Ricci-Notation Tensor Framework for Model-based Approaches to Imaging

TL;DR

The paper develops a Ricci-notation tensor (RT) framework for model-based imaging, unifying inner, entrywise, and outer tensor operations under a dual-variant index notation and Einstein summation. It provides a MATLAB-native RT algebra and the RTToolbox, designed for efficient numeric tensor computations without external dependencies, including pagewise kernels and FFT acceleration. The imaging example, inspired by exoplanet coronagraphy, demonstrates how a scalar SSE and its gradient and Hessian can be derived and computed efficiently by exploiting (inverse) 2D DFT structure, enabling practical optimization. Compared with prior NT-based approaches, RT offers a simpler, more expressive index formalism with extended outer operations, delivering practical computational advantages and a readily shareable MATLAB toolkit.

Abstract

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.

Figures (7)

• Figure 1: Source (left) and ground-truth (right) images (original scale). The $401 \times 401$ pixel source is a CC0-licensed image, Airy_disk_D65Wikipedia2022, after conversion to grayscale and division by 255. The ground-truth image, shown after pixel values are squared, is the entrywise square root of the source image, after masking (occulting) symmetric inner and outer regions, defined by two circles (left overlays), and after adding spots to simulate twin exoplanets.
• Figure 2: Input (left) and output (right) images of an enhancement. A model-based approach addresses an unknown phase aberration in the pupil plane (Fourier domain) by minimizing an image-plane SSE. Nonzero background, and negative foreground, pixels define the SSE (right overlay). Diffraction rings and twin exoplanets become visible in the annular foreground when the 2D phase aberration is adequately corrected. Before display, pixel values are squared.
• Figure 3: Compute time for imaging example (multiple scales). For an $M \times N$ problem, computing first (gradient) and second (HMF) derivatives of the scalar SSE, with respect to an $M \times N$ matrix (phase aberration in Fourier domain), is asymptotically equivalent to computing the SSE alone, thanks to time efficiencies of a model-based solution derived with the RT algebra.
• Figure 4: Memory space for imaging example (multiple scales). For an $M \times N$ problem, computing first (gradient) and second (HMF) derivatives of the scalar SSE, with respect to an $M \times N$ matrix (phase aberration in Fourier domain), is asymptotically equivalent to computing the SSE alone, thanks to space efficiencies of a model-based solution derived with the RT algebra.
• Figure 5: Model-based SSE and gradient. Developed for the imaging example, this MATLAB function's output arguments are the SSE, $E$, the SSE gradient, $\mathbf{\nabla}E$, and a corrected image, $\mathbf{X}^\text{t}$. Its input arguments are a phase aberration, $\mathbf{\Phi}$, the aberrated image, $\mathbf{X}^\text{a}$, and the occultation (background) mask, $\mathbf{W}^\text{b}$.