Convolving, or filtering, an image is a CPU-intensive operation. As the convolution kernel and image size increases in size, convolution becomes more and more processor intensive.

Convolution in Coordinate Space (the image) is equivalent to multiplication in Frequency space. Using Fourier mathematics, therefore, greatly decreases the computational CPU expense required to perform the filtering operation.

Here is a page from one of my previous courses describing convolution.
Thus, the method of *Image Restoration* is to transform the image (and PSF) into Frequency Space, do the multiplication, and restore the original image information using the inverse mathematical operation.

The process uses the Fourier Transformation to generate the frequency terms; in essence, deconstructing a spatial image into infinite component frequencies. In a digital image, spatial (image) features can be faithfully represented as a series of sine and cosine functions, and mapped in frequency space (using polar coordinates). This page explains the concept.

Fourier transformation in microscopy is used to restore voxel intensity information from a blurry Z-series. This method of microscopy is called **Restoration or Deconvolution Microscopy**.

Fluorescence image of a PI-stained onion epidermal peel

Fourier transform of the onion image