Linear Algebra & Geometry why is linear algebra useful in computer vision?

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1 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

2 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

3 Agenda 1. Basics definitions and properties 2. Geometrical transformations 3. SVD and its applications

4 Vectors (i.e., 2D or 3D vectors) P = [x,y,z] p = [x,y] 3D world Image

5 Vectors (i.e., 2D vectors) x2 v θ P Magnitude: x1 If, Is a UNIT vector Is a unit vector Orientation:

6 Vector Addition v+w v w

7 Vector Subtraction v v-w w

8 Scalar Product v av

9 Inner (dot) Product v α w The inner product is a SCALAR!

10 Orthonormal Basis x2 j i v θ P x1

11 Vector (cross) Product u w α v The cross product is a VECTOR! Magnitude:kuk = kv wk = kvkkwk sin Orientation:

12 Vector Product Computation i = (1,0,0) j = (0,1,0) k = (0,0,1) i = 1 j = 1 k = 1 i = j k j = k i k = i j = ( x2 y 3 x3 y ) i + 2 ( x3 y 1 x1 y3) j+ ( x1 y 2 x2 y1 ) k

13 Matrices A n m = 2 3 a 11 a a 1m a 21 a a 2m a n1 a n2... a nm Pixel s intensity value Sum: A and B must have the same dimensions! Example:

14 Matrices A n m = 2 3 a 11 a a 1m a 21 a a 2m a n1 a n2... a nm Product: A and B must have compatible dimensions!

15 Matrix Transpose Definition: C m n = A T n m c ij = a ji Identities: (A + B) T = A T + B T (AB) T = B T A T If A = A T, then A is symmetric

16 Matrix Determinant Useful value computed from the elements of a square matrix A det [ a 11 ] = a11 [ ] a11 a det 12 = a a 21 a 11 a 22 a 12 a a 11 a 12 a 13 det a 21 a 22 a 23 = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 31 a 32 a 33 a 13 a 22 a 31 a 23 a 32 a 11 a 33 a 12 a 21

17 Matrix Inverse Does not exist for all matrices, necessary (but not sufficient) that the matrix is square AA 1 = A 1 A = I [ ] 1 A 1 a11 a = 12 = 1 a 21 a 22 det A If det A = 0, A does not have an inverse. [ a22 a 12 a 21 a 11 ], det A 0

18 2D Geometrical Transformations

19 2D Translation P P t

20 2D Translation Equation ty y P t P x tx

21 2D Translation using Matrices ty P t P y x tx

22 Homogeneous Coordinates Multiply the coordinates by a non-zero scalar and add an extra coordinate equal to that scalar. For example,

23 Back to Cartesian Coordinates: Divide by the last coordinate and eliminate it. For example, NOTE: in our example the scalar was 1

24 2D Translation using Homogeneous Coordinates ty P t P y t P x tx

25 Scaling P P

26 Scaling Equation s y y P y P x s x x

27 Scaling & Translating P P P =S P P =T P P =T P =T (S P)=(T S) P = A P

28 Scaling & Translating A

29 Translating & Scaling = Scaling & Translating?

30 Rotation P P

31 Rotation Equations Counter-clockwise rotation by an angle θ y y P x θ x P

32 Degrees of Freedom R is 2x2 4 elements Note: R belongs to the category of normal matrices and satisfies many interesting properties:

33 Rotation+ Scaling +Translation P = (T R S) P If s x =s y, this is a similarity transformation!

34 Transformation in 2D -Isometries -Similarities -Affinity -Projective

35 Transformation in 2D Isometries: [Euclideans] - Preserve distance (areas) - 3 DOF - Regulate motion of rigid object

36 Transformation in 2D Similarities: - Preserve - ratio of lengths - angles -4 DOF

37 Affinities: Transformation in 2D

38 Transformation in 2D Affinities: -Preserve: - Parallel lines - Ratio of areas - Ratio of lengths on collinear lines - others - 6 DOF

39 Transformation in 2D Projective: - 8 DOF - Preserve: - cross ratio of 4 collinear points - collinearity - and a few others

40 Eigenvalues and Eigenvectors A eigenvalue λ and eigenvector u satisfies where A is a square matrix. Au = λu Multiplying u by A scales u by λ

41 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. The eigenvalues are the roots of this determinant which is polynomial in λ. Substitute the resulting eigenvalues back into Au = λu and solve to obtain the corresponding eigenvector.

42 Singular Value Decomposition UΣV T = A Where U and V are orthogonal matrices, and Σ is a diagonal matrix. For example:

43 Singular Value decomposition that

44 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 V T to produce a contribution to the columns of A

45 An Numerical Example + = Each product of (column i of U) (value i from Σ) (row i of V T ) produces a

46 An Numerical Example We can look at Σ to see that the first column has a large effect while the second column has a much smaller effect in this example

47 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

48 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

49 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 different between samples. This minimal representation makes machine learning and other algorithms much more efficient

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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

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