# High-Pass Sharpening Filters and Their Frequency Domain Characteristics (Especially for GATE-Geospatial 2022)

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Simply subtracting the low-frequency image resulting from a low pass filter from the original image can enhance high spatial frequencies. High-frequency information allows us either to isolate or to amplify the local detail. If the high-frequency detail is amplified by adding back to the image some multiple of the high-frequency component extracted by the filter, then the result is a sharper, de-blurred image.

## High-Pass Convolution Filters

**High-pass convolution** filters can be designed by representing a PSF with positive centre weight and negative surrounding weights. A typical 3x3 Laplacian filter has a kernel with a high central value, 0 at each corner, and -1 at the centre of each edge. Such filters can be biased in certain directions for enhancement of edges.

High-Pass filtering can be performed simply based on the mathematical concepts of derivatives, i.e.. , gradients in DN throughout the image. Since images are not continuous functions, calculus is dispensed with and instead derivatives are estimated from the differences in the DN of adjacent pixels in the x, y or diagonal directions. Directional first differencing aims at emphasizing edges in the image.

## Frequency Domain Filters

The Fourier transform of an image, as expressed by the amplitude spectrum is a breakdown of the image into its frequency or scale components. Filtering of these components uses frequency domain filters that operate on the amplitude spectrum of an image and remove, attenuate or amplify the amplitudes in specified wavebands. The frequency domain can be represented as a 2-dimensional scatter plot known as a Fourier Spectrum, in which lower frequencies fall at the centre and progressively higher frequencies are plotted outward.

## Steps in Filtering in the Frequency Domain

Filtering in the frequency domain consists of 3 steps:

- Fourier transforms the original image and computes the Fourier spectrum
- Select an appropriate filter transfer function (equivalent to the OTF of an optical system) and multiply by the elements of the Fourier spectrum.
- Perform an inverse Fourier transform to return to the spatial domain for display purposes.