Linear Algebra in Computer Vision. Lecture2: Basic Linear Algebra & Probability. Vector. Vector Operations
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1 Linear Algebra in Computer Vision CSED441:Introduction to Computer Vision (2017F Lecture2: Basic Linear Algebra & Probability Bohyung Han CSE, POSTECH Mathematics in vector space Linear algebra is frequently used in Representation of points and features Computation of similarities and distances Finding algebraic solution Transformations Optimization 2 Vector Geometric object that has both a magnitude and direction Notation Representation! R %! =! = ' (, ',, ' % Magnitude of a vector ' ( ' ' % 0! = ' ( ' ' % 1! = ' (, ',, ' %! Dot product (inner product 2 = 3 (, 3,, 3 % 4 = 5 (, 5,, 5 % Properties of dot product Distributiveness: Vector Operations 2 4 = 2 4 cos : = 3 ( 5 ( % 5 % = 2 4 = ; = 2 ; 4 ; 2 (4 ; = ; Linearity: >2 4 = 2 >4 = >(2 4 Symmetry: 2 4 = 4 2 Non-negativity: 2 0, 2 2 > = 0 2 = 0 3 4
2 0 Vector Operations Vector Norm 5 Outer Product 2 = 3 (, 3,, 3 D 4 = 5 (, 5,, 5 % 24 = 3 ( 3 5 (, 5,, 5 % = 3 D 3 ( 5 ( 3 ( 5 3 ( 5 % 3 5 ( % RD % 3 D 5 ( 3 D 5 3 D 5 % 6 Intuitively, it means the length of a vector. A norm on a vector space Ω is a function H: Ω R with the following properties: Positive scalability: H 3! = 3 H(! Triangle inequality: H! L H! H(L If H! = 0, then! = 0. Examples of norm 1-norm: 2-norm: %! ( = N ' O! = N ' O Infinity norm:! Q = max ' (, ',, ' % % U-norm! V = % N ' V O (/V 2-norm revisited! = ' (, ',, ' % L = X (, X,, X %! % = N ' O % Vector Norm = ' ( ' ' % =!!! L = N ' O X O =! L! L =! 2! L L Linear Dependency Given a set of vectors [ = {! (,!,,! % } Linear combination of vectors is written as `(! ( `! `%! %! O [ is linearly dependent if it can be written as a linear combination of [\! O.! (!! (! a! a linearly dependent! linearly independent 7 8
3 Basis A basis is a linearly independent set of vectors that spans the whole space. Any vector can be represented by a linear combination of basis vectors. Standard basis (a.k.a. unit vectors {d O R % d O = 0,, 0,1,0,, 0 } e.g.,! = (3,2,5 = 3d ( 2d 5d a Projection of a vector! e d O =! d O = d O!! = (! d (,! d,! d a Orthogonal d f d g = 0 Normalized Orthonormal d f d O = 1 d ( d a d Change of Basis Suppose that we have a basis i = {4 (, 4,, 4 % }, 4 O R D.!j = 4 ( e! 4 e! 4 % e! = k = 4 (! 4! 4 %! 4 ( 4 4 % = 4 ( 4 4 %! = k! Vector in the new basis is represented with a matrix-vector multiplication. Identity matrix: matrix maps a vector to itself Rotation matrix d ( 4 a d a!!j 4 ( d Matrix Rectangular (2D array of numbers Notation l R D % l = 3 (( 3 ( 3 (% 3 ( 3 3 % 3 D( 3 D 3 D% l Og = 3 Og Special matrices Square matrix: matrix with (# of rows = (# of columns Identity matrix: matrix with diagonal elements ones and non-diagonal elements zeros lm = ml = l Addition Matrix Operations Commutative: l k = k l Associative: l k n = l (k n Subtraction Not-commutative, not-associative Multiplication Associative: lk n = l(kn Distributive: l k n = lk ln Non-commutative: lk kl Transposition (l Og = l go (l = l (lk = k l 11 12
4 Rank of Matrix Definition The number of linearly independent rows or columns in a matrix Rank 1: Rank2: Properties l R D %, rank(l min(v, w l is full rank if rank l = min(v, w. rank l = rank l rank lk min(rank l, rank k rank l k rank l rank k A square matrix with full rank is not singular; otherwise, the matrix is singular. Matrix Inversion Inverse matrix Condition to have inverse matrix: square and non-singular Inverse matrix of l: l x( ll x( = l x( l = m 2D example: l = 3 5 } l x( = 1 det(l 5 } 3 Properties (lk x( = k x( l x( (l x( = (l x( For an orthonormal matrix, l x( = l For a diagonal matrix y = diag (,,, % R % %, y x( = diag x( (, x(, x( % Some properties det l = 0 iff l is singular. det lk = det l det k det l x( = det l x( det >l = > % det l Determinant How to compute 3 (( 3 ( 3 (% 3 (( 3 ( 3 (% 3 ( 3 3 % 3 ( 3 3 l = % 3 %( 3 % 3 %% 3 %( 3 % 3 %% Solving Linear Equations Finding the exact solution l! = 4, where l is an w w matrix, and! and 4 are w 1 vectors.! = l x( 4 Finding the least square solution l! = 4, where l is an v w (v > w matrix with at least w independent rows,! is an w 1 vector, and 4 is an v 1 vector. min l! 4 min l! 4 l! 4 2l (l! 4 = 0! = (l l x( l
5 Eigenvector and eigenvalue Eigen-Decomposition Defined for a square matrix, i.e., l Non-zero vector for which there is a > R such that l! = >!. Eigenvector:! Eigenvalue: > How to compute Solve (l >m! = 0 for! and >. Equivalently, find > for det(l >m = 0, and compute!. e.g., l = Get {! O, > O },,% as a solution when l is an w w matrix. Eigen-Decomposition A square and symmetric matrix l can be decomposed as l = ÅyÅ = Ç ( Ç Ç % 0 > 0 0 > ( > % where is Å an orthonormal matrix of l s eigenvectors and y is a diagonal matrix of the associated eigenvalues. Properties A square matrix is singular if any of eigenvalues is zero. A square matrix l is positive definite (negative definite if all eigenvalues are positives (negatives.! 1,! l! > 0 (! l! < 0 Semi-definiteness! 1,! l! 0 (! l! 0 Ç ( Ç É Ç % Ç O Ç f = 1 Ç O Ç g = Computing inverse matrix Use of Eigen-Decomposition Problem: lü = m Decomposition: l = ÅyÅ Problem revisited: ÅyÅ Ü = m Solution: Ü = Åy x( Å Solving optimization problems Objective function: min l! Find the eigenvector of l á l corresponding to the least eigenvalue! min l! = min >! Probability in Computer Vision Probability is used in computer vision to model uncertainties for the following problems: Visual tracking Human pose estimation Object detection and recognition Image segmentation and many more the least eigenvalue corresponding eigenvector 19 20
6 0 Random Variables Definition & properties A function that assigns a number to each point in a sample space S A variable whose possible values are numerical outcomes of a random phenomenon Discrete or continuous Involves probability distribution Coin toss example X 0 (Head 1 (Tail Probability Probability Axioms Sample space (à: the set of all the outcomes Event space (â: a power set whose element is a subset of à If ä â, then ä à. Probability measure a function å: â R with the following conditions å ä 0, ä â å à = 1 N å ä O O = å(ä ( ä ä a if ä O s are disjoint. X is a discrete random variable. S = {Head, Tail} is sample space Some Properties in Probability Set inclusion ä i å(ä å(i Union å ä i = å ä å i å(ä i Complement å à\ä = 1 å(ä Independence If ä and i are independent, å ä i = å ä å i. Notation å ä i å ä, i Conditional Probability Conditional probability of ä given i å(ä, i å ä i = å(i å(ä, i = å ä i å(i Independence å(ä, i å ä i = å(i = å ä å(i å(i = å(ä Conditional independence ä and i are conditionally independent given ë. å ä, i ë = å ä ë å(i ë å ä i, ë = å ä ë 23 24
7 0 0 Chain Rule Computation of joint probability using conditional probabilities e.g., U ä %, ä %x(, ä %x,, ä ( = U ä % ä %x(, ä %x,, ä ( U ä %x(, ä %x,, ä ( = U ä % ä %x(, ä %x,, ä ( U ä %x( ä %x,, ä ( U ä %x,, ä ( å ä í, ä a, ä, ä ( = å ä í ä a, ä, ä ( å ä a, ä, ä ( = å ä í ä a, ä, ä ( å ä a ä, ä ( å(ä, ä ( = å ä í ä a, ä, ä ( å ä a ä, ä ( å(ä ä ( å(ä ( Revisit conditional probabilities U ä i = Bayes theorem posterior Bayes Theorem å(ä, i å(i likelihood U i ä = prior å(ä, i å(ä å ä, i = U ä i U i = U i ä U(ä U ä i = å i ä å(ä å(i (Uìîïñóòìò (öóõñöóhìì (Uòóìò Characteristics Normal distribution A unimodal distribution Easy and fast to compute Limited to model variations Represented with a mean and a covariance Gaussian Distribution Hù '; ü, = 1 2 exp ' ü 2 General case Multivariate Gaussian Distribution Hù!;, = 1 2 / (/ exp 1 2! x(! where! = ' (, ',, ' = ü (, ü,, ü and Independent case Hù!;, = 1 exp ' O ü O 2 O 2 O = (
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