sparse image representation via combined transforms - Convex ...

168 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
So ρ is
Note in this case,
∫
ρ(x) =
ˆρ(ξ)e 2π√−1ξx dξ = −
sin(πx)
2πx(x 2 − 1) .
ρ(0) = 1,ρ(1) = ρ(−1) = 1 , and ρ(i) =0,i= ±2, ±3,... .
2
Figure B.8 shows the raised cosine function and the corresponding ρ in the time
domain.
(a) ρ
2.5
(b) ρ_hat is raised cosine
1
0.8
0.6
0.4
0.2
0
−0.2
2
1.5
1
0.5
0
−0.5
−0.4
−5 0 5
x
−1
−2 0 2
ξ
Figure B.8: Raised cosine function in (b) and its counterpart in time domain in (a).
3. In the first case, there is no tapering. In the second case, the tapering rate is 100%. It
is interesting to look at something in between. Suppose p (0