Accurate extraction of the image feature directions is essential to image smoothing and other image processing tasks. We show that the gradient-based feature direction extraction method can be very erroneous. The gradient is too local, and it cannot detect oscillations. We have developed two new methods: the Hessian method, an approach using higher order differentiations, and the Gabor (1946) method, an approach using space-frequency analysis.

The optimal reconstruction is an advanced decoding technique for the transform-quantization image coding schemes such as JPEG. From the same encoded data, the image decoded by the optimal reconstruction has better quality than the conventional one. In particular, the optimal reconstruction does not suffer from the Gibbs artifact. The optimal reconstruction is formulated as a constrained optimization problem and solved by an evolution-projection algorithm.

A multiscale Canny edge detection is equivalent to finding the local maxima of a wavelet transform. The authors study the properties of multiscale edges through the wavelet theory. For pattern recognition, one often needs to discriminate different types of edges. They show that the evolution of wavelet local maxima across scales characterize the local shape of irregular structures. Numerical descriptors of edge types are derived. The completeness of a multiscale edge representation is also studied. The authors describe an algorithm that reconstructs a close approximation of 1-D and 2-D signals from their multiscale edges. For images, the reconstruction errors are below visual sensitivity. As an application, a compact image coding algorithm that selects important edges and compresses the image data by factors over 30 has been implemented.>

A second generation image coding algorithm based on multiscale edges is described. Edges are detected from the local maxima of the image wavelet transform. The original image can be reconstructed from the corresponding multiscale edge map. For coding purposes, the edges that are important for the image visualization are selected, and an efficient encoding of the corresponding data is described. High compression ratio can be obtained for images where the texture information is not important. Examples with compression ratios of around 30 are given.>

The edges of an image can be detected at different scales from the local maxima of its wavelet transform. an algorithm is described that reconstructs images from their edges at dyadic scales. The wavelet maxima representation is a novel reorganization of the image information that makes it possible to develop algorithms uniquely based on edges for solving image processing and computer vision problems. The evolution of the wavelet maxima across scales gives a precise characterization of the edge type which can be used for pattern recognition. A coding algorithm is described that selects the most important image edges in order to obtain a compact representation.>

An algorithm that reconstructs one-dimensional signals and images from their sharper variation points at dyadic scales is described. This algorithm exactly reconstructs images from their multiscale edges. It is proved that the evolution across scales of the wavelet maxima characterizes the local shape of the sharp variations of the signal. One can thus not only detect edges but also classify them. The wavelet maxima representation is a new reorganization of the image information that makes it possible to develop algorithms uniquely based on edges for solving image processing problems.>