Multilinear Time Invariant System Theory

In biological and engineering systems, structure, function and dynamics are highly coupled. Such interactions can be naturally and compactly captured via tensor based state space dynamic representations. However, such representations are not amenable to the standard system and controls framework which requires the state to be in the form of a vector. In order to address this limitation, recently a new class of multiway dynamical systems has been introduced in which the states, inputs and outputs are tensors. We propose a new form of multilinear time invariant (MLTI) systems based on the Einstein product and even-order paired tensors. We extend classical linear time invariant (LTI) system notions including stability, reachability and observability for the new MLTI system representation by leveraging recent advances in tensor algebra.Comment: 8 pages, SIAM Conference on Control and its Applications 2019, accepted to appea

Evaluation of Iowa asphalt pavement joint cracking

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Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems

In this paper, we provide a system-theoretic treatment of certain continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor algebra. In particular, if a system of homogeneous polynomial differential equations can be represented by an orthogonally decomposable (odeco) tensor, we can construct its explicit solution by exploiting tensor Z-eigenvalues and Z-eigenvectors. We refer to such HPDS as odeco HPDS. By utilizing the form of the explicit solution, we are able to discuss the stability properties of an odeco HPDS. We illustrate that the Z-eigenvalues of the corresponding dynamic tensor can be used to establish necessary and sufficient stability conditions, similar to these from linear systems theory. In addition, we are able to obtain the complete solution to an odeco HPDS with constant control. Finally, we establish results which enable one to determine if a general HPDS can be transformed to or approximated by an odeco HPDS, where the previous results can be applied. We demonstrate our framework with simulated and real-world examples.Comment: 8 pages, 4 figure

Unsupervised Lesion Detection via Image Restoration with a Normative Prior

Unsupervised lesion detection is a challenging problem that requires accurately estimating normative distributions of healthy anatomy and detecting lesions as outliers without training examples. Recently, this problem has received increased attention from the research community following the advances in unsupervised learning with deep learning. Such advances allow the estimation of high-dimensional distributions, such as normative distributions, with higher accuracy than previous methods.The main approach of the recently proposed methods is to learn a latent-variable model parameterized with networks to approximate the normative distribution using example images showing healthy anatomy, perform prior-projection, i.e. reconstruct the image with lesions using the latent-variable model, and determine lesions based on the differences between the reconstructed and original images. While being promising, the prior-projection step often leads to a large number of false positives. In this work, we approach unsupervised lesion detection as an image restoration problem and propose a probabilistic model that uses a network-based prior as the normative distribution and detect lesions pixel-wise using MAP estimation. The probabilistic model punishes large deviations between restored and original images, reducing false positives in pixel-wise detections. Experiments with gliomas and stroke lesions in brain MRI using publicly available datasets show that the proposed approach outperforms the state-of-the-art unsupervised methods by a substantial margin, +0.13 (AUC), for both glioma and stroke detection. Extensive model analysis confirms the effectiveness of MAP-based image restoration.Comment: Extended version of 'Unsupervised Lesion Detection via Image Restoration with a Normative Prior' (MIDL2019