DFT, Transform Pairs, Action of Filters on Images, Magnitude and Phase (3.15 - 3.2.3)

What is the Discrete Fourier Transform (DFT) for images?

The DFT is a computational method for analyzing discrete images in the frequency domain. For an image of size M \times N, the forward DFT is given by:

F[u, v] = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} f[x, y] e^{-j2\pi\left(\frac{ux}{M} + \frac{vy}{N}\right)}

The corresponding Inverse DFT (IDFT) reconstructs the spatial image:

f[x, y] = \frac{1}{MN} \sum_{u=0}^{M-1} \sum_{v=0}^{N-1} F[u, v] e^{j2\pi\left(\frac{ux}{M} + \frac{vy}{N}\right)}

These calculations are typically implemented using the Fast Fourier Transform (FFT) for efficiency.

What is the Fourier Transform of a 1D rectangular impulse?

A 1D rectangular impulse, f(x) = 1 for |x| < \frac{X}{2} and f(x) = 0 elsewhere, transforms to the frequency domain as:

F(u) = X \cdot \text{sinc}(\pi X u), \quad \text{sinc}(x) = \frac{\sin(x)}{x}

This results in a sinc function with periodic oscillations that decay away from the origin. The width of the rectangle in the image domain is inversely proportional to the width of the sinc function in the frequency domain, illustrating the inverse relationship between the two domains.

What is the Fourier Transform of a 2D rectangular impulse (cuboid in 3D space)?

A 2D rectangular impulse, with sides of length X and Y, transforms to the Fourier domain as:

F(u, v) = XY \cdot \text{sinc}(\pi X u) \cdot \text{sinc}(\pi Y v)

Here, \text{sinc}(x) = \frac{\sin(x)}{x}. The result is a 2D sinc function with ripples decaying in both the u and v directions. Frequencies are lowest at the center of the Fourier domain and increase outward.

Why does the rectangular impulse transform to a sinc function?

A rectangular impulse contains sharp edges, requiring many high-frequency components to describe it. The sinc function in the frequency domain reflects this, with energy distributed broadly across frequencies. This relationship demonstrates the principle that sharp edges in the spatial domain correspond to broad energy distributions in the frequency domain.

What is the Fourier transform of a Gaussian centered at the origin?

A Gaussian in the image domain f(r) = \frac{1}{2\pi\sigma^2} e^{-\frac{r^2}{2\sigma^2}}, where r^2 = x^2 + y^2, transforms into another Gaussian in the Fourier domain F(u, v) = F(\rho) = e^{-2\pi^2\rho^2\sigma^2}, where \rho^2 = u^2 + v^2. The relationship between the two follows an inverse dependency on spread: as the standard deviation increases in one domain, it decreases in the other.

What is the Fourier transform of a circular disk centered at the origin?

A circular disk in the image domain with radius a is defined as

f(x, y) = \begin{cases} 1 & |r| < a \\ 0 & |r| \geq a \end{cases}.

Its Fourier transform is F(u, v) = F(\rho) = \frac{a J_1(\pi a \rho)}{\rho}, where J_1 is the Bessel function of the first kind. The Fourier domain representation exhibits a 2D rotational symmetry similar to a sinc function.

How does the Fourier transform describe the action of filters on real images?

The Fourier transform shows that low-pass filters suppress high-frequency components, leaving smooth content, while high-pass filters suppress low-frequency components, highlighting edges and details. This behavior can be seen in Fig. 3.10, where filtering operations zero out specific frequency bands and transform them back into the spatial domain for effect.

How is Fourier filtering used in forensic fingerprint analysis?

Fourier filtering is applied to remove distracting periodic patterns, such as fabric textures, from fingerprint impressions. In the Fourier domain, these patterns appear as periodic peaks. By filtering out these peaks, the texture is suppressed, enhancing the visibility of the fingerprint for analysis.

What is the role of the Fourier transform in image stitching?

In image stitching, the Fourier transform helps to suppress repetitive artifacts like vignetting from overlapping image sections. For example, in lunar survey images with vertical rippling, the Fourier domain shows vertical frequency patterns. By suppressing these patterns, seamless transitions between stitched sections are achieved.

How does the Fourier domain identify repetitive patterns in images?

Repetitive patterns in an image, such as fabric textures or vignetting, manifest as periodic peaks in the Fourier domain. These peaks correspond to specific spatial frequencies and can be selectively removed to suppress the patterns in the spatial domain.

What is the significance of the Fourier magnitude and phase in image reconstruction?

The magnitude |F(u, v)| represents the strength of spatial frequencies, while the phase \angle F(u, v) encodes the spatial structure and positional information. Although magnitude intuitively indicates frequency content, the phase carries critical information for reconstructing recognizable images.

What happens when the Fourier phase of one image is combined with the magnitude of another?

When the magnitude of one image and the phase of another are swapped, the reconstructed image retains the structural features of the image from which the phase was taken, demonstrating the dominant role of phase in preserving perceptual information.

How does Fourier phase affect image details?

The phase determines the arrangement and relationships between pixel intensities. If the phase is randomized or altered, the reconstructed image loses its recognizable structure, even if the magnitude is preserved.