2.1.1 On smoothing

In the calculations detailed above (Equations 6 to 10), was obtained by convoluting the measured image by the smoothing function , with the implication that it would be calculated by a Fourier transform and subsequent point-wise multiplication of the two terms. However, if this were the case we would have the subtle problem of Equation 10 being

$$\begin{eqnarray*} \mathbf{F}(i_{0}) & \cong & \frac{\mathbf{F}(R)}{\mathbf{F}(o\... ...dot \mathbf{F}(r)}\\ & = & \frac{\mathbf{F}(i)}{\mathbf{F}(o)} \end{eqnarray*}$$

That is, it has reduced back to Equation 4, where the smoothing function is not needed.

However, the reason for 's existence is that we can smooth in real space, not Fourier space. This is achieved using a mask operation on , for example, using the convolution matrix of

$$\left[ \begin{array}{ccc} 1 & 1 & 1\\ 1 & 5 & 1\\ 1 & 1 & 1 \end{array}\right]$$

and subsequently scaling the pixel values by , or, for a more aggressive smoothing, the convolution matrix of

$$\left[ \begin{array}{ccccc} 1 & 1 & 1 & 1 & 1\\ 1 & 5 & 5 &... ...1\\ 1 & 5 & 5 & 5 & 1\\ 1 & 1 & 1 & 1 & 1 \end{array}\right]$$

scaling this time by .

Again, this procedure will decrease the accuracy somewhat, but it will increase the speed of the algorithm substantially as this method is localized for each point, requiring only a constant amount of computational time per point, which is faster than any Fourier transform (refer to Section 4.1 for a complete analysis of the time complexity of the algorithm).