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3.2. WAVELETS AND POINT SINGULARITIES 63
2-D Wavelets
The conclusion that the 2-D wavelet is good at processing point singularities in an image
is a corollary of the 1-D result.
We consider a 2-D wavelet as a tensor product of two 1-D wavelets. It is easy to observe
that the 2-D wavelet is a spatially localized function. Utilizing the idea from the filter bank
and multi-resolution analysis, we can derive a fast algorithm for 2-D wavelet transform.
The fast algorithm is also based on a two-scale relationship.
Using a similar argument as in the 1-D case, we can claim that for an N × N image,
no more than O(log N) significant 2-D wavelet coefficients are needed to represent a point
singularity in an image. Hence the wavelet transform is good at processing point singularities
in images.
Figure 3.6 shows some 2-D wavelet basis functions. We see that wavelet functions look
like points in the image domain.
Figure 3.6: Two-dimensional wavelet basis functions. These are 32 × 32 images. The upper
left one is a tensor product of two scaling functions. The bottom right 2 by 2 images and
the (2, 2)th image are tensor products of two wavelets. The remaining images are tensor
products of a scaling function with a wavelet.