Applied Geometrical Applied Geometrical Matrix Computations Matrix Computations Alan Edelman Alan Edelman Dept of Mathematics: MIT Dept of Mathematics: MIT MIT Laboratory for Computer Science MIT Laboratory for Computer Science Householder Symposium XV June 21, 2002
Applied Geometrical Matrix Computations. Alan Edelman Dept of Mathematics: MIT MIT Laboratory for Computer Science. Householder Symposium XV June 21, 2002. Outline. Geometrical Matrix Computations Illustration with 2x2 matrices: - PowerPoint PPT Presentation
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Alan EdelmanAlan EdelmanDept of Mathematics: MITDept of Mathematics: MIT
MIT Laboratory for Computer ScienceMIT Laboratory for Computer Science
Householder Symposium XV
June 21, 2002
OutlineOutline
• Geometrical Matrix ComputationsGeometrical Matrix Computations• Illustration with 2x2 matrices:Illustration with 2x2 matrices:• Excursions into eigenland (or why tangency Excursions into eigenland (or why tangency
and curvature matter!!)and curvature matter!!)• Where do matrix factorizations come from?Where do matrix factorizations come from?• Application to Color ScienceApplication to Color Science• Matrix AnimationsMatrix Animations
Working definition:Working definition:•Concerns geometry of matrix space Concerns geometry of matrix space
(n(n2 2 dimensions rather than n)dimensions rather than n)•Involves numerical computation (probably MATLAB)Involves numerical computation (probably MATLAB)•Relates to an NLA problemRelates to an NLA problem
Lippert, Ma, Mahony, Malyshev, Sepulchre, Lippert, Ma, Mahony, Malyshev, Sepulchre, Tisseur, Trefethen, Van DoorenTisseur, Trefethen, Van Dooren
Vector Space DiagramsVector Space Diagrams•Points are vectors Points are vectors (not matrices!)(not matrices!)•Geometric relationships for vectors, Geometric relationships for vectors, subspaces, and linear transformationssubspaces, and linear transformations
OutlineOutline• Geometrical Matrix ComputationsGeometrical Matrix Computations• Illustration with 2x2 matrices:Illustration with 2x2 matrices:• Excursions into eigenland (or why tangency Excursions into eigenland (or why tangency
and curvature matter!!)and curvature matter!!)• Where do matrix factorizations come from?Where do matrix factorizations come from?• Application to Color ScienceApplication to Color Science• Matrix AnimationsMatrix Animations
Eigenland (in 2d)Eigenland (in 2d)
2 2x z
x -zM = z -x
xx
zz
Isoeig surfaces are hyperbolasIsoeig surfaces are hyperbolas
matrix space eig (w/singularity)spectral portrait2nC
OutlineOutline
• Geometrical Matrix ComputationsGeometrical Matrix Computations• Illustration with 2x2 matrices:Illustration with 2x2 matrices:• Excursions into eigenland (or why tangency Excursions into eigenland (or why tangency
and curvature matter!!)and curvature matter!!)• Where do matrix factorizations come from?Where do matrix factorizations come from?• Application to Color ScienceApplication to Color Science• Matrix AnimationsMatrix Animations
Circle/Hyperbola Tangency = High DensityCircle/Hyperbola Tangency = High Density
have eigenvalue distributions with
2 spikes.
eigenvalueeigenvalue
frequency
frequency
have eigenvalue distributions with
4 spikes.
eigenvalueeigenvalue
frequency
frequency
have eigenvalue distributions with
3 spikes.
frequency
frequency
eigenvalueeigenvalue
•Circles tangent to 2 hyperbolas…
* *
•Circles tangent to 4 hyperbolas…
*
**
*
•Circles tangent to 3 hyperbolas…
*
* *
Radius of Curvature = Highest Radius of Curvature = Highest DensityDensity•Circles are tangent to 3
hyperbolas when two tangency points collide
•The circle also shares a radius of curvature with
the hyperbola at this point
•This is even better than tangency, which means a
higher spike
eigenvalueeigenvalue
frequency
frequency
frequency
frequency
eigenvalueeigenvaluefr
equency
frequency
eigenvalueeigenvalue
*
* *
OutlineOutline
• Geometrical Matrix ComputationsGeometrical Matrix Computations• Illustration with 2x2 matrices:Illustration with 2x2 matrices:• Excursions into eigenland (or why tangency Excursions into eigenland (or why tangency
and curvature matter!!)and curvature matter!!)• Where do matrix factorizations come from?Where do matrix factorizations come from?• Application to Color ScienceApplication to Color Science• Matrix AnimationsMatrix Animations
Where do Matrix Factorizations Where do Matrix Factorizations Come From?Come From?
• ApplicationsApplications– Engineering: A factorization is useful if someone can use itEngineering: A factorization is useful if someone can use it
– Mathematics: The useful factorizations are characterized by an Mathematics: The useful factorizations are characterized by an abstract criterionabstract criterion
Ideas to GeneralizeIdeas to Generalize
E = (antisymmetric) + (symmetric)E = (antisymmetric) + (symmetric)
Algorithm: H Algorithm: H H .* H .* (W’(W’VV)./(W’)./(W’WHWH))
W W W .* W .* ((VVH’)./(H’)./(WHWHH’)H’)
Original Application: EigenfacesOriginal Application: Eigenfaces
Another Example: Color ScienceAnother Example: Color Science
OutlineOutline
• Geometrical Matrix ComputationsGeometrical Matrix Computations• Illustration with 2x2 matrices:Illustration with 2x2 matrices:• Excursions into eigenland (or why tangency Excursions into eigenland (or why tangency
and curvature matter!!)and curvature matter!!)• Where do matrix factorizations come from?Where do matrix factorizations come from?• Application to Color ScienceApplication to Color Science• Matrix AnimationsMatrix Animations
Color Science: Light Spectra from Color Science: Light Spectra from filmfilm
Reds Greens Blues
Grays
wavelength vs densitywavelength vs density
Film Recording and Film Recording and measurementsmeasurements
Reds
• Solid colors sent to film recorder, e.g. redsSolid colors sent to film recorder, e.g. reds
• Negative is produced: film appears as cyansNegative is produced: film appears as cyans
• Negative sent through projector to spectrometerNegative sent through projector to spectrometer
• Energy data at each Energy data at each wavelengthwavelength• Log ratio with no film (only Log ratio with no film (only bulb)bulb)
film density =film density =
log(no film / with film)log(no film / with film)
The DataThe Data• Inputs (r,g,b) for 1Inputs (r,g,b) for 1r,g,b r,g,b 10 scaled (1000 10 scaled (1000 frames)frames)• Output Space: Densities at 400:3:700 nm’sOutput Space: Densities at 400:3:700 nm’s• Data Structure: 101 x 1000 matrix “A”Data Structure: 101 x 1000 matrix “A”• Compute SVD(A)Compute SVD(A)
indexindex
svd
svd
•Project onto best 3 spaceProject onto best 3 space
Three significantThree significant singular valuessingular values
SVD Basis = no physical SVD Basis = no physical meaningmeaning
OutlineOutline• Geometrical Matrix ComputationsGeometrical Matrix Computations• Illustration with 2x2 matrices:Illustration with 2x2 matrices:• Excursions into eigenland (or why tangency Excursions into eigenland (or why tangency
and curvature matter!!)and curvature matter!!)• Where do matrix factorizations come from?Where do matrix factorizations come from?• Application to Color ScienceApplication to Color Science• Matrix AnimationsMatrix Animations
Incorporate 3d graphics tools directly into Matrix Incorporate 3d graphics tools directly into Matrix computations. Include geometry of matrix space.computations. Include geometry of matrix space.
How should this look?How should this look?
Generalize everything and incorporate into softwareGeneralize everything and incorporate into software