Envelope Scales Research Articles

Gabor filter design for multiple texture segmentation

A method is presented for the design of a single Gabor filter for the segmentation of multitextured images. Earlier methods were limited to filters designed for one or two textures or to filters selected from a predetermined filter bank. Our proposed method yields new insight into the design of Gabor filters for segmenting multitextured images and lays an essential foundation for the design of multiple Gabor filters. In the method, Rician statistics of filtered textures at two different Gabor-filter envelope scales are used to efficiently generate probability density estimates for each filtered texture over an extensive set of candidate filter parameters. Variable degrees of postfiltering and the accompanying effect on postfilter output statistics are also included in the design procedure. The result is a unified framework that analytically relates the texture power spectra, Gabor-filter parameters, postfiltering effects, and image-segmentation error. Finally, the resulting filter design is based on all constituent textures and is not constrained to a limited set of candidate filters.

Universal properties of the transition from quasi-periodicity to chaos in dissipative systems

An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C 1 conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency. When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.