Learning Filters for the 2D Wavelet Transform

We propose a new method for learning filters for the 2D discrete wavelet transform. We extend our previous work on the 1D wavelet transform in order to process images. We show that the 2D wavelet transform can be represented as a modified convolutional neural network (CNN). Doing so allows us to learn wavelet filters from data by gradient descent. Our learned wavelets are similar to traditional wavelets which are typically derived using Fourier methods. For filter comparison, we make use of a cosine measure under all filter rotations. The learned wavelets are able to capture the structure of the training data. Furthermore, we can generate images from our model in order to evaluate the filters. The main findings of this work is that wavelet functions can arise naturally from data, without the need for Fourier methods. Our model requires relatively few parameters compared to traditional CNNs, and is easily incorporated into neural network frameworks.