Multiple View Geometry in Computer Vision

Transcription

1 Multiple View Geometry in Computer Vision

2 2

3 3

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5 5

6 Visual 3D Modeling from Images 6

7 AutoStitch 7

8 Image Composite Editor redmond/groups/ivm/ice/ 8

9 Deep Zoom & HD View 9

10 Photosynth 10

11 PhotoCity 11

12 Projective Transformations

13 Homogeneous coordinates Homogeneous representation of lines ax by c 0 a,b,c T ka a,b,c T ~ ka,b,c T ( ) x ( kb) y kc 0, k 0 equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3 (0,0,0) T forms P 2 Homogeneous representation of points x xy,,1 on l a,b,c T if and only if ax by c 0 T x,y, 1 a,b,c x,y, 1 l 0 T T x, y,1 ~ kx, y,1, k 0 The point x lies on the line l if and only if x T l=l T x=0 Homogeneous coordinates x, y, z T Inhomogeneous coordinates x, y T but only 2DOF 13

14 Points from lines and vice-versa Intersections of lines The intersection of two lines l and l' is x l l' Line joining two points The line through two points x and x' is l x x' Example y 1 x 1 14

15 Ideal points and the line at infinity Intersections of parallel lines l a, b, c T and l' a, b, c' T l l' b, a,0 T Example b, a a,b tangent vector normal direction x 1 x 2 Ideal points x, 2,0 T 1 x 0,0,1 Line at infinity T l P 2 R 2 l Note that in P 2 there is no distinction between ideal points and others 15

16 A model for the projective plane exactly one line through two points exactly one point at intersection of two lines 16

17 Duality x l x T l 0 l T x 0 x l l' l x x' Duality principle: To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem 17

18 Projective 2D Geometry 18

19 Projective transformations Definition: A projectivity is an invertible mapping h from P 2 to itself such that three points x 1,x 2,x 3 lie on the same line if and only if h(x 1 ),h(x 2 ),h(x 3 ) do. Theorem: A mapping h:p 2 P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 reprented by a vector x it is true that h(x)=hx Definition: Projective transformation x' x' x' h h h h h h h h h x x x or x' H x 8DOF projectivity=collineation=projective transformation=homography 19

20 Mapping between planes central projection may be expressed by x =Hx (application of theorem) 20

21 Removing projective distortion select four points in a plane with know coordinates x' 1 h11x h12y h13 x' 2 h21x h22y h x' y' x' h x h y h x' h x h y h x' y' h31x h32y h33 h11x h12y h13 h31x h32y h33 h21x h22y h (linear in h ij ) (2 constraints/point, 8DOF 4 points needed)

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23 More Examples 23

24 Transformation for lines For a point transformation x' H x Transformation for lines l' H -T l 24

25 Isometries x' y' 1 cos sin 0 sin cos 0 tx x t y y orientation preserving: orientation reversing: 1 1 x' H E x R T 0 t x 1 R T R I 3DOF (1 rotation, 2 translation) special cases: pure rotation, pure translation Invariants: length, angle, area 25

26 Similarities cos sin sin cos 1 ' ' y x t s s t s s y x y x x 0 x x' 1 t T R H s S I R R T also know as equi-form (shape preserving) metric structure = structure up to similarity (in literature) 4DOF (1 scale, 1 rotation, 2 translation) Invariants: ratios of length, angle, ratios of areas, parallel lines

27 Affine transformations ' ' y x t a a t a a y x y x x 0 x x' 1 t T A H A non-isotropic scaling! (2DOF: scale ratio and orientation) 6DOF (2 scale, 2 rotation, 2 translation) Invariants: parallel lines, ratios of parallel lengths, ratios of areas DR R R A D

28 Projective transformations 28 x v x x' v P T t A H Action non-homogeneous over the plane 8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity) Invariants: cross-ratio of four points on a line (ratio of ratio) T 2 1, v v v v v s P A S T T T T v t v t A I K R H H H H T RK tv A s K 1 det K upper-triangular, decomposition unique (if chosen s>0)

29 Overview Transformations Projective 8dof Affine 6dof Similarity 4dof Euclidean 3dof h h h a a 0 sr sr 0 r11 r h h h a a r r sr sr h13 h 23 h 33 tx t y 1 tx t y 1 tx t y 1 Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l Ratios of lengths, angles. The circular points I,J lengths, areas. 29

30 Line at infinity v x v x v x x x x v A A T t x x x x v A A T t Line at infinity becomes finite, allows to observe vanishing points, horizon, Line at infinity stays at infinity

31 The line at infinity 31 v 1 v 2 l 1 l 2 l 4 l 3 l v 1 v 2 l l l v l l v The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity l t 0 l l A A H T T A

32 Affine properties from images projection rectification H PA 1 0 l1 0 1 l H l 3 A T l l l, l 0 l

33 Affine Rectification v 1 v 2 l l 1 l 3 l 2 l 4 v v l l1 2 l3 4 v 1 v 2 1 l 2 l 33

34 Projective 3D geometry Projective 15dof A T v t v Intersection and tangency Affine 12dof A T 0 t 1 Parallellism of planes, Volume ratios, centroids, The plane at infinity π Similarity 7dof s R T 0 t 1 The absolute conic Ω Euclidean 6dof R 0 T t 1 Volume 34

35 Camera Calibration

36 Pinhole Camera Model 36 T T Z fy Z fx Z Y X ) /, / ( ),, ( Z Y X f f Z fy fx Z Y X Z Y X f f Z fy fx

37 Pinhole Camera Model x K I 0X cam K f f px p y 1 Principal point offset Calibration Matrix 37

38 Internal Camera Parameters 38

39 Camera rotation and translation X ~ R X~ -C ~ cam RC ~ X R Y X cam 0 1 Z 1 R 0 RC X 1 x K I 0X cam x KR I C ~ X x PX P K R t t RC ~ 39

40 Camera Parameter Matrix P 40

41 CCD Cameras 41

42 Correcting Radial Distortion of Cameras 42

43 43

44 Calibration Process 44

45 x c ( x c )(1 k r k r k r )cos u x d x 1 d 2 d 3 d ( y c )(1 k r k r k r )sin d y 1 d 2 d 3 d y c ( x c )(1 k r k r k r )sin u y d x 1 d 2 d 3 d ( y c )(1 k r k r k r )cos d y 1 d 2 d 3 d x y p p m x m y m m x 0 u 1 u 2 6 u m7yu 1 m x m y m m x 3 u 4 u 5 6 u m7yu 1 45

46 OpenCV alibration_and_3d_reconstruction.html 46

47 Camera Calibration Toolbox 47

48 Camera Calibration 48

49 Calibrating a Stereo System 49

50 Robust Multi-camera Calibration 50

51 Multi-Camera Self Calibration 51

52 Multi-Camera Self Calibration 52

53 ARToolKit Camera Calibration 53

54 Epipolar Geometry and 3D Reconstruction.

55 Introduction Computer vision is concerned with the theory behind artificial systems that extract information from images. The image data can take many forms, such as video sequences, views from multiple cameras. Computer vision is, in some ways, the inverse of computer graphics. While computer graphics produces image data from 3D models, computer vision often produces 3D models from image data. Today one of the major problems in Computer vision is correspondence search. 55

56 Epipolar Geometry C,C,x,x and X are coplanar 56

57 Epipolar Geometry epipole 57

58 Epipolar Geometry All points on p project on l and l 58

59 Epipolar Geometry Family of planes p and lines l and l Intersection in e and e 59

60 Epipolar Geometry epipoles e,e = intersection of baseline with image plane = projection of projection center in other image = vanishing point of camera motion direction an epipolar plane = plane containing baseline (1-D family) an epipolar line = intersection of epipolar plane with image (always come in corresponding pairs) 60

61 The Fundamental Matrix F x' H π x l' e' x' e' H x Fx π H : projectivity=collineation=projective transformation=homography 61

62 Cross Products 62

63 Epipolar Lines 63

64 Three Questions (i) Correspondence geometry: Given an image point x in the first view, how does this constrain the position of the corresponding point x in the second image? (ii) Camera geometry (motion): Given a set of corresponding image points {x i x i }, i=1,,n, what are the cameras P and P for the two views? (iii) Scene geometry (structure): Given corresponding image points x i x i and cameras P, P, what is the position of (their pre-image) X in space? 64

65 Parameter estimation 2D homography Given a set of (x i,x i ), compute H (x i =Hx i ) 3D to 2D camera projection Given a set of (X i,x i ), compute P (x i =PX i ) Fundamental matrix Given a set of (x i,x i ), compute F (x i T Fx i =0) 65

66 The fundamental matrix F Algebraic Derivation X λ F e' P P' P x λc l' e' x' e' H x Fx l' P'C P'P x π P x P P I X λ (note: doesn t work for C=C F=0) 66

67 The fundamental matrix F Correspondence Condition The fundamental matrix satisfies the condition that for any pair of corresponding points x x in the two images x' T Fx 0 67

68 The Fundamental Matrix Song 68

69 The Fundamental Matrix 69

70 Epipolar geometry: basic equation 70 0 Fx x' T separate known from unknown 0 ' ' ' ' ' ' f yf xf f y yf y xf y f x yf x xf x 0,,,,,,,,,1, ',, ', ' ',, ', ' T f f f f f f f f f y x y y y x y x y x x x (data) (unknowns) Af 0 0 f 1 ' ' ' ' ' ' 1 ' ' ' ' ' ' n n n n n n n n n n n n y x y y y x y x y x x x y x y y y x y x y x x x

71 the NOT normalized 8-point algorithm f f f f f f f f f y x y y y y x x x y x x y x y y y y x x x y x x y x y y y y x x x y x x n n n n n n n n n n n n ~10000 ~10000 ~10000 ~10000 ~100 ~100 1 ~100 ~100! Orders of magnitude difference Between column of data matrix least-squares yields poor results

72 the normalized 8-point algorithm Transform image to ~[-1,1]x[-1,1] (0,500) (700,500) (-1,1) (0,0) (1,1) (0,0) (700,0) (-1,-1) (1,-1) Least squares yields good results (Hartley, PAMI 97) 72

73 The fundamental matrix F F is the unique 3x3 rank 2 matrix that satisfies x T Fx=0 for all x x (i) Transpose: if F is fundamental matrix for (P,P ), then F T is fundamental matrix for (P,P) (ii) Epipolar lines: l =Fx & l=f T x (iii) Epipoles: on all epipolar lines, thus e T Fx=0, x e T F=0, similarly Fe=0 (iv) F has 7 d.o.f., i.e. 3x3-1(homogeneous)-1(rank2) (v) F is a correlation, projective mapping from a point x to a line l =Fx (not a proper correlation, i.e. not invertible) 73

74 Fundamental matrix for pure translation 74

75 Fundamental matrix for pure translation 75

76 Fundamental matrix for pure translation e' e' F H 1 H K RK Example: 0 e' 1,0,0 T F 0 0 x' T Fx 0 y y' x x' PX K[I 0]X -1 P'X K[I t] K x Z ( X,Y,Z ) T -1 K x/z x' x Kt/Z motion starts at x and moves towards e, faster depending on Z pure translation: F only 2 d.o.f., x T [e] x x=0 auto-epipolar 76

77 General Motion x' x' T T x' e' Hx 0 e' xˆ 0 K'RK -1 x K't/Z 77

78 Projective transformation and invariance Derivation based purely on projective concepts xˆ x Hx, xˆ' H'x' Fˆ PX H' -1 PH H X Pˆ Xˆ -1 P'H H X Pˆ' Xˆ -T FH -1 F invariant to transformations of projective 3-space x' P'X P, P' F F P,P' unique not unique Canonical form P [I 0] P' [M m] F m M F e' P' P 78

79 Projective ambiguity of cameras given F 79 previous slide: at least projective ambiguity this slide: not more! Show that if F is same for (P,P ) and ~ (P,P ), ~ there exists a projective transformation H so that P=HP and P =HP P [I 0] P' [A a] F a A ~ a lemma: ~ a A ~ ka A ~ k 1 A av ~ ~ P [I 0] P' [A ~ ~ a] T af a a A 0 ~ af rank 2 T a A ~ a a ka ~ - A 0 ka ~ - A av A ~ 1 H k I 0 1 T k v k 1 P'H [A a] k I 1 k v T 0 k [ k ~ ~ 1 ~ T A -av ka] P' ~ a ka (22-15=7, ok)

80 Canonical cameras given F F matrix corresponds to P,P iff P T FP is skew-symmetric X T P' T FPX 0, X F matrix, S skew-symmetric matrix P [I 0] [SF e' ] T Possible choice: P [I 0] P' [SF e' ] T T F[I 0] F S F T e' F P' [[e'] F (fund.matrix=f) e' ] T 0 F S 0 0 T F 0 0 Canonical representation: P [I 0] P' [[e'] F e' v T λe'] 80

81 The Essential matrix ~ Fundamental matrix for calibrated cameras (remove K) t T E R R[R t] xˆ ' T E Exˆ K' T 0 FK xˆ 5 d.o.f. (3 for R; 2 for t up to scale) K -1 x; xˆ' K -1 x' E is essential matrix if and only if two singularvalues are equal (and third=0) T E Udiag(1,1,0)V 81

82 3D Reconstruction 82

83 Epipolar Geometry Underlying structure in set of matches for rigid scenes l T 1 l 2 T m F m1 2 0 Fundamental matrix (3x3 rank 2 matrix) C1 m1 l m1 1 e 1 e 2 P m2 l2 m2 C 2 M L2 L1 Canonical representation: P [I 0] P' [[e'] F e' v T λe'] 1. Computable from corresponding points 2. Simplifies matching 3. Allows to detect wrong matches 4. Related to calibration 83

84 3D reconstruction of cameras and structure Reconstruction Problem: given x i x i, compute P,P and X i xi PX i x i PX i for all i without additional informastion possible up to projective ambiguity 84

85 Outline of 3D Reconstruction (i) (ii) (iii) Compute F from correspondences Compute camera matrices P,P from F Compute 3D point for each pair of corresponding points computation of F use x i Fx i =0 equations, linear in coeff. F 8 points (linear), 7 points (non-linear), 8+ (least-squares) computation of camera matrices T use P [I 0] P' [[e'] F e' v λe' ] triangulation compute intersection of two backprojected rays 85

86 Reconstruction ambiguity: similarity x i PX i -1 PH H X S S i PH -1 S K[R t] R' 0 T - R' λ T t' K[RR' T RR' T t' λt] 86

87 Reconstruction ambiguity: projective x i PX i -1 PH H X P P i 87

88 Terminology x i x i Original scene X i Projective, affine, similarity reconstruction = reconstruction that is identical to original up to projective, affine, similarity transformation Literature: Metric and Euclidean reconstruction = similarity reconstruction 88

89 The projective reconstruction theorem If a set of point correspondences in two views determine the fundamental matrix uniquely, then the scene and cameras may be reconstructed from these correspondences alone, and any two such reconstructions from these correspondences are projectively equivalent x i x i P P P H P,P ', P,P ', 1 1 X1i P theorem from last class 2 2 X2i X except :Fx xf P 1H 2 HX1-1 HX1 i P1 H HX1 i P1 X1 i xi P2 2i 2 X along same ray of P 2, idem for P 2 two possibilities: X 2i =HX 1i, or points along baseline key result : allows reconstruction from pair of uncalibrated images i i 89

90 90

91 Stratified reconstruction (i) (ii) (iii) Projective reconstruction Affine reconstruction Metric reconstruction 91

92 Projective to affine Remember 2-D case 92

93 Projective to affine P,P', π T H - A, B, C, D T 0,0,0, 1 T X i π I 0 H π 0,0,0,1 T (if D 0) theorem says up to projective transformation, but projective with fixed p is affine transformation can be sufficient depending on application, e.g. mid-point, centroid, parallellism 93

94 Translational motion points at infinity are fixed for a pure translation reconstruction of x i x i is on p F [e] [e' ] P [I 0] P [I e'] 94

95 Scene constraints Parallel lines parallel lines intersect at infinity reconstruction of corresponding vanishing point yields point on plane at infinity 3 sets of parallel lines allow to uniquely determine p remark: in presence of noise determining the intersection of parallel lines is a delicate problem remark: obtaining vanishing point in one image can be sufficient 95

96 Scene constraints 96

97 Scene constraints Distance ratios on a line known distance ratio along a line allow to determine point at infinity (same as 2D case) 97

98 The infinity homography P [M m] P' [M' m' ] -1 H M' M unchanged under affine transformations P [M m] A a 0 1 H M' AA -1 M -1 [MA Ma m] X X ~,0 T affine reconstruction P [I 0] P [H e] x MX ~ x' M' X ~ 98

99 One of the cameras is affine According to the definition, the principal plane of an affine camera is at infinity to obtain affine recontruction, compute H that maps third row of P to (0,0,0,1) T and apply to cameras and reconstruction e.g. if P=[I 0], swap 3 rd and 4 th row, i.e. H

100 Affine to metric identify absolute conic transform so that : X Y Z 0, on π then projective transformation relating original and reconstruction is a similarity transformation 2 in practice, find image of W image w back-projects to cone that intersects p in W 2 2 * note that image is independent of particular reconstruction * 100

101 Affine to metric given P [M m] ω possible transformation from affine to metric is H A AA T T M ωm 1 (cholesky factorisation) proof: P ω M * M -1 ω PH M -1 M M -1 M -T [MA m] T M AA MAA T T M T * I

102 Orthogonality vanishing points corresponding to orthogonal directions T v ωv2 1 0 vanishing line and vanishing point corresponding to plane and normal direction l ωv 102

103 Correspondence and RANSAC Algorithm.

104 Correspondence Search 104

105 Feature Matching To match the points in one image to the points in the other image by exhaustive search(to match one point in one image to all the points in the other image) is a difficult and long process so some constraints are applied. As geometric constraint to minimize the search area for correspondence. The geometric constraints is provided by the epipolar geometry 105

106 Harris Detector: Mathematics Change of intensity for the shift [x,y]: E( x, y) w( u, v)[ I( x u, y v) I( x, y)] uv, 2 Window function Shifted intensity Intensity Window function w(u,v) = or 1 in window, 0 outside Gaussian 106

107 E x y I x y I x x y y 2 (, ) [ (, ) (, )] u x,v y w x I( x, y) I( x, y) [ I x( x, y) I y( x, y)] w y x [ I x( x, y) I y( x, y)] w y 2 2 ( I x( x, y)) I x( x, y) I y( x, y) w w x y x 2 I x( x, y) I y( x, y) ( I y( x, y)) y w w x x yc( x, y) y 2 107

108 Harris Detector: Mathematics For small shifts [u,v] we have a bilinear approximation: u E( u, v) u, v M v where M is a 22 matrix computed from image derivatives: M 2 I x I xi y w( x, y) 2 xy, I xi y I y 108

109 ( max ) -1/2 ( min) -1/2 Harris Detector: Mathematics Intensity change in shifting window: eigenvalue analysis u E( u, v) u, v M v 1, 2 eigenvalues of M If we try every possible orientation n, the max. change in intensity is 2 Ellipse E(u,v) = const 109

110 Harris Detector: Mathematics Classification of image points using eigenvalues of M: 2 Edge 2 >> 1 Corner 1 and 2 are large, 1 ~ 2 ; E increases in all directions 1 and 2 are small; E is almost constant in all directions Flat region Edge 1 >>

111 Harris Detector: Mathematics Measure of corner response: trace 2 R det M k M det M trace M (k empirical constant, k = ) 111

112 Harris Detector: Mathematics R depends only on eigenvalues of M R is large for a corner R is negative with large magnitude for an edge R is small for a flat region 2 Edge R < 0 Corner R > 0 Flat R small Edge R <

113 Harris Detector The Algorithm: Find points with large corner response function R Take the points of local maxima of R 113

114 Harris Detector : Workflow 114

115 Compute corner response R Harris Detector : Workflow 115

116 Find points with large corner response: R>threshold Harris Detector : Workflow 116

117 Take only the points of local maxima of R Harris Detector : Workflow 117

118 Harris Detector : Workflow 118

119 Correlation for Correspondence Search i 0,0 j 7,7 Left image: 1. From the left feature point image we select one feature point. 2. Draw the window(n*n) around, with feature point in the center. 3. Calculate the normalized window using the formula s 1 N N i1 j1 w 1 ( i, j)* w 1 ( i, j) N is the size of window w 1( nor) w ( i, j) 1 ( i, j) s 119 1

120 Correlation Algorithm Select one feature point from the first image. Draw the window across it(7*7). Normalize the window using the given formula. s w ( i, j)* w ( i, j) N is the size of window i1 j1 w 1( nor) N N ( i, j) w1 ( i, j) s 1 Find the feature in the right image, that are to be considered in the first image,(this should be done by some distance thresholding) After finding the feature point the window of same dimension should be selected in the second image. The normalized correlstion measure should be calculated using the formula N N s w ( i, j)* w ( i, j) i1 j1 cormat N N i1 j1 s 2 w ( i, j)* w ( i, j)

121 RANSAC 121

122 RANSAC Select sample of m points at random 122

123 RANSAC Select sample of m points at random Calculate model parameters that fit the data in the sample 123

124 RANSAC Select sample of m points at random Calculate model parameters that fit the data in the sample Calculate error function for each data point 124

125 RANSAC Select sample of m points at random Calculate model parameters that fit the data in the sample Calculate error function for each data point Select data that support current hypothesis 125

126 RANSAC Select sample of m points at random Calculate model parameters that fit the data in the sample Calculate error function for each data point Select data that support current hypothesis Repeat sampling 126

127 RANSAC Select sample of m points at random Calculate model parameters that fit the data in the sample Calculate error function for each data point Select data that support current hypothesis Repeat sampling 127

128 RANSAC ALL-INLIER SAMPLE RANSAC time complexity k number of samples drawn N number of data points t M time to compute a single model m S average number of models per sample 128

129 Feature-space outliner rejection Can we now compute H from the blue points? No! Still too many outliers What can we do?

130 Matching features What do we do about the bad matches?

131 RAndom SAmple Consensus Select one match, count inliers

132 RAndom SAmple Consensus Select one match, count inliers

133 Least squares fit Find average translation vector

134 RANSAC for estimating homography RANSAC loop: 1. Select four feature pairs (at random) 2. Compute homography H (exact) 3. Compute inliers where SSD(p i, H p i) < ε 4. Keep largest set of inliers 5. Re-compute least-squares H estimate on all of the inliers

135 RANSAC for Fundamental Matrix Step 1. Extract features Step 2. Compute a set of potential matches Step 3. do Step 3.1 select minimal sample (i.e. 7 matches) Step 3.2 compute solution(s) for F (verify hypothesis) Step 3.3 determine inliers until (#inliers,#samples)<95% Step 4. Compute F based on all inliers Step 5. Look for additional matches Step 6. Refine F based on all correct matches (generate hypothesis)

136 RANSAC 136

137 RANSAC 137

138 Example: Mosaicking with homographies 138

139 Recognizing Panoramas 139

140 Recognizing Panoramas 140

141 THANK YOU!! 141

142 9/28 10/5 10/12 10/19 Liu Chang photosynth create account Take pictures Construct photosynth Markus Photomodeller Select one small object Make 3d model data Yang Yun PhotoCity Select one small object Make 3d model data Wang Ping PhotoCity Presentation file How to install & use Cygwin, bunder, Matlab CC Liu Pengxin Zhu Zi Jian 142 Wang Yuo Remove Perspective distortion Load image Choose feature points SIFT & ASIFT OpenCV CC PhotoTourism ARToolkit Camera Calibration PhotoTourism Matlab CC OpenCV CC Search others photosynth works Select one point in DSU, take pictures Projector Price? Matlab CC Lab Image Data SIFT Fundamental Matrix Remove PT OpenCV CC PT PhotoTourism ARToolkit Camera Calibration Fundamental Matrix M C C led lamp

143 Topics in Image Processing

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149 illisis 149

150 LYYN 150

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152 faceapi 152

153 Total Immersion - D Fusion Pro 153

154 D Fusion Pro - Markless Tracking 154

155 155

156 Monitoring System 156

157 Scenarios Clear Vision - Denoising Motion Deblurring SuperResolution Panoramic View Background Modeling 157

158 Around View Monitor 158

159 Projective Transformations

160 Projective 2D Geometry 160

161 Projective transformations Definition: A projectivity is an invertible mapping h from P 2 to itself such that three points x 1,x 2,x 3 lie on the same line if and only if h(x 1 ),h(x 2 ),h(x 3 ) do. Theorem: A mapping h:p 2 P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 reprented by a vector x it is true that h(x)=hx Definition: Projective transformation x' x' x' h h h h h h h h h x x x or x' H x 8DOF projectivity=collineation=projective transformation=homography 161

162 Mapping between planes central projection may be expressed by x =Hx (application of theorem) 162

163 Removing projective distortion select four points in a plane with know coordinates x' 1 h11x h12y h13 x' 2 h21x h22y h x' y' x' h x h y h x' h x h y h x' y' h31x h32y h33 h11x h12y h13 h31x h32y h33 h21x h22y h (linear in h ij ) (2 constraints/point, 8DOF 4 points needed)