Abstract

Spectral reconstruction (SR) algorithms attempt to recover hyperspectral information from RGB camera responses. Recently, the most common metric for evaluating the performance of SR algorithms is the Mean Relative Absolute Error (MRAE)—an relative error (also known as percentage error). Unsurprisingly, the leading algorithms based on Deep Neural Networks (DNN) are trained and tested using the MRAE metric. In contrast, the much simpler regression-based methods (which actually can work tolerably well) are trained to optimize a generic Root Mean Square Error (RMSE) and then tested in MRAE. Another issue with the regression methods is—because in SR the linear systems are large and ill-posed—that they are necessarily solved using regularization. However, hitherto the regularization has been applied at a spectrum level, whereas in MRAE the errors are measured per wavelength (i.e., per spectral channel) and then averaged. The two aims of this paper are, first, to reformulate the simple regressions so that they minimize a relative error metric in training—we formulate both and relative error variants where the latter is MRAE—and, second, we adopt a per-channel regularization strategy. Together, our modifications to how the regressions are formulated and solved leads to up to a 14% increment in mean performance and up to 17% in worst-case performance (measured with MRAE). Importantly, our best result narrows the gap between the regression approaches and the leading DNN model to around 8% in mean accuracy.

Highlights

  • IntroductionA consumer RGB camera captures light signals with three types of color sensors

  • The mean results show that Linear Regression (LR) trained using all of our three new approaches outperform the standard LS, among which the Relative Error Least Absolute Deviation (RELAD) method performs the best—returning 14% lower Mean Relative Absolute Error (MRAE) compared to LS

  • Likewise, observing all other regression models, we found that the best minimization criterion in terms of mean MRAE is either Relative Error Least Squares (RELS) or RELAD depending on the model

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Summary

Introduction

A consumer RGB camera captures light signals with three types of color sensors. The light is physical radiance—a continuous spectral function of wavelength—which, intuitively, can hardly be described by a 3-dimensional color representation. A simple mathematical model of RGB formation is written as [69] x= Z Ω r (λ)s(λ) dλ , (1). Where r (λ) denotes the radiance spectrum, s(λ) = [s R (λ), sG (λ), s B (λ)]T is the set of three spectral sensitivities of the color sensors, and x = [ R G B]T is the derived color vector (the superscript T denotes the transpose operator). In the case of color imaging, the range of integration, Ω, is the visible range. We can approximate the integration in Equation (1) by inner products [69]:

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