The problem of quantizing noisy signals in an optimal manner is addressed. Quantizer designs relying on training sets, "clean" probability statistics, or composite statistics of source and noise, will yield designs which are sub-optimal and possibly detrimental to system performance. The concept of the quantizer as an estimator is used in conjunction with a risk function to produce a minimum risk quantizer for noisy sources. In particular, the minimum-risk quantizer design theory for the case of independent, identically distributed source with additive (i.i.d.) noise is developed. The minimum-risk quantizer criteria for the specific problem of a source signal with gaussian statistics corrupted by additive gaussian noise and the squared-error cost function is produced. It is further shown that the Max-Lloyd optimal quantizer criteria is a subset of the minimum-risk criteria for noiseless conditions.

In this paper, the problem of minimizing the quantization distortion when the source is a sequence of images is considered from a different perspective. The quantizer that is considered is part of a motion compensated image coding system. In this application, the number of levels is not fixed but varies from one block of pixels to another. The implication of this is that the calculation of the quantizer levels and threshold for each block when the distortion measure is the MSE is prohibitive from a computation standpoint. This problem is circumvented by choosing a distance criterion rather than the MSE. Additionally, the minimum distortion procedures of Max and Lloyd are not utilized. Rather, a minimum risk (minimum expected cost) procedure which employs the absolute value cost function is formulated. The procedure yields a quantizer with a lower variance error distribution.

In this paper we present a method for motion compensated image coding based upon a two-step displacement estimation procedure. The first step utilizes a maximum a posteriori (MAP) estimator to determine the best integer displacement, while the second step requires solving for the regression coefficients that supply the same information as the noninteger portion of the displacement. This approach is a different and simplified procedure in that the integer displacement is measured first, and then a linear combination of only four values from the previous image, shifted by the measured integer displacement, is used. This procedure differs both from the ones which measure the displacement vectorDfirst and interpolate the previous image, and from the ones which use only linear prediction. This method is derived and results are presented for two separate 40-frame digital image sequences. A sum of absolute error distortion measure is used to determine the optimal structure of the residual quantizer.

A new concept for examining shapes as vectors in a shape space is described. The shape space is defined in terms of its properties and the importance of the independence of the size variable to the shape vectors. Also, two theorems essential to the process of comparing partial shapes to the complete shape are stated and proved. A new method for detecting the points on a shape that appear to dominate visual perception is described. This method, called the adaptive line of sight method, detects the dominant points on a shape even though they do not always occur on points of high curvature. The critical points determined by this method are based on a set of axes that are dependent on the shape itself. Therefore, the points determined are independent of size, rotation, or relative displacement. The line of sight of a point concept is also introduced and subsequently utilized to extract features from a shape. These features are then compared to the features of other shape by a syntactic procedure for the purpose of recognizing whether a shape is a partial shape or a shape in its own right.

In this paper we present a motion compensated coding procedure based on a time modified autoregressive image source model. The procedure has the advantage that it does not depend on exact knowledge of the displacement vector. Instead, a two step procedure for determining the displacement is employed. In the first step the nearest integer displacement for any particular block is determined using a maximum a posteriori (MAP) estimate. The second step utilizes regression techniques that determine the best estimate for the autoregressive parameters of the model. This supplies knowledge equivalent to the non-integer portion of the displacement. The results of applying the algorithm to a forty one frame sequence of images is discussed.

In this paper we present a method of achievement dimensionality reduction on a set of data that is corrupted by Gaussian noise prior to observation. The noise is additive in the original space (domain) in which the pattern class is defined. In conjunction with the dimensionality reduction method, a pattern clustering technique is presented and employed in the range space. The p-dimensional observation is taken into a smallerl-dimensional space by a linear transformation T. The transformation is derived using well-known variational calculus techniques after the optimumality criteria have been defined. It is shown that the dimensionality reduction technique has the properties of the discrete Wiener filter and therefore possesses well defined optimality properties in the mean square sense.

A new concept for examining shapes as vectors in a shape space is described. The shape space is defined in terms of its properties and the importance of the independence of the size variable to the shape vectors defined on this shape space is stressed. Also, two theorems essential to the process of comparing partial shapes to the complete shape are given. A new method for detecting the points on a shape which appear to dominate visual perception is described. This method, called the Adaptive Line of Sight Method detects the dominant points of high curvature. The critical points, or the dominant points of the shape, which are determined by this method are based on a set of axes that are dependent on the shape itself. Therefore, the points determined are independent of size, rotation, or relative displacement. This concept is utilized to extract features from a shape. It is demonstrated that the feature vectors determined by this procedure are independent of size rotation, and displacement.

In this correspondence it is demonstrated that the derivative of the angle with respect to an arbitrary independent parameter cannot in general be used to find peaks and valleys of a shape. It is then shown, using concepts from differential geometry, that curvature corresponds to the derivative of the angle with respect to an independent variable if and only if the angle is deflned as a single variable. It is further shown that peaks and valleys detected using curvature as a criterion are insufficient to completely describe a shape.