Deep Scattering Network with Fractional Wavelet Transform

Deep convolutional neural networks (DCNNs) have recently emerged as a powerful tool to deliver breakthrough performances in various image analysis and processing applications. However, DCNNs lack a strong theoretical foundation and require massive amounts of training data. More recently, the deep scattering network (DSN), a variant of DCNNs, has been proposed to address these issues. DSNs inherit the hierarchical structure of DCNNs, but replace data-driven linear filters with predefined fixed multiscale wavelet filters, which facilitate an in-depth understanding of DCNNs and also offer the state-of-the-art performance in image classification. Unfortunately, DSNs suffer from a major drawback: they are suitable for stationary image textures but not non-stationary image textures, since 2D wavelets are intrinsically linear translation-invariant filters in the FT domain. The objective of this paper is to overcome this drawback using the fractional wavelet transform (FRWT) which can be viewed as a bank of linear translation-variant multiscale filters and thus may be well suited for non-stationary texture analysis. We first propose the fractional wavelet scattering transform (FRWST) based upon the FRWT. Then, we present a generalized structure for the DSN by cascading fractional wavelet convolutions and modulus operators. Basic properties of the generalized DSN are derived, followed by a fast implementation of the generalized DSN as well as their practical applications. The theoretical derivations are validated via computer simulations.