Digital communication systems are frequently operated over nonlinear channels with memory. The analysis of the performance of these systems is difficult and no complete analytical treatment of the problem has been obtained before. Several recent efforts have been directed toward the computation of error probabilities via Monte-Carlo simulation using a complete system model. These simulations require excessively large sample sizes and are not practical for estimating very low values of error probabilities. This paper presents a modified Monte-Carlo simulation technique for estimating error probabilities in digital communication systems operating over nonlinear channels. An importance-sampling technique is used to modify the probability density function of the noise process in a way to make simulation possible. Theoretical results as well as realistic examples are presented, showing that the number of samples needed for simulation is reduced considerably.

Edge detection and enhancement are widely used in image processing applications. In this paper we consider the problem of optimizing spatial frequency domain filters for detecting edges in digital pictures. The filter is optimum in that it produces maximum energy within a resolution interval of specified width in the vicinity of the edge. We show that, in the continuous case, the filter transfer function is specified in terms of the prolate spheroidal wave function. In the discrete case, the filter transfer function is specified in terms of the sampled values of the first-order prolate spheroidal wave function or in terms of the sampled values of an asymptotic approximation of the wave function. Both versions can be implemented via the fast Fourier transform (FFT). We show that the optimum filter is very effective for detecting blufred and noisy edges. Finally, we compare the performance of the optimum edge detection filter with other edge detection filters using a variety of input images.

In the above correspondence,1 Ahmed et al. proposed a discrete cosine transform (DCT) and compared its performance with the Karhunen-Loeve transform (KLT). They offered empirical evidence showing that the DCT and the KLT compare closely for a number of digital signal processing applications. The purpose of this note is to show that the DCT and the KLT are indeed asymptotically equivalent if the data covariance matrix is a Toeplitz matrix.

Transform image data compression consists of dividing the image into a number of nonoverlapping subimage regions and quantizing and coding the transform of the data from each subimage. Karhunen-Loève, Hadamard, and Fourier transforms are most commonly used in transform image compression. This paper presents a new discrete linear transform for image compression which we use in conjunction with differential pulse-code modulation on spatially adjacent transformed subimage samples. For a set of thirty-three 64 × 64 images of eleven different categories, we compare the performancea of the discrete linear transform compression technique with the Karhunen-Loève and Hadamard transform techniques. Our measure of performance is the mean-squared error between the original image and the reconstructed image. We multiply the mean-squared error with a factor indicating the degree to which the error is spatially correlated. We find that for low compression rates, the Karhunen-Loève outperforms both the Hadamard and the discrete linear basis method. However, for high compression rates, the performance of the discrete transform method is very close to that of the Karhunen-Loève transform. The discrete linear transform method performs much better than the Hadamard transform method for all compression rates.

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.

Of the several methods that have been proposed for imagery data compression, the Karhunen-Loève procedure minimizes the meansquare error between the original and reconstructed imagery data. In spite of its optimality property, the Karhunen-Loève procedure has not been widely used because of its computational complexity. The main difficulty is in the computation of the eigenvectors and the eigenvalues of the covariance matrix of the imagery data since the dimension of the covariance matrix is usually large. A computationally short procedure for calculating the eigenvalues and eigenvectors of the covariance matrix is presented. We show that the eigenvalues and eigenvectors of the N × N bisymmetric covariance matrix can be obtained from the eigenvalues and eigenvectors of two N/2 × N/2 submatrices. Since the eigenvector calculations are proportional to the third power of the matrix dimension, the proposed procedure reduces the computations by a factor of four.

Pattern recognition problems involving learning with a bad teacher or learning without a teacher require the updating of the conditional densities of unknown parameters using a mixture of probability density functions. Mixtures of density functions in general are not reproducing and hence the computations are infeasible. For learning without a teacher, a computationally feasible scheme has been suggested by Agrawala [1]. The learning procedure proposed by Agrawala makes use of a probabilistic labeling scheme. The probabilistic labeling scheme is extended to allow the use of reproducing densities for a large class of problems, including the problem of learning with an imperfect teacher.

Supervised learning in pattern recognition problems takes place through the use of a set of labeled sample patterns, the labels being provided by a "teacher." In most of the procedures for learning with a teacher, it is commonly assumed that the teacher is perfect, i. e., the labels of the sample patterns are always correct. However, there are many circumstances in which the patterns used for learning are occasionally mislabeled. A procedure for learning with an imperfect teacher who occasionally mislabels some of the learning patterns is developed. The proposed error correction scheme is based on a nonparametric learning scheme. The error correction scheme questions and attempts to correct the labels provided by the imperfect teacher using a threshold in the correction scheme. The use of threshold facilitates control over the amount of correction and provides a simple method for combining the knowledge acquired by the learning scheme with that provided by the teacher. Expressions for the threshold are derived, and the properties of the proposed error correction scheme are discussed. Through computer simulations the performance of the proposed error correction scheme is compared with that of an identical learning scheme without error correction.