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

Publisher Summary This chapter discusses the maximum-likelihood heavy-atom parameter refinement for multiple isomorphous replacement (MIR) and multiwavelength anomalous diffraction (MAD) The chapter describes its extension to probability distributions incorporating anomalous diffraction effects, as well as measurement error and nonisomorphism Integrating these distributions in a whole complex plane leads to likelihood functions that can be used for heavy-atom detection and refinement and for producing phase-probability distributions The current implementation of this formalism in the computer program statistical heavy-atom refinement and phasing (SHARP) is also described in the chapter Likelihood functions can be used for the final phasing and calculation of Hendrickson–Lattman coefficients Numerical tests have been performed for three types of common refinements—namely, single isomorphous replacement, multiple isomorphous replacement with anomalous scattering (MIRAS), and MAD—and the results are summarized in the chapter A key feature of SHARP is its ability to refine lack-of-isomorphism parameters along with all the others

[27] Maximum-likelihood heavy-atom parameter refinement for multiple isomorphous replacement and multiwavelength anomalous diffraction methods

An algorithm is proposed for solving the stereoscopic matching problem. The algorithm consists of five steps: (1) Each image is filtered at different orientations with bar masks of four sizes that increase with eccentricity; the equivalent filters are one or two octaves wide. (2) Zero-crossings in the filtered images, which roughly correspond to edges, are localized. Positions of the ends of lines and edges are also found. (3) For each mask orientation and size, matching takes place between pairs of zero-crossings or terminations of the same sign in the two images, for a range of disparities up to about the width of the mask’s central region. (4) Wide masks can control vergence movements, thus causing small masks to come into correspondence. (5) When a correspondence is achieved, it is stored in a dynamic buffer, called the 2½-D sketch.

A Computational Theory of Human Stereo Vision

The axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers. It was established that the jet was truly self-preserving some 70 diameters downstream of the nozzle and most of the measurements were made in excess of this distance. The quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity. The r.m.s. values of the various velocity fluctuations differ from those measured previously as a result of lack of self-preservation and insufficient frequency range in the instrumentation of the previous investigations. It appears that Taylor's hypothesis is not applicable to this flow, but the use of convection velocity of the appropriate scale for the transformation from temporal to spatial quantities appears appropriate. The energy balance was calculated from the various measured quantities and the result is quite different from the recent measurements of Sami (1967), which were obtained twenty diameters downstream from the nozzle. In light of these measurements some previous hypotheses about the turbulent structure and the transport phenomena are discussed. Some of the quantities were obtained by two or more different methods, and their relative merits and accuracy are assessed.

Some measurements in the self preserving jet

The form of the wavefunction psi for a semiclassical regular quantum state (associated with classical motion on an N-dimensional torus in the 2N-dimensional phase space) is very different from the form of psi for an irregular state (associated with stochastic classical motion on all or part of the (2N-1)-dimensional energy surface in phase space). For regular states the local average probability density Pi rises to large values on caustics at the boundaries of the classically allowed region in coordinate space, and psi exhibits strong anisotropic interference oscillations. For irregular states Pi falls to zero (or in two dimensions stays constant) on 'anticaustics' at the boundary of the classically allowed region, and psi appears to be a Gaussian random function exhibiting more moderate interference oscillations which for ergodic classical motion are statistically isotropic with the autocorrelation of psi given by a Bessel function.

https://iopscience.iop.org/article/10.1088/0305-4470/10/12/016/pdf

S. O. Rice

Papers

Mathematical Analysis of Random Noise-Conclusion