Linear Algebra Simplified

Transcription

1 Linear Algebra Simplified

2 Readings for camera geometry, , for rotation representation

3 Inner (dot) Product v w α ),, ( y x y x y x y y y x x x w v T + + =!!! " # $ $ $ % & =

4 Inner (dot) Product v w α ),, ( y x y x y x y y y x x x w v T + + =!!! " # $ $ $ % & = w v w v T = 0

5 Cross product X is normal to both v and w

6 Cross Product

7 Cross Product + -

8 Cross Product + -

9 Cross Product + -

10

11 Example

12 Cross product as a matrix operator v u = ˆv u ˆv = x 3 x 2 x 3 0 x 1 x 2 x

13 Lines and Points

14 Line Representation a line is is the distance from the origin to the line is the norm direction of the line It can also be written as

15 Example of Line

16 Example of Line (2) 0.42*pi

17 Line Homogeneous Representation

18 Homogeneous representation Line in Is represented by a point in : But correspondence of line to point is not unique We define set of equivalence class of vectors in R^3 - (0,0,0) As projective space

19 Point

20 Homogenous representation of point A point lies on a line: A point in P^2 is defined by the equivalence class of k(x,y,1)

21 Homogeneous Coordinates Homogeneous coordinates represent coordinates in 2 dimensions with a 3-vector & $ % x y #! " & $ '' homogeneous '''' coords $ $ % x y 1 #!!!" A point:

22 Homogeneous -> Real Coordinates divide the third number out: (x, y, w) represents a point at location (x/w, y/w) (x, y, 0) represents a point at infinity (in direction x,y) (0, 0, 0) is not allowed y 2 1 (2,1,1) or (4,2,2) or (6,3,3) Convenient coordinate system to represent many useful transformations 1 2 x

23 Example of Point >> 0.6*42+0.8* 141 ans = 138

24 Line passing two points

25 Line passing through two points Two points: x Define a line l is the line passing two points Proof:

26 Line passing through two points More specifically,

27 Matlab codes function l = get_line_by_two_points(x, y) x1 = [x(1), y(1), 1]'; x2 = [x(2), y(2), 1]'; l = cross(x1, x2); l = l / sqrt(l(1)*l(1)+l(2)*l(2));

28 Example of Line Verify p2 lies on the line >> 218*0.6+13*0.8 P2 ans = (close) P1

29 Example of Line Test line: P2 >> p1 = [42,142,1]; >> p2=[218,13,1]; >> l = cross(p1,p2) l = P1 >> l = l/sqrt(l(1)^2+l(2)^2) l =

30 >> x = [221;23]; >> y = [218;268]; >> l = get_line_by_two_points(x,y); >> l l =

31 Point passing two lines

32 Intersection of lines Given two lines:, Define a point l X is the intersection of the two lines

33 Intersection of lines More specifically,

34 Matlab codes function x0 = get_point_by_two_line(l, l1) x0 = cross(l, l1); x0 = [x0(1)/x0(3); x0(2)/x0(3)];

35 Example of Lines Intersection

36 Point and Line at infinity

37 Point at infinity Example: Consider two parallel horizontal lines: Intersection = Point at infinity in the direction of y

38 Point at infinity, Ideal points Intersection: Any point (x 1,x 2,0) is intersection of lines at infinity

39 Points at infinity Under projective transformation, All parallel lines intersects at the point at infinity One point at infinity ó one parallel line direction Where are the points at infinity in the image plane?

40 Line at infinity A line passing all points at infinity: Because :

41 Projective Transformation

42 From Plane to Plane

43 Projective transformation Goal: study geometry of image projection from one plane to another plane(the image plane). Facts: 1) parallel lines intersect, 2) circle becomes ellipses, 3) straight line is still straight

44 Projective transformation x 1 h(x 1 ) x 2 h(x 2 ) x 3 h(x 3 ) Definition: Line remains a line! Projective transform is an invertible mapping from to itself, such that three points lies on a same line iff do.

45 Theorem: Projective transformation Check what happened to lies on line l? Line Mapping:

46 y x y x = How many independent para? Can we always set h33 = 1?

47 Classes of 2D projective transformations

48 Special case: Similarity Transformation Similarity

49 Special Case: Affine Transformation

50 Special case: Projective transformation

51

52 Example

53 Example Projective

54 Affine Example

55 Similarity Example

56 Matlab codes function img_now = projective_transform(img, H); px = [0 1]; py = [0 1]; tform = maketform('projective', H'); [img_now, xdata, ydata]= imtransform(img, tform, 'udata',... px, 'vdata', py, 'size', size(img));

57 Extra slides

58 Points, Lines & Projective Transformation

59 Lines under Projective Transformation With homogenous coordinates: Point: Line:

60 Before transformation

61 After transformation

62 Just checking Verify:

63 Vanishing point, revisited

64 Points at infinity: Revisited Where are the points at infinity in the image plane? The point at infinity can be in the FINITE region of the image!

65 Example

66 Seeing vanishing point vanishing point of horizontal direction:

67 Line at infinity: Revisited A line passing all points at infinity: In the image plane:

68 Line of infinity line of infinity is: Notice: this is the last column of

69 The line of infinity