1. Purdue University Optimization of Sensor Response Functions for Colorimetry of Reflective 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.

2. Purdue University Overall Goal Design components (color filters) for an inexpensive device to perform colorimetric measurements from surfaces of different types

3. Purdue University Device Operation Highlights Output: XYZ tristimulus values 3 modes of operation D65 Reflective/D65 EE Reflective/EEEmissive n n n

4. Purdue University Computation of Tristimulus Values Stimulus Vector – n Emissive Mode Reflective Mode 31 samples taken at 10 nm intervals 400 700 n

5. Purdue University Tristimulus Vector Tristimulus vector Color matching matrix – A m (3 x 31) Effective stimulus

6. Purdue University Color Matching Matrix x y z 3 x 31 matrix of color matching functions

7. Purdue University Device Architecture LED’s Detectors Filters

8. Purdue University Computational Model TmTm

9. Purdue University Estimate of Tristimulus Vector Estimate Channel matrix  emissive mode  reflective modes

10. Purdue University Error Metric Tristimulus error CIE uniform color space

11. Purdue University Error Metric (cont.) Linearize about nominal tristimulus value t = t 0 Linearized error norm

12. Purdue University Error Metric (cont.) Consider ensemble of 752 real stimuli n k Rearrange and sum over k

13. Purdue University Regularization Filter feasbility  Roughness cost Design robustness  Effect of noise and/or component variations  Augment error metric

14. Purdue University Design Problem Overall cost function Solution procedure  For any fixed F = [f 1, f 2, f 3, f 4 ] T determine optimal coefficient matrices T EM, T EE, and T D65 as solution to least-squares problem  Minimize partially optimized cost via gradient search

15. Purdue University Experimental Results Optimal filter set for K r = 0.1 and K s = 1.0

16. Purdue University Experimental Results (cont.) Effect of system tolerance  on mean- squared error

17. Purdue University Experimental Results (cont.) Error performance in true L*a*b* for set of 752 spectral samples

18. Purdue University Experimental Results (cont.) Emissive mode L*a*b* error surface

19. Purdue University Approximation of Color Matching Matrix

20. Purdue University 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