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142 APPENDIX A. DIRECT EDGELET TRANSFORM
So the number of edgelets at scale j is
#{edgelets at scale j} =6· 2 2n−2l − 4 · 2 2n−j−l .
is
If we only consider the scale between s l and s u , l ≤ s l ≤ s u ≤ n, the number of edgelets
#{edgelets} = 6(s u − s l +1)· 2 2n−2l − 4
s u ∑
j=s l
2 2n−j−l
= 6(s u − s l +1)· 2 2n−2l − 4 · 2 2n−l (2 −(s l−1) − 2 −su ).
If s l = l +1,s u = n, the total number of edgelets is
#{edgelets} = 6(n − l) · 2 2n−2l − 4
n∑
j=l+1
2 2n−j−l
= 6(n − l) · 2 2n−2l − 4 · 2 2n−l (2 −l − 2 −n )
= 6(n − l) · 2 2n−2l − 2 2n−2l+2 +2 n−l+2
= 6(n − l)
(N/2 l) 2
− 4
(N/2 l) 2 (
+4 N/2 l) .
A.3.3
Ordering
In this subsection, we discuss how to order the edgelet coefficients. The ordering has three
layers:
1. order the scales;
2. within a scale, order dyadic squares;
3. within a dyadic square, order edgelets.
There is a natural way to order scales. The scale could go from the lowest s l to the highest
s u , s l ≤ s u .
In the next two subsubsections, we describe how we order dyadic squares within a scale
and how we order edgelets within a dyadic square.