Modern trends of education in photogrammetry & remote sensing

Bibliographic data

Monograph

Persistent identifier:: 856467936

Title:: Modern trends of education in photogrammetry & remote sensing

Sub title:: ISPRS Commission VI Symposium, September 13 - 16, 1990, Rhodes Island, Greece

Scope:: 1 Online-Ressource (251 Seiten)

Year of publication:: 1990

Place of publication:: Athens

Publisher of the original:: Technical Chamber of Greece

Identifier (digital):: 856467936

Illustration:: Diagramme

Language:: English

Usage licence:: Attribution 4.0 International (CC BY 4.0)

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2016

Document type:: Monograph

Collection:: Earth sciences

Chapter

Title:: Education (WG VI/2 and WG VI/7).

Document type:: Monograph

Structure type:: Chapter

Chapter

Title:: ON THE IMPORTANCE OF PROJECTIVE GEOMETRY FOR ANALYTICAL AND DIGITAL PHOTOGRAHMETRIC RESTITUTION. Gerhard Brandstatter.

Document type:: Monograph

Structure type:: Chapter

189 The Inhomogeneous coordinates of the transformation p* —> p" are therefore Mi u* ’ u 1 " = . Mo + (mi~Mo)u 1 ’ + (m2~mo)u 2 ’ + (m3~mo)u 3 1 1-Ums 1 -um3 2 UM3 1 UM3 : Mo- » Mi “ « M2- 1 —U3 1 —U3 2 U3 1 U3 2 and the affine coordinates 1n the map of an art) trary point of the image may be calculated from Those formulas hold 1f and only 1f all coordinates u 1 refer to the system of the basic points Gh . Therefore the measured coordinates v 1 must be reduced to this system using the spatial homo geneous affine transformation u=Av, where A = 10 0 0 aio ai 1 ai 2 ai3 a2o a2i 322 a23 a30 aai a32 a33 and the aij can be computed from the measured coordinates of the basic points by Avi = ei 1=0,1,2,3 ei 1s the homogeneous “unit vector" with 1 at the positions 0 and i. 1.4 Transformation between protective planes The transformation between projective planes meets the well-known rectification of linear images of plane objects. The algebraic-projective procedure, which represents a very elegant proof of this method, may easily be derived from the relations of section 1.3 omitting all fourth components. Its spedai treatment is given in {Brandstatter, 1989). The dimension of the projective spaces "map" and "Image" is n=2, the required number of control points n+2=4. Three of them (fig.2) define the respective affine systems. As in the map all points have orthonormal coordinates, the homogeneous Fig. 2: The affine coordinate systems of map and linear image position vectors Xi T = (1,Xi 1 ,Xi 2) have to be converted to the affine system of the map by AmXì = = um . The components of Am result from AmXi=0ì, 1=0,1,2, with the determinant D = j X11 X12 - X01 X02 + X01 X02 j X21 X22 X21 X22 X11 X12 as 1 ‘ D 0 0 Am = ~ X02X21-X22X01 X22-X02 X02-X12 , D X01X12~Xi1X02 X01-X21 X11-X01 • and the fourth control point now is um.i = AmX3 . In the image space the elements of the affine trans formation have quite the same shape replacing Xjk by vjk and the fourth point 1s U3 = AV3. According to (1.3.4) the factors pi are Miu 1 um 1 = MO + (pi~Mo)u 1 +(M2-Mo)u 2 (1.4, Flnaly its orthonormal coordinates are derived fr Am ' 1 um =X, where the components of Am -1 may determined from AM _1 ei=Xi as Am* 1 1 0 0 X01 X11-X01 X21-X01 X02 X12-X02 X22-X02 1.5 Optical rectification The procedure of 1.4 could be used for the comput tional rectification of digital images.But also t traditional optical rectification is a projectl method, where the transformation is realized by t laws of geometric optics. By means of the foe length f of an ideal lens the projection is defin by the regular projective matrix Pf=- f 0 0 1 0 f 0 0 -1 or Pf _ 1 = 1 0 0 -1/f 0 10 0 0 0 f 0 0 0 G f f 0 0 1 0 0 0 0 1 Introducing orthonormal coordinates X,Y,Z (= systi of rectifier, fig.3) in order to get metric settii elements, the homogeneous position vectors are i = (XO ,X1 ,X2 ,X3 ) , X0 = const. (X=X1 /xo , Y=X2/XO ,Z = Xj/X| and the projection is performed by x ’ =Pfl Therewith the contents of an image plane P : h T x = hoxo + hixi + 112x2 + h3X3 = 0 (1.5: are to be rectified into a map Pm: h’ T x’= ho’xo’+ hi’xi’+ h2’x2’+ h3’x3*= 0. The coefficients h’ depend on h because of x=Pr'i and therefore h T Pf 1 x’=0, by the simple relation h’ T = h T Pr ‘ 1 , (1.5.1 yielding the equation fhoxo’+ fhixi’+ fh2X2*+ (fh3-ho)x3’= 0 (1.5.1 of the plane Pm. For X3 = X3’=0 this equation ar (1.5.1) produce the same homogeneous line S hoxo + hixi + h2X2 = 0, in the principal plane H of the lens, proving ' the condition of Scheimpflug. In order to achiei sharp projection over the whole area of the recti fication this condition must be satisfied. The Hi S, which may be given by the equations B T u = 0 in P and Bm t um = 0 in Pm , must be parallel to : the image of the horizon (fig. 3), represented P by the homogeneous equation Mo ♦ (pi - mo )u 1 + (M2 - ¡Jo ) u 2 = a T u = 0 from denominators in (1.4.1) and the var equatior 1/Mo+O/ which r (1.3.2) two coef Bi =ai , Bm 1 = om 1 , In analc transfor an T =a T K' As the yields u Qm 0, qm from whi Bmo = 1 + By means plane cc can be con sum in section the X-ax setting performei usual 1 y 1974). T lines G: ho+h2' and the ■

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