Tikhonov's regularization and multiresolution analysis

If w_j^(I) are the wavelet coefficients of the image I at the scale j, we have:

 

$$\hat{w}_j^{(I)}(u,v) \hat{g}(2^{j-1}u, 2^{j-1}v) \prod_{i=j-2}^{i=0}\hat{h}(2^{i}u, 2^{i}v) \hat{I}(u,v)$$ =

$${\hat{\psi}(2^{j}u, 2^{j}v) \over \hat{\phi}(u,v)} \hat{P}(u,v) \hat{O}(u,v)$$ = (14.106)

$$\hat{w}_j^{(P)} \hat{O}(u,v)$$ =

where w_j^(P) are the wavelet coefficients of the PSF at the scale j. The wavelet coefficients of the image I are the product of convolution of object O by the wavelet coefficients of the PSF.

To deconvolve the image, we have to minimize for each scale j:

 

$$\parallel {\hat \psi(2^ju, 2^jv)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat w_j^{(I)}(u,v)\parallel^2$$ (14.107)

and for the plane at the lower resolution:

 

$$\parallel {\hat \phi(2^{n-1}u, 2^{n-1}v)\over \hat\phi(u, v)} \hat P(u,v) \hat O(u,v) - \hat c_{n-1}^{(I)}(u,v)\parallel^2$$ (14.108)

n being the number of planes of the wavelet transform ((n-1) wavelet coefficient planes and one plane for the image at the lower resolution). The problem has not generally a unique solution, and we need to do a regularization [40]. At each scale, we add the term:

 

$$\gamma_j \parallel w_j^{(O)} \parallel^2 \mbox{ min }$$ (14.109)

This is a smoothness constraint. We want to have the minimum information in the restored object. From equations 14.107, 14.108, 14.109, we find:

$$\hat D(u,v) \hat O(u,v) = \hat N(u,v)$$ (14.110)

with:

$$\begin{eqnarray*}\hat D(u,v) = \sum_j \mid \hat\psi(2^ju, 2^jv) \mid^2 (\mid\hat... ... \gamma_j) + \mid \hat \phi(2^{n-1}u,2^{n-1}v)\hat P(u,v) \mid^2 \end{eqnarray*}$$

and:

$$\begin{eqnarray*}\hat N(u,v) = \hat\phi(u, v) [ \sum_j \hat P^*(u,v)\hat\psi^*(2... ... \hat P^*(u,v) \hat\phi^*(2^{n-1}u,2^{n-1}v) \hat c_{n-1}^{(I)}] \end{eqnarray*}$$

if the equation is well constrained, the object can be computed by a simple division of $\hat N$ by $\hat D$. An iterative algorithm can be used to do this inversion if we want to add other constraints such as positivity. We have in fact a multiresolution Tikhonov's regularization. This method has the advantage to furnish a solution quickly, but optimal regularization parameters $\gamma_j$ cannot be found directly, and several tests are generally necessary before finding an acceptable solution. Hovewer, the method can be interesting if we need to deconvolve a big number of images with the same noise characteristics. In this case, parameters have to be determined only the first time. In a general way, we prefer to use one of the following iterative algorithms.