Frequency analysis

Some characteristics of texture in an image are not easily identifiable from the spatial domain. This includes the major repetitive components within the image. The need to distinguish these components leads to the mathematical representation of the image in terms of their orthogonal basis functions. The Fourier transform, for example, uses an infinite series of trigonometric functions at different frequencies. In the Fourier space image, each point represents a particular spatial frequency contained in the real domain image. These spatial frequency contents can reflect the texture of an image. For example, high spatial frequencies correspond to the presence of the sharp transitions in the intensity whereas low frequencies correspond to the absence of edges or regions of approximately uniform grey levels. The orientation of a spatial frequency corresponds to the orientation of an edge in the image. That is, the frequency contents of the image can be a measure of the coarseness and the directionality of textures. The Fourier transform of a signal f(t) is defined as

F(ω) =

∞ −∞

f(t) exp(−ωtˆ )dt, (3.5)

where ˆ=√−1 and ω is the frequency of the transform.

The Discrete Fourier Transform (DFT) is the sampled Fourier Transform and there- fore does not contain all frequencies forming an image, but only a set of samples which is large enough to fully describe the real domain image. The number of frequencies corresponds to the number of pixels in the real domain image; the image in the real and Fourier space are of the same size. For a square image of size N ×N, the two- dimensional DFT is given by F(k, l) = 1 N2 N−1 y=0 N−1 x=0 I(x, y) exp −ˆ2π kx N + ly N , (3.6)

where I(x, y) is the real domain image and the exponential term is the basis function to decompose the image into its sine and cosine components. The Fourier transform of each spatial frequency ω = (k, l) is obtained by a process of multiplying the image with the corresponding basis function and summing the result. By using the same basis function, the real domain image can be derived from the Fourier image. The

3.2. TEXTURAL FEATURES 80 inverse Fourier Transform is given by

I(x, y) = 1 N2 N−1 k=0 N−1 l=0 F(k, l) exp ˆ 2π kx N + ly N . (3.7)

The Fourier transform has the advantage of shift invariance. Strictly speaking, the magnitude of the Fourier transform is the same no matter what portion of the texture is considered. Changing the position only changes the phase of the transform. Figure 3.32 shows the Fourier image of the rain radar and the satellite images for the four type events in logarithmic scale. The Fourier images are shifted in such a way that the

F(0,0) is displayed in the centre of the images. From the figure, it is noticeable that there are cross-shaped ripples and rectangular grids that overshadow other frequency responses.

IR WV VIS Rain

(a)

IR WV VIS Rain

(b)

Figure 3.32: (a) Amplitudes and (b) phases of the Fourier transforms of the 4 events. Each row represents each event: (from top to bottom) convective, stratiform, at vor- tex’s centre and frontal.

The cross-shaped ringing artifacts, known as the Gibbs phenomenon, are a conse- quence of treating an image as if it is part of a periodically replicated array of identical images extending horizontally and vertically to infinity. The discontinuity at edges of each virtually-tiling image aggregates frequency responses along the x and y axes of the Fourier image and therefore governs other features (Duhamel and Vetterli, 1990). These edge effects can be significantly reduced by windowing the image with a func- tion that slowly tapers off at the edge such as a Gaussian function shown in Figure

3.2. TEXTURAL FEATURES 81 3.33. In Figure 3.34, the Fourier transform of the windowed image can significantly reduce the white cross in the middle of the images.

On the other hand, the rectangular grids in the transformed images are a conse- quence of interpolating the satellite images to align with the finer resolution of the rain radar images. The effect is illustrated in a sequence of the images in Figure 3.34. From left to right, the images becomes smoother and their Fourier transforms produce more black rectangular grids. The grids are quite obvious in Figure 3.34 (c) when the white crosses that dominate the transforms are removed by the Gaussian windows.

A A 0 120 0 0.5 1 Pixels

Figure 3.33: A Gaussian window for masking and image and its cross-section.

(a) Input images

(b) Without windowing

(c) With windowing

Figure 3.34: Fourier transforms of the smoothed images before and after using the Gaussian window.

3.2. TEXTURAL FEATURES 82 The Fourier transforms when a Gaussian window is applied to the satellite and rain radar images are shown in Figure 3.35. The value at zero frequency F(0,0) at the centre of the images are by far the largest components. Using the Gaussian window can recover the direction of edges in the rain radar images. However, such a direction could not be observed from the satellite images. Therefore, Sobel’s filter previously introduced is applied to the frontal cloud image prior to the Fourier transform to enhance structural information. The Fourier transforms of the filtered images and their corresponding principal components are shown in Figure 3.36. The principal components are defined by 20 pixels with the highest Fourier transforms. These pixels are represented as white in the lower panel of the figure. The illustration shows that the IR and VIS images can well extract the principal directions of the cloud.

In addition to the derived directions of the principal components, energy and en- tropy of the distribution in the Fourier space can be used as a rotation-invariant signature of texture characteristics.

IR WV VIS Rain

Figure 3.35: Amplitude of the windowed Fourier transform of the 4 events. Each row represents each event: (from top to bottom) convective, stratiform, at vortex’s centre and frontal.

3.2. TEXTURAL FEATURES 83

IR WV VIS Rain

Figure 3.36: Amplitude (upper) and principal directions (lower) of the windowed Fourier transform after applying Sobel’s filter to the frontal cloud.