The ICM-DCT reconstruction algorithm

Previous : The MCMCML estimation algorithm Next : Estimation results

The ICM-DCT reconstruction algorithm

This deterministic algorithm is based on the posterior modified Geman & Yang sampler, whose random part is suppressed.

We use the same half-quadratic form of :

auxiliary variable b, auxiliary images B^x and B^y (gradients w.r.t. x and y)
alternated minimization w.r.t. X and B^x, B^y

The criterion to optimize can be rewritten as follows :

with the new regularization term :

The ICM-DCT reconstruction algorithm

Initialization :

Let us first calculate the Fourier transform F[h⁴ ] and DCT[Y], and W : The initialization is done using X⁰= Y (observed image).

Minimization of the criterion w.r.t. Bx,By :

The pixels b of B^x and B^y are independent :
et

Minimisation of the criterion w.r.t. X :

The criterion is a quadratic form w.r.t. X :
$\rightarrow$ diagonalization by a cosine transform (DCT)
which enables to fulfill symmetric boundary conditions

Example (64x64 image) :

X⁰ initialization X⁰ = Y

X $\rightarrow$ B

B^{x 0}, B^{y 0}

B $\rightarrow$ X iteration 1

X¹

X $\rightarrow$ B

B^{x 1}, B^{y 1}

B $\rightarrow$ X iteration 2

X²

etc. N iterations before convergence usually N<10

Example : 64x64 synthetic image = 0.25 = 5		Example : 64x64 image = 0.45 = 10
0 1 2 3 4 5 6 7 8 9 10		0 1 2 3 4 5 6 7 8

André Jalobeanu - 24 Aug 1998

X⁰		initialization X⁰ = Y
X $\rightarrow$ B
B^{x 0}, B^{y 0}
B $\rightarrow$ X		iteration 1
X¹
X $\rightarrow$ B
B^{x 1}, B^{y 1}
B $\rightarrow$ X		iteration 2
X²
	etc. N iterations before convergence usually N<10