What does the r(lambda) function look like? Is it a delta function that extracts out the amount of pure red wavelength from Phi(lambda)?
whdawn
Why negative red?
whdawn
So this is a new color space?
kayvonf
@lucida. $r(\lambda)$ is not a spectrum, its the amount of red primary (it's magnitude) used to match a monochromatic color of the given wavelength $\lambda$. These curves are the result of the color matching experiments where a human is given three monochromatic sources and is asked to turn the intensity dials of the three sources until the mixture of the sources looks exactly like the reference monochromatic light of unknown wavelength.
lucida
@kayvonf, isn't the amount of red primary used to match the monochromatic color of the given wavelength represented by $c_{r}$ here on this slide? $r(\lambda)$ is being convolved with the spectrum $\Phi(\lambda)$ to yield $c_{r}$ which seems to imply that $r(\lambda)$ is a delta function used to extract the amount of pure red wavelength frequency from the spectrum $\Phi(\lambda)$?
kayvonf
Ah... I see how I'm confusing people.
First, a direct response to your comment...
$c_r$ is the amount of red primary needed to match the given spectrum $\Omega(\lambda)$ which may not be a monochromatic spectrum. $r(\lambda)$ is the amount of the red primary needed to match a monochromatic spectrum of wavelength $\lambda$. Therefore $c_r$ = $r(\lambda)$ only when $\Omega(\lambda)$ is a monochromatic spectrum of unit power in wavelength $\lambda$.
Now, a more intuitive explanation...
Imagine you were trying to match a monochromatic light A of unit power in wavelength $\lambda_A$.
The color matching experiments tell you that in order to match the appearance of this source, you need to turn your knobs to: $r(\lambda_A)$, $g(\lambda_A)$, $b(\lambda_A)$.
Now imagine you were trying to match a monochromatic light B of unit power in wavelength $\lambda_B$.
Again, the color matching experiments tell you that in order to match the color, you need to turn your knobs to: $r(\lambda_B)$, $g(\lambda_B)$, $b(\lambda_B)$.
Now, human color perception happens to be linear... (something we took for granted in this lecture, but is empirically be true, as defined by Grassmann's Laws).
Therefore, if there was a new light C formed from A+B (unit power in two wavelengths), then Grassmann's Laws and the color matching experiments tell us we'd have to turn our dials to: $r(\lambda_A)+r(\lambda_B)$, $g(\lambda_A)+g(\lambda_B)$, $b(\lambda_A)+b(\lambda_B)$.
Now follow this reasoning out to matching a full spectrum with non-zero power for all wavelengths, or in other words, the spectrum is a weighted combination of our unit power monochromatic lights, with the weights given by the spectral power distribution $\Omega(\lambda)$. This spectrum is just a combination of pure wavelengths, and so a weighted summation of the color matching values tells you how to turn the knobs to match the spectrum. That weighted summation is the integral on this slide.
You'll get confused if you think of the integral on this slide as integrating a spectrum against a spectral response function, as we did on slide 18. These integrals compute the values $c_r$, $c_g$, and $c_b$ that we need to set the color matching knobs to.
What does the r(lambda) function look like? Is it a delta function that extracts out the amount of pure red wavelength from Phi(lambda)?
Why negative red?
So this is a new color space?
@lucida. $r(\lambda)$ is not a spectrum, its the amount of red primary (it's magnitude) used to match a monochromatic color of the given wavelength $\lambda$. These curves are the result of the color matching experiments where a human is given three monochromatic sources and is asked to turn the intensity dials of the three sources until the mixture of the sources looks exactly like the reference monochromatic light of unknown wavelength.
@kayvonf, isn't the amount of red primary used to match the monochromatic color of the given wavelength represented by $c_{r}$ here on this slide? $r(\lambda)$ is being convolved with the spectrum $\Phi(\lambda)$ to yield $c_{r}$ which seems to imply that $r(\lambda)$ is a delta function used to extract the amount of pure red wavelength frequency from the spectrum $\Phi(\lambda)$?
Ah... I see how I'm confusing people.
First, a direct response to your comment...
$c_r$ is the amount of red primary needed to match the given spectrum $\Omega(\lambda)$ which may not be a monochromatic spectrum. $r(\lambda)$ is the amount of the red primary needed to match a monochromatic spectrum of wavelength $\lambda$. Therefore $c_r$ = $r(\lambda)$ only when $\Omega(\lambda)$ is a monochromatic spectrum of unit power in wavelength $\lambda$.
Now, a more intuitive explanation...
Imagine you were trying to match a monochromatic light A of unit power in wavelength $\lambda_A$.
The color matching experiments tell you that in order to match the appearance of this source, you need to turn your knobs to: $r(\lambda_A)$, $g(\lambda_A)$, $b(\lambda_A)$.
Now imagine you were trying to match a monochromatic light B of unit power in wavelength $\lambda_B$.
Again, the color matching experiments tell you that in order to match the color, you need to turn your knobs to: $r(\lambda_B)$, $g(\lambda_B)$, $b(\lambda_B)$.
Now, human color perception happens to be linear... (something we took for granted in this lecture, but is empirically be true, as defined by Grassmann's Laws).
Therefore, if there was a new light C formed from A+B (unit power in two wavelengths), then Grassmann's Laws and the color matching experiments tell us we'd have to turn our dials to: $r(\lambda_A)+r(\lambda_B)$, $g(\lambda_A)+g(\lambda_B)$, $b(\lambda_A)+b(\lambda_B)$.
Now follow this reasoning out to matching a full spectrum with non-zero power for all wavelengths, or in other words, the spectrum is a weighted combination of our unit power monochromatic lights, with the weights given by the spectral power distribution $\Omega(\lambda)$. This spectrum is just a combination of pure wavelengths, and so a weighted summation of the color matching values tells you how to turn the knobs to match the spectrum. That weighted summation is the integral on this slide.
You'll get confused if you think of the integral on this slide as integrating a spectrum against a spectral response function, as we did on slide 18. These integrals compute the values $c_r$, $c_g$, and $c_b$ that we need to set the color matching knobs to.
Ah that makes a lot of sense now, thanks!