ECE 638: Principles of Digital Color Imaging Systems

1. ECE 638: Principles ofDigital Color Imaging Systems Lecture 10: Discrete Wavelength Models Characterization of Human Visual Subspace

2. Synopsis • Review of discrete wavelength model

3. Discrete-wavelength trichomatic model • Stimulus • Sensor response • Response of the i-th channel • Define • Span(S) defines HVS subspace • Stack sensor outputs

4. Definition of projection operator • Projection operator is a linear operator that extracts the fundamental component of the stimulus , i.e. • Equivalent forms of projection operator

5. Extraction of fundamental component of stimulus: an imaging systems interpretation • A complete imaging system can be thought of as a capture device followed by a display device. • The process of extracting the fundamental component of the stimulus can be viewed in this way Sensor Display

6. Questions about the human visual subspace • Question 1: Are there real stimuli that are invisible? • Answer: “No” • Question 2: Are there real stimuli for which ? • Not yet answered • Question 3: What colors in the fundamental space are physically realizable with the addition of a black space component, i.e. what colors have a physically realizable metamer? • Fundamental component must be a non-negative linear combination of the columns of the projector operator

7. Orthonormal basis for R • It can be shown (see Appendix A of Wolski’s paper) that the projection operator can be expressed as • where is a 31x3 matrix whose columns form an orthonormal basis for the human visual subspace • Thus the fundamental component of any stimulus can be expressed as

8. Transformation between color matching matrix AXYZ and basis functions F 31x3

9. Orthonormal basis functions F and a realizable metamer • text Realizable metamer to F F spectral basis set

10. Columns of R (fundamental components of monochromatic stimuli) in F-space Columns of R in F-space Normalized to unit length Columns of R in F-space Non-normalized Points indicate fundamental component of spectral locus Solid arcs show octant of unit sphere. Dashed line is “purple line”

11. Elements of fundamental space realizable without a black-space component – start with cone functions Vos and Walraven cone functions in F-space Cone functions reported by Vos and Walraven Since cone functions are non-negative and span human subspace by definition, they are realizable without a black-space component. Points in convex hull of correspond to stimuli realizable with no null-space component

12. Expansion of set of directly realizable colors by region-growing Spectra corresponding to labeled points on left F-space representation of all realizable colors Colors inside inner dotted region are realizable without a black-space component. Colors outside this region, but inside outer dotted region are realizable only with a black-space component.