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1.1 Theory

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

o(-\Delta \vec{x})
\end{displaymath} (1)

This represents the signal contribution of a point located an offset \( \Delta \vec{x} \) from the point the system is measuring. For a perfect system, \( o(-\Delta \vec{x})=\delta (\vec{x}) \), where \( \delta (\vec{x}) \) is the Dirac-Delta function. However, for real systems of finite resolution, \( o(-\Delta \vec{x}) \) can be measured experimentally by recording the image \( i(\vec{x}) \) of a very small, point-like source, acting essentially as a baseline signal. The OTF is then given by \( o(-\Delta \vec{x})=i(\vec{x}_{0}+\Delta \vec{x}) \) where \( \vec{x}_{0} \) 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 \( -\Delta \vec{x} \) because this means that any measured image \( i(\vec{x}) \) derives from the original (``true'') image \( i_{0}(\vec{x}) \) by convolution with the optical transfer function.

i(\vec{x})=i_{0}(\vec{x})\otimes o(-\Delta \vec{x})
\end{displaymath} (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)
\end{displaymath} (3)

where the boldface \( \mathbf{F} \) indicates a Fourier transform and the Fourier transform of the optical transfer function \( \mathbf{F}(o) \) is sometimes referred to as the point spread function or psf. Quantitatively sharpening the image \( i(\vec{x}) \) to obtain \( i_{0}(\vec{x}) \) involves the deconvolution of equation 2, which now becomes the inversion of equation 3
\end{displaymath} (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 \( n_{p} \) due to a photon \( p \) and the optical instrument noise \( n_{o} \) and obtain

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

for the collected noise term \( n=n_{p}\otimes +n_{o} \). However, \( n \) cannot be known and so calculating Equation 5 is no longer possible. Further, considering small \( \mathbf{F}(i) \) in Equation 4 we see that neglecting \( n \) will introduce major disturbances and artifacts in the restored image.

next up previous
Next: 1.1.1 Exact image enhancement Up: 1 Introduction Previous: 1 Introduction
Kevin Pulo