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Optimization of Sensor Response Functions for Colorimetry of Reflective and Emissive Objects

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## Optimization of Sensor Response Functions for Colorimetry of Reflective and Emissive Objects

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**Optimization ofSensor Response Functionsfor Colorimetry**ofReflective and Emissive Objects Mark Wolski*, Charles A. Bouman, Jan P. Allebach Purdue University, School of Electrical and Computer Engineering, West Lafayette, IN 47907 Eric Walowit Color Savvy Systems Inc., Springboro, OH 45066 *now with General Motors Research and Development Center, Warren, MI 48090-9055.**Overall Goal**Design components (color filters) for an inexpensive device to perform colorimetric measurements from surfaces of different types**Device Operation Highlights**• Output: XYZ tristimulus values • 3 modes of operation Emissive Reflective/EE Reflective/D65 EE D65 n n n**n**700 400 l Computation of Tristimulus Values • Stimulus Vector – n • Emissive Mode • Reflective Mode 31 samples taken at 10 nm intervals**Tristimulus Vector**• Tristimulus vector • Color matching matrix – Am (3x31) • Effective stimulus**z**x y l Color Matching Matrix 3x31 matrix of color matching functions**Device Architecture**Detectors LED’s LED’s Filters**Estimate of Tristimulus Vector**• Estimate • Channel matrix • emissive mode • reflective modes**Error Metric**• Tristimulus error • CIE uniform color space**Error Metric (cont.)**• Linearize about nominal tristimulus value t = t0 • Linearized error norm**Error Metric (cont.)**• Consider ensemble of 752 real stimuli nk • Rearrange and sum over k**Regularization**• Filter feasbility • Roughness cost • Design robustness • Effect of noise and/or component variations • Augment error metric**Design Problem**• Overall cost function • Solution procedure • For any fixed F =[f1, f2, f3, f4]T determine optimal coefficient matrices TEM, TEE, and TD65 as solution to least-squares problem • Minimize partially optimized cost via gradient search**Experimental Results**• Optimal filter set for Kr = 0.1 and Ks = 1.0**Experimental Results (cont.)**• Effect of system tolerance W on mean-squared error**Experimental Results (cont.)**• Error performance in true L*a*b* for set of 752 spectral samples**Experimental Results (cont.)**• Emissive mode L*a*b* error surface**Conclusions**• For given device architecture, it is possible to design components that will yield satisfactory performance • filters are quite smooth • device is robust to noise • excellent overall accuracy • Solution method is quite flexible • independent of size of sample ensemble • Vector space methods provide a powerful tool for solving problems in color imaging