1.1 Theory

An optical system's finite resolution is described by its optical transfer function or OTF

$$o(-\Delta \vec{x})$$ (1)

This represents the signal contribution of a point located an offset from the point the system is measuring. For a perfect system, , where is the Dirac-Delta function. However, for real systems of finite resolution, can be measured experimentally by recording the image of a very small, point-like source, acting essentially as a baseline signal. The OTF is then given by where is the (known) position of the point-like source.

In addition, as the OTF depends only on the physical nature of the instruments, it can be analysed to give a theoretical OTF for the system. This is as good as the OTF obtained experimentally and as shown in Section 2.1.2 has several practical advantages over an experimental OTF.

Equation 1 has been defined in terms of because this means that any measured image derives from the original (``true'') image by convolution with the optical transfer function.

$$i(\vec{x})=i_{0}(\vec{x})\otimes o(-\Delta \vec{x})$$ (2)

Now when working in Fourier space, we can use the identity that convolution is equivalent to multiplication of Fourier transforms to get

$$\mathbf{F}(i)=\mathbf{F}(i_{0})\cdot \mathbf{F}(o)$$ (3)

where the boldface indicates a Fourier transform and the Fourier transform of the optical transfer function is sometimes referred to as the point spread function or psf. Quantitatively sharpening the image to obtain involves the deconvolution of equation 2, which now becomes the inversion of equation 3

$$\mathbf{F}(i_{0})=\frac{\mathbf{F}(i)}{\mathbf{F}(o)}$$ (4)

Unfortunately, real optical systems also suffer from the addition of noise by the instruments involved in the system. We modify equation 2 to account for the noise due to a photon and the optical instrument noise and obtain

$$i = (i_{0}+n_{p})\otimes o+n_{o}$$ (5)

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

$$= i_{0}\otimes o+n$$

for the collected noise term . However, cannot be known and so calculating Equation 5 is no longer possible. Further, considering small in Equation 4 we see that neglecting will introduce major disturbances and artifacts in the restored image.