Sensors and signal processing hardware and algorithms are under increasing pressure to accommodate ever larger and higher-dimensional data sets; ever faster capture, sampling, and processing rates; ever lower power consumption; communication over ever more difficult channels; and radically new sensing modalities. This four-hour course presents the fundamental theory and selected applications of Compressive Sensing, a new approach to data acquisition in which analog signals are digitized for processing not via uniform sampling but via inner products with random test functions. Unlike Nyquist-rate sampling, which completely describes a signal by exploiting its bandlimitedness, Compressive Sensing reduces the number of measurements required to completely describe a signal by exploiting its compressibility. The implications are promising for many applications and enable the design of new kinds of analog-to-digital converters, imaging systems and cameras, and radar systems, among others.

Wavelets and conventional filter banks have become popular because of their convenient representation and ability to isolate characteristic elements of an input in a compact subband form, often for the purposes of analysis, denoising, and compression. Recent research in this area has focused on perceptually relevant feature extraction, curve and shape characterizations, orientation selective processing, and efficient analysis-reconstruction. This four-hour course presents the fundamentals of wavelets and filter banks from a vector space point of view (in the first 2 hours) followed by a discussion of applications to image/signal processing and terrain modeling. History from FFTs evolving to treat non-stationary signal processing such as second order Time-Frequency joint representation instantaneous frequency representation and recently to the first order linear wavelet representation will be briefly reviewed. Prototypical algorithms for compression, denoising, and registration of images will be discussed and evaluated. Application of multiscale methods to terrain modeling and surface compression will be presented.