Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS 205 class 1
Types of Errors
Covered in class: 4, 6, 7 1. When doing integer calculations one can many times proceed exactly, except of course in certain situations, e.g. division 5/2=2.5. However, when doing floating point calculations rounding errors
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 17
Covered in class: All
Reading: 9.3.9
1. Backward (Implicit) Euler
yk 1 yk
f (t k 1 , yk 1 )
h
a. 1st order accurate
b. Backward Euler applied to the model equation y y is yk 1 yk h yk 1
i. So yk 1 (1 h )1 yk and yk (1 h ) k yo
ii. The erro
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205  Class 16
Readings: 9.3
Covered in Class: 1, 2, 3, 4, 5, 6
ODEs (Continued)
1. Model ODE Problems
a. Scalar ODE y ' f (t , y ) and the
i. linear model ODE is y ' y which solution is y y0 e (t t0 )
ii. Only three kinds of solutions
20
17.5
15
12.5
1
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 13
Covered in class: 1, 3, 5
Readings: 6.7, 7.2 to 7.3.3
1. Interpolation
a. polynomial of degree n y c1 c2 x c3 x 2 cn 1 x n
i. Monomial basis y c11 c22 c33 cn 1n 1 where the basis
function are j ( x ) x j 1 for j 1,2, n 1
1. Polynomial inter
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS 205 Class 12
Readings: Same as last
Covered in class: All
1. finding the Aorthogonal directions with GramSchmidt
a. given a vector Vk, construct sk by subtracting out the Aoverlap of Vk with s1 to sk 1 so that
sk Asi 0 for i=1,k1
k 1 V k As j
b. we
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 11
Covered in class: Everything
Readings: Shewchuk Paper on course web page and Heath 473478
1. Conjugate Directions
a. The goal is to choose a sequence of search directions s0 , s1 , that are all orthogonal. Then only one
step is needed in e
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 10
Covered in class: All
Reading: Shewchuk Paper on course web page
1. Conjugate Gradient Method this covers more than just optimization, e.g. well use it later as an iterative
solver to aid in solving pdes
2. Lets go back to linear systems of
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 8
Covered In Class: 1, 3, 4, 5, 6
Reading: Heath Chapter 6
1. Optimization given an objective function f , find relative maxima or minima. Note that since max f =
min f it is enough to only consider minima.
a. Well start with scalar functions
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205  Class 7
Readings: Heath 5.6
Systems of Nonlinear Equations
1. Lets turn our attention back to systems of nonlinear equations, i.e. A(x)=b or F(x)=0.
a. Here the Jacobian matrix, J(x), is rather useful as a linearization of the nonlinear problem.
b
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 6
Reading: Heath 3.6 (p137143), 4.7 (p202)
Singular Value Decomposition (SVD) contd.
1. SVD is a transformation into a diagonal axis aligned space.
a. Transform b into the space spanned by U T , U T UV T x V T x U T b b . No information is lo
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 5
Covered in class: 1, 3, 4, 5.
Reading: Heath Chapter 4.
Eigenvalues / Eigenvectors Continued
1. The Power Method allows one to compute the largest eigenvalue and eigenvector. Starting from a nonzero
vector x0 , iterate with xk 1 Axk .
a. To
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205  Class 4
1. As a review, all the matrices A we have looked at up to this point in the class have been full rank. a. For matrices with full rank, the first thing to consider is whether or not it is square. i. If the matrix is square, it is invertibl
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205  Class 3
1. So far we have discussed solving Ax=b for square n n matrices A. For more general m n matrices, there
are a variety of scenarios.
a. When m < n, the problem is underdetermined since there is not enough information to determine a
unique
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 2
Linear Systems Continued
Covered in class: all sections 1. When constructing M k we needed to divide by ak which is the element on the diagonal. This could pose difficulties if the diagonal element was zero. For example, consider the matrix
Mathematical Methods for Computer Vision, Robotics, and Graphics
CS 205

Spring 2010
CS205 Class 18
Covered in Class: 1, 2, 3, 4
Readings: Heath 11.111.2
Partial Differential Equations
1. There are three types of PDEs
a. Elliptic, Hyperbolic, Parabolic
2 p
f is the model elliptic equation
2
i 1 xi
For PDEs we often use subscript notatio