Basic probabilistic models form the basis of modern deep learning largely by defining the final activation layer of a network and allowing the networks outputs to be interpreted through a probabilistic lens. By increasing the complexity of these final activation layers to include features such as spatial regularisation, allow for these models to improve in performance without requiring significantly more data. The challenge however is to ensure that these more complex probabilistic graphical models adapt well to being used in tandem with deep learning, that is, ensuring they converge and have meaningful derivatives. The goal of this paper is to show two algorithms with provable convergence properties to allow this type of regularisation for image and signal classification models. Through both rigorous mathematical analysis and in silico experiments, we can show the differing performance between these solution types.
Figures & tables
Fig. 1 : Example hierarchy used in the brain tissue segmentation experiment. The leaf-labels include the background, external cerebral spinal fluid, cortical gray matter, subcortical gray matter, white matter, ventricles, and a combined label for the cerebellum and brain stem. The additional branch-labels are the within-skull label, the brain, and the sub-cortex.
Fig. 2 : Label ordering for random hierarchical model experiment containing 5 leaf labels and 3 branches (i.e. ways of grouping leaf labels) along with the source, S, representing all labels.
Fig. 3 : Comparision of marginal distributions of variables between solution types ( red: Parallel loopy belief propagation, blue: Iterative mean field estimation, and green: Bias term only with no regularisation) in terms of Root mean squared error (RMSE) and expected KL Divergence ( E[DKL] ). Each point cloud includes 100 simulations.
Fig. 4 : Comparision of marginal distributions of variables between solution types ( red: Parallel loopy belief propagation, blue: Iterative mean field estimation, and green: Bias term only with no regularisation) in terms of Root mean squared error (RMSE) and expected KL Divergence ( E[DKL] ). Each point cloud includes 50 simulations.
Fig. 5 : Example image (first row) and segmentation (second row) pairs of synthetic images for binary segmentation.
Fig. 6 : Network structure for simulated learning experiment
Fig. 7 : Dice scores for synthetic image segmentation across 10 replications for each network depth using the negative log-likelihood loss function.
Fig. 8 : Distribution of paired differences in Dice scores for synthetic image segmentation across 10 replications for each network depth using the negative log-likelihood loss function.
Fig. 9 : Dice scores for synthetic image segmentation across 10 replications for each network depth using the Dice loss function.
Fig. 10 : Distribution of paired differences in Dice scores for synthetic image segmentation across 10 replications for each network depth using the Dice loss function.
Fig. 11 : Example images with two-region background and three-nested-region foreground object.
Fig. 12 : Dice scores (above) and differences (below) for the brighter (left) and darker (right) background component in the synthetic image segmentation across 10 replications for each network depth using the negative log-likelihood loss function.
Fig. 14 : Dice scores (above) and differences (below) for the brightest and widest foreground component (left) and for the darker and narrower foreground component (right) in the synthetic image segmentation across 10 replications for each network depth using the negative log-likelihood loss function.