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

Transcription

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 Matrices Transpose: If A is symmetric Examples: Symmetric? No! Symmetric? Yes!

16 Matrices Determinant: A must be square Example:

17 Matrices Inverse: A must be square Example:

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

41 Eigenvalues and Eigenvectors The eigenvalues of A are the roots of the characteristic equation diagonal form of matrix Eigenvectors of A are columns of S

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

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45 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

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

47 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

48 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

49 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

50 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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52 HW 0.1: Compute eigenvalues and eigenvectors of the following transformations