1996 Research Summaries for the Ptolemy Project

Designing Two-Dimensional Filter Banks Based on Geometric Decompositions of the Frequency Domain


Researcher:William Chen
Advisor:Edward A. Lee
Sponsors:NSF Graduate Fellowship and the Ptolemy Project

A filterbank consists of an analysis section that breaks an input signal into N subband channels followed by a synthesis section that combines the N subband channels into a delayed approximation of the original signal. By coding the information on the subband channels, filterbanks have been successful at medium bit-rate compression of speech, audio, and image signals.

Two-dimensional filterbanks have more degrees of freedom than in the one-dimensional case. In two dimensions, the frequency domain is periodic in each dimension. That is, a fundamental tile is replicated to fill or pave the plane. In a critically sampled 2-D filter bank, the subband channels cover the fundamental tile, but do not overlap.

In this project, we apply computational geometry techniques to the design of non-separable analysis/synthesis filterbanks for two-dimensional signals. First, there is no unique way to divide the frequency plane. So, we start our search space by specifying a canonical set of filters with 3-, 4-, 5-, and 6-sided convex passbands. That is, we look for all such triangles, trapezoids, pentagons, and hexagons, and then design the filterbank minimizing phase, amplitude, and aliasing distortion. Using these canonical filterbanks, we can build a tree structure to create more complex and concave geometries. Second, 2-D filterbanks can be designed with either separable or nonseparable filters. While separable filters can be decomposed to a 1-D filter design problem [2], nonseparable filters are often used because they better reflect the property of 2-D signals. Furthermore, nonseparable filters can be designed to have frequency responses that better match a particular class of 2-D signals, e.g. directional sensitivity [3].

Some possible applications of 2-D filterbanks include subband image coding, image enhancement, and image analysis.

  1. Shankar Venkatarman, ``Structure and Design of 2-D Non-Separable Orthogonal Perfect Reconstruction Filter Banks and Wavelets with Applications to Image Compression,'' Ph.D. Thesis, University of California, Davis, CA, 1995.
  2. P. P. Vaidyanathan, ``Multirate Systems and Filter Banks,'' Prentice-Hall, 1993.
  3. Robert Bamberger, ``The Directional Filter Bank: A Multirate Filter Bank for the Directional Decomposition of Images,'' Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, November,1990.

Send comments to William Chen at williamc@eecs.berkeley.edu.