Outline • Notation and Basics • Motivation • Linear systems of equations Linear Algebra for Computer – Gauss Elimination, LU decomposition Vision • Linear Spaces and Operators – Addition, scalar multiplication, scalar product, Introduction transformation, operator, basis CMSC 828 D • Eigenvalues, Eigenvectors • Solvability conditions (“alternative theorem”) – Adjoint, null space, orthogonality Motivation Outline • Fundamental to representation and numerical solution of almost all problems including those in • Euclidean space R3 vision and computational statistics. – Solving equations for calibration, stereo, tracking, … – distance, angles, rotations • Geometry is fundamental to vision. However one • Metric Space way of doing geometry is via algebra. – Distance, angles, rotations – Intersections of lines, points, planes. Determining angles. Determining orthogonal projections … • Least Squares • Modern computer vision is formulated in terms of • Singular Value Decomposition “projective geometry”. Most results in projective geometry are stated algebraically and require • Other Matrix decompositions knowledge of concepts such as rank, null space, constraints Vectors Applications • A vector x of dimension d represents a point in a d dimensional space •Examples • Rectification of images – A point in 3D Euclidean space [x,y,z] or 2D image space [u,v] – A point in a projective space P3 [X,Y,Z,W] or in projective space • Calibrating cameras P2 [U,V,W] • Transforming color spaces – Point in color space [r,g,b] or [y, u, v] – Point in an infinite dimensional functional space on a Fourier • Tracking motion of a rigid body basis – Vector of intrinsic parameters for a camera (focal length, skew • Applying constraints from multiple views ratio, …) • Parametrizing fundamental matrix and • Essentially a short-hand notation to denote a grouping of trifocal tensor. points – No special structure yet 1 Vectors and Matrices Determinant • Determinant of a 2x2 matrix m11m22-m12m21 •d dimensional column vector x and its transpose • For a higher dimensional matrix we have a recursive definition • d×n dimensional matrix M and its transpose Mt • Transpose indicated with a superscript t or a prime ′ Matrix basics Determinant: Remarks • Square matrix: number of rows = number of columns • Determinant determines “magnitude” of • Symmetric matrix Aij=Aji . matrix. Matrix with determinant =0 is called • Skew symmetric matrix Aij=-Aji . singular. δ • Identity I ij= ij • Determinant is important in theorems δ ≠ δ – Kronecker delta ij=0 if i j ij=1 if i=j • Practically the way to compute the • Lower triangular Upper triangular determinant is not this way. aacd00"" %# %# • Homework problem -- determine number of cb0 b operations for recursive algorithm. #"%0 #%%e fgd""00 d Matrix vector product Linear systems of equations ++= ax11 1 ax 12 2 ax 13 3 b 1 × • Systems of equations ++= • m n dimensional matrix M multiplies by a ax21 1 ax 22 2 ax 23 3 b 2 • Can be written as a n dimensional vector x to produce a m ax++= ax ax b matrix vector product 31 1 32 2 33 3 3 dimensional vector • Can change or scale rows Ax= b • Solved via Gauss aaa11 12 13 x 1 b 1 elimination === Axbaaa21 22 23,, x 2 b 2 – Reduce system aaa x b to product of 31 32 33 3 3 lower or upper triangular matrix and x • O(N) operations to solve triangular system • O(N3) operations to perform Gauss elimination 2 LU decomposition Linear/Vector Spaces • Any matrix can be written as a product of a lower • Previous stuff was somewhat mechanical. triangular and upper triangular matrix. • In vision we have to answer questions when • Most used algorithm in linear algebra. – Models provide equations that are singular or degenerate. What can we say about the solutions? Can A=LU we restrict them? • Practically implemented by reordering equations and – Number of unknowns may be more or less than the scaling them so that loss of accuracy is minimized. number of observations. Can we still obtain a 01 meaningful solution? A=PLU 10 – How “far” is an approximation from a solution? How • Scaled LU decomposition with “partial pivoting” do we measure this distance? • When A is symmetric positive definite xAx t > 0 for all x – Matrices are operators that take one vector into another. What can we say about the properties of the operator? “Cholesky decomposition” When is an equation involving an operator solvable? A=LLt Operators Vector Space • Function, Transformation, Operator, Mapping: synonyms • A collection of points that obey certain rules • A function takes elements x defined on its “Domain” D to – Commutative, existence of a zero element elements y in its “Range” R which is part of E E uvvu+=+;()() u + vw + = uv + + w 2u ∃+=∀0,uuuuu 0 ; +−=() 0 R u+v D – Scalar multiplication u • If for each y in R there is exactly one x in D the function is αβ()()uuuu== αβ ;1 O one-to-one. In this case an inverse exists whose domain is R ()αβ+=+uuu α β; α () uvuv +=+ α α v and whose range is D •Let u , …, u be a set of vectors: • We are interested in situations where R and D are finite- 1 k Linear combination -u dimensional linear spaces Manifold spanned ααuu++" 11 kk by u Dependence and dimensionality Distances/Metrics and Norms • We would like to measure distances and directions in the • A set of vectors is dependent if for some scalars α , …, α not 1 k vector space the same way that we do it in Euclidean 3D all zero we can write αα++" = 11uukk 0 • Otherwise the vectors are independent. • Distance function d(u,v) makes a vector space a metric • If the zero vector is part of a set of vectors that set is space if it satisfies dependent. If a set of vectors is dependent so is any larger set – d(u,v)>0 for u,v different which contains it. – d(u,u)=0, d(u,v)=d(v,u) • A linear space is n dimensional if it possesses a set of n – d(u,w)≤ d(u,v)+d(v,w) (triangle inequality) independent vectors but every n+1 dimensional set is • Norm (“length”). dependent. –||u|| >0 for u not 0, ||0||=0 … • A set of vectors b1, , bk is a basis for a k dimensional space – || α u||=| α| ||u||, ||u+v || ≤ ||u|| + ||v|| X if each vector in X can be expressed in one and only one • Normed linear space is a metric space with the metric way as a linear combination of b , …, b 1 k defined by d(u,v)=||u-v|| and ||u||=d(u,0) • One example of a basis are the vectors (1,0,…,0), (0,1,…,0), …, (0,0, …, 1) 3 Dot Product Matrices as operators • Dot product of two vectors with same dimension • Matrix is an operator that takes a vector to <x,y> = another vector. • Dot product space behaves like Euclidean R3 • Dot product defines a norm and a metric. – Square matrix takes it to a vector in the space of • Parallelogram law the same dimension. ||u+v||2+||u-v||2 = 2||u||2 + 2 ||v||2 • Dot product provides a tool to examine • Orthogonal vectors <u,v>=0 matrix properties • Angle between vectors – Adjoint matrix < Au,v> = <u,A*v> θ cos =<x,y>/||x|| ||y|| – Square Matrix fully defined as result of its • Orthonormal basis -- elements have norm 1 and are operation on members of a basis. perpendicular to each other • Other distances and products can also define a space: Aij = < Abj,bi> – Mahalnobis distance Eigenvalues and Eigenvectors • Square matrix possesses its own natural basis. Motivation: Stereo • Eigen relation • Point (x,y) on the image plane lies on a line Au=λu in the world that passes Scene point •MatrixA acts on vector u and produces a scaled version of (xi , yi , 0) (x , y , z ) the vector. through the image point s s s y • Eigen is a German word meaning “proper” or “specific” and center of x • u is the eigenvector while λ is the eigenvalue. projection C O z –If u is an eigenvector so is αu –If||u||=1 then we call it a normal eigenvector •Image of this – λ is like a measure of the “strength” of A in the direction of u f • Set of all eigenvalues and eigenvectors of A is called the line in the world will “spectrum of A” form a line in another camera = ∑mmijij F 0 Epipolar Constraint Eight point algorithm: Determining the • Point in one image lies on the “epipolar Fundamental matrix line” in the other image • Given a set of matching points in the • Algebraic statement of geometry images, Determine F – Equation of line in the other image is Fm – Condition that the point m′ lies on this line is fff 11 12 13 u m′⋅Fm=0 []′′ = uv10fff21 22 23 v • F is the “fundamental matrix” ff 1 1 31 32 ′′()()()+++ +++++= • Estimating the fundamental matrix is an ufufvf11 12 13 vfufvf21 22 23 31 32 f10 f important problem in vision 4 Determining F • Write expression as an equation in the unknown elements. f 11 • If we have eight points we can solve f12 for elements of F, (e.g. via LU) f 13 f21 [][]uu′′′′′′ uv u vu vv v 11=− 1 f 22 • If we have more than eight points f23 we can use a least squares f 31 formulation f32 5.