sparse image representation via combined transforms - Convex ...

A.3. DETAILS 141
(a) Lenna image
(b) Filtered lenna image
(c) Edgelet trans. coeff.(best−5000/428032)
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(d) Largest−1000 coeff.(log scale)
(e) Largest−2000 coeff.
(f) Largest−4000 coeff.
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Figure A.4: Edgelet transform of Lenna image: (a) is the original Lenna image; (b) is the
filtered version; (c) is the sorted largest 5, 000 coefficients out of 428032. (d), (e) and (f)
are the reconstructions based on the largest 1000, 2000 and 4000 coefficients, respectively.
A.3.2
Cardinality
In this subsection, we discuss the size of edgelet coefficients.
For fixed scale j (l ≤ j ≤ n), the number of dyadic squares is
N/2 j × N/2 j =2 n−j × 2 n−j =2 2n−2j .
In each 2 j × 2 j dyadic square, every side has 1 + 2 j−l vertices; hence the total number of
edgelets is
1
{
}
4 · (2 · 2 j−l − 1) + 4 · (2 j−l − 1)(3 · 2 j−l − 1) =6· 2 j−l · 2 j−l − 4 · 2 j−l .
2