  A novel global energy model for multi-class semantic image segmentation is
proposed that admits very efficient exact inference and derivative calculations
for learning. Inference in this model is equivalent to MAP inference in a
particular kind of vector-valued Gaussian Markov random field, and ultimately
reduces to solving a linear system of linear PDEs known as a reaction-diffusion
system. Solving this system can be achieved in time scaling near-linearly in
the number of image pixels by reducing it to sequential FFTs, after a linear
change of basis. The efficiency and differentiability of the model make it
especially well-suited for integration with convolutional neural networks, even
allowing it to be used in interior, feature-generating layers and stacked
multiple times. Experimental results are shown demonstrating that the model can
be employed profitably in conjunction with different convolutional net
architectures, and that doing so compares favorably to joint training of a
fully-connected CRF with a convolutional net.
