On the dice loss gradient and the ways to mimic it

In the past few years, in the context of fully-supervised semantic segmentation, several losses -- such as cross-entropy and dice -- have emerged as de facto standards to supervise neural networks. The Dice loss is an interesting case, as it comes from the relaxation of the popular Dice coefficient; one of the main evaluation metric in medical imaging applications. In this paper, we first study theoretically the gradient of the dice loss, showing that concretely it is a weighted negative of the ground truth, with a very small dynamic range. This enables us, in the second part of this paper, to mimic the supervision of the dice loss, through a simple element-wise multiplication of the network output with a negative of the ground truth. This rather surprising result sheds light on the practical supervision performed by the dice loss during gradient descent. This can help the practitioner to understand and interpret results while guiding researchers when designing new losses.Comment: Currently under revie

Curriculum semi-supervised segmentation

This study investigates a curriculum-style strategy for semi-supervised CNN segmentation, which devises a regression network to learn image-level information such as the size of a target region. These regressions are used to effectively regularize the segmentation network, constraining softmax predictions of the unlabeled images to match the inferred label distributions. Our framework is based on inequality constraints that tolerate uncertainties with inferred knowledge, e.g., regressed region size, and can be employed for a large variety of region attributes. We evaluated our proposed strategy for left ventricle segmentation in magnetic resonance images (MRI), and compared it to standard proposal-based semi-supervision strategies. Our strategy leverages unlabeled data in more efficiently, and achieves very competitive results, approaching the performance of full-supervision.Comment: Accepted as paper as MICCAI 2O1

Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensions

This study investigates the optimization aspects of imposing hard inequality constraints on the outputs of CNNs. In the context of deep networks, constraints are commonly handled with penalties for their simplicity, and despite their well-known limitations. Lagrangian-dual optimization has been largely avoided, except for a few recent works, mainly due to the computational complexity and stability/convergence issues caused by alternating explicit dual updates/projections and stochastic optimization. Several studies showed that, surprisingly for deep CNNs, the theoretical and practical advantages of Lagrangian optimization over penalties do not materialize in practice. We propose log-barrier extensions, which approximate Lagrangian optimization of constrained-CNN problems with a sequence of unconstrained losses. Unlike standard interior-point and log-barrier methods, our formulation does not need an initial feasible solution. Furthermore, we provide a new technical result, which shows that the proposed extensions yield an upper bound on the duality gap. This generalizes the duality-gap result of standard log-barriers, yielding sub-optimality certificates for feasible solutions. While sub-optimality is not guaranteed for non-convex problems, our result shows that log-barrier extensions are a principled way to approximate Lagrangian optimization for constrained CNNs via implicit dual variables. We report comprehensive weakly supervised segmentation experiments, with various constraints, showing that our formulation outperforms substantially the existing constrained-CNN methods, both in terms of accuracy, constraint satisfaction and training stability, more so when dealing with a large number of constraints

Constrained-CNN losses for weakly supervised segmentation

The final publication is available at Elsevier via https://doi.org/10.1016/j.media.2019.02.009. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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. (2015a) 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. (2015a), 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, e.g., invariant shape moments (Klodt and Cremers, 2011) and other region statistics (Lim et al., 2014). Therefore, it has the potential to close the gap between weakly and fully supervised learning in semantic medical image segmentation. Our code is publicly available.This work is supported by the National Science and Engineering Research Council of Canada (NSERC), discovery grant program, and by the ETS Research Chair on Artificial Intelligence in Medical Imagin

Polystyrene: the Decentralized Data Shape That Never Dies

International audienceDecentralized topology construction protocols organize nodes along a predefined topology (e.g. a torus, ring, or hypercube). Such topologies have been used in many contexts ranging from routing and storage systems, to publish-subscribe and event dissemination. Since most topologies assume no correlation between the physical location of nodes and their positions in the topology, they do not handle catastrophic failures well, in which a whole region of the topology disappears. When this occurs, the overall shape of the system typically gets lost. This is highly problematic in applications in which overlay nodes are used to map a virtual data space, be it for routing, indexing or storage. In this paper, we propose a novel decentralized approach that maintains the initial shape of the topology even if a large (consecutive) portion of the topology fails. Our approach relies on the dynamic decoupling between physical nodes and virtual ones enabling a fast reshaping. For instance, our results show that a 51,200-node torus converges back to a full torus in only 10 rounds after 50% of the nodes have crashed. Our protocol is both simple and flexible and provides a novel form of collective survivability that goes beyond the current state of the art

Differentiable Boundary Point Extraction for Weakly Supervised Star-shaped Object Segmentation

Although Deep Learning is the new gold standard in medical image segmentation, the annotation burden limits its expansion to clinical practice. We also observe a mismatch between annotations required by deep learning methods designed with pixel-wise optimization in mind and clinically relevant annotations designed for biomarkers extraction (diameters, counts, etc.). Our study proposes a first step toward bridging this gap, optimizing vessel segmentation based on its diameter annotations. To do so we propose to extract bound- ary points from a star-shaped segmentation in a differentiable manner. This differentiable extraction allows reducing annotation burden as instead of the pixel-wise segmentation only the two annotated points required for diameter measurement are used for training the model. Our experiments show that training based on diameter is efficient; produces state-of-the-art weakly supervised segmentation; and performs reasonably compared to full supervision. Our code is publicly available: https://gitlab.com/radiology/aim/carotid-artery-image-analysis/diameter-learnin