ECE 638: Principles of Digital Color Imaging Systems Lecture 8: Discrete Wavelength Models Primaries

Synopsis Review of discrete wavelength model Definition of primary set for discrete wavelength model Color matching condition and graphical interpretation Fundamental component of primary set Computation of primary match amount Transformation between two different primary sets Color matching matrix

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

Primaries Discrete-wavelength representation of a single primary Primary set Stimuli generated using primary set consist of linear combinations of columns of

Color matching condition Stimulus is matched by primary combination if Graphical illustration of color matching condition for two wavelength samples, a monochromatic sensor, and a single primary Since , All possible primaries lie on this line

Fundamental component of primary set Recall that the fundamental component of a stimulus is determined by projecting onto the span of So we have where Considering the primary set as a set of three stimuli (columns of ), we can write its fundamental component as Then for any arbitrary primary combination , we have

Span of P* Can match any stimulus seen by sensor with some combination of the primary set , i.e. , and provided In this case, we say that (or ) is a set of visually independent primaries. Any stimulus in this 2-D space can be matched by some primary weight which will lie on the line shown.

Determining the primary mixture amount Color matching condition is given by If is non-singular, then primary mixture amount is given by Is nonsingular? If is visually independent, can write that for some invertible 3x3 matrix But is invertible, which implies that is invertible, since it is a product of two invertible matrices

Transforming between primaries Consider a fixed stimulus and two visually independent primary sets and . Suppose we know the mixture amount of primary that yields a match to the stimulus . Can we find the mixture amount of primary that also matches , if we just know , , and ; and we do not know directly? Since both primaries match the stimulus, we have that Also, since the primary set is visually independent, we have that

Color matching matrix Recall that (i-th monochromatic stimulus) Then (31x31 identity matrix) Define according to Concatenating these equations side-by-side for yields or , where (3x31) 31x3 matrix is color matching matrix whose i-th row is amount of primary needed to match i-th wavelength i-th column

Equivalence between human visual subspace and span of color matching matrix Theorem: Proof: This means that we can freely transform among color matching functions and the HVS sensitivity matrix

Example transformations between color matching functions We earlier introduced the 1931 CIE RGB color space with color matching matrix , and discussed the fact that the CIE also specified an alternative space – the CIE 1931 XYZ space with the properties that the the color matching functions are all non-negative and the color matching function is the relative luminous efficiency function. Since the color matching functions and represent primary amounts needed to match monochromatic stimuli with the two respective primary sets and , the matrix that transforms between these color matching functions also transforms between arbitrary coordinates (mixture amounts) in these two color spaces.

Primaries and color matching functions for 1931 CIE RGB and XYZ color spaces text