| Abstract: |
A "signal representation" (or "transform") is one of the fundamental concepts in signal processing. A good signal representation one which compacts most of the signal content into relatively few transform coefficients is critical for performing tasks such as compression, estimation in the presence of noise ("denoising"), and solving inverse problems (e.g. "deblurring" or tomographic imaging).
Much of the advance made by practical algorithms in these areas over the past 15 years is owed to the tremendous progress made in signal representations (fueled by the "wavelet revolution"). Parallel to this advance in applications has been the development of a theoretical understanding coming from the area of applied mathematics known as Computational Harmonic Analysis of the indispensable role that representation plays in signal processing.
In this talk, we will show that having a good representation also allows us to acquire a signal or image from a small number of indirect measurements. The main result is that the number of measurements we need to faithfully reconstruct a signal depends more on its sparsity (how well it can be approximated by a superposition using a small number of transform coefficients) than the desired resolution of the reconstruction. For example, if a 1 million pixel image can be closely approximated by a sum of 10,000 wavelet coefficients, then the number of measurements needed to acquire this image is much closer to 10,000 than 1 million.
An interesting consequence of this theory is that the Shannon-Nyquist sampling theorem (that the sampling rate must be twice as high as the highest frequency contained in the signal) is far too pessimistic if the signal is spectrally sparse. We will see rather that the number of samples is determined by the number of active frequecies, even though which frequencies are active is a priori unknown. |