sparse image representation via combined transforms - Convex ...

4.4. LAGRANGE MULTIPLIERS 87
ρ( x )
a Convex Measure of Sparsity
1_
−
λ
^
ρ( c)
Feasible Area
Distortion
||y||
2
||y-
Σ
c
i=1,...,N
i φ i ||
2
Figure 4.1: Quasi-sparsity and distortion curve.
¯ρ is a convex 1-D function. Note this is the idea of quadratic penalty-function method in
solving the following optimization problem with exact constraints:
minimize
x
ρ(x), subject to y =Φx. (4.3)
To better explain the connections we raise here, we introduce a concept called Quasisparsity
& distortion curve. “Quasi-sparsity” refers to the small values in the quasi-sparsity
measurement ρ(x). Figure 4.1 gives a depiction. The horizontal axis is the distortion
measure ‖y − Φx‖ 2 2 . The vertical axis is a measure of quasi-sparsity, in our case ρ(x).
Allowing some abuse of the terminology “sparsity”, we call this plane a distortion-sparsity
(D-S) plane. If there exists x such that (u, v) =(‖y − Φx‖ 2 2 ,ρ(x)), then we say the point
(u, v) on the D-S plane is feasible. We know ‖y − Φx‖ 2 2 is a quadratic form. When ρ(x) isa
convex function of x, all the feasible points on the D-S plane form a convex set; we call this
convex set a feasible area. For a fixed value of distortion, there is a minimum achievable
value for ρ(x). If all the points like this form a continuous curve, we call it a Quasi-Sparsity