Image processing: effect of illuminant on sRGB image

[Number2Pencil]

Homework Statement

Assume an image file is sRGB under D65. Generate the image as it would appear under illuminant A ( a 171x1 matrix).

Homework Equations

effect of illuminant:

B = LA

where A is the a color matching function, L is a diagonalized version of illuminant A, and B is the color matching function as a product of the illuminant. CMF's have the dimensions mxn where n is the number of primaries, and m is the sample count of the function. n is usually 3 because it's related to the 3 types of cones in the eye.

t = A'g

where g is the sampled spectrum, and t is the "tristimulus value". If using the color matching function for the CIEXYZ color-space, the tristimulus value has the form [X;Y;Z]. There are some manipulations to transform XYZ into other color-spaces (often utilizing a "white point", such as CIERGB, CIELab, etc).


The effect of the spectrum due to illuminant:

g = Lr

The Attempt at a Solution

I'm not sure where to really start here. I have sRGB with a white point D65, but since this is a tristimulus value, I really only know how to convert it to other tristimulus values. I could transform each pixel from sRGB to XYZ, multiply by the inverse XYZ CMF to convert back to a spectrum, adjust the XYZ CMF to account for the illuminant, adjust the spectrum for the illuminant, and use these to get a new XYZ tristimulus value, and convert back to sRGB...

The kink in that plan was that the CMF, A, is not square, so I can't take the inverse. So either there is some linear algebra magic I'm not seeing, or there is something I can do with this illuminant vector to the tristimulus values.

 

[Number2Pencil]

Haha...not a lot of image processing gurus out there I suppose?

I think I may have found a clue to this problem, it is in this formula:

B' = (A'Q)^-1 A'

Where B is a new color matching functions with associated primaries Q by using color matching function A (which is associated with Primaries P). There is a note that says the columns of A'Q are the amount of the new primaries Q required to match the old primaries P, and that it will be a 3x3 matrix.

Another formula says that if d is the color vector (tristimulus value?) associated with primaries P and d is the color vector associated with primaries Q, then

d = (A'Q)^-1c

So I have A, and I can get B using the luminance formula from the first post, and I have P because the primaries of RGB are well known...if I can find Q I think I can get it using the last formula.

I'm not really sure how to solve this matrix equation for Q:

B' = (A'Q)^-1 A'

If A'Q is really a 3x3 matrix, and A' is a 3x171, then that means Q needs to be a 171x3 to get a 3x3 output.