3D Computer Vision II. Reminder Projective Geometry, Transformations. Nassir Navab. October 27, 2009

Transcription

1 3D Computer Vision II Reminder Projective Geometr, Transformations Nassir Navab based on a course given at UNC b Marc Pollefes & the book Multiple View Geometr b Hartle & Zisserman October 27, 29

2 2D Transformations 2

3 2D Transformations Scaling Isotropic Scaling Non-isotropic Scaling Rotation Translation Euclidean Transformation Metric Transformation (Similarit) Affine Transformation Projective Transformation 3

4 2D Transformations Isotropic Scaling 4

5 2D Transformations Isotropic Scaling Same scaling factor in both dimensions: ' s s ' s s 5

6 2D Transformations Non-Isotropic Scaling 6

7 2D Transformations Non-Isotropic Scaling Different scaling factors in the two dimensions: ' s s ' s 2 s 2 7

8 2D Transformations Rotation 8

9 2D Transformations Rotation Rotation around origin b angle θ Rotation Matri R cosθ sinθ sinθ cosθ Transformed Point p ' Rp where ' cosθ sinθ ' sinθ cosθ 9

10 2D Transformations Rotation β ' v *cos( α β ) v *cos( α)*cos( β ) v *sin( α)*sin( β ) ' cosα sinα sinα ' v *sin( α β ) v *cos( α)*sin( β ) v *sin( α)*cos( β ) ' cosα

11 2D Transformations Translation

12 2D Transformations Translation Original point Transformed point p p ' Translation Transformation ' ' v p ' v p t t p T p ' 2

13 2D Transformations Translation Homogeneous Coordinates: into the matri- How to get the translation transformation vector-product form p Ap? p ' p t t Homogeneous point: p Transformation Matri: v p ' t t t t 3

14 Homogeneous Coordinates of Points λ λ λ λ Real point [,] (homogeneous representation): 4

15 2D Transformations Affine Transformations 5

16 2D Transformations Affine Transformations A ' d c b a d c b a ( ) ( ) ( ) φ φ θ R D R R A 2 λ λ D 6

17 Affine Transformations: Homogenous Coordinates ' A ' ' d c b a d c b a λ λ 7

18 Homogeneous Coordinates of Points λ λ λ λ λ Real point [,] (homogeneous representation): Points at infinit: Eample: 8

19 Affine Transformations: Points at Infinit ' ' ' a b λ c d a b λ c d 9

20 General Affine Transformations (6 Parameters) ' A ' ' f e d c b a f e d c b a λ λ 2

21 2D Transformations Projective Transformations

22 ' H ' ' h g f e d c b a h g f e d c b a λ λ 2D Transformations Projective Transformations

23 ' ' ' ' h g f e d h g c b a h g f e d c b a λ ' 2D Transformations Projective Transformations

24 Affine Transformations: Points at Infinit h g e d b a h g f e d c b a λ λ ' ' ' 24

25 h g e d h g b a h g e d b a ' ' ' ' λ H ' 2D Transformations Projective Transformations

26 2D Projective Space The Projective Plane A line in plane can be represented as: abc We can represent an line l b three parameters a, b, and c The same line l can be represented b multiples of a, b and c (ka, kb and kc for a scalar k) since: kakbkck*abc A line is represented b a set of vectors, which differ onl in scale k (k non zero) This equivalence set of vectors is known as a Homogeneous Vector An particular vector of the class is a representative of the equivalence class The set of all these equivalence classes (i.e. homogeneous vectors) of vectors in R³ (ecluding (,,)) is called the Projective Space P² 26

27 A Model for the Projective Plane We ll use this intuition throughout the course. 27

28 A Model for the Projective Plane eactl one line through two points eactl one point at intersection of two lines 28

29 Homogeneous Coordinates Points p T T (,,) ~ k(,,), k A point is represented in P² b a simple vector Remember that ever vector in P² is just a representative of an equivalence class of vectors Hence a point is represented b an vector of the equivalence class 29

30 Homogeneous Coordinates Lines In P² lines as well are represented as vectors Cecause of the equivalence class of vectors, an vector of the class is representative a b c ( a,b,c) T ( ka) ( kb) kc, k ( a,b,c ) T ~ k( a,b,c) T Note that Points and Lines are represented both in P² as vectors 3

31 Degrees of Freedom (DOF) DOF refers to the number of parameters needed to specif an object in the respective space Eamples: line in 2D (2dof) line in 3D (4dof) 3

32 Homogeneous Vectors (in P²) have 2 DOF An homogeneous vector (in P²) has 3 coordinates but onl 2 DOF since it is up to scale hence onl the two ratios : 3 and 2 : 3 between the coordinates are significant. Homogeneous vector : (, 2, 3 ) T k (, 2, 3 ) T The ratios :3 and 2:3: : 3 3 k k : 3 3 k k

33 Incidence of Points and Lines The point lies on the line l if and onl if T ll T (,,) T l ( a,b,c) T (,,) ( a,b,c) T T l on if and onl if a b c 33

34 Points from Lines - Intersections of Lines The intersection point of two lines l and l 2 satisfies: t l t l 2 l l 2 Eample 34

35 Points from Lines - Intersections of Lines Eample t l t l 2 l l 2 The intersection point of two lines l and l 2 satisfies: 35

36 Alternative Notation for the Cross Product The cross product of two vectors a and b (ab) can be written as a product of an anti-smmetric matri [a] and the vector b a b [ a] b [ b] a The appropriate anti-smmetric matri [a] is: [ a] a3 a 2 a a 3 a 2 a 36

37 Points from Lines Intersections of Lines The intersection of two lines l and l' is the point l l' [ l] l' [ l' ] l Eample 37

38 Lines from Points Line Joining two Points The line l joining two points and 2 satisfies: t l 2 t l l 2 38

39 Lines from Points Line Joining two Points The line through two points and ' is the line l ' [] ' [' ] 39

40 Lines from Points Line Joining two Points The line through two points and is the line ] [' ' [] ' l ' Eample 4

41 Ideal Points and the Line at Infinit ( ) T,, l' l a b Intersections of parallel lines are the points at infinit (ideal points) ( ) ( ) T T and ',, l',, l c b a c b a Ideal points ( ) T, 2, Line at infinit ( ) T,, l l 2 2 R P Note that in P 2 there is no distinction between ideal points and others ) ' ( ) ' ( ' ' ] [ a b c c a c c b c b a a b a c b c l l alternative definition of P²: 4

42 Ideal Points and the Line at Infinit an Eample Eample 2 42

43 Dualit T l l T l l l' l ' Dualit principle: To an theorem of 2-dimensional projective geometr there corresponds a dual theorem, which ma be derived b interchanging the role of points and lines in the original theorem. 43

44 Summar Projective Space P² Homogeneous Coordinates Points Lines Points and Lines at the Infinit Intersection of Lines: Points Connection of Points: Lines Dualit 44