Identification of Kernels in a Convolutional Neural Network: Connections Between Level Set Equation and Deep Learning...

Area of Science:

• Medical imaging
• Computational mathematics
• Artificial intelligence

Background:

• Medical image segmentation, particularly for liver segmentation in CT scans, is a manual and time-intensive process crucial for hepatocellular carcinoma treatment planning.
• Existing segmentation methods like level set methods and convolutional neural networks (CNNs) utilize convolutions and nonlinearities for feature extraction.
• A theoretical link exists between CNNs using specific convolution kernels and the numerical analysis of differential equations.

Purpose of the Study:

• To investigate the relationship between CNNs and differential equation solvers by constructing a level set network.
• To analyze whether standard CNNs learn finite difference stencils during training for image segmentation.
• To compare the segmentation accuracy of a level set network with traditional level set equation solving and standard CNN architectures.

Main Methods:

• Developed a level set network, a CNN architecture designed to mirror the expansion of level sets.
• Treated forward propagation in the level set network as solving the level set equation.
• Analyzed learned convolution kernels in standard CNNs to identify patterns related to finite difference stencils.

Main Results:

• The level set network achieved segmentation accuracy comparable to solving the level set equation directly.
• The level set network did not reach the segmentation accuracy of a common CNN architecture.
• Analysis of standard CNNs revealed learned kernels that deviated from those typically used in level set equation solvers, particularly in deeper layers.

Conclusions:

• Establishing an explicit relationship between CNNs and numerical analysis provides a framework for analyzing CNN stability and accuracy.
• While CNNs can perform image segmentation, standard architectures may learn kernels that differ from those optimized for level set methods.
• Further research into learned CNN kernels is needed to understand their departure from finite difference stencils and potential implications for segmentation accuracy.