6.1. van-Cittert iterations

First, we take a look at the Van-Citter-Algorithm for Deconvolution to get a better feeling for the task at Hand.

Example 6.1 (Van-Citter-Algorithm for Deconvolution)

As we have an image \(f\) which is convoluted we can look at the following approach:

\[f \quad = \quad \phi * u \quad \Leftrightarrow \quad \hat{f} \quad = \quad \hat{\phi} \cdot \hat{u},\]

with \(\hat{\phi}\) being the Filteroperator, which can be displayed as the convolution of the image \(u\) with \(\phi\). This mean the goal of deconvolution, computing the deconvoluted image \(g\), in this scenario can be described as:

\[\hat{u} \quad = \quad \hat{\phi}^{-1}\cdot \hat{f}.\]

This yields to be problematic if \(\hat{\phi} \approx 0\), which can be avoided via the auxiliary function \(\hat{q} := 1 - \hat{\phi}\), which yields \(\hat{\phi} = 1-\hat{q}\). To simplify our with another auxiliary function \(g(\hat{q}) := \frac{1}{1-\hat{q}} = \frac{1}{\hat{\phi}}\). This makes sense because \(\hat{u} = g(\hat{q}) \cdot \hat{f}\). Since \(f\) is already given we just need to approximate \(\hat{\phi}\) via \(g(\hat{q})\) and what better way to approximate a function is there than with Taylor Approximation. We approximate in \(0\):

\[\begin{aligned} g'(\hat{q}) &= \left(\frac{1}{1 - \hat{q}} \right) = \frac{1}{(1 - \hat{q})^2},, \\ g''(\hat{q}) &= \left(\frac{1}{(1 - \hat{q})^2} \right) = \frac{2}{(1 - \hat{q})^3} \\ &\vdots \\ g^{(n)}(\hat{q}) &= \frac{n}{(1 - \hat{q})^n}. \end{aligned}\]

This gives us a very simple Taylor series for \(g(\hat{q})\):

\[T_g(\hat{q};0) = \sum_{n=0}^{+ \infty} \hat{q}^n .\]

In total this gives us the approximation scheme:

\[\hat{u} = \frac{\hat{f}}{\hat{\phi}} = \frac{\hat{f}}{1 - \hat{q}} = g(\hat{q})\hat{f} \approx (1 + \hat{q} + \hat{q}^ 2 + \hat{q}^3 \dots) \cdot f\]

With \(f = \phi * u = \Leftrightarrow \hat{f} = \hat{\phi} \cdot \hat{u}\) this leaves us with the iterative scheme of the Van-Citter-Algorithm:

\[\begin{aligned} u_0 &= f, \\ u_1 &= f + q*u_0 = f + q*f = (Id + q)*f, \\ u_2 &= f + q*u_1 = f + q*(f + q*f) = f + q*f + q^2*f = (Id + q + q^2)*f, \\ &\vdots \\ u_{n} &= (Id + q + q^2 + \dots q^{n})*f. \end{aligned}\]

So the Van-Citter-Algorithm can compactly be defined as:

\[\begin{aligned} u_0 &= f, \\ u_{n} &= (Id + q + q^2 + \dots q^{n})*f. \end{aligned}\]