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

187 where coordl of B®. ON THE IMPORTANCE OF PROJECTIVE GEOMETRY FOR ANALYTICAL AND DIGITAL PHOTOGRAHMETRIC RESTITUTION Gerhard Brandstatter Institute of Applied Geodesy and Photogrammetry Graz University of Technology, Austria X» = ï gives The rc recipr vector systerr unit v yE mus Fig. 1 vector of the y’-u 1 b Abstract Projective geometry is an already ancient scope of mathematics, generally treated synthetically, because during the period of analog restitution instruments there was no need for projective calculation techniques. But the use of analytic and digital methods in photogrammetry and videometry causes a new interest in those methods, in parti cular, when non-metric cameras are used. For prac tical purposes algebraic relations of projective methods based on linear transformations of homo geneous geometric eLements are required. The paper Will give a short treatment of this "algebraic projective geometry", of its application to photo- grammetric problems ( rectification and realtive orientation ) and the consequences regarding uni versity education in photogrammetry, videometry, and digital image processing. 0. Preface Analytical and digital photogrammetry start from image coordinates, which are measured by means of optomechanical or optoelectronic instruments as monocomparators, stereocomparators, analytical plotters, digitizers, photodetectors, videocameras or CCD-cameras. None of these instruments can a priori ensure that its coordinates are related to an orthogonal and isometric system. Such systems are to be called here orthonormal and correspond, of course, to rectangular cartesian systems. Every manufacturer of instruments will try to approach this ideal state but will never attain it exactly because of too great expenses in production and quality checks. So, in order to simulate ideal conditions, usually the systems must be calibrated by the producer or by the user, and the corrections may be stored in tables or matrices. But often it 1s more expedient to accept the fact, that in reality all -coordinates are oblique and hetero metric, or in one word, affine. In this case the working methods of photogrammetry may be adapted to the principles of algebraic projective geometry, which 1s based on linear transformations of homo geneous vectors. 1. Projective transformations 1.1 General linear homogeneous trans formation It is well-known, that a general 4x4-matrix ■ • poo Poi PO 2 P0 3 yo Po T y P10 P1 1 Pi 2 P13 yi = pi T y P20 P21 P22 P23 Y2 P 2 T y P3 0 P3 1 P3 2 P3 3 ya P3 T y defines a projective transformation (Hohenberg- Tschupik, 1972). If P is regular (rank(P)=4), the relation exists between two tridimensional (order n+1) vector spaces as e.g. for optical imaging of ideal lenses.If P is singular,the resulting vectors fill a projection plane P* (rank=3) or a straight line (rank=2). The linear image P 2 is the ideal foundation of all photogrammetric theories, the projective line sometimes appears in connection with architectural and engineering applications. (1.1.1) contains 15 Independent parameters so that five or generally n+2 noncomplanar spatial points define the relation between two projective spaces. These points may be four arbitrary points Pi and a unit point Pe, which determines affine units along the other four position vectors , yielding the base B(bo,bi,b2,b3) of the related vector space. 1.2 Determination of transformat ion elements The basis vectors depend on the position vectors yi by the linear relation bi=Uiyi, B* = (poyo piyi U2 y2 H3 y3 ) (1.2.1) and must meet the conditions 3 3 I bi = I myi = yE or Yu = yE \ =0 1=0 (Fuchs, 1988), from which the vector u T = (uo ,m ,M2 ,M3) of the unknowns results according to M = Y~ 1 yE with 3 Ui = (y*) T yE = I y 1 jyEJ , (1.2.2) j=0 where the y 1 are the row vectors of Y _1 or the so- called reciprocal vectors of the y». Any other vector y may be combined by means of the basic vectors in the form 3 y = l X 1 bi = B*x B , (1.2.3) 1 sO with u U = Fig. 1 Becausi B® Uu = follows 1 u 1 U2 u 3 In ordi Image s by Its wo Wo W2 - = U 3 Wo so that and the through The bas

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