Data Compression: The Complete Reference

5.13 The Laplacian Pyramid 593⋄ Exercise 5.7: Given the data vector x = (112, 97, 85, 99, 114, 120, 77, 80), use Equation(5.24) to calculate its forward and inverse integer wavelet transforms.The same concepts can be applied to the case where the number N of data itemsis odd. We first define k by N =2k + 1, then define the forward and inverse integertransforms byy 2i+1 = x 2i+1 −⌊(x 2i + x 2i+2 )/2⌋, for i =0, 1,...,k− 1,⎧⎨ x 2i + ⌊y 2i+1 /2⌋, for i =0,y 2i = x 2i + ⌊(y 2i−1 + y 2i+1 )4⌋, for i =1, 2,...,k− 1,⎩x 2i + ⌊y 2i−1 /2⌋, for i = k.⎧⎨ y 2i −⌊y 2i+1 /2⌋, for i =0,z 2i = y⎩ 2i −⌊(y 2i−1 + y 2i+1 )/4⌋, for i =1, 2,...,k− 1,y 2i −⌊y 2i−1 /2⌋, for i = k,z 2i+1 = y 2i+1 + ⌊(z 2i + z 2i+2 )/2⌋, for i =0, 1,...,k− 1.Notice that the IWT produces a vector y i where the detail coefficients and theweighted averages are interleaved. The algorithm should be modified to place the averagesin the first half of y and the details in the second half.The extension of this transform to the two-dimensional case is obvious. The IWT isapplied to the rows and the columns of the image using any of the image decompositionmethods discussed in Section 5.10.5.13 The Laplacian PyramidThe main feature of the Laplacian pyramid method [Burt and Adelson 83] is progressivecompression. The decoder inputs the compressed stream section by section, and eachsection improves the appearance on the screen of the image-so-far. The method usesboth prediction and transform techniques, but its computations are simple and local(i.e., there is no need to examine or use values that are far away from the current pixel).The name “Laplacian” comes from the field of image enhancement, where it is used toindicate operations similar to the ones used here. We start with a general description ofthe method.We denote by g 0 (i, j) the original image. A new, reduced image g 1 is computedfrom g 0 such that each pixel of g 1 is a weighted sum of a group of 5×5 pixels of g 0 .Image g 1 is computed [see Equation (5.25)] such that it has half the number of rows andhalf the number of columns of g 0 ,soitisone-quarterthesizeofg 0 . It is a blurred (orlowpass filtered) version of g 0 . The next step is to expand g 1 to an image g 1,1 the sizeof g 0 by interpolating pixel values [Equation (5.26)]. A difference image (also called anerror image) L 0 is calculated as the difference g 0 − g 1,1 , and it becomes the bottom levelof the Laplacian pyramid. The original image g 0 can be reconstructed from L 0 and g 1,1 ,and also from L 0 and g 1 . Since g 1 is smaller than g 1,1 , it makes sense to write L 0 andg 1 on the compressed stream. The size of L 0 equals that of g 0 , and the size of g 1 is 1/4of that, so it seems that we get expansion, but in fact, compression is achieved, sincethe error values in L 0 are decorrelated to a high degree, and so are small (and therefore