Keywords: front-page

# DFT: Filtering in the Frequency Domain

[Laboratory]

We want to use the properties of the DFT Spectrum as a recipe to filter images by selecting which cosine waveforms (primitives) pass through the filter and which get attenuated. It is known that the inverse DFT reconstructs an original image without any loss if all primitives are used. Recall, the DFT is a weighted sum of the different primitives. We can amplify or attenuate these primitives and this process is known as filtering. Using this principle, it is possible to select which cosine components we want to utilize in a filtering process of an image. The processing element that modifies the primitive siganals is know as a filter. There are various types of filters.

Shown below is our original image (128x128)

Original image

### Low-Pass Filter

For our experiment we want to filter out all primitives greater than 8 pixels. This can be accomplished by multiplying the complex DFT of the image by a circular filter of radius 128/8 = 16, or diameter of 31. This process is shown below

a)original spectrum; b)filter spectrum; c)result of multiplication
a) | b) | c)

Taking the inverse DFT of the results yields

Low-pass result

This type of filter is called an "ideal" low-pass filter.

### High-Pass Filter

In this experiment we want to filter out all frequencies with a period less than 8 pixels. The filtering process is depicted below

a)original spectrum; b)filter spectrum; c)result of multiplication
a) | b) | c)

The inverse DFT of the result is

High-pass result

This type of filter is called an "ideal" high-pass filter.

### Band-Pass Filter

We can select cosine components with periods between 7 and 8 pixels to pass (7 >= T >= 8). This type of filter consists of a "ring" frequency filter. That is, a ring of external radius of 128/7 = 18 (diameter 35) and internal radius of 128/8 = 16 (diameter 31-2). The filtering process is shown below

a)original spectrum; b)filter spectrum; c)result of multiplication
a) | b) | c)

The inverse DFT of the result is

Band-pass result

This type of filter is know as "ideal" band pass filter.

- Lab: DFT: Filtering in the Frequency Domain