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Model-based approaches to imaging, like specialized image enhancements in astronomy, facilitate explanations of relationships between observed inputs and computed outputs. These models may be expressed with extended matrix-vector (EMV) algebra, especially when they involve only scalars, vectors, and matrices, and with n-mode or index notations, when they involve multidimensional arrays, also called numeric tensors or, simply, tensors. While this paper features an example, inspired by exoplanet imaging, that employs tensors to reveal (inverse) 2D fast Fourier transforms in an image enhancement model, the work is actually about the tensor algebra and software, or tensor frameworks, available for model-based imaging. The paper proposes a Ricci-notation tensor (RT) framework, comprising a dual-variant index notation, with Einstein summation convention, and codesigned object-oriented software, called the RTToolbox for MATLAB. Extensions to Ricci notation offer novel representations for entrywise, pagewise, and broadcasting operations popular in EMV frameworks for imaging. Complementing the EMV algebra computable with MATLAB, the RTToolbox demonstrates programmatic and computational efficiency via careful design of numeric tensor and dual-variant index classes. Compared to its closest competitor, also a numeric tensor framework that uses index notation, the RT framework enables superior ways to model imaging problems and, thereby, to develop solutions., Comment: 39 pages, 7 figures, 5 tables

Researchers are increasingly incorporating numeric high-order data, i.e., numeric tensors, within their practice. Just like the matrix/vector (MV) paradigm, the development of multi-purpose, but high-performance, sparse data structures and algorithms for arithmetic calculations, e.g., those found in Einstein-like notation, is crucial for the continued adoption of tensors. We use the example of high-order differential operators to illustrate this need. As sparse tensor arithmetic is an emerging research topic, with challenges distinct from the MV paradigm, many aspects require further articulation. We focus on three core facets. First, aligning with prominent voices in the field, we emphasise the importance of data structures able to accommodate the operational complexity of tensor arithmetic. However, we describe a linearised coordinate (LCO) data structure that provides faster and more memory-efficient sorting performance. Second, flexible data structures, like the LCO, rely heavily on sorts and permutations. We introduce an innovative permutation algorithm, based on radix sort, that is tailored to rearrange already-sorted sparse data, producing significant performance gains. Third, we introduce a novel poly-algorithm for sparse tensor products, where hyper-sparsity is a possibility. Different manifestations of hyper-sparsity demand their own approach, which our poly-algorithm is the first to provide. These developments are incorporated within our LibNT and NTToolbox software libraries. Benchmarks, frequently drawn from the high-order differential operators example, demonstrate the practical impact of our routines, with speed-ups of 40% or higher compared to alternative high-performance implementations. Comparisons against the MATLAB Tensor Toolbox show over 10 times speed improvements. Thus, these advancements produce significant practical improvements for sparse tensor arithmetic., Comment: To appear in SIAM Journal on Scientific Computing