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

We show how computation of geometric properties of a region represented by a linear quadtree can be speeded up by about a factor of p by using a p -processor CREW PRAM model of parallel computation. Similar speedups are obtained for computing the union and intersection of two regions, and the complement of a region, using linear quadtree representations.

Parallel processing of regions represented by linear quadtrees

Parallel web automata (PWA?a generalization of cellular arrays to arbitrary graph structures?are defined It is shown that such automata make poor acceptors (ie, they are not good recognizers of their own graph structures) However, they can be used to perform parallel local pattern matching (with respect to patterns of bounded size) on themselves, so that they have potential usefulness as models for ``semantic memory'' They can also be used to find greatest compatible sets of graph labelings subject to given local constraints; this problem arises, for example, in the analysis of scenes containing three-dimensional objects

Networks of Automata: Some Applications

Texture discrimination by projective invariants

Determining Simplicity and Computing Topological Change in Strongly Normal Partial Tilings of R2 or R3

Blum's medial axis transformation (MAT) for binary pictures yields medial axis points that lie midway between opposite borders of a region or along angle bisectors. This note discusses a generalization of the MAT in which a score is computed for each point P of a grayscale picture based on the gradient magnitudes at pairs of points that have P as their midpoint. These scores are high at points that lie midway between pairs of antiparallel edges or along angle bisectors, so that they define a MAT-like ``skeleton,'' which we may call the GRADMAT. However, this skeleton is rather sensitive to the presence of noise edges or to irregularities in the region edges, and it also is subject to artifacts created by pairs of edges belonging to different objects.

Azriel Rosenfeld

Papers

Parallel processing of regions represented by linear quadtrees

Networks of Automata: Some Applications

Texture discrimination by projective invariants

Determining Simplicity and Computing Topological Change in Strongly Normal Partial Tilings of R2 or R3

A Medial Axis Transformation for Grayscale Pictures