Colors:
Desktop Monitor to
the Big Screen (& back)
Alan Edelman
Dept of Mathematics: MIT
MIT Laboratory for Computer Science
Frank Wang
Arun Rao (Pixar)
Graphics Lunch
April 25, 2003
Other Topics Not Covered Today
• Parallel MATLAB
• The Bohemian Dome
Horizontal
A=QQT
Vertical
A=QQ
Villarceau
A=QR
Hopf Fibration
Outline
• The Data
• The Problem
• Step 1: Finding the right three dimensional
basis
• Step 2: Inverting onto that basis
• Step 3: Forming the model
Outline
• The Data
• The Problem
• Step 1: Finding the right three dimensional
basis
• Step 2: Inverting onto that basis
• Step 3: Forming the model
The data (101 points x 1000 frames)
wavelength vs density
Reds
Greens
Grays
film density =
log(no film / with film)
Blues
Film Recording and measurements
• Solid colors sent to film recorder, e.g. reds
• Negative is produced: film appears as cyans
• Negative sent through projector to spectrometer
• Energy data at each wavelength
Reds
• Log ratio with no film (only bulb)
film density =
log(no film / with film)
Outline
• The Data
• The Problem
• Step 1: Finding the right three dimensional
basis
• Step 2: Inverting onto that basis
• Step 3: Forming the model
Movie Making
Step I: The artists PixarArtists choose colors on their desktop
computer monitors
Step II: Color Recording Digital Images Recorded on Film
1.
http://www.pixar.com/companyinfo/press/1999/pr99-0204a.html
2.
Film Developed
Step III: Color Reproduction Film Projected Onto Screen at a
movie theatre near you
Problem:
Colors on the big screen just do not look the
same.
The Two Stages
color recording stage
color reconstruction stage
The Two Stages
color recording stage
color reconstruction stage
Models, Algorithms, Numerics
•
physically based models
•
numerical techniques
To invert the color reproduction &
recording steps.
Outline
• The Data
• The Problem
• Step 1: Finding the right three dimensional
basis
• Step 2: Inverting onto that basis
• Step 3: Forming the model
SVD of the data
•
•
•
•
Inputs (r,g,b) for 1r,g,b 10 scaled (1000 frames)
Output Space: Densities at 400:3:700 nm’s
Data Structure: 101 x 1000 matrix “A”
Compute SVD(A) •Project onto best 3 space
svd
Three significant
singular values
index
SVD Basis = no physical meaning
Orthogonality Constraint too strong
The NNMF Basis = primary colors
Non-Negative Matrix Factorization
•The NNMF (Lee, Seung 1999)
•V  WH Input: Vij>0
Output: Wij>0 Hij>0 (low rank)
Algorithm:
H  H .* (W’V)./(W’WH)
W  W .* (VH’)./(WHH’)
•Original Application: Eigenfaces
•New Algorithm: Project SVD Into Cone using
Convex Hull Algorithm
Errors of two NNMF implementations
• More than 1Mflop per NNMF iteration. Error flattens after 50000
iteration.
• Accuracy of new algorithm improves as samples increase, but not
for NNMF.
• NNMF can easily generalize to higher dimension.
SVD with a geometry tweak
Compare bases extracted by the two
methods
Projection of 1000 spectra onto the basis
101x1000  3x1000

10x10x10
Input and output for stage 1
• Find a functional relationship between laser input and
output of concentration vectors by either interpolation
or a physical model.
Outline
• The Data
• The Problem
• Step 1: Finding the right three dimensional
basis
• Step 2: Inverting onto that basis
• Step 3: Forming the model
Color matching functions
CIERGB
CIEXYZ
chromaticity diagram (XYZ)
spectrum locus
purple line
CIELAB color space
Obtain coefficients from Color
• Given a color as (x,y,z) in CIEXYZ coordinates
compute c1,c2,c3 such that
(x,y,z)=∫λ (x(λ),y(λ),z(λ)) I0(λ) *
-(c1b1(λ)+c2b2(λ)+c3b3(λ))
e
dλ
Newton’s Method
The Physical Model
I out
c
   
I in
Bear’s Law
I out  I inc   m    y  
c1
c2
c3
I out ( )
d ( )   log(
)  c1b1 ( )  c2b2 ( )  c3b3 ( )
I in ( )
 c1 


 b1 ( ) b2  ) b 3 ( )c2 
c3 
Outline
• The Data
• The Problem
• Step 1: Finding the right three dimensional
basis
• Step 2: Inverting onto that basis
• Step 3: Forming the model
The Two Stages
color recording stage
color reconstruction stage
The Two Stages
color recording stage
color reconstruction stage
The tri-pack structure of color film
Exposing Color Light
yellow filter
blue-sensitive layer
yellow filter
green-sensitive layer
red-sensitive layer
yellow filter
Film development process
HD curve of film: density v.s. exposure
Physical effects motivates co-linear fit
• Inter-layer effects are at play: cross-layer
inhibition, cross-layer exposure and crosslayer absorption.
• Possible diminishing cross layer exposure
effect motivates a bilinear basis in the model.
(1,•logThe
R, log model
G, log B, log
G, logsquare
G log B, log
log the
R, logdata
R log G log B)
is Ralog
least
fitBof
involving only co-linear bases:
Summary
• Expose 1000 frames of color film to 1000 colors sampled from a RGB
color cube.
• Collect 1000 spectra by measuring the output color light of the 1000
frames of film.
• Invert the second stage:
– From the spectra data, extract three bases, i.e. the absorption functions of
three dye layers using either NNMF or a geometrical approach involving SVD.
– From a given intended color specified in XYZ color coordinates, solve for
density vectors using Newton's method.
• Invert the first stage:
– Compute all 1000 concentration vectors of the 1000 spectra.
– Build a functional relationship between the 1000 colors from a RGB cube and
the 1000 density vectors using either interpolation or a physical model.
• Solve this function for a set of RGB inputs that will give the density
vector obtained from previous stage.