse engineering. In this paper, a deflation algorithm that integrates an adaptive mesh and physical constraint model is presented for the 3D reconstruction of geometric-closed shape (genus 0) from unorganized data points. First, an initial mesh is formed using the Delaunay algorithm. Second, an asymptotic deforming performance is accomplished to deflate initial mesh towards the local concave boundary step-by-step. In this phase, a physical constraint model of coupled particle systems based on particle dynamics and Newtonian law of motion is constructed, and the model dynamically controls mesh deformation as a behavior constraint. To guarantee that the resultant mesh is homeomorphous to the original surface of data points, a continuously deforming mechanism, visibility cone and collision-detecting criterions are designed. At last, experimental results in reverse engineering which supports the usefulness of this method for reconstruction .

Blind source estimation of FIR channels for binary sources: a grouping decision approach

This paper proposes a novel grouping decision approach for blind source estimation of FIR (finite impulse response) channels with binary sources. First, solvability is discussed for single-input systems and multi-input systems. Necessary and sufficient conditions for recoverability are derived. For single-input systems, a new deterministic algorithm based on grouping and decision is proposed to recover the source up to a delay. The algorithm is easy to implement and has several advantages. For instance, when the solvability conditions are satisfied, it can be applied to cases in which: (i) the channel has zeros on the unit circle or outside of the unit circle; (ii) there are fewer sensors than sources; (iii) the source is temporarily dependent. To improve noise tolerance and reduce computational cost, the algorithm is further elaborated for highly noisy channels and high-order FIR channels, respectively. For the channels with high unimodal noise, fewer peaks appear in the probability density function (pdf) of the outputs compared to the pdf of the outputs of channels with a higher SNR. After the peaks representing cluster centers are estimated using a maximum likelihood (ML) approach, the deterministic algorithm can be used. Similar to highly noisy channels, the algorithm is also effective for high-order, exponentially decaying channels after fewer cluster centers are estimated. Furthermore, blind source estimation for multi-input systems also can be carried out as with the case of single input systems. Two deflation algorithms are presented for temporarily dependent sources and i.i.d. sources. Based on the source estimation and deflation algorithms, the sources can be obtained one by one. Finally, the validity and performance of the algorithms are illustrated by several simulation examples.

Direction-of-Arrival Tracking Scheme for DS/CDMA Systems: Direction Lock Loop

In this paper, a new direction-of-arrival (DOA) tracking algorithm, direction lock loop (DiLL), is proposed for wireless direct-sequence code-division multiple-access systems. It has a similar concept to the delay lock loop that is used for timing synchronization. It may track the DOA of sources by iterations. Its computational requirements are NK+O(K) for coherent DiLL and 2NK+O(K) for noncoherent DiLL, where N is the number of antenna elements and K is the number of signal sources, which is less than that of the projection approximation subspace tracking with deflation algorithm. The DOA tracking accuracy is demonstrated by analysis and computer simulations.

A deflation algorithm for the blind source-factor separation of MIMO-FIR channels driven by colored sources

The letter proposes a new iterative deflation algorithm to solve the blind source-factor separation for the outputs of multiple-input-multiple-output finite-impulse response (MIMO-FIR) channels driven by source signals that are temporally correlated but spatially uncorrelated. Using the proposed algorithm, filtered versions of the source signals, each of which is the contribution of each source signal to the outputs of MIMO-FIR channels, are extracted one by one from the mixtures of source signals.

Blind Deconvolution Algorithms for MIMO-FIR Systems Driven by Fourth-order Colored Signals

This paper deals with the blind deconvolution problem of MIMO-FIR channels driven by source signals which are temporally second-order uncorrelated but fourth-order correlated and spatially second- and fourth-order uncorrelated. Recently, we proposed a criterion for solving the blind deconvolution problem [8], [9]. However, we have not yet shown an algorithm to implement our proposed criterion. One of our objectives in the present paper is to propose an iterative algorithm for achieving the blind deconvolution by using this criterion. To achieve this objective, we consider an approach called the non-deflation method. We propose an iterative algorithm based on the method. Moreover, we present a deflation algorithm using this criterion for the blind separation of MIMO constant channels.

Convergence behaviour of deflated GMRES(m) algorithms on AP3000

GMRES(m) method, the restarted version of the GMRES (generalized minimal residual) method, is one of the major iterative methods for numerically solving large and sparse nonsymmetric problems of the form Ax=b . However, the information of some eigenvectors that compose the approximation disappears and then the good approximate solution cannot be obtained, because of this restart. Recently, in order to improve such a weak point, some algorithms which named MORGAN, DEFLATION and DEFLATED-GMRES algorithm, have been proposed. Those algorithms add the information of eigenvectors that can be obtained in the previous restart frequency. In this paper, we study those algorithms and compare their performances. From the numerical experiments on the distributed memory machine Fujitsu AP3000, we show that DEFLATED-GMRES( m, k ) method performs the good reduction of residual norms in these algorithms.

Super-exponential algorithms for multichannel blind deconvolution

Multichannel blind deconvolution has been receiving increasing attention. Shalvi and Weinstein proposed an attractive approach to single-channel blind deconvolution called the super-exponential methods. The objective of this correspondence is to extend the Shalvi and Weinstein (1993, 1994) approach to the multichannel case and present super-exponential algorithms for multichannel blind deconvolution. We propose three approaches to multichannel blind deconvolution. In the first one, we present a multichannel super-exponential algorithm. In the second one, we present a super-exponential deflation algorithm. In the third one, we present a two-stage super-exponential algorithm.

A new adaptive algorithm for minor component analysis

In this paper, a new adaptive algorithm for minor component analysis is proposed. The (PASTd) projection approximation subspace tracking with deflation algorithm for principal component analysis by Yang is modified by putting the minus sign to the adaptive gain with orthonormalization. Using the ODE (ordinary differential equation) approach, the convergent point of the algorithm is shown to be the desired one. Simulation results show that the proposed algorithm has better estimation accuracies than the existing adaptive algorithm.

The objective function of partial least squares regression

A simple objective function in terms of undeflated X is derived for the latent variables of multivariate PLS regression. The objective function fits into the basic framework put forward by Burnham et al. (J. Chemometrics, 10, 31–45 (1996)). We show that PLS and SIMPLS differ in the constraint put on the length of the X-weight vector. It turns out that PLS does not penalize the length of the part of the weight vector that can be expressed as a linear combination of the preceding weights, whereas SIMPLS does. By using artificial data sets, it is shown that it depends on the data which of the two methods explains the larger amount of variance in X and Y. The objective function framework adds insight to the nature of PLS and SIMPLS and how they relate to other methods. In addition, we present an implicit deflation algorithm for PLS, explain why PLS and SIMPLS become equivalent when Ychanges from multivarite to univariate, and list some geometrical results that may also prove useful in the study of other latent variable methods. © 1998 John Wiley & Sons, Ltd.

An implicit deflation method for ill-conditioned Sylvester and Lyapunov equations

This paper describes an effective algorithm for solving ill-conditioned Sylvester or Lyapunov equations. These equations can be solved by conventional methods for ill-conditioned linear systems. However, such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data. For these ill-conditioned matrix equations, an implicit deflation algorithm is proposed to implement a certain SVD-based minimum norm least-squares approximate solution. The method is practical since it relies only on the ability to solve a Sylvester or Lyapunov equation. Certain practical details are also discussed.

Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations

The Behavior of the Newton-Raphson method at the singular roots has been studied by a number of authors and the convergence of the Newton-Raphson sequence has been shown to be linear. In this paper a new method with symbolic and numerical manipulations, termed the modified deflation algorithm, is proposed for the singular root of a system of nonlinear algebraic equations. The basic idea of the present method is to replace a part of the original equations by a set of new equations which pass through the singular root. According to the method, both convergency and accuracy can greatly be improved. In addition it is often possible to obtain analytically the singular root from the new equations. In order to show the effectiveness of the present method two illustrative examples are solved.

Modified deflation algorithm for the solution of singular problems. II. Nonlinear multipoint boundary value problems

Various kinds of iterative methods have been proposed for the solution of nonlinear multipoint boundary-value problems MPBVP's. However, it is necessary for these methods that the adjusting matrix, which corresponds to the Jacobian of nonlinear equations, is nonsingular at the solution. In this paper an algorithm for the singular solution of nonlinear MPBVP's, which is an extension of the modified deflation algorithm for the singular root of nonlinear algebraic equations developed by the author is presented. According to the present method, the singular solution can ultimately be reduced to the usual simple solution and both convergency and accuracy can greatly be improved. The effectiveness of the present method is shown by solving two illustrative examples.

Deflation algorithm for the multiple roots of a system of nonlinear equations

T. Ojika, S. Watanabe, and T. Mitsui (in preparation) have been developing a subroutine package NAES (Nonlinear Algebraic Equations Solver) for the numerical solutions of the system of nonlinear equations. An algorithm, in the package, termed the deflation algorithm, for determining multiple roots for a system of nonlinear equations, is presented and its effectiveness is shown by solving a numerical example.

The computation of Kronecker's canonical form of a singular pencil

We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil. This problem can be viewed as a generalization of the well-known eigenvalue problem of pencils of the type λ I− A. We first show that the elementary divisors ( λ − α) i of a regular pencil λ B− A can be retrieved with a deflation algorithm acting on the expansion ( λ − α) B − ( A − αB). This method is a straightforward generalization of Kublanovskaya's algorithm for the determination of the Jordan structure of a constant matrix. We also show how to use this method to determine the structure of the infinite elementary divisors of λ B− A. In the case of singular pencils, the occurrence of Kronecker indices—containing the singularity of the pencil—somewhat complicates the problem. Yet our algorithm retrieves these indices with no additional effort, when determining the elementary divisors of the pencil. The present ideas can also be used to separate from an arbitrary pencil a smaller regular pencil containing only the finite elementary divisors of the original one. This is shown to be an effective tool when used together with the QZ algorithm.

On Newton’s Method for Singular Problems

Sufficient conditions are given for Newton’s method to converge when the derivative is singular at the root.

A modified hotelling deflation scheme for inverting matrices

The Hotelling Deflation Algorithm can be modified to generate, with almost no additional effort, the inverse of a matrix simultaneously as its eigenvectors and eigenvalues are being determined. The method is particularly appropriate for small computers (or programmable calculators) with limited memory because the modified matrix occupies the same memory space used for the original matrix.