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Geometry of two cameras G.Georgiev1, N.Danailova2, V.Radulov3 Key words: projective coordinates, projection matrix. Scientific area: mathematics (projective geometry) ABSTRACT The investigation of image formation and object modeling is widely developed in the last thirty years. The formation of planar images of our three dimensional world plays an important role in our century of communications through computers. The geometry of multiple images provides us the description of the geometric laws for creation of different views of a scene. The inverse problem of reconstruction of the object by its images is studied also for the needs of photogrametry. The central projection is may be the most used projection because of its closeness to the human perceptions. Although the laws of the central projection and perspective are known for many years, the mathematicians investigate them now in the framework of the projective, affine and Euclidian geometry to obtain more efficient and elegant results. In this context we present here the main law for central projection in projective coordinate system and applications. The system of two cameras is studied also, which is the subject of the so called epipolar geometry. Some applications are given in the context of the projective and Euclidean geometry from algebraic viewpoint. 1. Introduction The planar image creation of the three dimensional world is investigated lately in the framework of the projective, affine or Euclidian geometry (see [1], [2] and [3]). Some correspondences can be established between two or three and more images, which properties can be applied. We introduce projective coordinate space system as well as projective homogeneous and non-homogeneous coordinates, which are the most common case of coordinates. The projective coordinates have a nice property - they do not change 1 G.Georgiev, Asso.Prof., PhD,Shumen University, partially supported under grant RD-05-289/11.03.2009, g.georgiev@fmi.shu-bg.net 2 N.Danailova, Asso. Prof., PhD, UACEG-Sofia, dichevan@yahoo.com 3 V.Radulov, Assis. Prof., UACEG-Sofia, vradulov@yahoo.com after central projection. The affine and Euclidian case will be considered as particular cases. We define a well-known cross ratio of 4 points on a line as follows: ABCD AC AD ABC : BC BD ABD Let five points be fixed in the space (Fig.1), called basic points, so that no four of them lie on a plane. We call U, V, W – excluded points, О – a zero point and Е – a unit point. Let denote the intersection points by EU OU (VEW ) , EV OV (UEW ) and EW OW (UEV ). If А is an arbitrary point, not lying in the plane (UVW), then we can get by analogy points A U, AV, AW. The numbers U A (UOEU AU ), V A (VOEV AV ), W A (WOE W AW ) are called non-homogeneous projective coordinates of the point А. We compare to the basic points the following coordinates; U(∞; 0; 0), V(0; ∞; 0), W(0; 0; ∞), О(0; 0; 0), Е(1; 1; 1). The defined by this way coordinate system К(U, V, W, О, Е) compares to each point (except those lying in the plane (UVW)) an ordered triple of real numbers and conversely - for each ordered triple of numbers corresponds a point. In order of points on (UVW) have coordinates, we introduce homogeneous projective coordinates: Homogeneous projective coordinates of the point А is called the ordered four of numbers (u A , v A , wA , t A ), as follows: - if А is a point, not lying in (UVW), then where t A u A U A t A , v A V A t A , wA WA t A , 0 is an arbitrary real number. - if А is a point in the plane (UVW), then t A 0 , u A 0, v A 0, w A 0 . The geometric sense of the homogeneous projective coordinates of a point A is, that they are proportional to the ratios of the distances from the points A and E to the walls of the coordinate tetrahedron, namely: uA a a1 a a , vA 2 , w A 3 , t A 4 . e1 e2 e3 e4 2. Analytical model of central projection Let 5 points U, V, W, O, E be given, so that no 4 of them lie in a plane (fig.2), and let consider the projective coordinate system K(U, V, W, O, E). If А is an arbitrary point, then uA A, (OVW ) a1 . The first projective coordinate is e1 E, (OVW ) equal to the relation of the distances from А and Е to the plane (OVW). If we draw the lines АА1 and ЕЕ1, orthogonal to (VWO), and A1 , E1 (VWO) , then ' uA A, (VWO) E , (VWO) AA1 EE1' . Let take an arbitrary plane π as a projective one, containing the point Е, and the projection center be the point О. Denoting by U ' , V 'W ' , A' the central projections of the corresponding points, and E E'. The A' A1' is orthogonal to the plane (VWO). The projecting ray OAA' is projected ' orthogonally on (VWO) in the line OA1 A1 . From the similarity of the triangles OAA1 AA1 OA ' A , or AA1 A . A' A1' . Let draw in and OA' A1 it follows that A' A1' OA' the plane π the perpendiculars A'a' and Ee' to V 'W '. From ' ' A' A1 (VWO), a' A1 (VWO) it follows, that the triangle A' A1' a' is ' ' rectangular with right angle A1 . By analogy we consider the triangle EE1e'. The ' ' rectangular triangles A' A1 a' and EE1e' are similar, since A' a' // Ee' ' ' (perpendiculars to V 'W ' ) and A' A1 \ \ EE1 (perpendiculars to OVW). Hence, line A' A1' A' a' . In this way we have EE1' Ee' uA where AA1 A . A' A1' A . A' a A u A' , ' ' Ee' EE1 EE1 u A' is the first projective coordinate of’ A' in the projective coordinate system K ' U ' ,V ' ,W ' , E'. By analogy v A A .v A' , w A A .wA' . If the projection plane does not pass through Е, then the coefficient of the proportion between the space projective homogenous coordinates of А and the plane projective homogenous coordinates of its central projection A' is A , E where A OA is the ratio of the OA' distances from the projection center to the given point and to its central projection.. We obtain, that if A(u A , v A , wA , t A ) is an arbitrary point with projective homogenous coordinates in projective coordinate system К(U, V, W, O, E)), and A' (u A' , v A' , wA' ) is the central projection of А, so that we project on arbitrary projection plane with projection center O, and the projective homogenous coordinates of А’ are in K ' U ' ,V ' ,W ' , E' – the central projections of the points U, V, W and E, then the following relations are true: A ' uA E v A A v A' E A ' wA wA . E uA E uA A v' A E v A A E w' A wA . A u' A , which is equivalent to Using matrix equation, this correspondence can be given by PA A' , 1 P0 0 0 0 0 1 0 0 , 1 0 0 A . E (1) The matrix P is called projection matrix of size (3 x 4) and rang (P) = 3. 3. Projection matrix and arising correspondences Let us consider an arbitrary projective coordinate system with origin – the 0 0 point C , which is the projection centre also, and an arbitrary projection plane ω. 0 1 u v To each point in the space M there corresponds a point – its central projection w t u ' m v ' CM . This correspondence is given by PM = m, where the projection w' matrix is given by (1). 1 0 0 1 0 T Let consider the matrix P 0 1 . It arises a correspondence 0 0 0 0 0 between the set of lines g ' in ω and the set of planes ' , containing the center С. a Really, let g '≡ au ' + bv ' + cw '= 0, or g ' b . Then c a 0 0 1 a b 0 1 . b . This vector corresponds to the plane 0 c c 0 0 0 a b c ' u v w 0t 0, or ' au bv cw 0t 0. Evidently 1 T P g' 0 0 0 that C(0,0,0,1) belongs to it. In this way to each line g' in ω there corresponds a plane π, passing through the projection center. Let Р+ be a matrix of size (4 x 3), so that Р.Р+ = Е3. It is easily seen that 0 0 P 00 a b 0 0 , where a, b, c are arbitrary real numbers. The matrix Р+ is related c with the plane au bv cw t 0, which does not pass through the projection center, since 0. So, let consider this correspondence a b P (2) c Theorem 1. To each point m the matrix P compares the point M . u' Proof: Let m v ' . It follows, that w' u ' 0 0 u u' 0 0 v ' v P m . v' w M . We check that 00 w' a b c w' au 'bv'cw' t au bv cw t au 'bv'cw' (au 'bv'cw' ) 0. M . 0 0 a Now let consider the matrix P 0 0 b . 0 0 c T Тheorem 2. The matrix P determines a correspondence between the set T of planes, different from π, and the set of lines in ω, comparing to each plane α ≠ π a line l ' , which is the image (central projection) of the line l . A B Proof Let be an arbitrary plane, α ≠ π, and let l . We have C D A 0 0 a A aD B PT 0 0 b B bD l ' 0 0 c C C cD D l ' ( A aD)u ( B bD)v ( C cD) w 0 (3) is a line in ω. We shall show, that it is the image of l . Really, if m1 m2 l , then these coordinates satisfy the equations М m3 m 4 Am1 Bm 2 Cm3 Dm 4 0 am1 bm2 cm3 m4 0 The central projection is m1 m m2 CM . These coordinates satisfy the equation (3), and m l '. m 3 Theorem 3: The above matrices have the following properties: 1 1. PP T 2. T P 0, P.C 0 . 3. 4. 5. 6. 2 E3 . ( E4 P P) C T . 1 P T P 2 ( E 4 C T ) . 1 P T P T E 4 C T . T P P E3 T 4. Double central projection and geometry of two cameras. Let С and С1 be two different projection centers, ω and ω1 be the corresponding projection planes, and let P, Q be the projection matrices (fig .3). Take М to be an arbitrary point with central projections m CM and m1 C1M 1 - the two images. The corresponding projection matrices are P and Q: PМ = m and QМ = m1. With the matrix P the plane π is related so that P m M , M . It follows QP m QM m1 . Let us denote the matrix QP H . Theorem 4. The matrix Нπ, related to the plane π, arises one-to-one and invertible correspondence between ω and ω1. The converse correspondence is 1 determined by the matrix H P.Q . 0 0 0 0 Proof: Let m be a point in ω. Consider the matrix P , where a, b, c 0 0 a b c are arbitrary real numbers. The plane π is defined au bv cw t 0, There exists a point М in π, for which P m M . But the point M is projected on 1 by QM m1 1. A point in ω1 is compared to each point in ω: 0 0 u ' qu ' u' q 0 0 0 0 0 If m v ' , then m1 QP m 0 q 0 0 v' qv' . w' 0 0 q 0 0 0 w' qw' a b c u' ' q' ' u ' ' ' If m' v' ' is another point in ω, by analogy m1 q ' ' v' ' . w' ' q' ' w' ' qu ' q' ' u ' ' If we assume that m1 m1' , then qv' q' ' v' ' or qw' q' ' w' ' u ' v' w' q' ' / q k . It means that u ' ku' ' , v' kv' ' , w' kw' ' , or u ' ' v' ' w' ' m m' . This contradiction shows, that different points in ω are compared to different points in ω1. The correspondence QP H is one-to-one correspondence. It is determined by matrix of size (3 x 3) and rang H = 3, therefore it is convertible, or 1 the matrix H Property 3 of Th. 3 confirms, that It follows, that there exists. From QP H we obtain QP P H P. H P Q( E 4 H PQ QQ 1 P P E4 1 1 C T . C T ) . Multiply to the right: QC T Q . Since QQ E3 (Property 1 Th. 3) and Q 0 (Property 2 Th. 3), we have H PQ E3 T 1 or H PQ . This is the matrix of the converse correspondence from ω1 to ω. BIBLIOGRAPHY 1. Faugeras O. , Luoung Q. T., ed. T. Papadopoulo, The Geometry of Multiple Images, MIT Press, 2001. 2. G. Georgiev On the shapes of the images of space curves under orthogonal axonometry. Applications of Mathematics in Engineering and Economics, vol 32 Softtrade, Sofia, (2007), 11-20. 3. Hartley R.,. Zisserman A., Multiple View geometry in Computer Vision (Second edition), Cambridge University Press , 2003. 4. Смирнов С. А., Стереоперспектива в фотограметрии, Москва, “Недра”,1982г. 5. Радулов В.,