Frequency Domain Transformation
Frequency domain transformation operator, when applied to images, decompose them from gray-level of the spatial domain to the components in fundamental frequencies in frequencies domain
Discrete Fourier Transform (DFT)
The DFT, when applied to image $f(k,l)$ with finite elements $N × M$, it provides coefficient of transform $F(u,v)$ of dimension $N × M$ as below :
\[F(u,v) = \sum\limits_{k=0}^{N-1}\sum\limits_{l=0}^{M-1}\, f(k,l)\,B(k,l;\,u,v)\]where \(B(k,l;u,v)\) indicates the images forming the base of the frequency space identified by the \(u-v\) system, each with dimension \(k × l\) Therfore, the transformation process quantifies the decomposition of the input image \(f(k,l)\) in the weighted sum of the base images, where the coefficients \(F(u,v)\) are precisely the weights.
- The values of the frequencies near the origin of the system \((u,v) →\) low frequencies.
- The value farthest from the origin \(\to\) high frequencies
The input image \(f(k,l)\) can be reconstructed in the spatial domain through the coefficients of the transform \(F(u,v)\) with the equation of the inverse Fourier transform, i.e,
\[f(k,l) = F^{-1}(F(u,v)) =\sum\limits_{u=0}^{N-1}\sum\limits_{v=0}^{M-1} F(u,v)B^{-1}(k,l;\,u,v)\]- When the basic images of transformation are represented by sine and cosine functions, the transformation of image \(f(k,l)\) is given by :
where function \(F(u,v)\) represents the frequency content of the image \(f(k,l)\), which is complex and periodic in both \(u\) and \(v\) with period \(2\pi\). Here, cosine represents the real part and sine is the complex part, thus the general expression can be written as :
\[F(u,v) = R_{e}(u,v) + jI_{m}(u,v)\]DFT - Magnitude, Phase Angle and Power Spectrum
However, the real and imaginary components are not effective representation, the more effective representation of coefficient \(F(u,v)\) is through magnitude \(\lvert F(u,v) \rvert\) and Phase Angle \(Φ (u,v)\) as :
\[\lvert F(u,v) \rvert = \sqrt{R_{e}^2\,(u,v) + I_{m}^2\,(u,v)}\] \[Φ (u,v) = tan^{-1}\left[\frac{I_{m}(u,v)}{R_{e}(u,v)}\right]\]Then, Fourier Transform, in terms of its magnitude and phase can be written as :
\[F(u,v) = R_{e}(u,v) + jI_{m}(u,v) = \lvert F(u,v) \rvert e^{j Φ (u,v)}\]and power spectrum or spectral density \(P(u,v)\) of an image is defined as :
\[P(u,v) = \lvert F(u,v) \rvert^2 = R_{e}^2(u,v) + I_{m}^2(u,v)\]Fourier Transform when applied to the trigonometric form, the Euler relation \(e^{jx} = \text{cos} x + j \text{sin} x\) becomes
\[F(u,v) = \frac{1}{\sqrt{NM}}\sum\limits_{k=0}^{N-1}\sum\limits_{l=0}^{M-1} f(k,l) ⋅ e^{-2π j\left[u\frac{k}{N} + v\frac{l}{M}\right]}\]Similarly, the Inverse DFT, then becomes :
\[f(k,l) = \frac{1}{NM}\sum\limits_{u=0}^{N-1} \sum\limits_{v=0}^{M-1} F(u,v) e^{2 π j\left[k\frac{u}{N} + l\frac{v}{M}\right]}\]