**2.4 Spectral feature extraction through wavelet-based PCA and ICA**

Spectral decomposition of geo-images provides a unique way for feature extraction in the frequency domain. Although the Fourier transform is a powerful tool for image decomposition, it does not represent abrupt changes efficiently due to the infinite oscillation of the periodic function in any given direction. In contrast, wavelets are localized in space and have finite durations. Therefore, the output of wavelet decomposition effectively reflects the sharp changes in images, and that makes it an ideal tool for feature extraction [10–12].

Mathematically, the 2D CWT of an image *I* (*x*, *y*) is defined as a decomposition of that image (*I*) on a translated and dilated version of a mother wavelet *ψ* (*x*, *y*). Thus, the 2D CWT coefficients are given by:

$$\mathcal{C}\_{s} = (b\_{1}, b\_{2}, a) = \frac{1}{\sqrt{|a|}} \iint (\boldsymbol{x}, \boldsymbol{y}) \boldsymbol{\mu}^{\*} \left(\frac{\boldsymbol{x} - b\_{1}}{a}, \frac{\boldsymbol{y} - b\_{2}}{a}\right) d\boldsymbol{x} d\boldsymbol{y} \tag{14}$$

Where *b*<sup>1</sup> and *b*<sup>2</sup> are controlling the spatial translation, *a* > 1 is the scale, and *ψ*\* is the complex conjugate of the mother wavelet *ψ* (*x*, *y*).

Analysis of isolated sources in potential field data with CWT was introduced by Moreau [13]. Moreau showed that the maxima lines of the CWT indicate the location of the potential field sources [10]. He also showed that the maxima lines bear the highest signal-to-noise ratios, allowing the treatment of the noisy data sets [14]. An example of a CWT on a geophysical image in which the mother wavelet is shifting and scaling in one direction is shown in **Figure 4**. The scale is inversely

**Figure 4.**

*(a) CWT on a geophysical image. CWT increases the dimensionality depending on the choice of scales (*S*) and directions (*D*). (b) CWT Mexican Hat mother wavelet in five scales (*S*1–*S*5). (c) CWT with Cauchy mother wavelet in six directions (*D*1–*D*<sup>6</sup> at* S*4).*

proportional to frequency. Large-scale factors are corresponding to largely expanded mother wavelets (low frequencies).

As can be seen in different frequencies, different features are detectable (**Figure 4b**). Scaling and shifting the mother wavelet in other directions also reveal other sets of features (**Figure 4c**). If the mother wavelet is isotropic, there is no dependence on the angle in the CWT. The Mexican Hat mother wavelet used in **Figure 4b** is an example of isotropic wavelets.

On the other hand, an anisotropic mother wavelet is dependent on the angle in the analysis; therefore, the CWT acts as a local filter for an image in scale, position, and angle. The Cauchy wavelet is an example of an anisotropic wavelet (**Figure 4c**). To better see the effect of directional transformation, we can use different anisotropic mother wavelets and compare the results.
