sparse image representation via combined transforms - Convex ...

4.6. HOMOTOPY 91
A way to interpret the above result is that for the residual r = y − Φˆx, the maximum
amplitude of the analysis transform of the residual ‖Φ T r‖ ∞ is upper bounded by λ 2 . Hence
if λ is small enough and if Φ T is norm preserving—each column of Φ has almost the same l 2
norm—then the deviation of the reconstruction based on ̂x, Φ̂x, from the image y is upper
bounded by a small quantity (literally λ 2
) at each direction given by the columns of Φ. So
if we choose a small λ, the corresponding reconstruction cannot be much different from the
original image.
After choosing ρ and λ, we have the Hessian and gradient of the objective function (later
denoted by f(x)). For fixed vector x k =(x k 1 ,xk 2 ,... ,xk N )T ∈ R N , the gradient at x k is
⎛
g(x k )=−2Φ T y +2Φ T Φx k + λ ⎜
⎝
¯ρ ′ (x k 1 )
⎞
.
⎟
⎠ , (4.7)
¯ρ ′ (x k N )
where x k i is the i-th element of vector x k , i =1, 2,... ,N, and the Hessian is the matrix
⎛
H(x k )=2Φ T Φ+λ ⎜
⎝
¯ρ ′′ (x k 1 ) ⎞
. .. ⎟
⎠ , (4.8)
¯ρ ′′ (x k N )
where ¯ρ ′′ is the second derivative of function ¯ρ.
4.6 Homotopy
Based on the previous choice of ρ and ¯ρ, when the parameter γ goes to +∞, ρ(x, γ) =
∑ N
i=1 ¯ρ(x i,γ) goes to function ‖x‖ 1 . Note our ultimate goal is to solve the minimum l 1
norm problem (P 1 ). Considering the Lagrangian multiplier method, for a fixed λ, wesolve
minimize ‖y − Φx‖ 2
x
2 + λ‖x‖ 1 . (4.9)
When γ goes to +∞, will the solution of the problem in (4.2) converge to the solution of
the problem in (4.9)?
We prove the convergence under the following two assumptions. The first one is easy to
satisfy. The second one seems too rigorous and may not be true for many situations that we