Data Compression: The Complete Reference

2.19 Context-Tree Weighting 1611b 3 b 4 b 5 b 6 b 710b 3 b 6 b 1 b 4 b 2 b 50100b 3 b 60b 7b 7b 3 b 6b 1 b 2b 1 b 410b 2 b 51b 7b 1 b 4 b 2 b 5(a)(4,3) P w=7/2048(2,1)1/16(2,2)9/128(1,0) 1/2 (1,1) 1/8 (2,0) 5/16 (0,2)3/8(1,0)1/2(1,1) (1,0)1/8 1/2(b)(1,0)1/2(0,2)3/8Figure 2.81: (a) A Context Tree. (b) A Weighted Context Tree.The last step that needs to be described is the way the context tree is updatedwhen the next bit is input. Suppose that we have already input and encoded the stringb 1 b 2 ...b t−1 . Thus, we have already constructed a context tree of depth d for this string,we have used Eqs. (2.12) and (2.13) to calculate weighted probabilities for the entiretree, and the root of the tree already contains a certain weighted probability. We nowinput the next bit b t and examine it. Depending on what it is, we need to update thecontext tree for the string b 1 b 2 ...b t−1 b t . The weighted probability at the root of thenew tree will then be sent to the arithmetic encoder, together with bit b t , and will beused to encode b t .If b t = 0, then updating the tree is done by (1) incrementing the a s counts for allnodes s, (2) updating the estimated probabilities P e (a s ,b s ) for all the nodes, and (3)updating the weighted probabilities P w (a s ,b s ) for all the nodes. If b t = 1, then all the b sshould be incremented, followed by updating the estimated and weighted probabilitiesas above. Figure 2.82a shows how the context tree of Figure 2.81b is updated whenb t =0.Figure 2.82b shows how the context tree of Figure 2.81b is updated when b t =1.