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Purdue ECE 638
Color Science
Color Basic Concepts
- Trichromatic theory of color: interaction of stimulus and 3 different types of receptors in eyes.
- Color matching experiment: half of the area display with $C_{Target}$, the other half display with $C_{match} = C_{1} + C_{2} + C_{3}$ for additive system. Let examer to match the two colors
- Spectral representation of color
- Stimulus for eyes: $S(\lambda) = R(\lambda)\cdot I(\lambda)$, where $R$ is reflectance of the surface, $I$ is the illuminant ray shoot on the surface
- Have to be normal, due to angular dependence
- Trichromatic sensor model, where $(R_{s}, G_{s}, B_{s})$ is stimulus vector, $Q$ is spectral response functions of the sensor (eyes/HVS, cameras, etc.)
- $R_{s} = \int S(\lambda)Q_{R}(\lambda)d\lambda$
- $G_{s} = \int S(\lambda)Q_{G}(\lambda)d\lambda$
- $B_{s} = \int S(\lambda)Q_{B}(\lambda)d\lambda$
- Metamerism: stimulus $S_{1}(\lambda)$ and $S_{2}(\lambda)$ are metameric with respect to $(Q_{R}(\lambda), Q_{G}(\lambda), Q_{B}(\lambda))$ is they have identical responses
- Sensor response: R,G,B axies are no need to be perpendicular to each other. We force the color plane constructed with $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Scaling of color will only affect lightness.
- Sensor space: $R + G + B = K$ will appears equally light/bright; $R + G + B = 1$ for chromaticity diagram
- Chromaticity diagram
- $C = (r,g,b)$ is a chromaticity coordinates
- $r = \frac{R}{R+G+B}$, $g = \frac{G}{R+G+B}$, $b = \frac{B}{R+G+B}$, thus, $r+g+b=1$
- The chromaticity diagram formed with $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ is an equilateral triangle with $\sqrt{2}$ for each side.
- The saturation is the distance from the origin to the color chromaticity coordinates, the hue is the angle between the G axis to the vector
- Spectral locus: the resulting curve from the response of a sensor to an arbitrary stinulus in the chromaticity diagram.
- Chromaticity gamut: for a sensor, chromaticity gamut is the set of all points in the chromaticity space that correspond to the response of that sensor to some real stimulus.
- Primaries: a combined stimulis of $P(\lambda)=p_{R}P_{R}(\lambda)+p_{G}P_{G}(\lambda)+p_{B}P_{B}(\lambda)$, where $P_{R}$, $P_{G}$, and $P_{B}$ are primaries. Here, $R$, $G$, and $B$ are just arbitrary spectral densities
- Additive devices: CRT monitor, rear projection TV, LCD
- Sensor response to additive primary
- Snesor response to stimuli p: $\vec{C_{p}}=A\vec{p}$, wehre $\vec{C_{p}}=\begin{bmatrix} R_{P}&G_{P}&R_{P}\end{bmatrix}^{T}$, amount of each primary $\vec{p}=\begin{bmatrix} p_{R}&p_{G}&p_{B}\end{bmatrix}^{T}$
- Color matching function: $A = [a_{i,j}]$, $a_{i,j}=\int p_{j}(\lambda)Q_{i}(\lambda)d\lambda$, where $i,j\in {R, G, B}$, $i$ row is the response of channel $i$ to 3 primaries, $j$ column is the response of 3 channels to primary $j$, $Q$ is spectral response function of the sensor
- Relation between sensor and color matching function
- Color matching function is a sensor subspace: $\begin{bmatrix} \bar{r}(\mu) \\ \bar{g}(\mu) \\ \bar{b}(\mu)\end{bmatrix} = A^{-1}\begin{bmatrix} \bar{Q_{R}}(\mu) \\ \bar{Q_{G}}(\mu) \\ \bar{Q_{B}}(\mu)\end{bmatrix}$
- Color matching functions are a linear combination of the sensor response functions
- Color matching functions can be directly measured
- CIE RGB Observer
- Color matching functions: $\hat{r}$ is negative in range of 435.8~546.1mm
- Chromaticity diagram: the shape distorted to negative in R direction
- CIE XYZ Observer
- Linear transformation from the color matching functions of CIE RGB observer
- Color matching functions $[\bar{x}(\lambda), \bar{y}(\lambda), \bar{z}(\lambda)]$ are non-negative at all wavelengths
- The chromaticity coordinates of all realizable stimuli are non-negative
- $\bar{y}(\lambda)$ color matching function is equal to the relative luminous efficiency function $V(\lambda)$
Discrete-Wavelength Trichromatic Model
- Sensor output
- Sensor output: $\vec{q}=S^{T}\vec{n}$, where, $S$ is sensor response for channels (HVS subspace, size of $31\times 3$), $\vec{n}$ sampled stimulus (size of $31\times 1$)
- Color matching condition: $\vec{n_{1}}=\vec{n_{2}} \iff \vec{q_{1}}=\vec{q_{2}} \iff \vec{n_{1}^{\star}}=\vec{n_{2}^{\star}}$
- The fundamental component (projection on $span(s)$): $\vec{n^{*}} = S[S^{T}S]^{-1}S^{T}\vec{n}$
- Introducing Primary set
- $P = [\vec{p_{1}}, \vec{p_{2}}, \vec{p_{3}}]$ (size of $31\times 3$)
- Stimuli generated using primary set $P$: $P\vec{a}$, where $\vec{a}=[a_{1}, a_{2}, a_{3}]^{T}$ is primary mixture amount
- The fundamental component of $P$: $P^{*} = S[S^{T}S]^{-1}S^{T}\vec{P}$
- Sensor output: $\vec{q}=S^{T}\vec{n} = S^{T}P^{*}\vec{a} \implies \vec{a} = (S^{T}P)^{-1}S^{T}\vec{n}$
- Color matching matrix
- For monochromatic stimulus, let $I$ of size $31\times 31$ identity matrix to represent all levels of monochrom colors
- $S^{T}PA^{T} = S^{T}I$, where $A^{T} = [\vec{a_{1}},\cdots,\vec{a_{31}}]$ (size of $3\times 31$) is color matching matrix whose $i$-th row is amount of primary $P$ neede to match $i$-th wavelength
- $span(A)=span(S)$ This means that we can freely transform among color matching functions and the HVS sensitivity matrix $S$
- Transform between CIE RGB and RGB color spaces: $A_{XYZ}=T_{RGB-XYZ}A_{RGB}$, where $A_{XYZ}$ is color matching matrix of CIE XYZ color space.
- The fundamental component (projection on $span(A)$): $\vec{n^{*}} = A[A^{T}A]^{-1}A^{T}\vec{n}$
- The Projection operator is $R=A[A^{T}A]^{-1}A^{T}=S[S^{T}S]^{-1}S^{T}=P^{\star}(P^{\star T}P^{\star})^{-1}P^{\star T}$
- An image system interpretation
- Considering an image system with a capture device followed by a display device: stimulus $\vec{n} \rightarrow$ sensor $A^{T} \rightarrow$ Display $P^* \rightarrow fundamental component of the given stimulus \vec{n^{*}}$
- $\vec{q}=S^{T}PA^{T}=S^{T}I \implies S^{T}P^{\star}A^{T}=S^{T}R\implies P^{\star}A^{T}=R \implies \vec{n^{\star}}=P^{\star}\vec{a} \implies \vec{a}=A^{T}\vec{n}$
- A stimulus $\vec{n}$ has a physical realizable metamer $\iff$ its fundamental component can be written as a non-negative linear combination of the columns of $R$
Color Opponency
Characterization of Illuminants
- Black-body radiator
- Color temperature
- Non-linear response of human visual system
- Weber’s Law
- Gamma correction
- MacAdam ellipse
- Uniform color space (ex. CIE Lab)
Color Management
- CRT display
- Forward model
- Get gamma value
- Get T
White point adjustment
- Desired reference white point
- From Source RGB to white balanced RGB
- From girven RGB to CIE XYZ
Image Capture Devices
Camera
Three chips
Foven sensor
Image Output Devices
General Model for Scanning and Sampling
- Line-continuous scanning; focal plane array
- Aperture effects
- Sampling effect
General Model for Display and Printing
- Ideal reconstruction
- Zero-order hold reconstruction
- Aliasing artifacts
Digital halftoning
- Error diffusion
- Direct binary search (DBS)
- Tone correction
- Cluster dot periodic
- Supercell approach
- Hybrid screen
- HVS and error image
- Dual interpretation
- Screen frequency
- Media-colorant-light interaction
- Print modeling
- Measuring the accuracy of the color imaging pipeline
- Perceptual model-color device
Image Quality
Computational Color