sparse image representation via combined transforms - Convex ...

62 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES
of the signal (namely N), O(log N) can be treated as a constant. Hence, we say that the
wavelet transform is good at processing point singularities.
Figure 3.5 illustrates a multiresolution analysis of a time singularity. The top left curve
is a function that contains a time singularity—it is generated by a Laplacian function. The
other plots in the left column show its decompositions into the coarsest scale subspace
V 0 (made by dilation and translation of a scaling function) and the wavelet subspace W 0
through W 5 . The right column contains illustrations of the DWT coefficients corresponding
to different functional subspaces at different scales. Note that at every scale, there are only
a few significant coefficients. Compared with the length of the signal (which is N), the total
number of the significant DWT coefficients (O(log N)) is relatively small.
signal
Wavelet Coefficients
V 0
W 0
W 1
W 2
W 3
W 4
W 5
Figure 3.5: Multiresolution analysis of a point singularity with Haar wavelets.