PROJECTS
• Image Exploitation
• DSP Algorithm Development
• Other Projects of Interest
Mathematical Image Exploitation
M. An and R. Tolimieri have generalized traditional Fourier methods in digital computational applications to nonabelian group harmonic analysis models. As real-time digital image processing tools, algorithms have been developed for designing noncommutative matched filters for automatic and robust detection and location of targets in severe noise.
The placement of group structure on the data indexing set plays two important roles in applications: properties of the matched filters are predictable and can be controlled by algebraic manipulations, and the computational complexity of the orthogonal projection is on the order of N log N, where N is the size of data.
This project was supported by the US Air Force until April 2002. Main results and their application to image processing of this project thus far have been documented in the recent text Group Filters and Image Processing, Psypher Press, 2003.
This is a continuing project of interest, current applications including texture determination and analysis.
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DSP Algorithm Development
R. Tolimieri was the director of the Center for Large Scale Computing at CUNY for 6 years until 1991. Among the many projects he conducted at the center are "DSP algorithm development for for massively parallel and embeddable systems," (funded by DARPA), direct methods in X-ray crystallography and time frequency representations and analysis. In tandem with developing algorithms, optimized implementations have been generated and used in many application areas including computational electro-magnatics, fluid dynamics and radar image processing.
M. An has developed symmetry specific FFT algorithms in collaboration with J. Cooley. The symmetries addressed include all crystallographic space groups. In addition to savings in computations, these algorithms greatly reduce memory requirements, resulting in up to hundred-fold speed-up in processing time. She has designed and implemented similar algorithms in hyperspectral, tomographic image reconstruction problems with comparable speed-up in processing time. These algorithms have moved off-line, remote processing procedures to on-line, on-board procedures. This is also a continuing project of interest, current efforts centering on the design of signals, waveforms and filters in mixed time-frequency domain.
Results of these effort have been documented in three graduate level textbooks and elsewhere.
• Time-Frequency Representations , Birkhauser 1998.
• Mathematics of Multidimensional Fourier Transform Algorithms, Springer, 2nd Ed., 1997.
• Algorithms for Discrete Fourier Transform and Convolution, Springer, 2nd Ed., 1997.
• "Discrete Fast Fourier Transform Algorithms: A Tutorial Survey," Adv. in Electronics and Electron Phys., 80, Academic Press, 1991.
• "Group Invariant Fourier transform," Adv. in Electronics and Electron Phys., 93, Academic Press, 1995.
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Other Current Projects
Psypher scientists recently attended the NATO ASI on Computational Noncommutative Algebra and Applications in Tuscany as representives of Prometheus Inc. Richard Tolimieri and Myoung An gave a presentation on abelian and non-abelian DSP routines covered in Group Filters and Image Processing. Prometheus is administering the publication of papers from this Advanced Study Institute.
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