Modern trends of education in photogrammetry & remote sensing

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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

188 where x B 1s the vector of the contravarisnt coordinates related to B and the br are the columns of B®. The Inverse form of (1.2.3), namely x" = By = (B*)- 1 y, (1.2.4) gives the transformation of y Into the system B. The rows of B contain, 1n analogy to (1.2.2), the reciprocal vectors b 1 of the system bi . As the vectors myi establish the contravariant coordinate system, their transformation into 1t will yield the unit vectors ei 8 = By* and the transformation of yE must produce the unit point (e* B ) T = (1,1,1,1). Fig.1 illustrates, that the image y’= py of the vector y may be represented by affine coordinates of the image space B as y’“U 1 bi +u 2 b2+u 3 b3 + ( 1-u 1 -u 2 -u 3 )bo = B®Uu (1.2.5) with u T =(1,u 1 ,u 2 ,u 3 ), 1 -1 -1 -1 1111 0 10 0 and U' 1 = 0 10 0 0 0 10 0 0 10 0 0 0 1 0 0 0 1 ■ . therefore results from B* = (UP) -1 1f, as usual in photogrammetry, the vector u contains a constant homogenizing component and two or more Inhomo geneous affine cordmates of the image space. l»-3—Ir.s<! 1 ^ format ion .between two different projective spaces A mam problem of photogrammetry is to find the transformation u" - Pu’ from a projective space P’ (e.g. model) into a reference space P" (system of object), by means of control points. As it was shown, generally n+2 points (=5 in three dimensions) are required. From (1.2.6) and (1.2.4) u" = p"U _1 B"y and u’ = M ? U' 1 B’y . may be derived. The second equation yields 1 y = - (B’)- 1 Uu\ (1.3.1) p’ and thence by means of the first one M" u" = - U* 1 B"(B’)- 1 Uu’ = Mu’. (1.3.2) p’ Using four of the position vectors of the control points as base vectors bi", the base of P’ (compare section 1.2) will be bi’ = ui’bi" and therefor B'B” = b° " T b 1 ” T b 2 " T b 3 " T ^ Mo * bo " M1 ’ bi ” M2 ’ b2 ” M3 ’ b3 " j . Fig. 1: Unit tetrahedron of a base B(bi). The Gi are the so-called basic points of the projective space P 3 Because of (1.2.3) equation B*Uu = pB*x B , Uu = mx b , u = pU* 1 x B (1.2.6) follows and thus 1 Zx 1 u 1 = p X 1 u 2 x 2 u 3 x 3 . In order to get Inhomogeneous coordinates of the image space the vector w of (1.1.1) must be divided by Its own first (homogenizing) component wo PO T y W1 Pi T y — = 1 = — , — = u 1 = , Wo P0 T y wo po T y W2 P2 T y W3 P3 T y — = U 2= , — = u 3 = , Wo po T y wo PO T y so that at the left of (1.1.1) u may substitute w and the projective matrix P depends on the base B through the relation P=U _1 B. The base of any regular projective transformation According to (1.2.3) and (1.2.4) the product b’ T bi of a base vector and its reciprocal results 1n 1, the other products yield 0 and because of this fact the product of the matrices must be M" - B' B” p’ po Pi P2 P3 (1.3.3) where pi=p"pi’/p’. These four unknowns can be determined from the coordinates of the fifth point using the expanded form 1 po pi-po P2-pO P3~P0 1 u 1 " z 0 Ml o 0 u 1 ’ u 2 " 0 0 M2 0 u 2 ’ u 3 " 0 0 0 M3 u 3 ’ . » . of (1.3.2) giving the simple relations ) 1 1 c r c r c GJ u 1 " cu 3 J 1 { u 3 " i M1 - » H4. • 1 , M3- • 1-u 1 ’-u 2 ’-u 3 ’ u 1 ’ u 2 ’ u 3 ’ (1.3.4) From the first term 1t is seen, that the fifth point does not dare to coincide with the plane u 1 ‘+u 2 ’ + u 3 ’ = 1 ( = plane through G1,G2,G3 in fig. 1) nor with one of the affine coordinate axis as show the other terms.

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