Image Frequency Domain Filtering

This document explores four fundamental concepts in image frequency domain processing with practical implementation approaches:

1. Fourier Transform of Images and Spectrum Centralization: The 2D Fast Fourier Transform (FFT) converts spatial domain images to frequency domain representations. The fftshift function is commonly used to center the low-frequency components by rearranging the quadrants of the Fourier transform output. In MATLAB implementations, this involves using fft2() for transformation followed by fftshift() for spectrum centering, enabling better visualization and filtering operations.

2. Rotational Property of Fourier Transform: This property states that rotating an image in the spatial domain corresponds to an equivalent rotation in the frequency domain. This characteristic is leveraged in applications like image registration and rotational invariant pattern recognition. Code implementation typically involves comparing rotated versions of images through their Fourier magnitude spectra.

3. Ideal Low-Pass Filter: This filter perfectly preserves frequencies below a specified cutoff frequency while completely eliminating higher frequencies. Implementation involves creating a circular mask in the frequency domain with radius D0 (cutoff frequency) where values inside the circle remain unchanged and outside become zero. The filtering process follows: FFT → multiply with ideal mask → inverse FFT.

4. Butterworth High-Pass Filter: A more gradual transition filter defined by its order n and cutoff frequency D0. The transfer function is H(u,v) = 1 / [1 + (D0/D(u,v))^(2n)], where D(u,v) is the distance from the frequency origin. Higher orders create steeper transitions. This filter preserves high-frequency details like edges while suppressing low-frequency content, making it ideal for image sharpening applications.

Mastering these concepts provides essential knowledge for frequency-domain image processing, with practical implementations involving FFT operations, filter design in the frequency domain, and inverse transformations to obtain filtered images.