Bert W. Rust

Comparing Real and Synthetic Images: Some Ideas About Metrics

This paper explores numerical techniques for comparing real and synthetic luminance images. We introduce components of a perceptually based metric using ideas from the image compression literature. We apply a series of metrics to a set of real and synthetic images, and discuss their performance. Finally, we conclude with suggestions for future work in formulating image metrics and incorporating them into new image synthesis methods.

Residual periodograms for choosing regularization parameters for ill-posed problems

Consider an ill-posed problem transformed if necessary so that the errors in the data are independent identically normally distributed with mean zero and variance 1. We survey regularization and parameter selection from a linear algebra and statistics viewpoint and compare the statistical distributions of regularized estimates of the solution and the residual. We discuss methods for choosing a regularization parameter in order to assure that the residual for the model is statistically plausible. Ideally, as proposed by Rust (1998 Tech. Rep. NISTIR 6131, 2000 Comput. Sci. Stat. 32 333–47 ), the results of candidate parameter choices should be evaluated by plotting the resulting residual along with its periodogram and its cumulative periodogram, but sometimes an automated choice is needed. We evaluate a method for choosing the regularization parameter that makes the residuals as close as possible to white noise, using a diagnostic test based on the periodogram. We compare this method with standard techniques such as the discrepancy principle, the L-curve and generalized cross validation, showing that it performs better on two new test problems as well as a variety of standard problems.

Variable projection for nonlinear least squares problems

The variable projection algorithm of Golub and Pereyra (SIAM J. Numer. Anal. 10:413–432, 1973) has proven to be quite valuable in the solution of nonlinear least squares problems in which a substantial number of the parameters are linear. Its advantages are efficiency and, more importantly, a better likelihood of finding a global minimizer rather than a local one. The purpose of our work is to provide a more robust implementation of this algorithm, include constraints on the parameters, more clearly identify key ingredients so that improvements can be made, compute the Jacobian matrix more accurately, and make future implementations in other languages easy.

Confidence Intervals for Inequality-Constrained Least Squares Problems, with Applications to Ill-Posed Problems

Computing confidence intervals for functions

Exponential Data Fitting and its Applications

\phi (x) = w^T x

Fitting nature's basic functions. I. Polynomials and linear least squares

, where

The fast Fourier transform for experimentalists, part IV: autoregressive spectral analysis

Kx = y + e

Constrained Least Squares Interval Estimation

and e is a normally distributed error vector, is a standard problem in multivariate statistics. In this work, we develop an algorithm for solving this problem if additional information,

Fitting nature's basic functions. IV. The variable projection algorithm

x \geqq 0

Fitting nature's basic functions. Part III: exponentials, sinusoids, and nonlinear least squares

, is given. Applications to estimating solutions to integral equations of the first kind are given.