Signal processing on graphs is a theoretical framework that generalizes classical discrete signal processing from regular domains, such as lines and rectangular lattices, to arbitrary, irregular domains commonly represented by graphs. Different from network science, signal processing on graphs focuses on the interplay between the graph structure and the corresponding signals. The goal of this research is to build a theoretical foundation from the perspective of signal processing to handle practical data analysis tasks.

This area was started by Markus Püschel, whose goal is to formulate an algebraic framework for signal processing. Our current work focuses on understanding and formulating such a framework for filter banks and multiresolution transforms.

Frames are redundant representations which have become popular in recent years. Work here focuses on characterizing finite-dimensional frames, searching for useful frame families as well as applying frames to biomolecular and cellular imaging and biometrics.

We focus on developing automated systems for analysis and interpretation of biomedical images. Our current work focuses on delineating and recognizing tissues in histopathology images and diagnosis of otitis media.

We explore an indirect measurement approach for bridge structural health monitoring that collects sensed information from the dynamic responses of many vehicles travelling over a bridge and then makes extensive use of advanced signal processing techniques to determine information about the state of the bridge.

Correlation, as a pattern recognition tool, may be applied to texture features that have joint locality in space and frequency. Wavelets produce these types of discriminatory features, and we can prune wavelet packets to recover the best subspaces for correlation filter recognition.

This is a technique where the data is broken into several streams with some redundancy among the streams. When all the streams are received, one can guarantee low distortion at the expense of having a slightly higher bit rate than a system designed purely for compression. On the other hand, when only some of the streams are received, the quality of the reconstruction degrades gracefully, which is very unlikely to happen with a system designed purely for compression.

One of the first works on multidimensional filter banks and associated wavelet bases including the first examples of a regular irreducible two-dimensional wavelet as well as an orthonormal and symmetric two-dimensional wavelet basis. There is also a brief description on building local orthogonal bases in multiple dimensions as well as assorted applications such as HDTV representation and coding, use of three-dimensional filter banks with the FCO lattice and deinterlacing by successive approximation.

Local cosine bases, or, MDCT, have been shown to be very useful in audio and image coding. Some video works contain local cosine bases as well. For that reason, I investigated the local cosine bases in two dimensions. Moreover, a general framework was put into place leading to local orthogonal bases usable for audio, image and video coding.

One of the main goals of signal analysis in recent years has been to develop a mixed signal representation in terms of some elementary blocks well localized in time and frequency, where these blocks are known as time-frequency atoms. Each one of these blocks would reside mostly in a well-defined area (usually a rectangle) in the time-frequency plane. We discuss here several ways of building these arbitrary tilings, with particular emphasis on those obtainable from local orthogonal bases.

The most studied case of filter banks is the one with integer sampling factors. However, if one wants to analyze the signal into unequal subbands (such as in acoustics), rational sampling factors have to be allowed. You can find here about two solutions to the open problem of constructing nonuniform filter banks. The first is general while the second one is based on local orthogonal bases. As a result of the second one, we were able to build a critical-band filter bank for use in audio coding.

This work was motivated by the need to obtain even more powerful compression schemes than what is currently available. For subband systems, unsolved problems with large potential benefits are in the area of joint design of quantization and filtering. The work here is one of the few on the subject.

We propose a perceptually-based system for pattern retrieval and matching. We detect basic visual categories that people use in judgment of similarity, and design a computational model which accepts patterns as input, and depending on the query, produces a set of choices that follow human behavior in pattern matching.

We propose an efficient simplification method for regular meshes obtained with a binary subdivision scheme. Our mesh connectivity is constrained with a quadtree data structure. We propose a quadtree built especially for this class of meshes having a constant-time traversal property. We introduce a rate-distortion (RD) framework to decimate the mesh and build a progressive representation for the model. We apply our technique to a large dataset of terrains and give extensive experimental results.