On the rate of growth of condition numbers for convolution matrices

Abstract

When analyzing linear systems of equations, the most important indicator of potential instability is the condition number of the matrix. For a convolution matrix W formed from a series w (where Wij wi-, + ,, 1 5 ij + 1 5 k, W,j = 0 otherwise), this condition number defines the stabirity of the deconvolution process. For the larger convolution matrices commonly encountered in practice, direct computation of the condition number (e.g., by singular value decomposition) would be extremely time consuming. However, for convolution matrices, an upper bound for the condition number is defined by the ratio of the maximum to the minimum values of the amplitude spectrum of w. This bound is infinite for any series w with a zero value in its amplitude spectrum; although for certain such series, the actual condition number for W may in fact be relatively small. In this paper we give a new simple derivation of the upper bound and present a means of defining the rate of growth of the condition number of W for a bandlimited series by means of the higher order derivatives of the amplitude spectrum of w at its zeros. The rate of growth is shown to be proportional to m p , where m is the column dimension of Wand p is the order of the zero of the amplitude spectrum.

Analysis of multiframe super-resolution reconstruction for image anti-aliasing and deblurring

“…and if B is non-singular, Thus by using (15) and (27), we obtain that, if H is of full rank and unperturbed and B is non-singular,…”

“…It is known that B is generally ill conditioned, and its condition number has been well studied [8,15]. The upper bound on the condition number k 2 (B) is given by the ratio of the maximum to the minimum values of the amplitude spectrum of the PSF h [15], i.e.…”

“…It is known that B is generally ill conditioned, and its condition number has been well studied [8,15]. The upper bound on the condition number k 2 (B) is given by the ratio of the maximum to the minimum values of the amplitude spectrum of the PSF h [15], i.e. k 2 ðBÞ% maxjFðhÞj=minjFðhÞj; (27) where F(h) denotes the Fourier transform of h. In addition, the growth of k 2 (B) is generally fast as the HR image size increases [15].…”

A new Wiener filtering based detection scheme for time domain perceptual audio watermarking

“…Too large means roughly that log 10 (K) precision of matrix entries. numbers [5,6]. In this context, we introduce the spectrum flatness measure ν [7]:…”

Effects of ill-conditioned data on least squares adaptive filters

“…where the subscript v has been attached to the autocorrelation values of x (11) with the noise present. From (Sb) it is seen that the noise alters only the main diagonal of R in (4), producing a "noisy" set of equations given by 6 (R + (J'1~I )W p = p, (6) where w t7 is the solution to the noisy problem. TIle following section next quantifies the relative change from w to W t7 as a function of data conditioning.…”