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Two-dimensional wavelets offer a number of advantages over discrete wavelet transforms when processing rapidly varying functions and signals. In particular, they offer benefits for real-time applications such as medical imaging, fluid dynamics, shape recognition, image enhancement and target tracking. This book introduces 2-D wavelets via 1-D continuous wavelet transforms. The authors then describe the underlying mathematics before progressing to more advanced topics such as matrix geometry of wavelet analysis and three-dimensional wavelets. Practical applications and illustrative examples are employed extensively throughout, ensuring the book's value to engineers, physicists and mathematicians. Two-dimensional wavelets offer a number of advantages over discrete wavelet transforms, in particular, for analysis of real-time signals in such areas as medical imaging, fluid dynamics, shape recognition, image enhancement and target tracking.

The Workshop on Image Processing Applications for the Digital Battlefield was held at Clark Atlanta University, September 14-15, 1998. The workshop focused on the area of image processing applications as applied to the needs of the military. Presentations during the workshop covered topics in image and video compression, image analysis for automatic target detection/recognition, motion analysis, and target tracking. The speakers included distinguished representatives from academic, industry, and military sponsors.

The project aimed at developing new algorithms for tracking missiles and warheads. The initial theoretical foundation for this effort is multiresolution analysis, in particular, continuous multi-dimensional wavelet theory. In concert with this framework, new classes of motion-model techniques for target tracking were developed.

The two-dimensional continuous wavelet transform (CWT) is characterized by a rotation parameter, in addition to the usual translations and dilations. The CWT has been interpreted as space-frequency representation of two-dimensional signals, where the translation corresponds to the position variable, and the inverse of the scale and the rotation, taken together, correspond to the spatial-frequency variable. The integral of the CWT's squared modulus, with respect to all variables, gives the energy of the original signal. Therefore, an integration on a subset of the parameters gives an energy density in the remaining variables. This paper deals with the implementation of the two basic densities, that is, the position (or aspect-angle) and scale-angle densities., Prepared in cooperation with Georgia Inst. of Tech., School of Electrical Engineering, Atlanta, GA and Universite Catholique de Louvain, Institut de Physique Theorique, Louvain-la-Neuve, Belgium.

Both in 1D (signal analysis) and 2D (image processing), the wavelet transform (WT) has become by now a standard tool. Although the discrete version, based on multiresolution analysis, is probably better known, the continous WT (CWT) plays a crucial role for the detection and analysis of particular features in a signal, and we will focus here on the latter. In 2D however, one faces a practical problem. Indeed, the full parameter space of the wavelet transform of an image is 4D. It yields a representation of the image in position parameters (range and perception angle), as well as scale and anisotropy angle. The real challenge is to compute and visualize the full continuous wavelet transform in all four variables--obviously a demanding task. Thus, in order to obtain a manageable tool, some of the variables must be frozen. In other words, one must limit oneself to sections of the parameter space, usually 2D or 3D. For 2D sections, two variables are fixed and the transform is viewed as a function of the two remaing ones, and similarly for 3D sections. Among the six possible 2D sections, two play a privileged role. They yield respectively the position representation, which is the standard one, and the scale-angle representation, which has been proposed and studied systematically by two of us in a number of works. In this paper we will review these results and investigate the four remaining 2D representations. We will also make some comments on possible applications of 3D sections. The most spectacular property of the CWT is its ability at detecting discontinuities in a signal. In an image, this means in particular the sharp boundary between two regions of different luminosity, that is, a contour or an edge. Even more prominent in the transform are the corners of a given contour, for instance the contour of a letter. In a second part, we will exploit this property of the CWT and describe how one may design an algorithm for automatic character recognition (here we obviousl