Enhancing color vision by breaking binocular redundancy

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Enhancing color vision by breaking binocular redundancy ( enhancing-color-vision-by-breaking-binocular-redundancy )

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Methodology for counting metamers in β€œmore abstract” calculation: In this section, we describe the technique by which we estimated the reduction of the occurrence of metamerism using our vision enhancement device. Each spectrum 𝐼(πœ†) is mapped to LAB tristimulus values [𝐿, π‘Ž, 𝑏] via the CIE matching functions described earlier in the supplementary. Counting the number of metamers for a particular LAB reference point [𝐿0,π‘Ž0,𝑏0] amounts to counting the number of different spectra 𝐼(πœ†) that map to tristimulus values that are within a sphere in LAB space of radius Ξ”E of the reference point. The number of metameric spectra is infinite, so we instead compute a surrogate quantity. Roughly, we discretize each spectrum by wavelength so each spectrum can be abstracted as a point in a finite-dimensional space. We then count metamers by computing the volume that they occupy in this space. The details of the computation are described below. 1. Represent spectra by using 𝑁𝑆 equally spaced samples in wavelength. For example, 𝐼(πœ†) is represented as a vector [𝐼1, 𝐼2, ... , 𝐼𝑁𝑆 ], which corresponds to the intensities at the wavelengths [πœ†1,πœ†2,...,πœ†π‘π‘†]. 2. The map [𝐼1, ... , 𝐼𝑁𝑆 ] β†’ [𝐿, π‘Ž, 𝑏] from the discretized spectrum to LAB tristimulus values is smooth and nonlinear. Since the map is only being evaluated in a local neighborhood of the reference point, the map is well approximated by its first order Taylor expansion. This allows us to replace the nonlinear map with an affine function 𝑔(𝐼1, ... , 𝐼𝑁𝑆 ) = [𝐿, π‘Ž, 𝑏]. 3. Let 𝑆0 = {⁑[𝐿, π‘Ž, 𝑏]⁑|⁑(𝐿 βˆ’ 𝐿0)2 + (π‘Ž βˆ’ π‘Ž0)2 + (𝑏 βˆ’ 𝑏0)2 ≀ Δ𝐸2⁑} be the set of tristimulus values indistinguishable from the reference point. The set of metameric spectra is given by the image of 𝑆0 under the inverse map π‘”βˆ’1. 4. Since the inverse map π‘”βˆ’1 is affine and the set 𝑆0 is a sphere in LAB space, the image π‘”βˆ’1(𝑆0) is an ellipsoid in the discretized spectrum space [S1]. Note that this ellipsoid is degenerate; it will be infinite in the directions corresponding to the kernel of 𝑔. 5. We assume that we are counting β€œreflection metamers” or β€œtransmission metamers” under a certain illuminant. That is, we are excluding metamers generated by active emissive sources for the sake of this calculation, since including unbounded emissive sources complicates this calculation further. Under this assumption, the allowed intensities are not infinite, since every admissible spectrum has intensities bounded by the corresponding intensities of the illuminant. More formally, definethesetofadmissiblespectraas𝐢0 ={⁑[𝐼1,...,𝐼𝑁𝑆]⁑|⁑0β‰€πΌπ‘˜ β‰€πΌπ‘˜π·65⁑forβ‘π‘˜=1,...,𝑁𝑆},where πΌπ‘˜π·65 is the intensity at πœ†π‘˜ of the D65 illuminant. 6. The volume of metameric spectra is therefore the volume of the set π‘”βˆ’1(𝑆0) ∩ 𝐢0. 5

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