2D Computer Graphics Diego Nehab Summer 2020 IMPA 1 Color and - - PowerPoint PPT Presentation

2D Computer Graphics

Diego Nehab Summer 2020

IMPA 1

Color and compositing

The prism experiment

2

More than visible light

Visible light: prism experiment (Newton, 1666) Infrared light: thermometers (Herschel, 1800) Ultraviolet light: silver chloride (Ritter, 1801)

3

More than visible light

Visible light: prism experiment (Newton, 1666) Infrared light: thermometers (Herschel, 1800) Ultraviolet light: silver chloride (Ritter, 1801)

3

More than visible light

Visible light: prism experiment (Newton, 1666) Infrared light: thermometers (Herschel, 1800) Ultraviolet light: silver chloride (Ritter, 1801)

3

Full electromagnetic spectrum

4

Radiometry

Measurement of radiant energy in terms of absolute power Wave vs. particle

• Wavelength ( ), frequency (

c ), and amplitude (A)

• Energy (E

h , where h is Planck’s constant) and fmux ( ) Pure spectral light (monochromatic colors) Spectrometer

5

Radiometry

Measurement of radiant energy in terms of absolute power Wave vs. particle

• Wavelength (λ), frequency (ν = c

λ), and amplitude (A)

• Energy (E = hν, where h is Planck’s constant) and fmux (Φ)

Pure spectral light (monochromatic colors) Spectrometer

5

Radiometry

Measurement of radiant energy in terms of absolute power Wave vs. particle

• Wavelength (λ), frequency (ν = c

λ), and amplitude (A)

• Energy (E = hν, where h is Planck’s constant) and fmux (Φ)

Pure spectral light (monochromatic colors) Spectrometer

5

Radiometry

Measurement of radiant energy in terms of absolute power Wave vs. particle

• Wavelength (λ), frequency (ν = c

λ), and amplitude (A)

• Energy (E = hν, where h is Planck’s constant) and fmux (Φ)

Pure spectral light (monochromatic colors) Spectrometer

5

Colors are spectral distributions

6

Spectral representation

As a continuous function of c(λ) c : R>0 → R≥0, λ → Aλ As a discrete set of values c

i

c

1 2 n

R R

i

A

i

Light emitter has a spectrum, material properties modulate the refmected spectrum (Fluorescence is something else)

7

Spectral representation

As a continuous function of c(λ) c : R>0 → R≥0, λ → Aλ As a discrete set of values c(λi) c : {λ1, λ2, . . . , λn} ⊂ R>0 → R≥0 λi → Aλi Light emitter has a spectrum, material properties modulate the refmected spectrum (Fluorescence is something else)

7

Spectral representation

As a continuous function of c(λ) c : R>0 → R≥0, λ → Aλ As a discrete set of values c(λi) c : {λ1, λ2, . . . , λn} ⊂ R>0 → R≥0 λi → Aλi Light emitter has a spectrum, material properties modulate the refmected spectrum (Fluorescence is something else)

7

Black-body radiation

B(ν, T) = 2hν3 c2

• e

hν kBT − 1

−1, where kB is Boltsmann’s constant

2E+11 4E+11 6E+11 8E+11 500 1000 1500 2000 Spectral energy density / kJ/m3nm Wavelength / nm 3500K 4000K 4500K 5000K 5500K

8

Photometry

Measurement light in terms of perceived brightness to human eye Visible light 390nm 700nm approximately

9

Photometry

Measurement light in terms of perceived brightness to human eye Visible light λ ∈ [390nm, 700nm] approximately

9

Photometry

Measurement light in terms of perceived brightness to human eye Visible light λ ∈ [390nm, 700nm] approximately

Film or sensor Mirror Lenses Aperture Prism Viewfinder

9

Photopic vision

Well-lit conditions Cones: Three types of retinal cells with distinct spectral responses Highly concentrated on fovea Response curves S (short ), M (medium ), L (long )

• Peaks at

420nm, 534nm, and 564nm

• Overlap each other
• Not R, G, and B

What about the color-blind? Are there tetrachromats among us?

10

Photopic vision

Well-lit conditions Cones: Three types of retinal cells with distinct spectral responses Highly concentrated on fovea Response curves S (short ), M (medium ), L (long )

• Peaks at

420nm, 534nm, and 564nm

• Overlap each other
• Not R, G, and B

What about the color-blind? Are there tetrachromats among us?

10

Photopic vision

Well-lit conditions Cones: Three types of retinal cells with distinct spectral responses Highly concentrated on fovea Response curves S (short ), M (medium ), L (long )

• Peaks at

420nm, 534nm, and 564nm

• Overlap each other
• Not R, G, and B

What about the color-blind? Are there tetrachromats among us?

10

Photopic vision

Well-lit conditions Cones: Three types of retinal cells with distinct spectral responses Highly concentrated on fovea Response curves S (short λ), M (medium λ), L (long λ)

• Peaks at λ = 420nm, λ = 534nm, and λ = 564nm
• Overlap each other
• Not R, G, and B

What about the color-blind? Are there tetrachromats among us?

10

Photopic vision

Well-lit conditions Cones: Three types of retinal cells with distinct spectral responses Highly concentrated on fovea Response curves S (short λ), M (medium λ), L (long λ)

• Peaks at λ = 420nm, λ = 534nm, and λ = 564nm
• Overlap each other
• Not R, G, and B

What about the color-blind? Are there tetrachromats among us?

10

Photopic vision

Well-lit conditions Cones: Three types of retinal cells with distinct spectral responses Highly concentrated on fovea Response curves S (short λ), M (medium λ), L (long λ)

• Peaks at λ = 420nm, λ = 534nm, and λ = 564nm
• Overlap each other
• Not R, G, and B

What about the color-blind? Are there tetrachromats among us?

10

Scotopic vision

Low-light conditions Rods: One type of retinal cell Mostly peripheral 20 more numerous, 1000 more sensitive than cones Response curve

• R: peak at

498nm (between S and M) Things look “gray-bluish” at night

11

Scotopic vision

Low-light conditions Rods: One type of retinal cell Mostly peripheral 20 more numerous, 1000 more sensitive than cones Response curve

• R: peak at

498nm (between S and M) Things look “gray-bluish” at night

11

Scotopic vision

Low-light conditions Rods: One type of retinal cell Mostly peripheral 20 more numerous, 1000 more sensitive than cones Response curve

• R: peak at

498nm (between S and M) Things look “gray-bluish” at night

11

Scotopic vision

Low-light conditions Rods: One type of retinal cell Mostly peripheral 20× more numerous, 1000× more sensitive than cones Response curve

• R: peak at

498nm (between S and M) Things look “gray-bluish” at night

11

Scotopic vision

Low-light conditions Rods: One type of retinal cell Mostly peripheral 20× more numerous, 1000× more sensitive than cones Response curve

• R: peak at λ = 498nm (between S and M)

Things look “gray-bluish” at night

11

Scotopic vision

Low-light conditions Rods: One type of retinal cell Mostly peripheral 20× more numerous, 1000× more sensitive than cones Response curve

• R: peak at λ = 498nm (between S and M)

Things look “gray-bluish” at night

11

Human photoreceptor distribution

Temporal Eccentricity (degrees) Nasal 20 40 60 80 100 120 140 160 Receptor density (mm−2 × 103) 80 80 60 60 40 40 20 20 Cones Cones Rods Rods Optic disk 12

Luminous efficiency function

Spectral sensitivity V(λ) of human perception of brightness Different for photopic and scotopic vision Immense dynamic range 1 1010 (brightness adaptation) Convert radiant intensity (W/sr) to luminous intensity (cd) v c c V d

13

Luminous efficiency function

Spectral sensitivity V(λ) of human perception of brightness Different for photopic and scotopic vision Immense dynamic range 1 1010 (brightness adaptation) Convert radiant intensity (W/sr) to luminous intensity (cd) v c c V d

13

Luminous efficiency function

Spectral sensitivity V(λ) of human perception of brightness Different for photopic and scotopic vision Immense dynamic range 1 : 1010 (brightness adaptation) Convert radiant intensity (W/sr) to luminous intensity (cd) v c c V d

13

Luminous efficiency function

Spectral sensitivity V(λ) of human perception of brightness Different for photopic and scotopic vision Immense dynamic range 1 : 1010 (brightness adaptation) Convert radiant intensity (W/sr) to luminous intensity (cd) v(c) =

• λ

c(λ)V(λ)dλ

13

Photopic luminous efficiency function

14

Lightness

Nonlinear perceptual response to brightness Power law L 116 100 Y Y0

1 3

0 16 Weber law of just noticeable difference L Y 100

15

Lightness

Nonlinear perceptual response to brightness Power law L∗ ≈ 116 100 Y Y0 1

3

− 0.16 Weber law of just noticeable difference L Y 100

15

Lightness

Nonlinear perceptual response to brightness Power law L∗ ≈ 116 100 Y Y0 1

3

− 0.16 Weber law of just noticeable difference ∆L∗ ≈ Y 100

15

Gamma correction

Created to compensate for input-output characteristic of CRT displays Y = V2.5 = Vγ Today, due to remarkable coincidence, it is used for encoding effjciency

16

Gamma correction

Created to compensate for input-output characteristic of CRT displays Y = V2.5 = Vγ Today, due to remarkable coincidence, it is used for encoding effjciency

16

Gamma correction

Created to compensate for input-output characteristic of CRT displays Y = V2.5 = Vγ Today, due to remarkable coincidence, it is used for encoding effjciency

16

Gamma correction

Created to compensate for input-output characteristic of CRT displays Y = V2.5 = Vγ Today, due to remarkable coincidence, it is used for encoding effjciency

16

Illusion

17

Modeling color perception

1st attempt: Measure spectral distribution of stimulus

• Convex combinations of monochromatic colors
• Could use spectrophotometer to measure c(λ).
• But how to would you reproduce it?

2nd attempt: Measure optical nerve response

• Remove eye, attach wires to cones: The Matrix
• Re-inject signal to reproduce
• Painful, but only 3 values per color

18

Modeling color perception

1st attempt: Measure spectral distribution of stimulus

• Convex combinations of monochromatic colors
• Could use spectrophotometer to measure c(λ).
• But how to would you reproduce it?

2nd attempt: Measure optical nerve response

• Remove eye, attach wires to cones: The Matrix
• Re-inject signal to reproduce
• Painful, but only 3 values per color

18

Modeling color perception

3rd attempt: Linear algebra Measuring

• c

is the target color’s spectral distribution

• L

, M , and S the spectral sensitivities for the cones

• Inner-product functions f and g is

f g f g d

• The cone responses to c must be

Sc c S Mc c M and Lc c L

19

Modeling color perception

3rd attempt: Linear algebra Measuring

• c(λ) is the target color’s spectral distribution
• L

, M , and S the spectral sensitivities for the cones

• Inner-product functions f and g is

f g f g d

• The cone responses to c must be

Sc c S Mc c M and Lc c L

19

Modeling color perception

3rd attempt: Linear algebra Measuring

• c(λ) is the target color’s spectral distribution
• L(λ), M(λ), and S(λ) the spectral sensitivities for the cones
• Inner-product functions f and g is

f g f g d

• The cone responses to c must be

Sc c S Mc c M and Lc c L

19

Modeling color perception

3rd attempt: Linear algebra Measuring

• c(λ) is the target color’s spectral distribution
• L(λ), M(λ), and S(λ) the spectral sensitivities for the cones
• Inner-product functions f and g is

f, g = ∞

−∞

f(λ)g(λ)dλ

• The cone responses to c must be

Sc c S Mc c M and Lc c L

19

Modeling color perception

3rd attempt: Linear algebra Measuring

• c(λ) is the target color’s spectral distribution
• L(λ), M(λ), and S(λ) the spectral sensitivities for the cones
• Inner-product functions f and g is

f, g = ∞

−∞

f(λ)g(λ)dλ

• The cone responses to c must be

Sc = c, S, Mc = c, M, and Lc = c, L

19

Cone spectral sensitivities (not to scale)

20

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

Rc r Gc g Bc b S Sc Rc r Gc g Bc b M Mc Rc r Gc g Bc b L Lc Sr Sg Sb Mr Mg Mb Lr Lg Lb Rc Gc Bc Sc Mc Lc Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

Rc r Gc g Bc b S Sc Rc r Gc g Bc b M Mc Rc r Gc g Bc b L Lc Sr Sg Sb Mr Mg Mb Lr Lg Lb Rc Gc Bc Sc Mc Lc Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

Rc r Gc g Bc b S Sc Rc r Gc g Bc b M Mc Rc r Gc g Bc b L Lc Sr Sg Sb Mr Mg Mb Lr Lg Lb Rc Gc Bc Sc Mc Lc Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

       Rc r + Gc g + Bc b, S = Sc Rc r + Gc g + Bc b, M = Mc Rc r + Gc g + Bc b, L = Lc Sr Sg Sb Mr Mg Mb Lr Lg Lb Rc Gc Bc Sc Mc Lc Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

       Rc r + Gc g + Bc b, S = Sc Rc r + Gc g + Bc b, M = Mc Rc r + Gc g + Bc b, L = Lc ⇔    Sr Sg Sb Mr Mg Mb Lr Lg Lb       Rc Gc Bc    =    Sc Mc Lc    Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

       Rc r + Gc g + Bc b, S = Sc Rc r + Gc g + Bc b, M = Mc Rc r + Gc g + Bc b, L = Lc ⇔    Sr Sg Sb Mr Mg Mb Lr Lg Lb       Rc Gc Bc    =    Sc Mc Lc    Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Modeling color perception

Reproduction

• Assume 3 different stimuli colors r(λ), g(λ), and b(λ)
• Find stimuli intensities Rc, Gc and Bc that correspond to c
• I.e., intensities that reproduce responses Sc, Mc, and Lc

       Rc r + Gc g + Bc b, S = Sc Rc r + Gc g + Bc b, M = Mc Rc r + Gc g + Bc b, L = Lc ⇔    Sr Sg Sb Mr Mg Mb Lr Lg Lb       Rc Gc Bc    =    Sc Mc Lc    Stimuli must be linearly independent Result Rc, Gc, or Bc could be non-convex

• There is no negative light…

21

Space of visible colors

All convex combinations of visible monochromatic colors

• Could use entire spectrum

Unnecessary (most of the time) due to metamerism

• Different spectra result in same perceived color S

M L

• E.g., c and Rcr

Gcg Bcb Obtain Rc, Gc, and Bc directly from c and RGB color matching functions Rc c R Gc c G Bc c B How to measure color matching functions R, G, and B

22

Space of visible colors

All convex combinations of visible monochromatic colors

• Could use entire spectrum

Unnecessary (most of the time) due to metamerism

• Different spectra result in same perceived color
• S

M L

• E.g., c and Rcr + Gcg + Bcb

Obtain Rc, Gc, and Bc directly from c and RGB color matching functions Rc c R Gc c G Bc c B How to measure color matching functions R, G, and B

22

Space of visible colors

All convex combinations of visible monochromatic colors

• Could use entire spectrum

Unnecessary (most of the time) due to metamerism

• Different spectra result in same perceived color
• S

M L

• E.g., c and Rcr + Gcg + Bcb

Obtain Rc, Gc, and Bc directly from c and RGB color matching functions Rc = c, R Gc = c, G Bc = c, B. How to measure color matching functions R, G, and B

22

Space of visible colors

All convex combinations of visible monochromatic colors

• Could use entire spectrum

Unnecessary (most of the time) due to metamerism

• Different spectra result in same perceived color
• S

M L

• E.g., c and Rcr + Gcg + Bcb

Obtain Rc, Gc, and Bc directly from c and RGB color matching functions Rc = c, R Gc = c, G Bc = c, B. How to measure color matching functions R, G, and B

22

CIE 1931 RGB color matching functions

23

XYZ color matching functions

Visible colors always use non-negative coordinates Linear transformation to R, G, B Y is the photopic luminosity function Equal-energy radiator (constant SPD in visible spectrum, illuminant E) is at

1 3 1 3 1 3

Z ended up almost equal to S

24

XYZ color matching functions

Visible colors always use non-negative coordinates Linear transformation to R, G, B Y is the photopic luminosity function Equal-energy radiator (constant SPD in visible spectrum, illuminant E) is at

1 3 1 3 1 3

Z ended up almost equal to S

24

XYZ color matching functions

Visible colors always use non-negative coordinates Linear transformation to R, G, B Y is the photopic luminosity function Equal-energy radiator (constant SPD in visible spectrum, illuminant E) is at

1 3 1 3 1 3

Z ended up almost equal to S

24

XYZ color matching functions

Visible colors always use non-negative coordinates Linear transformation to R, G, B Y is the photopic luminosity function Equal-energy radiator (constant SPD in visible spectrum, illuminant E) is at

• 1

3 1 3 1 3

• Z ended up almost equal to S

24

XYZ color matching functions

Visible colors always use non-negative coordinates Linear transformation to R, G, B Y is the photopic luminosity function Equal-energy radiator (constant SPD in visible spectrum, illuminant E) is at

• 1

3 1 3 1 3

• Z ended up almost equal to S

24

XYZ color matching functions

25

CIE chromaticity diagram

Similar to RP2

• Given α > 0,
• αX

αY αZ

• have same chromaticity
• Different brightness

Separation of chromaticity and brightness x X X Y Z y Y X Y Z

26

CIE chromaticity diagram

Similar to RP2

• Given α > 0,
• αX

αY αZ

• have same chromaticity
• Different brightness

Separation of chromaticity and brightness x = X X + Y + Z y = Y X + Y + Z

26

CIE chromaticity diagram

27

CIE chromaticity diagram

Horseshoe shape Locus of monochromatic colors Locus of black-body colors Line of purples Color gamut Color calibration and matching

28

CIE chromaticity diagram

Horseshoe shape Locus of monochromatic colors Locus of black-body colors Line of purples Color gamut Color calibration and matching

28

CIE chromaticity diagram

Horseshoe shape Locus of monochromatic colors Locus of black-body colors Line of purples Color gamut Color calibration and matching

28

CIE chromaticity diagram

Horseshoe shape Locus of monochromatic colors Locus of black-body colors Line of purples Color gamut Color calibration and matching

28

CIE chromaticity diagram

Horseshoe shape Locus of monochromatic colors Locus of black-body colors Line of purples Color gamut Color calibration and matching

28

CIE chromaticity diagram

Horseshoe shape Locus of monochromatic colors Locus of black-body colors Line of purples Color gamut Color calibration and matching

28

Other color spaces

sRGB [IEC Project Team 61966, 1998]    R G B    =    γ(Rℓ) γ(Gℓ) γ(Bℓ)    ,    Rℓ Gℓ Bℓ    =    3.2406 −1.5372 −0.4986 −0.9689 1.8758 0.0415 0.0557 −0.2040 1.0570       XD65 YD65 ZD65    γ(u) =    12.92u u < 0.0031308 1.055u1/2.4 − 0.055

• therwise

Munsel (HSV and HSL) Additive (CMY and CMYK) TV (PAL YUV, NTSC YIQ) Perceptual (CIE L a b ) Opponent color models

29

Other color spaces

sRGB [IEC Project Team 61966, 1998]    R G B    =    γ(Rℓ) γ(Gℓ) γ(Bℓ)    ,    Rℓ Gℓ Bℓ    =    3.2406 −1.5372 −0.4986 −0.9689 1.8758 0.0415 0.0557 −0.2040 1.0570       XD65 YD65 ZD65    γ(u) =    12.92u u < 0.0031308 1.055u1/2.4 − 0.055

• therwise

Munsel (HSV and HSL) Additive (CMY and CMYK) TV (PAL YUV, NTSC YIQ) Perceptual (CIE L a b ) Opponent color models

29

Other color spaces

sRGB [IEC Project Team 61966, 1998]    R G B    =    γ(Rℓ) γ(Gℓ) γ(Bℓ)    ,    Rℓ Gℓ Bℓ    =    3.2406 −1.5372 −0.4986 −0.9689 1.8758 0.0415 0.0557 −0.2040 1.0570       XD65 YD65 ZD65    γ(u) =    12.92u u < 0.0031308 1.055u1/2.4 − 0.055

• therwise

Munsel (HSV and HSL) Additive (CMY and CMYK) TV (PAL YUV, NTSC YIQ) Perceptual (CIE L a b ) Opponent color models

29

Other color spaces

sRGB [IEC Project Team 61966, 1998]    R G B    =    γ(Rℓ) γ(Gℓ) γ(Bℓ)    ,    Rℓ Gℓ Bℓ    =    3.2406 −1.5372 −0.4986 −0.9689 1.8758 0.0415 0.0557 −0.2040 1.0570       XD65 YD65 ZD65    γ(u) =    12.92u u < 0.0031308 1.055u1/2.4 − 0.055

• therwise

Munsel (HSV and HSL) Additive (CMY and CMYK) TV (PAL YUV, NTSC YIQ) Perceptual (CIE L a b ) Opponent color models

29

Other color spaces

sRGB [IEC Project Team 61966, 1998]    R G B    =    γ(Rℓ) γ(Gℓ) γ(Bℓ)    ,    Rℓ Gℓ Bℓ    =    3.2406 −1.5372 −0.4986 −0.9689 1.8758 0.0415 0.0557 −0.2040 1.0570       XD65 YD65 ZD65    γ(u) =    12.92u u < 0.0031308 1.055u1/2.4 − 0.055

• therwise

Munsel (HSV and HSL) Additive (CMY and CMYK) TV (PAL YUV, NTSC YIQ) Perceptual (CIE L∗a∗b∗) Opponent color models

29

Other color spaces

sRGB [IEC Project Team 61966, 1998]    R G B    =    γ(Rℓ) γ(Gℓ) γ(Bℓ)    ,    Rℓ Gℓ Bℓ    =    3.2406 −1.5372 −0.4986 −0.9689 1.8758 0.0415 0.0557 −0.2040 1.0570       XD65 YD65 ZD65    γ(u) =    12.92u u < 0.0031308 1.055u1/2.4 − 0.055

• therwise

Munsel (HSV and HSL) Additive (CMY and CMYK) TV (PAL YUV, NTSC YIQ) Perceptual (CIE L∗a∗b∗) Opponent color models

29

Transparency

Seminal work by Porter and Duff [1984]

Semitransparent color f on top of opaque background color b

• Assume probability of light hitting f is α
• Refmected color (integrated over small area) is

f, α ⊕ b = αf + (1 − α)b

• This is what we call alpha blending or the over operator

30

Compositing

Now imagine f1, α1 on top of f2, α2 on top of b

• Refmected color is

f1, α1 ⊕ (f2, α2 ⊕ b) = α1f1 + (1 − α1)

• α2f2 + (1 − α2)b
• Can we combine f1

1 and f2 2 into a single material f

? f 1 b

1f1

1

1 2f2

1

2 b 1f1

1

1 2f2

1

1

1

2 b

So we have 1 b 1

1

1

2 b

f

1f1

1

1 2f2 1

1

1 2

f

1f1

1

1 2f2 31

Compositing

Now imagine f1, α1 on top of f2, α2 on top of b

• Refmected color is

f1, α1 ⊕ (f2, α2 ⊕ b) = α1f1 + (1 − α1)

• α2f2 + (1 − α2)b
• Can we combine f1, α1 and f2, α2 into a single material f, α?

αf + (1 − α)b = α1f1 + (1 − α1)

• α2f2 + (1 − α2)b
• = α1f1 + (1 − α1)α2f2 + (1 − α1)(1 − α2)b

So we have 1 b 1

1

1

2 b

f

1f1

1

1 2f2 1

1

1 2

f

1f1

1

1 2f2 31

Compositing

Now imagine f1, α1 on top of f2, α2 on top of b

• Refmected color is

f1, α1 ⊕ (f2, α2 ⊕ b) = α1f1 + (1 − α1)

• α2f2 + (1 − α2)b
• Can we combine f1, α1 and f2, α2 into a single material f, α?

αf + (1 − α)b = α1f1 + (1 − α1)

• α2f2 + (1 − α2)b
• = α1f1 + (1 − α1)α2f2 + (1 − α1)(1 − α2)b

So we have    (1 − α)b = (1 − α1)(1 − α2)b αf = α1f1 + (1 − α1)α2f2 ⇒    α = α1 + (1 − α1)α2 αf = α1f1 + (1 − α1)α2f2

31

Compositing

   (1 − α)b = (1 − α1)(1 − α2)b αf = α1f1 + (1 − α1)α2f2 ⇒    α = α1 + (1 − α1)α2 αf = α1f1 + (1 − α1)α2f2 Setting f f, f1

1f1,

and f2

2f2, we obtain 1

1

1 2

f f1 1

1 f2

This is what we call pre-multiplied alpha Blending becomes associative f1

1

f2

2

b f1

1

f2

2

b Should we blend front-to-back or back-to-front?

32

Compositing

   (1 − α)b = (1 − α1)(1 − α2)b αf = α1f1 + (1 − α1)α2f2 ⇒    α = α1 + (1 − α1)α2 αf = α1f1 + (1 − α1)α2f2 Setting ˜ f = αf, ˜ f1 = α1f1, and ˜ f2 = α2f2, we obtain    α = α1 + (1 − α1)α2 ˜ f = ˜ f1 + (1 − α1)˜ f2 This is what we call pre-multiplied alpha Blending becomes associative f1

1

f2

2

b f1

1

f2

2

b Should we blend front-to-back or back-to-front?

32

Compositing

   (1 − α)b = (1 − α1)(1 − α2)b αf = α1f1 + (1 − α1)α2f2 ⇒    α = α1 + (1 − α1)α2 αf = α1f1 + (1 − α1)α2f2 Setting ˜ f = αf, ˜ f1 = α1f1, and ˜ f2 = α2f2, we obtain    α = α1 + (1 − α1)α2 ˜ f = ˜ f1 + (1 − α1)˜ f2 This is what we call pre-multiplied alpha Blending becomes associative f1

1

f2

2

b f1

1

f2

2

b Should we blend front-to-back or back-to-front?

32

Compositing

   (1 − α)b = (1 − α1)(1 − α2)b αf = α1f1 + (1 − α1)α2f2 ⇒    α = α1 + (1 − α1)α2 αf = α1f1 + (1 − α1)α2f2 Setting ˜ f = αf, ˜ f1 = α1f1, and ˜ f2 = α2f2, we obtain    α = α1 + (1 − α1)α2 ˜ f = ˜ f1 + (1 − α1)˜ f2 This is what we call pre-multiplied alpha Blending becomes associative ˜ f1, α1 ⊕ (˜ f2, α2 ⊕ b) = (˜ f1, α1 ⊕ ˜ f2, α2) ⊕ b Should we blend front-to-back or back-to-front?

32

Compositing

   (1 − α)b = (1 − α1)(1 − α2)b αf = α1f1 + (1 − α1)α2f2 ⇒    α = α1 + (1 − α1)α2 αf = α1f1 + (1 − α1)α2f2 Setting ˜ f = αf, ˜ f1 = α1f1, and ˜ f2 = α2f2, we obtain    α = α1 + (1 − α1)α2 ˜ f = ˜ f1 + (1 − α1)˜ f2 This is what we call pre-multiplied alpha Blending becomes associative ˜ f1, α1 ⊕ (˜ f2, α2 ⊕ b) = (˜ f1, α1 ⊕ ˜ f2, α2) ⊕ b Should we blend front-to-back or back-to-front?

32

References

IEC Project Team 61966. Colour measurement and management in multimedia systems and equipment. IEC/4WD 61966-2-1, 1998. Part 2.1: Default RGB colour space — sRGB.

• B. MacEvoy. Hardprint: Color vision, 2015. URL

http://www.handprint.com/LS/CVS/color.html.

• T. Porter and T. Duff. Compositing digital images. Computer Graphics

(Proceedings of ACM SIGGRAPH 1984), 18(3):253–259, 1984.

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