Basics. Bilkent University. CS554 Computer Vision Pinar Duygulu

Transcription

1 1 Basics CS 554 Computer Vision Pinar Duygulu Bilkent University

2 2 Outline Image Representation Review some basics of linear algebra and geometrical transformations Slides adapted from Octavia Camps, Penn State and Stefan Roth, Brown University Introduction to Matlab Handling Images in Matlab

3 3 Image Representation Digital Images are 2D arrays (matrices) of numbers Each pixel is a measure of the brightness (intensity of light) that falls on an area of an sensor (typically a CCD chip)

4 4 Image Representation

5 5 Vectors Ordered set of numbers Example: coordinates of point

6 6 Vectors

7 7 Vectors Vector addition:

8 8 Vectors Vector subtraction:

9 9 Vectors Scalar Product:

10 10 Vectors Inner (dot) Product:

11 11 Vectors Projection of one point onto the other Different notations The shown segment has length <x,y>, if x and y are unit vectors

12 12 Linear Dependence

13 13 Basis

14 14 Bases

15 15 Bases Orthonormal basis:

16 16 Vectors Vector (cross) product: Note that vxw is not equal to wxv

17 17 Vectors Vector product computation:

18 18 Matrices

19 19 Matrices

20 20 Matrices

21 21 Matrices

22 22 Matrices

23 23 Matrices For singular matrices determinant is 0

24 24 Eigenvalues and Eigenvectors

25 25 Eigenvalues and Eigenvectors

26 26 Eigendecomposition

27 27 Singular Value Decomposition (SVD)

28 28 Singular Value Decomposition

29 29 Singular Value Decomposition (SVD) The columns of U are the eigenvectors of AAT and the (non-zero) singular values of A are the square roots of the (non-zero) eigenvalues of AAT

30 30 Geometrical Transformations 2D Translation:

31 31 Geometrical Transformations 2D Translation Equation:

32 32 Geometrical Transformations 2D Translation using matrices:

33 33 Geometrical Transformations Homogeneous Coordinates

34 34 Geometrical Transformations Back to Cartesian Coordinates

35 35 Geometrical Transformations 2D Translation using Homogeneous Coordinates

36 36 Geometrical Transformations Scaling

37 37 Geometrical Transformations Scaling Equation

38 38 Geometrical Transformations

39 39 Geometrical Transformations Scaling & Translating

40 40 Geometrical Transformations

41 41 Geometrical Transformations Rotation

42 42 Geometrical Transformations Rotation Equations:

43 43 Geometrical Transformations Rotation Equations: y' x = r.cosa y = r.sina y x' = r.cos(a+b) = r.cosa.cosb r.sina.sinb y' = r.sin(a+b) = r.sina.cosb + r.cosa.sinb B A x' x x' = x.cosb y.sinb y' = y.cosb + x.sinb [ x' ] = [cosb -sinb] [x] [ y' ] [sinb cosb ] [y] sin ( A + B ) = sin A cos B + cos A sin B cos ( A + B ) = cos A cos B - sin A sin B

44 44 Geometrical Transformations Scaling, Translating & Rotating:

45 45 Geometrical Transformations 3D Rotation of Points:

46 46 Geometrical Transformations 3D Translation of Points:

47 47 Introduction to Matlab One kind of object a rectangular numerical matrix Scalars : 1x1 matrices Vectors : matrices with only one row or one column a 3x3 matrix A = [1 2 3; 4 5 6; 7 8 9] is equal to A = [ ]

48 48 Introduction to Matlab A = [1 2; 3 4] N=5 v = [1 0 0] V = [1; 2; 3] v = v' v = 1:2:7 v = [] % creates a 2x2 matrix % a scalar % a row vector % a column vector % transpose of a vector % [start:stepsize:end] v = [ ] % empty vector

49 49 Introduction to Matlab m = zeros(2,3) v = ones(1,3) m = eye(3) v = rand(3,1) m = zeros(3) % creates a 2x3 matrix of zeros % creates a 1x3 matrix (row vector) of ones % identity matrix % randomly filled matrix % 3x3 matrix of zeros d = diag(a) % diagonal of matrix a

50 50 Introduction to Matlab v = [1 2 3] v(3) % access a vector element m = [ ; ; ; ] m(1,3) % access a matrix element (row #, column #) m(2,:) % access a whole matrix row m(:,1) % access a whole matrix column m(1,1:3) % access elements 1 through 3 of 1st row m(2:end, 3)

51 51 Introduction to Matlab m = [1 2 3; 4 5 6] size(m) % returns the size of a matrix m1 = zeros(size(m)) a = [ ]' 2*a a/4 b = [ ]' a+b a b a.^2

52 52 Introduction to Matlab a = [ ] sum(a) % sum of vector elements mean(a) % mean var(a) % variance std(a) % standard deviation max(a) % maximum min(a) % minimum a = [1 2 3; 4 5 6] mean(a) % mean of each column mean(a,2) % mean of each row max(max(a)) % maximum value of the matrix

53 53 Introduction to Matlab a = [1 2 3; 4 5 6; 7 8 9] inv(a) % matrix inverse eig(a) % vector of eigenvalues of a [V, D] = eig(a) % D:eigenvalues on diagonal %V :matrix of eigenvectors [U,S,V] = svd(a) % singular value decomposition of a % a = U * S * V'

54 54 Introduction to Matlab B = zeros(m,n) for i=1:m for j=1:n if A(i,j) > 0 B(i,j) = A(i,j) end end end B = zeros(m,n) ind = find(a>0) B(ind) = A(ind)

55 55 Introduction to Matlab x = [ ] plot(x, 2*x) xlabel('x') ylabel('2*x') title('dummy') bar(x)

56 56 Introduction to Matlab myfunction.m function y = myfunction(x) a = [ ] y = a + x;

57 57 Handling Images in Matlab I = imread('img.jpg'); imshow(i) imwrite(i, filename) % read a jpg image % shows an image % writes an image to file

58 58 Handling Images in Matlab figure imagesc(i) colormap gray; % display it as gray level image

59 59 Handling Images in Matlab size(img)

60 60 Handling Images in Matlab

61 61 Handling Images in Matlab

62 62 Handling Images in Matlab Histograms

63 63 Handling Images in Matlab Histograms