  We present Shape-Tailored Deep Neural Networks (ST-DNN). ST-DNN extend
convolutional networks (CNN), which aggregate data from fixed shape (square)
neighborhoods, to compute descriptors defined on arbitrarily shaped regions.
This is natural for segmentation, where descriptors should describe regions
(e.g., of objects) that have diverse shape. We formulate these descriptors
through the Poisson partial differential equation (PDE), which can be used to
generalize convolution to arbitrary regions. We stack multiple PDE layers to
generalize a deep CNN to arbitrary regions, and apply it to segmentation. We
show that ST-DNN are covariant to translations and rotations and robust to
domain deformations, natural for segmentation, which existing CNN based methods
lack. ST-DNN are 3-4 orders of magnitude smaller then CNNs used for
segmentation. We show that they exceed segmentation performance compared to
state-of-the-art CNN-based descriptors using 2-3 orders smaller training sets
on the texture segmentation problem.
