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Motivation
We do not cover all the math
Just the common basics (yellow triangle)
IVA IllVis Modeling Domain specific vis
Motivation We do not cover all the math Just the common basics - - PowerPoint PPT Presentation
Motivation We do not cover all the math Just the common basics - - PowerPoint PPT Presentation
Motivation We do not cover all the math Just the common basics (yellow triangle) IVA Modeling IllVis Domain specific vis 0 Vector spaces and linear systems Julius Parulek Vector spaces Vector Spaces Linear combination Vector Space A
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Vector spaces
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Vector Spaces
Linear combination Vector Space
A set for which linear combinations is defined Is closed under those linear combinations Ex:
a set of 2D vectors of real numbers a set of 2D vectors of positive real numbers is not! An arbitrary plane and an arbitrary line
Their union is not! Their intersection is!
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Vector Spaces
Linear dependence
There exists a linear relationship among vectors
Dimension of a linear space
The largest number of linearly independent vectors
Basis of a linear space
A set of n linearly independent vectors for a linear space of dimension n
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Vector Spaces
Column Space Null space of A
Which vectors transform to zero vector?
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Vector Spaces
Null Space examples
Do B-1, C-1 exist?
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Vector Spaces
Linear map
Defined by a matrix A, v=Au When A has n rows, m columns
A defines a map Vector v is in Rm, vector u is in Rn
A is a linear map when
Inverse matrix (map)
A has to be square! A-1 defines inverse matrix of A A A-1 = I A-1=AT if A is orthonormal
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Vector Spaces
Rank of a matrix
Number of linear independent vectors (columns) in a matrix A = [a1,a2,…,an], rank(A) = dimensions(a1,a2,…,an) rank(A)≤min(m,n) Non-full-rank matrices are called singular Full-rank matrices are invertible (have inverse matrix)
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Vector Spaces
Change of basis
v A B
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Vector Spaces
Example: Transformation to reference frame
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[Geomcell, parulek et al, 2009]
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Linear systems
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Linear Systems
System of equations
#equations = #unknows
Fit 2 points by a line Exact solution
#equations > #unknows
Approximate 3 points by a line Approximative solutions
#equations < #unknows
Fit 3 points by a curve of 4 Free variables
Example
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Linear Systems
A linear system of equations for unknowns u1,…,un
Abbreviated to Au=b
Au=b is solvable if b is in the column space of A
If invertible, then there is only one solution Solution, u = A-1b
u1a1+ u2a2=b b a1 a2
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Linear Systems
How to solve?
Gauss elimination Iterative solvers
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Linear Systems
Gauss elimination
Ex: Transforms matrix into an upper triangular matrix
Row exchange can help
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Linear Systems
Elimination matrix construction
Steps encoded into matrices
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Linear Systems
Elimination matrix construction
Steps encoded into matrices
C D D*(C*A)=(D*C)*A
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Linear Systems
Decomposition to lower and upper triangular matrix
A=LU
If A is symmetric
A=LDLT
Diagonal matrix of pivots!
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Linear Systems
Solution stability issues
Ill-conditioned (unstable) systems
Tiny change of input data results in drastic changes in the solution
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Linear Systems
Iterative system solver – I. (Gauss-Jacobi iteration) In case of many thousands of equations, Gauss elimination would be slow Often only few non-zero entries per row in the coefficient matrix (sparse)
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Linear Systems
Iterative system solver – II.
A = D + R, di,i=ai,i, di,j=0 if i!=j Iteration step Convergence
A is diagonally dominant
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Linear systems
Use-case for Gauss-Jacobi
Fluid Flow
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Linear systems
Data Fitting: Polynomial Interpolation
polynomial fit
f(t) = sqrt(t) (gray), evaluated at 6 points Approximated by p(t) of degree 4 (black)
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Linear Systems
Overdetermined Systems
Au=b, A is not square! Solution
ATA = symmetric matrix Pseudinverse matrix (ATA)-1AT
u=(ATA)-1ATb
… or Gauss elimination
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Linear systems
Solving the null space
Columns of A must be linearly depended
Otherwise x=[0,…,0]
Solving the inverse matrix
A[u1,…,un]=I
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Linear Systems
Other methods
Conjugate Gradients
For symmetric and positive definite matrices Iterative and fast method
Cramer’s rule
Using determinants Impractical for large matrices
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Linear Systems
Use case: manual registration between two images
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Linear Systems
Use case: manual registration between two images
Find transformation A (3x3)
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Literature
- G. Farin, D. Hansford: Mathematical Principles for Scientific
Computing and Visualization http://www.farinhansford.com/books/scv/teaching.html MIT Open Courses, Linear Algebra http://ocw.mit.edu/courses/mathematics/18-06-linear- algebra-spring-2010/video-lectures/
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