A Mathematical Investigation into the Design of Prefilters That Make Cameras More Colorimetric.

Yuteng Zhu,Graham D Finlayson

doi:10.3390/s20236882

Abstract

By placing a color filter in front of a camera we make new spectral sensitivities. The Luther-condition optimization solves for a color filter so that the camera’s filtered sensitivities are as close to being linearly related to the XYZ color matching functions (CMFs) as possible, that is, a filter is found that makes the camera more colorimetric. Arguably, the more general Vora-Value approach solves for the filter that best matches all possible target spectral sensitivity sets (e.g., any linear combination of the XYZ CMFs). A concern that we investigate here is that the filters found by the Luther and Vora-Value optimizations are different from one another. In this paper, we unify the Luther and Vora-Value approaches to prefilter design. We prove that if the target of the Luther-condition optimization is an orthonormal basis—a special linear combination of the XYZ CMFs which are orthogonal and are in unit length—the discovered Luther-filter is also the filter that maximizes the Vora-Value. A key advantage of using the Luther-condition formulation to maximize the Vora-Value is that it is both simpler to implement and converges to its optimal answer more quickly. Experiments validate our method.

Highlights

When the spectral sensitivities of the RGB sensors found in a camera and the XYZ color matching functions (CMFs) [1] are exactly a linear transform apart, the camera is said to be colorimetric [2].When this colorimetric condition is met, the RGBs measured by the camera—for all and any spectral stimuli—are the same linear transform from the corresponding XYZ coordinates
The key result, which we present in this paper, is to prove that if we choose to optimize the Luther condition using not the XYZ CMFs but a linear combination which is orthonormal, we are optimizing the Vora-Value
We presented a unifying filter design method based on the underlying relation between the Luther condition and the Vora-Value

Summary

Introduction

When the spectral sensitivities of the RGB sensors found in a camera and the XYZ color matching functions (CMFs) [1] are exactly a linear transform apart, the camera is said to be colorimetric [2].When this colorimetric condition is met, the RGBs measured by the camera—for all and any spectral stimuli—are the same linear transform from the corresponding XYZ coordinates. When the spectral sensitivities of the RGB sensors found in a camera and the XYZ color matching functions (CMFs) [1] are exactly a linear transform apart, the camera is said to be colorimetric [2]. For cross-illuminant color measurement—where we wish to measure XYZs under different lights and map the recorded tristimuli to a fixed reference illuminant, the best sensors are not linearly related to XYZ CMFs [6,7,8] It was recently proposed [9,10] that a color filter could be designed which, when placed in front of the camera, would make it more colorimetric, see Figure 1 (including for the cross-illuminant measurement case [10]). The filters can be selected from a group of commercially available filters [26,27,28,29,30,31]

Methods

Results

Conclusion