Signal Representation 2.0 2003/05/09 2003/05/09 Feng Qiao fqiao@rice.edu Rachael Milam rmilam@rice.edu Charlet Reedstrom charlet@rice.edu Rachael Milam rmilam@rice.edu Feng Qiao fqiao@rice.edu signal The idea of signal representation is to decompose a general signal into a linear combination of a fixed set of vectors. For any v 2 , we have v x 1 0 y 0 1 Suppose we have an orthonormal set of vectors v k t which spans a subspace V of L 2 , then f t V , we have f t k k f t v k t v k t In practical applications, signals are generally smooth or at least piecewise smooth (e.g., sound waves, image intensities). Such signals can be well approximated by using piecewise polynomials. A function is piecewise polynomial with K 1 pieces if p t i 0 K p i t w t w t 1 t i t t i + 1 0 where t 0 0 and t K + 1 T and p i t d 0 D a i , d t d are polynomials of max degree D. So when we try to project a polynomial signal f t t n to a subspace spanned by v k t , we will get f ^ t k k t n v k t v k t The term t n v k t t t n v k t is called the n th moment of v k t . In many cases we would like a "parsimonious" representation of the signal (e.g., compression of the signal). That is, we would like most of the coefficients to be zero. In the problem of polynomial signal representation, that is to say, we want the moments of the basis vector to be zero. This is called vanishing moments of v k t . How to choose v k t wisely so that we have as many vanishing moments as possible?