sparse image representation via combined transforms - Convex ...

4.5. HOW TO CHOOSE ρ AND λ 89
3
(a) Original
1.5
(b) First Derivative
2.5
1
2
0.5
ρ(x)
1.5
ρ’(x)
0
1
−0.5
0.5
−1
0
−3 −2 −1 0 1 2 3
x
−1.5
−3 −2 −1 0 1 2 3
x
6
(c) Second Derivative
5
4
ρ’’(x)
3
2
1
0
−3 −2 −1 0 1 2 3
x
Figure 4.2: Function ¯ρ (as ρ in figures) and its first and second derivatives. ¯ρ, ¯ρ ′ and ¯ρ ′′
are solid curves. The dashed curve in figure (a) is the absolute value. The dashed curve in
figure (b) is the signum function.
and
∂ 2
∂ 2 x ¯ρ(x, γ) =γe−γ|x| . (4.5)
It is easy to verify that ¯ρ is C 2 .
Figure 4.2 shows for fixed γ, the function ¯ρ and its first and second derivatives. Figure
4.3 shows the function ¯ρ at a neighborhood of the origin with different values of γ.
For fixed γ and function ¯ρ, the following result tells us how to choose λ.
Proposition 4.1 If ̂x is the solution to the problem in (4.2), then we have
‖Φ T (y − Φ̂x)‖ ∞ ≤ λ 2 . (4.6)