Burt and Adelson [3] introduced quad-pyramid image coding with the laplacian pyramid. In their scheme, the input image I0, is lowpass filtered to obtain G0. The difference between I0 and G0 is saved as L0, then G0 is downsampled by a factor of two horizontally and vertically to yield I1. I1 has one quarter of the number of pels in I0 (hence the term "quad-pyramid"). The process is then applied to I1, generating G1, L1, and I2. Finally, a third level is generated from I2, yielding G2, L2, and I3. The subbands I3, L2, L1, and L0 are transmitted. The decoder recovers the original image by successive upsamplings, interpolation and addition of difference images. The total num ber of points in the subband representation is 85/64 times the size of the input image (85/64 = 1 + 1/4 + 1/16 + 1/64), but the difference images have low entropy.
A laplacian pyramid may be represented as a quadtree wherein point (x,y) in Ln is regarded as the parent of the four points (2x, 2y), (2x+1, 2y), (2x, 2y+1), (2x+1, 2y+1) in Ln-1. The addition of a codeword to mark the end of a tree branch allows efficient coding of regions where all the values in higher subbands are zero. For example, if L0(4,6), L0(5,6), L0(4,7), L0(5,7) and L1(2,3) were all zero, an end-of-branch code sent instead of zero for L1(2,3) would tell the decoder that all L1(2,3)'s children were zero, and their values would not be transmitted at all.
A second form of quad-pyramid uses orthogonal wavelets to allow a representation with the same number of points as the original image [4-6]. The basic operation is filtering with pairs of quadrature mirror filters and downsampling. This yields four channels at each level: (1) LL: vertical lowpass, horizontal lowpass; (2) HL: vertical highpass, horizontal lowpass; (3) LH: vertical lowpass, horizontal highpass; (4) HH: vertical highpass, horizontal highpass. Each channel has one quarter the number of points in the original, and the process can be successively applied to the LL band. This form of decomposition results in 10 subbands for a 3-level pyramid: LL3, HL3, LH3, HH3, HL2, LH2, HH2, HL1, LH1, HH1. Again, the higher spatial frequency bands have low entropy and may be coarsely quantized. A quadtree-like structure can again be used to represent blocks of zeros efficiently, although the branching is slightly more complex [7]. Such a structure is becoming widely used in wavelet subband coders.
The third form of image pyramid, and the one used in BTPC, is the binary pyramid.
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