Here we see an 8-bit, greyscale image on the left and its DGT representation on the right. The decomposition was preformed with the basis set descibed in the previous pages.

• Representation is four dimensional: Each coefficient is indexed by 2 spatial indeces and 2 derivative order indeces. Subsampling of the 256x256 image by 8 results in 32x32 different spatial locations. The coefficients have been collected by like derivative order. For each derivative order (64 combinations of 8 derivatives in the x and y dimensions) we have collected the coefficients at the 32x32 locations. Thus, the upper left 32x32 block represents the coefficients of the basis functions which are Gaussian in both dimensions. The next block to the right represents the coefficients of the basis functions which are Gaussian in the y dimension and the third derivative in the x dimension.

• The coefficients generated by this transform are real and thus any finite precision representation will necessarily introduce quantization error in the reconstruction. For this image and basis set, when the coefficients are represented with 12 or more bits, the quantization error in the reconstruction falls below one LSB in the 8-bit reconstruction and the reconstruction is said to be a perfect reconstruction.

• It is this coefficient map from which both spatial and spectral information can be gathered for image analysis applications and to which spatially varying filters can be applied for image enhancement and manipulation applications.

• The grey level seen at the lower left of the coefficient map is very close to zero. We see that, in fact, about 3/4 of this coefficient map is close to zero as the majority of the image information is being compactly represented in the upper left quarter of the coefficients. This suggests an application of the DGT to image compression.

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