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7.2. MODIFYING EDGELET DICTIONARY 131
7.2 Modifying Edgelet Dictionary
By inspection of Figure 6.2, one sees that the existing edgelet dictionary can be further
improved. Indeed, in those figures, one sees that the wavelet component of the reconstruction
is carrying a significant amount of the burden of edge representation. It seems that
the terms in our edgelet dictionary do not have a width matched to the edge width, and
that in consequence, many fine-scale wavelets are needed in the representation. In future
experiments, we would try using edgelet features that have a finer width at fine scales and
coarser width at coarse scales. This intuitive discussion matches some of the concerns in
the paper [54].
The idea in [54] is to construct a new tight frame intended for efficient representation
of 2-D objects with singularities along curves. The frame elements exhibit a range of
dyadic positions, scales and angular orientations. The useful frame elements are highly
directionally selective at fine scales. Moreover, the width of the frame elements scales with
length according to the square of length.
The frame construction combines ideas from ridgelet and wavelet analysis. One tool is
the monoscale ridgelet transform. The other tool is the use of multiresolution filter banks.
The frame coefficients are obtained by first separating the object into special multiresolution
subbands f s , s ≥ 0 by applying filtering operation ∆ s f =Ψ 2s ∗ f for s ≥ 0, where
Ψ 2s is built from frequencies in an annulus extending from radius 2 2s to 2 2s+2 . To the s-th
subband one applies the monoscale ridgelet transform at scale s. Note that the monoscale
ridgelet transform at scale index s is composed with multiresolution filtering near index 2s.
The (s, 2s) pairing makes the useful frame elements highly anisotropic at fine scales.
The frame gives rise to a near-optimal atomic decomposition of objects which have
discontinuities along a closed C 2 curve. Simple thresholding of frame coefficients gives rise
to new methods of approximation and smoothing that are highly anisotropic and provably
optimal.
We refer readers to the original paper cited at the beginning of this section for more
details.
7.3 Accelerating the Iterative Algorithm
The idea of block coordinate relaxation can be found in [124, 125]. A proof of the convergence
can be found in [135]. We happen to know an independent work in [130]. The idea