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Linear Algebra & Geometry Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] – chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia I. Camps, Penn State University Why is linear algebra useful in computer vision? • Representation – 3D points in the scene – 2D points in the image • Coordinates will be used to – Perform geometrical transformations – Associate 3D with 2D points • Images are matrices of numbers – Find properties of these numbers Agenda 1. Basics definitions and properties 2. Geometrical transformations 3. SVD and its applications Vectors (i.e., 2D or 3D vectors) P = [x,y,z] p = [x,y] 3D world Image Vectors (i.e., 2D vectors) x2 P v θ x1 Magnitude: If , Is a UNIT vector Is a unit vector Orientation: Vector Addition v+w v w Vector Subtraction v-w v w Scalar Product av v Inner (dot) Product v α w The inner product is a SCALAR! Orthonormal Basis x2 P v j θ i x1 Vector (cross) Product u w α v The cross product is a VECTOR! Magnitude: u = v w = v w sin ↵ k k k ⇥ k k kk k Orientation: Vector Product Computation i = (1,0,0) || i ||= 1 i = j k ⇥ j = (0,1,0) || j||=1 j = k i ⇥ k = (0,0,1) || k ||= 1 k = i j ⇥ = − + − + − (x2 y3 x3 y2 )i (x3 y1 x1 y3 )j (x1 y2 x2 y1)k Matrices a11 a12 ... a1m a21 a22 ... a2m An m = 2 . 3 ⇥ . 6 7 6a a ... a 7 6 n1 n2 nm7 4 5 Pixel’s intensity value Sum: A and B must have the same dimensions! Example: Matrices a11 a12 ... a1m a21 a22 ... a2m An m = 2 . 3 ⇥ . 6 7 6a a ... a 7 6 n1 n2 nm7 4 5 Product: A and B must have compatible dimensions! Matrix Transpose Definition: T Cm×n = An×m cij = aji Identities: (A + B)T = AT + BT (AB)T = BT AT If A = AT , then A is symmetric Matrix Determinant Useful value computed from the elements of a square matrix A det a11 = a11 a11 a12 det = a11a22 − a12a21 a21 a22 2 3 a11 a12 a13 det 4a21 a22 a235 = a11a22a33 + a12a23a31 + a13a21a32 a31 a32 a33 − a13a22a31 − a23a32a11 − a33a12a21 Matrix Inverse Does not exist for all matrices, necessary (but not sufficient) that the matrix is square AA−1 = A−1A = I a a −1 1 a −a A−1 = 11 12 = 22 12 ; det A 6= 0 a21 a22 det A −a21 a11 If det A = 0, A does not have an inverse. 2D Geometrical Transformations 2D Translation P’ P t 2D Translation Equation P’ ty P t y x tx 2D Translation using Matrices P’ t ty P y x tx Homogeneous Coordinates • Multiply the coordinates by a non-zero scalar and add an extra coordinate equal to that scalar. For example, Back to Cartesian Coordinates: • Divide by the last coordinate and eliminate it. For example, • NOTE: in our example the scalar was 1 2D Translation using Homogeneous Coordinates P’ t ty P t y P x tx Scaling P’ P Scaling Equation P’ sy y P y x sx x Scaling & Translating P’’ P P’=S·P P’’=T·P’ P’’=T · P’=T ·(S · P)=(T · S)·P = A · P Scaling & Translating A Translating & Scaling = Scaling & Translating ? Rotation P’ P Rotation Equations Counter-clockwise rotation by an angle θ P’ y’ θ y P x’ x Degrees of Freedom R is 2x2 4 elements Note: R belongs to the category of normal matrices and satisfies many interesting properties: Rotation+ Scaling +Translation P’= (T R S) P If sx=sy, this is a similarity transformation! Transformation in 2D -Isometries -Similarities -Afnity -Projective Transformation in 2D Isometries: [Euclideans] - Preserve distance (areas) - 3 DOF - Regulate motion of rigid object Transformation in 2D Similarities: - Preserve - ratio of lengths - angles -4 DOF Transformation in 2D Afnities: Transformation in 2D Afnities: -Preserve: - Parallel lines - Ratio of areas - Ratio of lengths on collinear lines - others… - 6 DOF Transformation in 2D Projective: - 8 DOF - Preserve: - cross ratio of 4 collinear points - collinearity - and a few others… Eigenvalues and Eigenvectors A eigenvalue λ and eigenvector u satisfies Au = λu where A is a square matrix. I Multiplying u by A scales u by λ Eigenvalues and Eigenvectors Rearranging the previous equation gives the system Au − λu = (A − λI)u = 0 which has a solution if and only if det(A − λI) = 0. I The eigenvalues are the roots of this determinant which is polynomial in λ. I Substitute the resulting eigenvalues back into Au = λu and solve to obtain the corresponding eigenvector. Singular Value Decomposition T UΣV = A • Where U and V are orthogonal matrices, and Σ is a diagonal matrix. For example: Singular Value decomposition that An Numerical Example • Look at how the multiplication works out, left to right: • Column 1 of U gets scaled by the first value from Σ. • The resulting vector gets scaled by row 1 of VT to produce a contribution to the columns of A An Numerical Example + = • Each product of (column i of U)·(value i from Σ)·(row i of VT) produces a An Numerical Example We can look at Σ to see that the while the second first column has column has a a large efect much smaller efect in this example SVD Applications • For this image, using only the first 10 of 300 singular values produces a recognizable reconstruction • So, SVD can be used for image compression Principal Component Analysis • Remember, columns of U are the Principal Components of the data: the major patterns that can be added to produce the columns of the original matrix • One use of this is to construct a matrix where each column is a separate data sample • Run SVD on that matrix, and look at the first few columns of U to see patterns that are common among the columns • This is called Principal Component Analysis (or PCA) of the data samples Principal Component Analysis • Often, raw data samples have a lot of redundancy and patterns • PCA can allow you to represent data samples as weights on the principal components, rather than using the original raw form of the data • By representing each sample as just those weights, you can represent just the “meat” of what’s diferent between samples. • This minimal representation makes machine learning and other algorithms much more efcient .
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