Andrew Buck

Derek Anderson, Matthew Deardorff, Timothy Havens et al.

The fuzzy integral is a powerful parametric nonlin-ear function with utility in a wide range of applications, from information fusion to classification, regression, decision making,interpolation, metrics, morphology, and beyond. While the fuzzy integral is in general a nonlinear operator, herein we show that it can be represented by a set of contextual linear order statistics(LOS). These operators can be obtained via sampling the fuzzy measure and clustering is used to produce a partitioning of the underlying space of linear convex sums. Benefits of our approach include scalability, improved integral/measure acquisition, generalizability, and explainable/interpretable models. Our methods are both demonstrated on controlled synthetic experiments, and also analyzed and validated with real-world benchmark data sets.

Muhammad Aminul Islam, Bryce Murray, Andrew Buck et al.

While most deep learning architectures are built on convolution, alternative foundations like morphology are being explored for purposes like interpretability and its connection to the analysis and processing of geometric structures. The morphological hit-or-miss operation has the advantage that it takes into account both foreground and background information when evaluating target shape in an image. Herein, we identify limitations in existing hit-or-miss neural definitions and we formulate an optimization problem to learn the transform relative to deeper architectures. To this end, we model the semantically important condition that the intersection of the hit and miss structuring elements (SEs) should be empty and we present a way to express Don't Care (DNC), which is important for denoting regions of an SE that are not relevant to detecting a target pattern. Our analysis shows that convolution, in fact, acts like a hit-miss transform through semantic interpretation of its filter differences. On these premises, we introduce an extension that outperforms conventional convolution on benchmark data. Quantitative experiments are provided on synthetic and benchmark data, showing that the direct encoding hit-or-miss transform provides better interpretability on learned shapes consistent with objects whereas our morphologically inspired generalized convolution yields higher classification accuracy. Last, qualitative hit and miss filter visualizations are provided relative to single morphological layer.