A threshold selection method from gray level histograms

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

Texture is one of the important characteristics used in identifying objects or regions of interest in an image, whether the image be a photomicrograph, an aerial photograph, or a satellite image. This paper describes some easily computable textural features based on gray-tone spatial dependancies, and illustrates their application in category-identification tasks of three different kinds of image data: photomicrographs of five kinds of sandstones, 1:20 000 panchromatic aerial photographs of eight land-use categories, and Earth Resources Technology Satellite (ERTS) multispecial imagery containing seven land-use categories. We use two kinds of decision rules: one for which the decision regions are convex polyhedra (a piecewise linear decision rule), and one for which the decision regions are rectangular parallelpipeds (a min-max decision rule). In each experiment the data set was divided into two parts, a training set and a test set. Test set identification accuracy is 89 percent for the photomicrographs, 82 percent for the aerial photographic imagery, and 83 percent for the satellite imagery. These results indicate that the easily computable textural features probably have a general applicability for a wide variety of image-classification applications.

Textural Features for Image Classification

Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. >

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A theory for multiresolution signal decomposition: the wavelet representation

We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.

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Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images

Sparse, opaque three-dimensional texture. 2a: visibility

Approximation of waveforms and contours by one-dimensional pyramid linking☆

Given a noisy signal g , if we have a probabilistic model for the ensemble of possible (non-noisy) signals f and a probabilistic model for the noise, we can in principle find the f that most likely gave rise to g , using Bayes′ theorem; this most likely f is called the MAP ("maximum a posteriori") estimate of g . For discrete (digital) signals, the signal and noise probabilities can be arbitrary: it is not necessary to assume standard probability densities (e.g., Gaussian), which are unrealistic in many situations. However, MAP estimation is not commonly used even for digital signals, because the size of the set of possible f s is often exponential. In this paper we show, for piecewise constant f s, how the most likely f can be found using dynamic programming techniques that require only polynomial space and time. We also illustrate the performance of MAP estimation in recovering parameters of a simple signal or ensemble of signals in the presence of various types of noise.

MAP estimation of piecewise constant digital signals

Convexity properties of space curves

Nearly 40 years ago Hough showed how a point-to-line mapping that takes collinear points into concurrent lines can be used to detect collinear sets of points, since such points give rise to peaks where the corresponding lines intersect. Over the past 30 years many variations and generalizations of Hough's idea have been proposed, Hough's mapping was linear, but most or all of the mappings studied since then have been nonlinear, and take collinear points into concurrent curves rather than concurrent lines; little or no work has appeared in the pattern recognition literature on mappings that take points into lines.This paper deals with point-to-line mappings in the real projective plane. (We work in the projective plane to avoid the need to deal with special cases in which collinear points are mapped into parallel, rather than concurrent, lines.) We review basic properties of linear point-to-point mappings (collineations) and point-to-line mappings (correlations), and show that any one-to-one point-to-line mapping that takes collinear points into concurrent lines must in fact be linear. We describe ways in which the matrices of such mappings can be put into canonical form, and show that Hough's mapping is only one of a large class of inequivalent mappings. We show that any one-to-one point-to-line mapping that has an incidence-symmetry property must be linear and must have a symmetric matrix which has a diagonal canonical form. We establish useful geometric properties of such mappings, especially in cases where their matrices define nonempty conics.

Azriel Rosenfeld

Papers

Sparse, opaque three-dimensional texture. 2a: visibility

Approximation of waveforms and contours by one-dimensional pyramid linking☆

MAP estimation of piecewise constant digital signals

Convexity properties of space curves

Point-to-Line Mappings and Hough Transforms