In the equation from our discussion of the Haar transform: y=T⁢x⁢TT y T x T and to invert: x=TT⁢y⁢T x T y T we saw how a 1-D transform could be extended to 2-D by pre- and post-multiplication of a square matrix xx to give a matrix result yy. Our example then used 2×2 2 2 matrices, but this technique applies to square matrices of any size.

Hence the DCT may be extended into 2-D by this method.

E.g. the 8×8 8 8 DCT transforms a subimage of 8×8 8 8 pels into a matrix of 8×8 8 8 DCT coefficients.

The 2-D basis functions, from which xx may be reconstructed, are given by the n2 n 2 separate products of the columns of TT T with the rows of TT. These are shown for n=8 n 8 in (a) of Figure 1 as 64 subimages of size 8×8 8 8 pels.

The result of applying the 8×8 8 8 DCT to the Lenna image is shown in (b) of Figure 1. Here each 8×8 8 8 block of pels xx is replaced by the 8×8 8 8 block of DCT coefficients yy. This shows the 8×8 8 8 block structure clearly but is not very meaningful otherwise.

Part(c) of Figure 1 shows the same data, reordered into 64 subimages of 32×32 32 32 coefficients each so that each subimage contains all the coefficients of a given type - e.g: the top left subimage contains all the coefficients for the top left basis function from (a) of Figure 1. The other subimages and basis functions correspond in the same way.

We see the major energy concentration to the subimages in the top left corner. (d) of Figure 1 is an enlargement of the top left 4 subimages of (c) of Figure 1 and bears a strong similarity to the group of third level Haar subimages in (b) of this figure. To emphasise this the histograms and entropies of these 4 subimages are shown in Figure 2.

Comparing Figure 2 with this figure, the Haar transform equivalent, we see that the Lo-Lo bands have identical energies and entropies. This is because the basis functions are identical flat surfaces in both cases. Comparing the other 3 bands, we see that the DCT bands contain more energy and entropy than their Haar equivalents, which means less energy (and so hopefully less entropy) in the higher DCT bands (not shown) because the total energy is fixed (the transforms all preserve total energy). The mean entropy for all 64 subimages is 1.3622 bit/pel, which compares favourably with the 1.6103 bit/pel for the 4-level Haar transformed subimages using the same Qstep =15 Qstep 15 .

Figure 1: (a) Basis functions of the 8×8 8 8 DCT; (b) Lenna transformed by the 8×8 8 8 DCT; (c) reordered into subimages grouped by coefficient type; (d) top left 4 subimages from (c).

Figure 2: The probabilities pi pi and entropies hi hi for the 4 subimages from the top left of the 8×8 8 8 DCT ((d) of Figure 1).

Figure 3: (a) Mesh and (b) row plots of the entropies of the subimages of (c) of Figure 1.

Figure 4: Lenna transformed by the 4×4 4 4 DCT (a) and 16×16 16 16 DCT (b).

Figure 5: (a) Mesh and (b) row plots of the entropies of the 4×4 4 4 DCT in (a) of Figure 4.

Figure 6: (a) Mesh and (b) row plots of the entropies of the 16×16 16 16 DCT in (b) of Figure 4.

Figure 7: Comparison of the mean entropies of the Haar transform of Lenna at levels 1 to 4, and of the DCT for sizes from 2×2 2 2 to 16×16 16 16 pels with Qstep =15 Qstep 15 .

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Last edited by Elizabeth Gregory on Jun 8, 2005 10:09 am -0500.