Wavelet Scattering Transform

A wavelet scattering transform processes data in stages. The output of one stage becomes input for the next stage. Each stage consists of three operations.

The zeroth-order scattering coefficients are computed by simple averaging of the input. Here is a tree view of the algorithm:

The $$\left\{{\psi }_{j,k}\right\}$$ are wavelets, $${\varphi }_J$$ is the scaling function, and $$f$$ is the input data. In the case of image data, for each $${\psi }_{j,k}$$, there are a number of user-specified rotations of the wavelet. A sequence of edges from the root to a node is referred to as a path. The tree nodes are the scalogram coefficients. The scattering coefficients are the scalogram coefficients convolved with the scaling function $${\varphi }_J$$. The set of scattering coefficients are the low-variance features derived from the data. Convolution with the scaling function is lowpass filtering and information is lost. However, the information is recovered when computing the coefficients in the next stage.

To extract features from the data, first use waveletScattering (for time series) or waveletScattering2 (for image data) to create and configure the network. Parameters you set include the size of invariance scale, the number of filter banks, and the number of wavelets per octave in each filter bank. In waveletScattering2 you can also set the number of rotations per wavelet. To derive features from time series, use the waveletScattering object functions scatteringTransform or featureMatrix. To derive features from image data, use the waveletScattering2 object functions scatteringTransform or featureMatrix.

The scattering transform generates features in an iterative fashion. First, you convolve the data with the scaling function, $$f\ast {\varphi }_J$$ to obtain S[0], the zeroth-order scattering coefficients. Next, proceed as follows:

Repeat the process at every node.

The scatteringTransform function returns the scattering and scalogram coefficients. The featureMatrix function returns the scattering features. Both outputs can be made easily consumable by learning algorithms, as demonstrated in Wavelet Time Scattering for ECG Signal Classification or Texture Classification with Wavelet Image Scattering.

Invariance Scale

The scaling filter plays a crucial role in the wavelet scattering network. When you create a wavelet scattering network, you specify the invariance scale. The network is invariant to translations up to the invariance scale. The support of the scaling function determines the size of the invariant in time or space.

Quality Factors and Filter Banks

When creating a wavelet scattering network, in addition to the invariance scale, you also set the quality factors for the scattering filter banks. The quality factor for each filter bank is the number of wavelet filters per octave. The wavelet transform discretizes the scales using the specified number of wavelet filters.

This plot shows the wavelet filters in the network created by waveletScattering. The invariance scale is one second and sampling frequency is 200 Hz. The first filter bank has the default quality value of 8, and the second filter bank has the default quality factor of 1.

For image data, large quality factors are not necessary. Large values also result in significant computational overhead. By default waveletScattering2 creates a network with two filter banks each with a quality factor of 1. This plot shows the wavelet center frequencies for a wavelet image scattering network with two filter banks. The first filter bank has a quality factor of 2, and the second filter bank has a quality factor of 1. The number of rotations per filter bank is 6.

In Practice

With the proper choice of wavelets, the scattering transform is nonexpansive. Energy dissipates as you iterate through the network. As the order m increases, the energy of the m^th-order scalogram coefficients and scattering coefficients rapidly converges to 0 [3]. Energy dissipation has a practical benefit. You can limit the number of wavelet filter banks in the network with a minimal loss of signal energy. Published results show that the energy of the third-order scattering coefficients can fall below one percent. For most applications, a network with two wavelet filter banks is sufficient.

Consider the tree view of the wavelet time scattering network. Suppose that there are M wavelets in the first filter bank, and N wavelets in the second filter bank. The number of wavelet filters in each filter bank do not have to be large before a naive implementation becomes unfeasible. Efficient implementations take advantage of the lowpass nature of the modulus function and critically downsample the scattering and scalogram coefficients. These strategies were pioneered by Andén, Mallat, Lostanlen, and Oyallon [4] [6] [7] in order to make scattering transforms computationally practical while maintaining their ability to produce low-variance data representations for learning. By default, waveletScattering and waveletScattering2 create networks that critically downsample the coefficients.