2.1 Algorithm description

The algorithm consists of blurring the measured image with a regularizing smoothing function . There are several possible choices of the function , however this project isn't concerned with finding an optimal ; we will use the simple unitary Gaussian function in three dimensions²

$$r(\vec{x}) = e^{-\left\vert \vec{x}\right\vert ^{2}}$$ (6)

$$= e^{-x^{2}-y^{2}-z^{2}}$$

We now obtain the smoothed, measured image

$$R = i\otimes r$$

$$= ((i_{0}+n_{p})\otimes o+n_{o})\otimes r$$ (7)

Here we introduce the imperfection that lowers accuracy but increases speed, and it is to assume that all the noise was brought in before the convolution with (specifically the instrument noise term). Based on this assumption, Equation 5 is modified to become

$$i=(i_{0}+n_{p}+n_{o})\otimes o$$ (8)

Equation 7 now becomes

$$R = (i_{0}+n_{p}+n_{o})\otimes o\otimes r$$ (9)

$$= i\otimes r$$

Observing that this is simply the measured image after being smoothed. The actual method used to calculate the smoothed image will be examined in Section 2.1.1. We now obtain an approximation to the true image by deconvoluting Equation 9 by the smoothed optical transfer function

$$\mathbf{F}(i_{0})\cong \frac{\mathbf{F}(R)}{\mathbf{F}(o\otimes r)}$$ (10)

Now we have the problem that Equation 10 becomes unreliable when becomes small. The sophisticated way to avoid this problem is to use an averaging filter. A suitable averaging term is for an arbitrarily suitable weighting term. is the sum of the neighbourhood in , where usually is taken as the 4-element neighbourhood of and . The averaging procedure is then

$$\mathbf{F}'(i_{0}) = (1-\alpha )\frac{\mathbf{F}(R)}{\mathbf{F}(o\otimes r)}+\frac{\alpha }{4}\sum \mathbf{F}_{\sigma }(i_{0})\textrm{ for }\mathbf{F}(o\otimes r)\neq 0$$ (11)

$$\mathbf{F}'(i_{0}) = w_{0}\cdot \mathbf{F}(R)+\frac{1}{4}\sum \mathbf{F}_{\sigma }(i_{0})\textrm{ otherwise}$$

However, Equation 11 is only defined when exists, specifically, . So we sum the averaged elements instead, giving the recurrence relation

$$\mathbf{F}'(i_{0}) = (1-\alpha )\frac{\mathbf{F}(R)}{\mathbf{F}(o\otimes r)}+\frac{\al... ...}{4}\sum \mathbf{F}_{\sigma }'(i_{0})\textrm{ for }\mathbf{F}(o\otimes r)\neq 0$$ (12)

$$\mathbf{F}'(i_{0}) = w_{0}\cdot \mathbf{F}(R)+\frac{1}{4}\sum \mathbf{F}_{\sigma }'(i_{0})\textrm{ otherwise}$$

This definition requires a recursive method to be used, such as flood-filling.

In order to simplify things somewhat, we used the observation that the most common pattern under which is for thin, two dimensional spherical shells³. Thus, the flood-filling technique is excessive and would rarely come into play in normal images. Our solution will sacrifice some image quality in order to decrease the complexity of this stage of the algorithm.

We adopt basically the same strategy as above, except that we define Equation 11 to be valid for , for some suitably close to 0. We then process the image iteratively, not recursively, and if a position with is encountered we simply continue by assigning and continuing as in Equation 11.