Computer Graphics and Imaging UC Berkeley CS184/284A
Lecture 23:
Introduction to Color Science
credit: Science Media Group.
Physical Basis of Color
Ren NgCS184/284A
The Fundamental Components of Light
• Newton showed sunlight can be subdivided into a rainbow with a prism • Resulting light cannot be further subdivided with a second prism
Ren NgCS184/284A
Electromagnetic radiation
• Oscillations of different frequencies (wavelengths)
The Visible Spectrum of Light
Image credit: Licensed under CC BY-SA 3.0 via Commons https://commons.wikimedia.org/wiki/File:EM_spectrum.svg#/media/File:EM_spectrum.svg
Visible electromagnetic spectrum
Ren NgCS184/284A
Spectral Power Distribution (SPD)
Salient property in measuring light
• The amount of light present at each wavelength
• Units:
• radiometric units / nanometer (e.g. watts / nm)
• Can also be unit-less
• Often use “relative units” scaled to maximum wavelength for comparison across wavelengths when absolute units are not important
Ren NgCS184/284A
Daylight Spectral Power Distributions Vary
Blue sky Solar disk
[Brian Wandell]
Ren NgCS184/284A
Spectral Power Distribution of Light Sources
Describes distribution of energy by wavelength
Figure credit:
Ren NgCS184/284A
Superposition (Linearity) of Spectral Power Distributions[Brian W
andell]
Ren NgCS184/284A
What is Color?
• Color is a phenomenon of human perception; it is not a universal property of light
• Colors are the perceptual sensations that arise from seeing light of different spectral power distributions
• Technically speaking, different wavelengths of light are not “colors”
Biological Basis of Color
Ren NgCS184/284A
Anatomy of The Human Eye
Ren NgCS184/284A
Retinal Photoreceptor Cells: Rods and Cones
Rod cells Cone cellsRods are primary receptors in very low light (“scotopic” conditions), e.g. dim moonlight
• ~120 million rods in eye
• Perceive only shades of gray, no color
Cones are primary receptors in typical light levels (“photopic”)
• ~6-7 million cones in eye
• Three types of cones, each with different spectral sensitivity
• Provide sensation of colorhttp://ebooks.bfwpub.com/life.php Figure 45.18
Ren NgCS184/284A
Spectral Response of Human Cone CellsThree types of cone cells: S, M, and L (corresponding to peak response at short, medium, and long wavelengths)
Nor
mal
ized
resp
onse
Wavelength (nm)
Ren NgCS184/284A
Fraction of Three Cone Cell Types Varies Widely
Hof
er, H
. et a
l. J.
Neu
rosc
i. 20
05;2
5:96
69-9
679
Distribution of cone cells at edge of fovea in 12 different humans with normal color vision. Note high variability of percentage of different cone cell types. (false color image)
Measuring Light
Ren NgCS184/284A
A Simple Model of a Light DetectorCredit: M
arschner
Ren NgCS184/284A
Mathematics of Light Detection
Same math carries over to spectral power distributions
• Light entering the detector has its spectral power distribution, s(λ)
• Detector has its spectral sensitivity or spectral response, r(λ)
measured signal
input spectrum
detector’s sensitivity
Ren NgCS184/284A
If we think of s and r as discrete, sampled representations (vectors) rather than continuous functions, this integral operation is a dot product:We can also write this in matrix form:
Mathematics of Light Detection
X =⇥
s⇤2
64r
3
75
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Ren NgCS184/284A
Dimensionality Reduction From ∞ to 1
At the detector: • SPD is a function of wavelength (∞ - dimensional signal)
• Detector result is a scalar value (1 - dimensional signal)
Tristimulus Theory of Color
Ren NgCS184/284A
Spectral Response of Human Cone Cells
Instead of one detector as before, now we have three detectors (S, M, L cone cells), each with a different spectral response curve
Nor
mal
ized
resp
onse
Wavelength (nm)
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Ren NgCS184/284A
Spectral Response of Human Cone Cells
Instead of one detector as before, now we have three detectors (S, M, L cone cells), each with a different spectral response curve
Nor
mal
ized
resp
onse
Wavelength (nm)
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rL(�)<latexit sha1_base64="DSBcDilXdsDN8Zn3/L3UUooJZvk=">AAAB8nicbVBNS8NAFHypX7V+VT16WSxCeymJl+qt4EXBQwVjC00om82mXbrZhN2NUELBX+HFg4pXf403/42btgdtHVgYZubx3k6Qcqa0bX9bpbX1jc2t8nZlZ3dv/6B6ePSgkkwS6pKEJ7IXYEU5E9TVTHPaSyXFccBpNxhfFX73kUrFEnGvJyn1YzwULGIEayN5cnBb97iJh7gxqNbspj0DWiXOgtTajSco0BlUv7wwIVlMhSYcK9V37FT7OZaaEU6nFS9TNMVkjIe0b6jAMVV+Prt5is6MEqIokeYJjWbq74kcx0pN4sAkY6xHatkrxP+8fqajCz9nIs00FWS+KMo40gkqCkAhk5RoPjEEE8nMrYiMsMREm5oqpgRn+curxD1vXjbtO1PGDcxRhhM4hTo40II2XEMHXCCQwjO8wpuVWS/Wu/Uxj5asxcwx/IH1+QPVjpHc</latexit><latexit sha1_base64="DSBcDilXdsDN8Zn3/L3UUooJZvk=">AAAB8nicbVBNS8NAFHypX7V+VT16WSxCeymJl+qt4EXBQwVjC00om82mXbrZhN2NUELBX+HFg4pXf403/42btgdtHVgYZubx3k6Qcqa0bX9bpbX1jc2t8nZlZ3dv/6B6ePSgkkwS6pKEJ7IXYEU5E9TVTHPaSyXFccBpNxhfFX73kUrFEnGvJyn1YzwULGIEayN5cnBb97iJh7gxqNbspj0DWiXOgtTajSco0BlUv7wwIVlMhSYcK9V37FT7OZaaEU6nFS9TNMVkjIe0b6jAMVV+Prt5is6MEqIokeYJjWbq74kcx0pN4sAkY6xHatkrxP+8fqajCz9nIs00FWS+KMo40gkqCkAhk5RoPjEEE8nMrYiMsMREm5oqpgRn+curxD1vXjbtO1PGDcxRhhM4hTo40II2XEMHXCCQwjO8wpuVWS/Wu/Uxj5asxcwx/IH1+QPVjpHc</latexit><latexit sha1_base64="DSBcDilXdsDN8Zn3/L3UUooJZvk=">AAAB8nicbVBNS8NAFHypX7V+VT16WSxCeymJl+qt4EXBQwVjC00om82mXbrZhN2NUELBX+HFg4pXf403/42btgdtHVgYZubx3k6Qcqa0bX9bpbX1jc2t8nZlZ3dv/6B6ePSgkkwS6pKEJ7IXYEU5E9TVTHPaSyXFccBpNxhfFX73kUrFEnGvJyn1YzwULGIEayN5cnBb97iJh7gxqNbspj0DWiXOgtTajSco0BlUv7wwIVlMhSYcK9V37FT7OZaaEU6nFS9TNMVkjIe0b6jAMVV+Prt5is6MEqIokeYJjWbq74kcx0pN4sAkY6xHatkrxP+8fqajCz9nIs00FWS+KMo40gkqCkAhk5RoPjEEE8nMrYiMsMREm5oqpgRn+curxD1vXjbtO1PGDcxRhhM4hTo40II2XEMHXCCQwjO8wpuVWS/Wu/Uxj5asxcwx/IH1+QPVjpHc</latexit>
S = rS · sM = rM · sL = rL · s
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Written as vector dot products:
Matrix formulation:2
4SML
3
5 =
2
4rSrMrL
3
5
2
64s
3
75
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Ren NgCS184/284A
Dimensionality Reduction From ∞ to 3
At each position on the human retina: • SPD is a function of wavelength (∞ - dimensional signal)
• 3 types of cones near that position produce three scalar values (3 - dimensional signal)
What about 2D images?
• The dimensionality reduction described above is happening at every 2D position in our visual field
Ren NgCS184/284A
The Human Visual System
• Human eye does not measure and brain does not receive information about each wavelength of light
• Rather, the eye measures three response values only (S, M, L) at each position in visual field, and this is only spectral info available to brain
• This is the result of integrating the incoming spectrum against response functions of S, M, L cones
Kayvon Fatahalian
Metamerism
Ren NgCS184/284A
Metamers
Metameters are two different spectra (∞-dim) that project to the same (S,M,L) (3-dim) response.
• These will appear to have the same color to a human The existence of metamers is critical to color reproduction
• Don’t have to reproduce the full spectrum of a real world scene
• Example: A metamer can reproduce the perceived color of a real-world scene on a display with pixels of only three colors
Ren NgCS184/284A
MetamerismColor matching is an important illusion that is understood quantitatively
Brian Wandell
Ren NgCS184/284A
Metamerism is a Big EffectBrian W
andell
Ren NgCS184/284A
Pseudo-Geometric InterpretationWe are projecting a high dimensional vector (wavelength spectrum function) onto a low-dimensional subspace (SML visual response)
• Differences that are perpendicular to the basis vectors of the low-dimensional space are not detectable
Slide credit: Steve Marschner
Color Reproduction
Ren NgCS184/284A
Additive Color
• Given a set of primary lights, each with its own spectral distribution (e.g. R,G,B display pixels):
• We can adjust the brightness of these lights and add them together to produce a linear subspace of spectral distribution:
• The color is now described by the scalar values:
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RsR(�) +GsG(�) +B sB(�)<latexit sha1_base64="LVQIOiqJL9Q+qlLtKnelhba/mrM=">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</latexit><latexit sha1_base64="LVQIOiqJL9Q+qlLtKnelhba/mrM=">AAAClHicbZHLahRBFIZr2lscbxOzyMJNYRASTZruYFAXgZAgMRuJwTGB6aaprj6TKVKXpuq0ZtI0+Bw+jVt9A9/G6plBJhMPFPx85/x1+SsvpXAYRX86wa3bd+7eW7rfffDw0eMnveWnX5ypLIc+N9LYs5w5kEJDHwVKOCstMJVLOM0vDtr+6VewThj9GcclpIqdazEUnKFHWe/1SbJJXXaynkhvKtgGfUUPk02XHc6T/Zbs/yNZby0Ko0nRmyKeibW9je+kreNsuXOUFIZXCjRyyZwbxFGJac0sCi6h6SaVg5LxC3YOAy81U+DSevK8hr7wpKBDY/3SSCd03uG3sWzcXEfKubHKvVkxHLnFXgv/1xtUOHyb1kKXFYLm07OHlaRoaBsfLYQFjnLsBeNW+OtTPmKWcfQhdxMN37hRiuniZZ343DFntqkTW0kYbMVwmdZRuFNiU2+HO3DZNAuOkbFXc452Jp2NzoxN14cfL0Z9U/S3w3dh9Ml/whGZ1hJ5Rp6TdRKTN2SPfCDHpE84+UF+kl/kd7Aa7AYHwfvpaNCZeVbItQo+/gXhesl/</latexit><latexit sha1_base64="LVQIOiqJL9Q+qlLtKnelhba/mrM=">AAAClHicbZHLahRBFIZr2lscbxOzyMJNYRASTZruYFAXgZAgMRuJwTGB6aaprj6TKVKXpuq0ZtI0+Bw+jVt9A9/G6plBJhMPFPx85/x1+SsvpXAYRX86wa3bd+7eW7rfffDw0eMnveWnX5ypLIc+N9LYs5w5kEJDHwVKOCstMJVLOM0vDtr+6VewThj9GcclpIqdazEUnKFHWe/1SbJJXXaynkhvKtgGfUUPk02XHc6T/Zbs/yNZby0Ko0nRmyKeibW9je+kreNsuXOUFIZXCjRyyZwbxFGJac0sCi6h6SaVg5LxC3YOAy81U+DSevK8hr7wpKBDY/3SSCd03uG3sWzcXEfKubHKvVkxHLnFXgv/1xtUOHyb1kKXFYLm07OHlaRoaBsfLYQFjnLsBeNW+OtTPmKWcfQhdxMN37hRiuniZZ343DFntqkTW0kYbMVwmdZRuFNiU2+HO3DZNAuOkbFXc452Jp2NzoxN14cfL0Z9U/S3w3dh9Ml/whGZ1hJ5Rp6TdRKTN2SPfCDHpE84+UF+kl/kd7Aa7AYHwfvpaNCZeVbItQo+/gXhesl/</latexit>
R, G, B<latexit sha1_base64="QByaFQdX/vzs2ihQ1+egrnL8uh0=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4KGVXBPVW9KDeqri22C4lm2bb0CS7JFmhLP0XXjyoePXnePPfmLZ70NYHA4/3ZpiZFyacaeO6387C4tLyymphrbi+sbm1XdrZfdBxqgj1Scxj1QyxppxJ6htmOG0mimIRctoIB5djv/FElWaxvDfDhAYC9ySLGMHGSo93lXblyha66JTKbtWdAM0TLydlyFHvlL7a3ZikgkpDONa65bmJCTKsDCOcjortVNMEkwHu0ZalEguqg2xy8QgdWqWLoljZkgZN1N8TGRZaD0VoOwU2fT3rjcX/vFZqorMgYzJJDZVkuihKOTIxGr+PukxRYvjQEkwUs7ci0scKE2NDKtoQvNmX54l/XD2vurcn5dpNnkYB9uEAjsCDU6jBNdTBBwISnuEV3hztvDjvzse0dcHJZ/bgD5zPH5SVjxs=</latexit><latexit sha1_base64="QByaFQdX/vzs2ihQ1+egrnL8uh0=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4KGVXBPVW9KDeqri22C4lm2bb0CS7JFmhLP0XXjyoePXnePPfmLZ70NYHA4/3ZpiZFyacaeO6387C4tLyymphrbi+sbm1XdrZfdBxqgj1Scxj1QyxppxJ6htmOG0mimIRctoIB5djv/FElWaxvDfDhAYC9ySLGMHGSo93lXblyha66JTKbtWdAM0TLydlyFHvlL7a3ZikgkpDONa65bmJCTKsDCOcjortVNMEkwHu0ZalEguqg2xy8QgdWqWLoljZkgZN1N8TGRZaD0VoOwU2fT3rjcX/vFZqorMgYzJJDZVkuihKOTIxGr+PukxRYvjQEkwUs7ci0scKE2NDKtoQvNmX54l/XD2vurcn5dpNnkYB9uEAjsCDU6jBNdTBBwISnuEV3hztvDjvzse0dcHJZ/bgD5zPH5SVjxs=</latexit><latexit sha1_base64="QByaFQdX/vzs2ihQ1+egrnL8uh0=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4KGVXBPVW9KDeqri22C4lm2bb0CS7JFmhLP0XXjyoePXnePPfmLZ70NYHA4/3ZpiZFyacaeO6387C4tLyymphrbi+sbm1XdrZfdBxqgj1Scxj1QyxppxJ6htmOG0mimIRctoIB5djv/FElWaxvDfDhAYC9ySLGMHGSo93lXblyha66JTKbtWdAM0TLydlyFHvlL7a3ZikgkpDONa65bmJCTKsDCOcjortVNMEkwHu0ZalEguqg2xy8QgdWqWLoljZkgZN1N8TGRZaD0VoOwU2fT3rjcX/vFZqorMgYzJJDZVkuihKOTIxGr+PukxRYvjQEkwUs7ci0scKE2NDKtoQvNmX54l/XD2vurcn5dpNnkYB9uEAjsCDU6jBNdTBBwISnuEV3hztvDjvzse0dcHJZ/bgD5zPH5SVjxs=</latexit>
Ren NgCS184/284A
Recall: Real LCD Screen Pixels (Closeup)
iPhone 6S Galaxy S5iphonearena.com iphonearena.com
Notice R, G, B sub-pixel geometry. Effectively three lights at each (x,y) location.
Ren NgCS184/284A
Color Reproduction Problem
Goal: at each pixel, choose R, G, B values for display so that the output color matches the appearance of the target color in the real world.
Display outputs spectrumTarget real spectrum
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s(�)<latexit sha1_base64="d4GaUWH1wM1bJ2ypPOHpJ6cs7BY=">AAAB8XicbVBNSwMxFHzrZ12/qh69BItQL2UrgnoreNFbBdcWtkvJZrNtaDZZkqxQlv4MLx5UvPpvvPlvTNs9aOtAYJiZR96bKONMG8/7dlZW19Y3Nitb7vbO7t5+9eDwUctcEeoTyaXqRlhTzgT1DTOcdjNFcRpx2olGN1O/80SVZlI8mHFGwxQPBEsYwcZKga67PW7TMT7rV2tew5sBLZNmSWpQot2vfvViSfKUCkM41jpoepkJC6wMI5xO3F6uaYbJCA9oYKnAKdVhMVt5gk6tEqNEKvuEQTP190SBU63HaWSTKTZDvehNxf+8IDfJVVgwkeWGCjL/KMk5MhJN70cxU5QYPrYEE8XsrogMscLE2JZcW0Jz8eRl4p83rhve/UWtdVe2UYFjOIE6NOESWnALbfCBgIRneIU3xzgvzrvzMY+uOOXMEfyB8/kDft2QTQ==</latexit><latexit sha1_base64="d4GaUWH1wM1bJ2ypPOHpJ6cs7BY=">AAAB8XicbVBNSwMxFHzrZ12/qh69BItQL2UrgnoreNFbBdcWtkvJZrNtaDZZkqxQlv4MLx5UvPpvvPlvTNs9aOtAYJiZR96bKONMG8/7dlZW19Y3Nitb7vbO7t5+9eDwUctcEeoTyaXqRlhTzgT1DTOcdjNFcRpx2olGN1O/80SVZlI8mHFGwxQPBEsYwcZKga67PW7TMT7rV2tew5sBLZNmSWpQot2vfvViSfKUCkM41jpoepkJC6wMI5xO3F6uaYbJCA9oYKnAKdVhMVt5gk6tEqNEKvuEQTP190SBU63HaWSTKTZDvehNxf+8IDfJVVgwkeWGCjL/KMk5MhJN70cxU5QYPrYEE8XsrogMscLE2JZcW0Jz8eRl4p83rhve/UWtdVe2UYFjOIE6NOESWnALbfCBgIRneIU3xzgvzrvzMY+uOOXMEfyB8/kDft2QTQ==</latexit><latexit sha1_base64="d4GaUWH1wM1bJ2ypPOHpJ6cs7BY=">AAAB8XicbVBNSwMxFHzrZ12/qh69BItQL2UrgnoreNFbBdcWtkvJZrNtaDZZkqxQlv4MLx5UvPpvvPlvTNs9aOtAYJiZR96bKONMG8/7dlZW19Y3Nitb7vbO7t5+9eDwUctcEeoTyaXqRlhTzgT1DTOcdjNFcRpx2olGN1O/80SVZlI8mHFGwxQPBEsYwcZKga67PW7TMT7rV2tew5sBLZNmSWpQot2vfvViSfKUCkM41jpoepkJC6wMI5xO3F6uaYbJCA9oYKnAKdVhMVt5gk6tEqNEKvuEQTP190SBU63HaWSTKTZDvehNxf+8IDfJVVgwkeWGCjL/KMk5MhJN70cxU5QYPrYEE8XsrogMscLE2JZcW0Jz8eRl4p83rhve/UWtdVe2UYFjOIE6NOESWnALbfCBgIRneIU3xzgvzrvzMY+uOOXMEfyB8/kDft2QTQ==</latexit>
Ren NgCS184/284A
Pseudo-Geometric Interpretation of Color ReproductionSlide credit: Steve M
arschner
• The display can only produce a low-dimensional subspace of all possible (linear combinations of display primaries)
• In color reproduction, for a given spectrum s (high dimensional), we want to choose a spectrum s’ in the display’s low-dimensional subspace, such that s’ and s project to the same response in the low-dimensional subspace of the eye’s SML response
Ren NgCS184/284A
Example Primaries for CRT Display
•Curves determined by phosphor emission propertieswavelength (nm)
Em
issi
on (w
atts
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Ren NgCS184/284A
Example Primaries: LCD Display
•Curves determined by backlight and color filters
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Ren NgCS184/284A
Color Reproduction as Linear Algebra
2
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3
5
<latexit sha1_base64="ySbGeev8GB1wVQekFUh7gq+TFEw=">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</latexit><latexit sha1_base64="ySbGeev8GB1wVQekFUh7gq+TFEw=">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</latexit><latexit sha1_base64="ySbGeev8GB1wVQekFUh7gq+TFEw=">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</latexit>
What R, G, B values?Input spectrum s
Ren NgCS184/284A
Color Reproduction as Linear Algebra
Spectrum produced by display given values R,G,B:
sdisp(�) = RsR(�) +GsG(�) +B sB(�)
=)
2
64sdisp
3
75 =
2
64sR sG sB
3
75
2
4RGB
3
5
<latexit sha1_base64="18wLgN673vQKKZCyt4CRfdPEkds=">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</latexit><latexit sha1_base64="18wLgN673vQKKZCyt4CRfdPEkds=">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</latexit><latexit sha1_base64="18wLgN673vQKKZCyt4CRfdPEkds=">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</latexit>
Ren NgCS184/284A
Color Reproduction as Linear Algebra
What color do we perceive when we look at the display?2
4SML
3
5
disp
=
2
4rSrMrL
3
5
2
4sdisp
3
5
=
2
4rSrMrL
3
5
2
4sR sG sB
3
5
2
4RGB
3
5
=
2
4rS · sR rS · sG rS · sBrM · sR rM · sG rM · sBrL · sR rL · sG rL · sB
3
5
2
4RGB
3
5
<latexit sha1_base64="gUtbmUZoR17U/Bu8DCv/rO5/fu4=">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</latexit><latexit sha1_base64="gUtbmUZoR17U/Bu8DCv/rO5/fu4=">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</latexit><latexit sha1_base64="gUtbmUZoR17U/Bu8DCv/rO5/fu4=">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</latexit>
We want this displayed spectrum to be a metamer for the real-world target spectrum.
Ren NgCS184/284A
Color Reproduction as Linear Algebra
2
4SML
3
5
real
=
2
4rSrMrL
3
5
2
64s
3
75
<latexit sha1_base64="vpwXq4grUl/G2ryr3gQ4FdsVkSE=">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</latexit><latexit sha1_base64="Cgo6Dv3RPA01cjs8LCaOp+n5NJs=">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</latexit><latexit sha1_base64="Cgo6Dv3RPA01cjs8LCaOp+n5NJs=">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</latexit>
Color perceived for real scene spectra, s
How do we reproduce the color of s? Set these lines equal and solve for R,G,B as a function of s!
2
4SML
3
5
disp
=
2
4rSrMrL
3
5
2
64sR sG sB
3
75
2
4RGB
3
5
<latexit sha1_base64="c9XrMfzwWCr0TFZym5LUbsxONJ0=">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</latexit><latexit sha1_base64="Wtc3NRdxUn3t01iGA0VslKbl3WM=">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</latexit><latexit sha1_base64="Wtc3NRdxUn3t01iGA0VslKbl3WM=">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</latexit>
Color perceived for display spectra with values R,G,B
Ren NgCS184/284A
Color Reproduction as Linear Algebra
2
4SML
3
5
real
=
2
4rSrMrL
3
5
2
64s
3
75
<latexit sha1_base64="vpwXq4grUl/G2ryr3gQ4FdsVkSE=">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</latexit><latexit sha1_base64="Cgo6Dv3RPA01cjs8LCaOp+n5NJs=">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</latexit><latexit sha1_base64="Cgo6Dv3RPA01cjs8LCaOp+n5NJs=">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</latexit>
2
4SML
3
5
disp
=
2
4rSrMrL
3
5
2
64sR sG sB
3
75
2
4RGB
3
5
<latexit sha1_base64="c9XrMfzwWCr0TFZym5LUbsxONJ0=">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</latexit><latexit sha1_base64="Wtc3NRdxUn3t01iGA0VslKbl3WM=">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</latexit><latexit sha1_base64="Wtc3NRdxUn3t01iGA0VslKbl3WM=">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</latexit>
Solution:
=)
2
4RGB
3
5 =
0
B@
2
4rSrMrL
3
5
2
64sR sG sB
3
75
1
CA
�1 2
4rSrMrL
3
5
2
64s
3
75
<latexit sha1_base64="dlU4UJyiFCkxP8TewoVADb5Yn24=">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</latexit><latexit sha1_base64="dlU4UJyiFCkxP8TewoVADb5Yn24=">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</latexit><latexit sha1_base64="dlU4UJyiFCkxP8TewoVADb5Yn24=">AAAEg3iczVNdb9MwFPVCGCN8bIVHXqxNTBtiVTKYgIdJ03gYk0Aag7JJdagc97a15jiR7UBLFIlXfg2v8FP4NzhpOvoxaa9cKcrJ8T33nlzbUSq4Nr7/Z8m54d5cvrVy27tz99791bXGg086yRSDFktEos4jqkFwCS3DjYDzVAGNIwFn0cXrcv3sCyjNE/nRjFIIY9qXvMcZNZbqNJx1wmPbCLRHIuhzmUcxNYoPC++UkCNCDgnxCMjuP37fIwJ6ZsvDeF6CbZBBor5FVOFNrDr5h8K+LylbazHl3fUpb6dTyr4zhq50YX/ajPWXcBrXbXQnPy1KsFnCowqO8WGFJ26uqbZoSfH+wGx/zneC4r+d08S+LaXLXa4/F3SdtQ2/6VeBF0FQg42D7e+ojJNOY+mYdBOWxSANE1TrduCnJsypMpwJsJ0zDSllF7QPbQsljUGHeXWaC/zYMl3cS5R9pMEVO62wZRQdFbNUrPUojqzYmh7o+bWSvGqtnZneyzDnMs0MSDbu3csENgkubwvucgXMiJEFlClu7WM2oIoyY++URyR8ZUkcU9l9kk/GV+REZQLaOwEMw9xv7qWmyJ8192BYFHOKeqMmijInrFNrYTX8YH7Ui6C123zV9N/bTThG41hBj9A62kIBeoEO0Bt0glqIOT+cn84v57e77D51d93n41RnqdY8RDPh7v8FkKxtWQ==</latexit>
Ren NgCS184/284A
Color Reproduction as Linear Algebra
2
4RGB
3
5 =
0
B@
2
4rSrMrL
3
5
2
64sR sG sB
3
75
1
CA
�1 2
4rSrMrL
3
5
2
64s
3
75
<latexit sha1_base64="Wl0252l/Dqwpjl1qZoH92XkycxE=">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</latexit><latexit sha1_base64="Wl0252l/Dqwpjl1qZoH92XkycxE=">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</latexit><latexit sha1_base64="Wl0252l/Dqwpjl1qZoH92XkycxE=">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</latexit>
Solution (form #1):
1xNNx33xNNx31x3
1x33x3
RGB = (MSML MRGB)�1MSML s
<latexit sha1_base64="/6n86shPh1lUXOidu5AVGuU3I4U=">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</latexit><latexit sha1_base64="/6n86shPh1lUXOidu5AVGuU3I4U=">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</latexit><latexit sha1_base64="/6n86shPh1lUXOidu5AVGuU3I4U=">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</latexit>
Solution (form #2):
1x3 Nx3 3xN Nx3 1xN
Nx3
Ren NgCS184/284A
Color Reproduction as Linear Algebra
2
4RGB
3
5 =
2
4rS · sR rS · sG rS · sBrM · sR rM · sG rM · sBrL · sR rL · sG rL · sB
3
5�1 2
4rSrMrL
3
5
2
4s
3
5
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Solution (form #3):
Nx3
Ren NgCS184/284A
Color Reproduction Issue: No Negative Light
R,G,B values must be positive
• Display primaries can’t emit negative light
• But solution formulas can certainly produce negative R,G,B values
What do negative R,G,B values mean?
• Display can’t physically reproduce the desired color
• Desired color is outside the display’s color gamut (more on this later)
To Be Continued