The Theory And Application Of Fractional Wavelet Scattering Convolutional Network

The recent eruption in applications of deep convolutional neural networks has achieved breakthrough performance in various image analysis and processing tasks.However,deep convolutional neural networks lack strong theoretical grounding and require copious amounts of training data.More recently,the scattering convolutional network,a variation of deep convolutional neural networks,has been proposed to address these issues.Scattering convolutional networks inherit the hierarchical structure of deep convolutional neural networks,but replace data-driven linear filters with predefined fixed wavelet filters,which enable a theoretical understanding of deep convolutional neural networks and also show state-of-the-art performance in image classification.Unfortunately,scattering convolutional networks still suffer from a major drawback: they are suitable for stationary features but not for non-stationary features,since wavelets are intrinsically linear timeinvariant bandpass filters.Our objective of this paper is to overcome this drawback using the fractional wavelet transform which can be viewed as a bank of linear time-varying bandpass filters and thus may suit for non-stationary feature analysis.First,from the perspective of multi-resolution analysis,the impact of the signal profile and details obtained by the fractional wavelet analysis on the feature analysis is clarified.The former reflects the large-scale characteristics of the signal and is insensitive to external changes,which can be directly used for signal analysis,while the latter shows the small-scale characteristics of the signal,which is susceptible to interference and not conducive to signal analysis.In order to overcome the signal details that are susceptible to interference,a fractional wavelet scattering transform is proposed from the perspective of a fractional domain filter,which is defined as a series of cascades of modulus operation and fractional wavelet transform,which is also the output of the fractional wavelet scattering convolution network to be constructed.Secondly,based on the proposed fractional wavelet scattering transform,a fractional wavelet scattering convolutional network is constructed.A strict construction process is given mathematically,and the basic properties of the fractional wavelet convolutional network are analyzed,including non-expansiveness,energy conservation,translation invariance and deformation stability.These properties provide a theoretical guarantee forthe fractional order wavelet scattering convolutional network to obtain good signal analysis performance.In particular,the conventional wavelet scattering convolutional network can be regarded as a special case of the fractional wavelet scattering convolutional network.Finally,for the practical application of the fractional wavelet scattering convolutional network,the fast discrete algorithm and efficient filter implementation of the fractional wavelet scattering convolutional network are proposed.It also discusses the application of fractional wavelet scattering convolutional network in image classification.By comparing with other classification methods,it further demonstrates its superiority in non-stationary feature analysis.