sparse image representation via combined transforms - Convex ...

92 CHAPTER 4. COMBINED IMAGE REPRESENTATION
are interested in. We suspect that the convergence is still true when the second assumption
is relaxed. We leave the analysis for future research.
Assumption 1 For given y, Φandλ, the solution to the problem in (4.9) exists and is
unique.
Since the objective function is convex and its Hessian, as in (4.8), is always positive definite,
it is easy to prove that the above assumption is true in most cases.
Assumption 2 Φ T Φ is a diagonally dominant matrix, which means that there exists a
constant ɛ>0 such that
|(Φ T Φ) ii |≥(1 + ɛ) ∑ k≠i
|(Φ T Φ) ik |,
i =1, 2,... ,N.
As we mentioned, this assumption is too rigorous in many cases. For example, if Φ is a
concatenation of the Dirac basis and the Fourier basis, this assumption does not hold.
Theorem 4.1 For fixed γ, letx(γ) denote the solution to problem (4.2). Let x ′ denote the
solution to problem (4.9). If the previous two assumptions are true, we have
lim
γ→+∞ x(γ) =x′ . (4.10)
From the above result, we can apply the following method. Starting with a small γ 1 ,we
get the solution x(γ 1 ) of problem (4.2). Next we choose γ 2 >γ 1 ,setx(γ 1 ) as the initial guess,
and apply an iterative method to find the solution (denoted by x(γ 2 )) of problem (4.2). We
repeat this process, obtaining a parameter sequence γ 1 ,γ 2 ,γ 3 ,..., and a solution sequence
x(γ 1 ),x(γ 2 ),x(γ 3 ),.... From Theorem 4.1, the sequence {x(γ i ),i =1, 2,...} converges to
the solution of problem (4.9). This method may save computing time because when γ is
small, an iterative method takes a small number of steps to converge. After finding an
approximate solution by using a small valued γ, we then take it as a starting point for an
iterative method to find a more precise solution. Some iterations may be saved in the early
stage.