  Weakly-supervised learning based on, e.g., partially labelled images or
image-tags, is currently attracting significant attention in CNN segmentation
as it can mitigate the need for full and laborious pixel/voxel annotations.
Enforcing high-order (global) inequality constraints on the network output (for
instance, to constrain the size of the target region) can leverage unlabeled
data, guiding the training process with domain-specific knowledge. Inequality
constraints are very flexible because they do not assume exact prior knowledge.
However, constrained Lagrangian dual optimization has been largely avoided in
deep networks, mainly for computational tractability reasons. To the best of
our knowledge, the method of [Pathak et al., 2015] is the only prior work that
addresses deep CNNs with linear constraints in weakly supervised segmentation.
It uses the constraints to synthesize fully-labeled training masks (proposals)
from weak labels, mimicking full supervision and facilitating dual
optimization. We propose to introduce a differentiable penalty, which enforces
inequality constraints directly in the loss function, avoiding expensive
Lagrangian dual iterates and proposal generation. From constrained-optimization
perspective, our simple penalty-based approach is not optimal as there is no
guarantee that the constraints are satisfied. However, surprisingly, it yields
substantially better results than the Lagrangian-based constrained CNNs in
[Pathak et al., 2015], while reducing the computational demand for training. By
annotating only a small fraction of the pixels, the proposed approach can reach
a level of segmentation performance that is comparable to full supervision on
three separate tasks. While our experiments focused on basic linear constraints
such as the target-region size and image tags, our framework can be easily
extended to other non-linear constraints.
