Parallel Algorithms for the Two-Dimensional Discrete Wavelet Transform

We present in this paper two parallel algorithms for the two-dimensional discrete wavelet transform. A mathematical model for the computation vs. communication tradeoff for these algorithms is presented and the scalability of the algorithms is analyzed. The wormhole routing model was used to model the communication costs involved. The algorithms were implemented on the Connection Machine-5 (CM-5^{a}). The theoretical results obtained were validated by the simulations on the CM-5.

[1]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[2]  Geoffrey C. Fox,et al.  Scheduling regular and irregular communication patterns on the CM-5 , 1992, Proceedings Supercomputing '92.

[3]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[4]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[5]  Rajeev Thakur,et al.  All-to-all communication on meshes with wormhole routing , 1994, Proceedings of 8th International Parallel Processing Symposium.

[6]  D. Esteban,et al.  Application of quadrature mirror filters to split band voice coding schemes , 1977 .

[7]  Mohan Vishwanath,et al.  Time-frequency distributions: complexity, algorithms and architectures , 1993 .

[8]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[9]  T. T. Kwan,et al.  Communication and computation performance of the CM-5 , 1993, Supercomputing '93.

[10]  P. Vaidyanathan.  Quadrature mirror filter banks, M-band extensions and perfect-reconstruction techniques , 1987, IEEE ASSP Magazine.

[11]  Lionel M. Ni,et al.  A survey of wormhole routing techniques in direct networks , 1993, Computer.

[12]  John W. Woods,et al.  Subband coding of images , 1986, IEEE Trans. Acoust. Speech Signal Process..

[13]  Charles E. Leiserson,et al.  Fat-trees: Universal networks for hardware-efficient supercomputing , 1985, IEEE Transactions on Computers.

[14]  M. Vetterli.  Multi-dimensional sub-band coding: Some theory and algorithms , 1984 .

[15]  Vipin Kumar,et al.  Isoefficiency: measuring the scalability of parallel algorithms and architectures , 1993, IEEE Parallel & Distributed Technology: Systems & Applications.

[16]  O. Rioul,et al.  Wavelets and signal processing , 1991, IEEE Signal Processing Magazine.

[17]  D. S. Scott,et al.  Efficient All-to-All Communication Patterns in Hypercube and Mesh Topologies , 1991, The Sixth Distributed Memory Computing Conference, 1991. Proceedings.

[18]  Vipin Kumar,et al.  Performance and Scalability of Preconditioned Conjugate Gradient Methods on Parallel Computers , 1995, IEEE Trans. Parallel Distributed Syst..