Identification of kernels in a convolutional neural network: Connections between level set equation and deep learning for image segmentation

Medical image segmentation remains a difficult, time-consuming task; currently, liver segmentation from abdominal CT scans is often done by hand, requiring too much time to construct patient-specific treatment models for hepatocellular carcinoma. Image segmentation techniques, such as level set methods and convolutional neural networks (CNN), rely on a series of convolutions and nonlinearities to construct image features: neural networks that use strictly mean-zero finite difference stencils as convolution kernels can be treated as upwind discretizations of differential equations. If this relationship can be made explicit, one gains the ability to analyze CNN using the language of numerical analysis, thereby providing a well-established framework for proving properties such as stability and approximation accuracy. We test this relationship by constructing a level set network, a type of CNN whose architecture describes the expansion of level sets; forward-propagation through a level set network is equivalent to solving the level set equation; the level set network achieves comparable segmentation accuracy to solving the level set equation, while not obtaining the accuracy of a common CNN architecture. We therefore analyze which convolution filters are present in a standard CNN, to see whether finite difference stencils are learned during training; we observe certain patterns that form at certain layers in the network, where the learned CNN kernels depart from known convolution kernels used to solve the level set equation.