3.1.2 Searching for Minimum Cost Table 2: Concentration of Energy for Some Example Orthogonal Qfs

A procedure for designing exact reconstruction lter banks for tree-structured sub-band coders. where p(n) = jx(n)j 2 =kxk 2. Note that d = exp H(x) where H(x), deened in Equation 55 above, is the entropy of the sequence x. Beylkin in 1] observed earlier that computing the periodic discrete wavelet transform of all N circulant shifts of an N-point periodic signal requires computing only N log 2 N coeecients. If we build a complete binary tree with information cost tags computed from from appropriate subsets of the shifted coeecients, then the best complete branch will give a representation of the circulant shift which yields the lowest cost transform. After solving the technical problem of ties, the computed shift can be used as a registration point for the signal. The rst step is to build a binary tree of the information costs of the wavelet subspaces computed with all circulant shifts. We write the cost of a node of the tree into an auxiliary variable attached to the node, which will later be added together with the other nodes along the branch to give a \branch" cost. We also assume that the output array is at least q=2 elements long, to accommodate the intermediate outputs of convolution and decimation. The algorithm is implemented recursively as follows: shiftscosts(output y; input x; parameter q): Costs of circulant shifts If q 1 then return (this is the recursion termination condition). Convolve-decimate the q-periodic input sequence fx(1);:::;x(q)g to a q=2-periodic output sequence fy(1);::: ; y(q=2)g using the high-pass lter G. Compute the information cost of y and store it. Convolve-decimate the q-periodic input sequence fx(1);:::;x(q)g to a q=2-periodic output sequence fy(1);::: ; y(q=2)g using the low-pass lter H. Apply shiftscosts to the q=2-periodic sequence fy(1); y(2); : : : ; y(q=2)g. Apply shiftscosts to the q=2-periodic sequence fy(2); : : : ; y(q=2); y(1)g. The function shiftscosts can also be used to accumulate the costs of a a whole branch into its leaf at the same time that we compute the coeecients, as we descend. One of the inputs to the function is q, and we assume that the input sequence is q-periodic and registered at 0. Then the information cost of a 2 L-point discrete periodic wavelet transform shifted by T will be found in the node at level L whose block index is the bit-reverse of T. We can extract these values with …