sparse image representation via combined transforms - Convex ...

88 CHAPTER 4. COMBINED IMAGE REPRESENTATION
& Distortion (QSD) curve. Actually, the QSD curve is the lower boundary of the feasible
area, as shown in Figure 4.1. Note in the figure, the notation ∑ i=1,... ,N c iφ i serves the same
meaning as Φx in the above text.
A noteworthy phenomenon is that for fixed λ, the corresponding (u, v) point given by
the solution of (4.2) is actually the tangent point of a straight line having slope −1/λ with
the QSD curve. The tangent is the leftmost straight line having slope −1/λ and intersecting
with the feasible area. Moreover, the QSD curve is the pointwise upper bound of all these
tangents. We can see a similar argument on the rate and distortion curve (R&D curve) in
Information Theory [10, 33].
There are two limiting cases:
1. When the distortion ‖y − Φx‖ 2 2
is zero, the QSD curve intersects with the vertical
axis. The intersection point, which has the coordinates (0,ρ(ĉ)), is associated with
the solution ĉ to the exact constraint problem as in (4.3).
2. When the measure of sparsity ρ(x) is zero, because ρ(x) is convex, nonnegative, and
symmetric about zero, we may think of x as an all-zero vector. Hence the distortion
is equal to ‖y‖ 2 . The corresponding point on the D-S plane is (‖y‖ 2 , 0), and it is the
intersection of the QSD curve with the horizontal axis.
4.5 How to Choose ρ and λ
Since ρ(x) = ∑ N
i=1 ¯ρ(x i), in order to determine ρ, we only need to determine ¯ρ. We choose
¯ρ asaconvex,l 1 -like and C 2 function. We choose ¯ρ to be convex, so that the optimization
problem has a global solution.
We choose ¯ρ to be an l 1 -like function, for the reasons
mentioned in Section 4.3. We choose ¯ρ to be C 2 so that the problem in (4.2) is tractable
by standard Newton methods. Our choice of ¯ρ is
¯ρ(x, γ) =|x| + 1 γ e−γ|x| − 1 , for γ>0,
γ
where γ is a controlling parameter. Note when γ → +∞, ¯ρ(x, γ) →|x|. The derivatives of
¯ρ have the form:
{
∂
1 − e −γx
∂x ¯ρ(x, γ) = , x ≥ 0,
−1+e γx , x ≤ 0,
(4.4)