About

This is the home page of the SMART project (SMART stand for Signal Models, Algebra Representations, and Transforms, and for our hope to produce good work).

The objective of SMART is to explore the relationship between signal processing and (abstract) algebra, the theory of groups, rings, and fields. Our past and forthcoming results show that this relation is far deeper than previously understood.

In short, our goal is to work towards an
"Algebraic Signal Processing Theory."

Sponsor

SMART has been funded since 2000 by the National Science Foundation within the Signal Processing Systems (SYS) program and recently the Theoretical Foundations (TF) program under the awards

• CCF-0634967: Algebraic Signal Processing Theory: Towards Multiresolution Analysis (12/2006-08/2009)
• SYS-310941: The Algebraic Structure of Linear Transforms (09/2003-08/2006)
• SYS-9988296: Group Representations and Automatic Generation of Fast Algorithms for Discrete Signal Transforms (09/2000-08/2003).

PIs: Markus Püschel, Jelena Kovacevic and earlier José Moura

PhD student: Aliaksei Sandryhaila

Collaborators: Martin Rötteler and earlier Sebastian Egner

Disclaimer

Any opinions, findings, and conclusions or recommendations expressed on this website are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Overview of Results

• Algebraic Signal Processing Theory. We finished the first papers that develop an algebraic signal processing theory. The algebraic theory is a general approach to and an extension of linear signal processing. The theory encompasses all our previous results listed below. (More info)
• Discrete Lattice Transforms. We derived new signal transforms for 2-D signals given on a finite nonseparable hexagonal or quincunx lattice including fast algorithms for the transform's efficient computation. (More info)
• Cooley-Tukey FFT like DCT Algorithms. We have derived new algorithms for the DCT, type II and III that are in structure and in mathematical origin the analogue of the (arbitrary radix) Cooley-Tukey FFT. (More info)
• Algebraic Derivation of DCT/DST Algorithms. Similar to the DFT, also virtually all known DCT/DST algorithms can be derived using algebraic methods. The algebraic approach makes the derivation very concise (no tedious matrix entry manipulations), explains the existence of the algorithms, allows their classification, and enables the discovery of new algorithms (see bullet above). (More info)
• Automatic Generation of Transform Algorithms. Using methods from representation theory of groups and algebras, it is possible to automatically generate fast algorithms for discrete signal transforms directly from their definition. The method works for many different transforms and was the motivation for this research. (More info)
• Frames. We recently derived the first class of real, tight frames that are maximally robust to erasures. (More info)

Internal

Publications