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

An isothetic polygonal arc is one that has all its sides oriented in two orthogonal directions, so that all its angles are right angles. Such an arc is determined (up to congruence) by specifying a “code” sequence of the form α 1 A 1 α 2 … α m −1 A m −1 α m , where the α's are positive real numbers representing side lengths, and the A 's are single bits that specify whether the arc turns left or right between one side and the next. In this paper we develop basic properties of this code and show how to derive various geometric properties of the arc (or the region it bounds, if it is closed) directly from the code.

Contour codes of isothetic polygons

An approach is presented to 3-D primitive reconstruction that is independent of the selection of volumetric primitives used to model objects. The approach first takes an arbitrary set of 3-D volumetric primitives and generates a hierarchical aspect representation based on the projected surfaces of the primitives; conditional probabilities capture the ambiguity of mappings between levels of the hierarchy. The integration of object-centered and viewer-centered representations provides the indexing power of 3-D volumetric primitives, while supporting a 2-D matching paradigm for primitive reconstruction. Formulation of the problem based on grouping the image regions according to aspect is presented. No domain dependent heuristics are used; the authors exploit only the probabilities inherent in the aspect hierarchy. For a given selection of primitives, the success of the heuristic depends on the likelihood of the various aspects; best results are achieved when certain aspects are more likely, and fewer primitives project to a given aspect. >

Qualitative 3-D shape reconstruction using distributed aspect graph matching

In order to determine the physical properties of a rock sample represented digitally as a set of serial cross sections it is necessary first to decompose the sample into discrete objects and then to approximate each of those objects by another with well-defined mathematical properties. For decomposition, the convex enclosure is defined and shown to be a good approximation of the three-dimensional convex hull yet less complex to calculate. The convex enclosure approximates the convex hull of simple objects with error no more than twenty %, and the enclosure deficiency can be used as a measure of the compactness of the object. Using this measure, it is possible to determine whether an object is sufficienty complex to require its decomposition into a set of subobjects. The decomposition procedure continues recursively until each subobject is sufficiently compact. The subobjects are then approximated by ellipsoids. For each subobject, the axes of the approximating ellipsoid are given by the eigenvalues and eigenvectors of the matrix used in the computation of its convex enclosure.

Decomposition and approximation of three-dimensional solids

Strongly Normal Sets of Tiles in N Dimensions

A method of detecting natural ``plateaus'' (equals maximal intervals of approximately constant value) in a one-dimensional pattern is described. The method is based on examining neighborhoods of each point having a range of sizes; rejecting neighborhoods of each size for which the standard deviation (of pattern values over the neighborhood) is not a local minimum; and further rejecting any neighborhood N whose subintervals of conparable size do not have mean pattern values approximately equal to that of N.

Azriel Rosenfeld

Papers

Contour codes of isothetic polygons

Qualitative 3-D shape reconstruction using distributed aspect graph matching

Decomposition and approximation of three-dimensional solids

Strongly Normal Sets of Tiles in N Dimensions

Detecting Natural ``Plateaus'' in One-Dimensional Patterns