Gradient-based Filter Design for the Dual-tree Wavelet Transform

In this work we explore the task of learning filters for the dual-tree complex wavelet transform [9, 17]. This transform was introduced to address several shortcomings of the separable, real-valued wavelet transform algorithm. However, the dual-tree transform requires greater care when designing filters. The added complexity makes the transform a good candidate to replace the traditional filter derivations with learning. We demonstrate that it is possible to learn filters for the dual-tree complex wavelet transform in a similar fashion to [16, 15]. We show that very few changes to the original autoencoder framework are necessary to learn filters that overcome the limitations of the separable 2D wavelet transform. Wavelet representations have been shown to perform well on a variety of machine learning tasks. Specifically, wavelet scattering networks have shown stateof-the-art results despite the fact they use a fixed representation (in contrast to the learned representations of convolutional neural networks) [2, 11, 3, 12].