This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.

A Computational Approach to Edge Detection

Summary
The classic, multivariate technique of principal component analysis can be used to find and estimate the directions of lines and planes of best least-squares fit along the demagnetization path of a palaeomagnetic specimen, thereby replacing vector subtraction, remagnetization circles and difference vector paths with one procedure. Eigenvalues from the analysis are the variance of the data along each principal axis, and provide a relative measure of collinearity or coplanarity which may be used to define a general palaeomagnetic precision index. Demagnetization planes found with principal component analysis may be used in place of difference vector paths for locating Hoffman—Day directions, avoiding unnecessary vector subtraction and intensity truncation steps. Two methods are discussed for jointly estimating an average remanence direction from demagnetization lines and planes.

The least-squares line and plane and the analysis of palaeomagnetic data

This paper gives an exposition of linear prediction in the analysis of discrete signals The signal is modeled as a linear combination of its past values and present and past values of a hypothetical input to a system whose output is the given signal In the frequency domain, this is equivalent to modeling the signal spectrum by a pole-zero spectrum The major part of the paper is devoted to all-pole models The model parameters are obtained by a least squares analysis in the time domain Two methods result, depending on whether the signal is assumed to be stationary or nonstationary The same results are then derived in the frequency domain The resulting spectral matching formulation allows for the modeling of selected portions of a spectrum, for arbitrary spectral shaping in the frequency domain, and for the modeling of continuous as well as discrete spectra This also leads to a discussion of the advantages and disadvantages of the least squares error criterion A spectral interpretation is given to the normalized minimum prediction error Applications of the normalized error are given, including the determination of an "optimal" number of poles The use of linear prediction in data compression is reviewed For purposes of transmission, particular attention is given to the quantization and encoding of the reflection (or partial correlation) coefficients Finally, a brief introduction to pole-zero modeling is given

Linear prediction: A tutorial review

In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Spectrum estimation and harmonic analysis

A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.

Spectrum analysis&#8212;A modern perspective

Inferential statistical methods for estimating and comparing cosinor parameters.

The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of $\Psi$ is illustrated by a numerical example.

/pdf/an-antipodally-symmetric-distribution-on-the-sphere-5amn68q08d.pdf

An Antipodally Symmetric Distribution on the Sphere

The paper discusses the impact of the fast Fourier transform on the spectrum of time series analysis. It is shown that the computationally fastest way to calculate mean lagged products is to begin by calculating all Fourier coefficients with a fast Fourier transform and then to fast-Fourier-retransform a sequence made up of a_{k}^{2}+b_{k}^{2} (where a_{k}+ib_{k} are the complex Fourier coefficients). Also discussed are raw and modified Fourier periodograms, bandwidth versus stability aspects, and aims and computational approaches to complex demodulation. Appendixes include a glossary, a review of complex demodulation without fast Fourier transform, and a short explanation of the fast Fourier transform.

Modern techniques of power spectrum estimation

Replacement by a simple approximation of expected values of order statistics in the Shapiro-Francis (1972) W′ statistics, yield a statistics that is more srlitable for machine computation. We show that is equivalent to W′ and discuss its percentage points.

An Approximate Analysis of Variance Test for Non-Normality Suitable for Machine Calculation

Purpose: The extensive and unpredictable biliary excretion of CPT-11 and its metabolites, SN-38 and SN-38 glucuronide (SN-38G) may contribute to the wide interpatient variability reported in the disposition and gastrointestinal toxicity of CPT-11. We studied the role of P-glycoprotein (P-gp) in in vivo biliary excretion of CPT-11, SN-38 and SN-38G in mice lacking mdr1-type P-gp [mdr1a/1b(–/–)] in the presence of the multidrug resistance (MDR) reversal agent, PSC833. Methods: Wild-type (Wt) and mdr1a/1b(–/–) mice (n=3 or 4) were treated intragastrically with PSC833 (50 mg/kg) or vehicle 2 h prior to i.v. CPT-11 dosing (10 mg/kg), and bile samples were collected. Results and conclusions: P-gp was found to play an important role in CPT-11 biliary excretion, as there was a significant (40%, P<0.05) decrease in its biliary recovery in 90 min in mdr1a/1b(–/–) mice (6.6±0.6% dose) compared with Wt mice (11±1.2%). This also implied a major role of other undetermined non-P-gp-mediated mechanism(s) for hepatic transport of CPT-11, which was inhibited by PSC833 (1.8±0.8% with PSC833, 6.6±0.6% without PSC833) in mdr1a/1b(–/–) mice. SN-38 and SN-38G biliary transport was unchanged in mice lacking P-gp after vehicle treatment, indicating a lack of P-gp mediation in their transport. PSC833 significantly reduced (56–89%) SN-38 and SN-38G biliary transport in Wt and mdr1a/1b(–/–) mice, suggesting that PSC833 may be a candidate to modulate biliary excretion of SN-38 with potential use in reducing CPT-11 toxicity.

Christopher Bingham

Papers

Inferential statistical methods for estimating and comparing cosinor parameters.

An Antipodally Symmetric Distribution on the Sphere

Modern techniques of power spectrum estimation

An Approximate Analysis of Variance Test for Non-Normality Suitable for Machine Calculation

Biliary transport of irinotecan and metabolites in normal and P-glycoprotein-deficient mice.