Ideal Filter
This tool consists in the realization of ideal Filters for image processing. The creation of a low pass, high pass and band pass filter will be implemented to filter the given image with each of the three filter types. Furthermore, the modules and phase of the filter's impulse response are plotted according to its power spectrum.
The implantation of an Ideal Lowpass Filter (ILPF) diminishes or eliminates the high frequencies. This is represented mathematically as the individual multiplication of the frequency components by a nonincreasing function of ω such as:
If we establish:
- D(u,v) as the Euclidean Distance from any point (u, v) to the origin of the frequency plane:
- D0 is the cutoff frequency, a positive constant.
ILPF can be represented as:
In Image Processing, ILPF makes blurring results.
.png)
.png)
The implantation of an Ideal Highpass Filter (IHPF) enhance or amplify the high frequencies. This is represented mathematically as the individual multiplication of the frequency components by an increasing function of ω such as:
IHPF can be represented as:
In Image Processing, IHPF makes sharpening results.
.png)
.png)
The implementation of an IBPF consists of, subtracting a filtered image of low pass of radio D2 from a filtered image of low pass of radio D1, where D2<D1
.png)
.png)
Ideal Filters do not produce ideal results, a sharp cutoff in the frequency domain causes ringing in the spatial domain. This is due to the uncertainty relation, a small object in space has a large frequency extent and vice-versa.
