Data Compression: The Complete Reference

590 5. Wavelet Methodsimplying that all the scaling functions for different values of j and k are translations anddilations (scaling) of φ(x), the fundamental solution of the subdivision process. φ(x) isshown in Figure 5.54 for n =2,4,6,and8.1.00.50.0−0.5−4 −3 −2 −1 01 2 3 4−3 −2 −1 0 1 2 3 41.00.50.0−0.5−4 −3 −2 −1 01 2 3 4−3 −2 −1 0 1 2 3 4Figure 5.54: Scaling Functions φ j,k (x) for n =2, 4, 6, and 8.The main properties of φ(x) are compact support and smoothness (it can be nonzeroonly inside the interval [−(n − 1),n− 1]), but it also satisfies a refinement relation ofthe formn∑φ(x) = h l φ(2x − l),l=−nwhere h l are called the filter coefficients. A change of variables allows us to write thesame refinement relation in the formφ j,k (x) = ∑ lh l−2k φ j+1,l (x).The odd-numbered filter coefficients are the coefficients of the interpolating polynomialat its midpoint P n−1,j,k−(n/2−1) (0.5). The even-numbered coefficients are zero except