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