# Fourier Transforms in GIS: Spatial Frequencies in Fourier Transformation & Spatial Processing (Especially for GATE-Geospatial 2022)

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The Fourier Transform operates on a single band image. Its purpose is to break down the image into its scale components, which are defined to be sinusoidal waves with varying amplitudes, frequencies and directions. The coordinates of two-dimensional space are expressed in terms of frequency (cycles per basic interval) .

## Illustration of Fourier Transform

The function of the Fourier Transform is to convert a single-band image from its spatial domain representation to the equivalent frequency-domain representation and vice-versa.

- The dotted line is equal to the sum of the three solid lines.
- Spatial Frequency is defined as 1/ (periodic pattern) Units are 1/ (size of a pixel)
- The lowest spatial frequency is 0
- While the amplitude of the f = 0 is the average for the image

## Spatial Frequencies in Fourier Transformation

The idea underlying the Fourier Transform is that the grey-scale value forming a single band image can be viewed as a three-dimensional intensity surface, with the rows and columns defining two axes and the grey-level value at each pixel giving the third (z) dimension. The Fourier Transform thus provides details of

- The frequency of each of the scale components of the image
- The proportion of information associated with each frequency component

## Spatial Processing

**Spatial Filtering**: Spatial Filtering can be described as selectively emphasizing or suppressing information at different spatial scales over an image. Filtering techniques can be implemented through the Fourier transform in the frequency domain or in the spatial domain by convolution.

**Convolution Filters**: Filtering methods exists is based upon the transformation of the image into its scale or spatial frequency components using the Fourier transform. The spatial domain filters or the convolution filters are generally classed as either high-pass (sharpening) or as low-pass (smoothing) filters.

**Low-Pass (Smoothing) Filters**: Low-pass filters reveal underlying two-dimensional waveform with a long wavelength or low-frequency image contrast at the expense of higher spatial frequencies. Low-frequency information allows the identification of the background pattern and produces an output image in which the detail has been smoothed or removed from the original.

A 2-dimensional moving-average filter is defined in terms of its dimensions which must be odd, positive and integral but not necessarily equal, and its coefficients. The output DN is found by dividing the sum of the products of corresponding convolution kernel and image elements often divided by the number of kernel elements. A similar effect is given from a median filter where the convolution kernel is a description of the PSF weights. Choosing the median value from the moving window does a better job of suppressing noise and preserving edges than the mean filter.

Adaptive filters have kernel coefficients calculated for each window position based on the mean and variance of the original DN in the underlying image.