sparse image representation via combined transforms - Convex ...

B.6. MISCELLANEOUS 169
The function ρ is
ρ(x) =
=
=
∫
∫ p
−p
ˆρ(ξ)e 2π√ −1ξx dξ
cos(2πξx)dξ +
sin(πx) + sin(2πpx)
2πx
∫ 1/2
p
[
cos(2πξx) 1+cos
1
1 − (1 − 2p) 2 x 2 .
]
2π(ξ − p)
dξ
1 − 2p
When p is 1/2, we get the sinc function in case one. When p is 0, we get the ρ function
in case two. When p = 1 4
, the Figure B.9 shows how the function ρ and function ˆρ
look.
When p is 1/4 and function ρ(·) only takes integer values, unlike the two previous
cases, the function ρ does not have finite support.
(a) ρ when p = 0.25
2.5
(b) ρ_hat is 50% taped.
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.9: Fifty percent tapered window function in (b) and its counterpart in time domain
in (a).