Wavelet Scattering Transform

 

Fourier Vs Wavelet Transform

What Is Fourier Transform?

An idea you’ll find all over science, math and engineering is the Fourier Transform (FT). The FT decomposes a function into simple sines and cosines (i.e, waves). In theory, any function can be represented in this way — that is, as a sum of (possibly infinite) sine and cosine functions of different amplitudes and frequencies.

I’ve provided a toy example below. Here, I’ve translated the top signal from the time domain to the frequency domain. In other words, we change the x-axis from time to frequency. The way to interpret peaks in the bottom plot is that the original signal represents the sum of two simple sine waves with frequencies one and two Hz, respectively.

What Is a Wavelet Transform

A major disadvantage of the Fourier transform is it captures global frequency information, meaning frequencies that persist over an entire signal. This kind of signal decomposition may not serve all applications well, for example electrocardiography (ECG), where signals have short intervals of characteristic oscillation. An alternative approach is the wavelet transform, which decomposes a function into a set of wavelets.

 The basic idea is to compute how much of a wavelet is in a signal for a particular scale and location. For those familiar with convolutions, that’s exactly what this is. A signal is convolved with a set of wavelets at a variety of scales.

In other words, we pick a wavelet of a particular scale (like the blue wavelet below). Then, we slide this wavelet across the entire signal (i.e. vary its location), where at each time step we multiply the wavelet and signal. The product of this multiplication gives us a coefficient for that wavelet scale at that particular time step. We then increase the wavelet scale (e.g. the red and green wavelets) and repeat the process