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7 Projective Plane 68 7.1Motivation–Perspectiveprojectioninaffinespace Geometry of Computer Vision Translation of Euclid’s Elements by Adelardus Bathensis (1080–1152) http://en.wikipedia.org/wiki/File:Woman teaching geometry.jpg Tom´aˇsPajdla [email protected] Monday 2nd May, 2011 Contents 1Notation 1 2 Linear algebra 2 2.1 Change of coordinates induced by the change of basis . ..... 3 2.2Dualspaceanddualbasis........................ 6 2.3Operationswithmatrices........................ 8 3Affinespace 11 3.1Vectors.................................. 12 3.1.1 Geometricscalars......................... 13 3.1.2 Geometricvectors........................ 13 3.1.3 Boundvectors.......................... 14 3.2Linearspace................................ 15 3.3Freevectors................................ 17 3.4Affinespace................................ 18 3.5Coordinatesysteminaffinespace.................... 19 3.6Examplesofaffinespaces........................ 21 3.6.1 Affine space of solutions of a system of linear equations . 21 3.6.2 Smallaffinespaces........................ 22 3.7Incidenceaxiomsofaffinespaces.................... 24 4 Image coordinate system 27 5 Perspective camera 29 5.1Perspectivecameramodel........................ 30 5.2 Computing camera projection matrix from image of six points . 34 5.3Camerapose............................... 36 5.4Cameracalibrationandanglebetweenprojectionrays........ 42 5.5Calibratedcameraposecomputation.................. 44 6 Homography 53 6.1Homographybetweenimageswiththesamecenter.......... 54 6.2Homographybetweenimagesofaplane................ 56 6.2.1 Imageofaplane......................... 56 6.2.2 Twoimagesofaplane...................... 57 6.3 Spherical image . ...................... 60 6.4Homography–summary......................... 61 6.5Imageofimageofanimage....................... 63 6.6Computinghomographyfromimagematches............. 66 6.6.1 Generalperspectivecameras.................. 66 6.6.2 Calibratedcameras........................ 67 ii T. Pajdla. Geometry of Computer Vision 2011-5-2 ([email protected]) 7 Projective plane 68 7.1Motivation–perspectiveprojectioninaffinespace.......... 69 7.2Realprojectiveplane........................... 72 7.2.1 Geometricalmodeloftherealprojectiveplane........ 72 7.2.2 Algebraicmodeloftherealprojectiveplane.......... 73 7.2.3 Linesoftherealprojectiveplane................ 74 7.2.4 Idealline............................. 76 7.2.5 Homogeneouscoordinates.................... 77 7.2.6 Incidenceofpointsandlines.................. 78 7.2.7 Joinofpoints........................... 79 7.2.8 Meetoflines........................... 80 7.3 Line coordinates under homography . ............... 81 7.4Vanishingpoints............................. 82 7.5Vanishinglineandhorizon........................ 82 7.6Incidenceaxiomsofprojectiveplane.................. 83 8 Projective space 85 8.1Motivation–unionofidealpointsofallaffineplanes......... 85 8.2Perspectivecamerainprojectivespace................. 87 8.2.1 Parallelprojection........................ 88 9 Camera Auto-Calibration and Metrology 92 9.1Internalendexternalcalibration.................... 93 9.2Single-viewcameracalibration..................... 94 9.3Single-viewmetrology.......................... 94 9.4Multiple-viewcameracalibration.................... 94 Index 96 iii 1Notation À ... emptyset[1] exp U . the set of all subset of set U [1] U ¢ V . Cartesian product of sets U and V [1] Z ... wholenumbers[1] Q . rational numbers [2] R ... realnumbers[2] i . imaginary unit [2] Õ ÔS, , . space of geometric scalars A . affine space (space of geometric vectors) Õ ÔAo, , . space of geometric vectors bound to point o A2 ... realaffineplane A3 . three-dimensional real affine space P2 . real projective plane P3 . three-dimensional real projective space Õ ÔV, , . space of free vectors x . vector A ... matrix Aij ... ij element of A A  ... transposeof A A . determinant of A I ... identitymatrix R . rotation matrix . Kronecker product of matrices × β Öb1, b2, b3 . basis (an ordered triple of independent generator vectors) β¯ . the dual basis to basis β xβ . column matrix of coordinates of x w.r.t. the basis β  ¤ x ¤ y . Euclidean scalar product of x and y (x y xβ yβ in an orthonormal basis β) x ¢ y . cross (vector) product of x and y ¤ x . Euclidean norm of x ( x x x) orthogonal vectors . mutually perpendicular and all of equal length orthonormal vectors . unit orthogonal vectors P ¥ l ... point P is incident to line l P Q . line(s) incident to points P and Q k l . point(s) incident to lines k and l 1 2 Linear algebra M. C. Escher. Drawing Hands. http://www.worldofescher.com/gallery/A13.html 2 T. Pajdla. Geometry of Computer Vision 2011-5-2 ([email protected]) We rely on linear algebra [3, 4, 5, 6, 7, 8]. We recommend excellent text books [6, 3] for acquiring basic as well as more advanced elements of the topic. Monograph [4] provides a number of examples and applications and provides a link to numerical a computational aspects of linear algebra. We will next review the most crucial topics needed in this text. 2.1 Change of coordinates induced by the change of basis Let us discuss the relationship between the coordinates of a vector in a linear space, which is induced by passing from one basis to another. We shall derive the rela- tionship between the coordinates in a three-dimensional linear space, which is the most important when modeling the geometry around us. The formulas for all other n-dimensional spaces are obtained by passing from 3 to n. 3 § 1Coordinates Let us consider an ordered basis β b1 b2 b3 of V .Avector 3 x È V is uniquely expressed as a linear combination of the basic vectors by its real coordinates x, y, z, i.e. x x b1 y b2 z b3, and can be represented as an ordered  triple of coordinates, as a coordinate vector xβ xyz . ½ ½ ½ ½ § 2Twobases Having two ordered bases β b1 b2 b3 and β b1 b2 b3 leads to expressing one vector x in two ways as x x b1 y b2 z b3 and x ½ ½ ½ ½ ½ ½ ½ x b1 y b2 z b3. The vectors of the basis β can also be expressed in the basis β using their coordinates. Let us introduce ½ ½ ½ b1 a11 b1 a21 b2 a31 b3 ½ ½ ½ b2 a12 b1 a22 b2 a32 b3 (2.1) ½ ½ ½ b3 a13 b1 a23 b2 a33 b3 § 3 Change of coordinates We will next use the above equations to relate the coordinates of x w.r.t. the basis β to the coordinates of x w.r.t. the basis β ½ x x b1 y b2 z b3 ½ ½ ½ ½ ½ ½ ½ ½ ½ Ô Õ Ô Õ Ô Õ x a11 b1 a21 b2 a31 b3 y a12 b1 a22 b2 a32 b3 z a13 b1 a23 b2 a33 b3 ½ ½ ½ Ô Õ Ô Õ Ô Õ a11 x a12 y a13 z b1 a21 x a22 y a23 z b2 a31 x a32 y a33 z b3 ½ ½ ½ ½ ½ ½ x b1 y b2 z b3 (2.2) Since coordinates are unique, we get ½ x a11 x a12 y a13 z (2.3) ½ y a21 x a22 y a23 z (2.4) ½ z a31 x a32 y a33 z (2.5) Coordinate vectors xβ and xβ ½ are thus related by the following matrix multiplication x½ a11 a12 a13 x ½ y a21 a22 a23 y (2.6) z ½ a31 a32 a33 z 3 T. Pajdla. Geometry of Computer Vision 2011-5-2 ([email protected]) which we concisely write as ½ xβ A xβ (2.7) The columns of matrix A can be viewed as vectors of coordinates of basic vectors, b1,b2,b3 of β in the basis β ½ b1 b2 b3 (2.8) ½ ½ A β ½ β β and the matrix multiplication can be interpreted as the linear combination of columns of A by coordinates of x w.r.t. β ½ xβ x b1 y b2 z b3 (2.9) ½ ½ β ½ β β Matrix A plays such an important role here that it deserves its own name. Matrix A is very often called the change of basis matrix from basis β to β ½ or the transition matrix from basis β to basis β ½ [4, 9] since it can be used to pass from coordinates w.r.t. β to coordinates w.r.t. β ½ by Equation 2.7. However, literature [5, 10] calls A the change of basis matrix from basis β ½ to β, i.e. it (seemingly illogically) swaps the bases. This choice is motivated by the fact that A canalsobeusedtoobtainthevectorsofβ from vectors of β ½ using Equation 2.1 as ½ ½ ½ ½ ½ ½ b1 b2 b3 a11 b1 a21 b2 a31 b3 a12 b1 a22 b2 a32 b3 ½ ½ ½ a13 b1 a23 b2 a33 b3 (2.10) a11 a12 a13 ½ ½ ½ b1 b2 b3 a21 a22 a23 a31 a32 a33 ½ ½ ½ b1 b2 b3 A (2.11) or equivalently ¡1 ½ ½ ½ b1 b2 b3 b1 b2 b3 A (2.12) where the multiplication of a row of column vectors by a matrix from the right in Equation 2.11 has the meaning given by Equation 2.10 above. Yet another variation 1 of the naming appeared in [7, 8] where A ¡ was named the change of basis matrix from basis β to β ½ . We have to conclude that the meaning associated with the change of basis matrix varies in the literature and hence we will avoid this confusing name and talk about A as about the matrix transforming coordinates of a vector from basis β to basis β ½ . There is the following interesting variation of Equation 2.11 b1½ b1 ¡Â ½ b2 A b2 (2.13) b3½ b3 4 T. Pajdla. Geometry of Computer Vision 2011-5-2 ([email protected]) where the basic vectors of β and β ½ are understood as elements of column vectors. For instance, vector b1½ is obtained as ½ b1 a¯11 b1 a¯12 b2 a¯13 b3 (2.14) ¡Â × where Öa¯1, a¯2, a¯3 is the first row of A . § 4Example We demonstrate the relationship between vectors and bases on a concrete example. Consider two bases α and β represented by coordinate vectors, which we write into matrices 110 α a1 a2 a3 011 (2.15) 001 111 β b1 b2 b3 001 , (2.16) 011 and a vector x with coordinates w.r.t. the basis α 1 xα 1 (2.17) 1 We see that basic vectors of α can be obtained as the following linear combinations of basic vectors of β a1 1 b1 0 b2 0 b3 (2.18) ¡ a2 1 b1 1 b2 1 b3 (2.19) ¡ a3 1 b1 0 b2 1 b3 (2.20) (2.21) or equivalently ¡ 111 ¡ a1 a2 a3 b1 b2 b3 0 10 b1 b2 b3 A (2.22) 011 Coordinates of x w.r.t. β are hence obtained as 11¡1 ¡ xβ A xα, A 0 10 (2.23) 011 1 11¡1 1 ¡ ¡ 1 0 10 1 (2.24) 2 011 1 We see that α β A (2.25) 110 111 11¡1 011 001 0 ¡10 (2.26) 001 011 011 5 T.
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