either is parallel to the X-axis when projected on the XZ- surface from its best-fitting analytical solid. In Figure 2 it is This proc 
plane, whence Qi = Qo, Q2 = Qo, Or is tangential to the ~~ shown that, although image point q corresponds to actual following 
cylinder (one solution) which is of no practical interest. point Q, it is a point Q' on the fitted solid which is mapped 
instead. Spatial displacements QQ' depend on departures 4. Fore 
3.2 Investigation r and angle €. According to the residuals of fitting and the left (^ 
required accuracy, this consideration could further restrict 2. Next, 
Among the four solutions, the correct one is automatically the useful area of mapping from each single image. with ! 
selected given the concavity or convexity of the recorded 3. The | 
surface. Indeed, as seen from Figure 1, points Qo; and Qo 3.3 Implementatlon 4. Henc 
do not belong to ray p: they are detected as not satisfying 5. Fore 
the collinearity equations. If OC » R, then points Q1 or Q2 The basic steps of monoplotting cylindrical surfaces may the o 
(OQ? » OQ.) are retained for convexity or concavity, res- be summarized as follows: 6. Back 
pectively. Other possibilities — the perspective centre O — 1, Fitting of the analytical solid to sparce XYZ data. leads 
lies within the cylinder and the imaged part is closer or — 2. Transformation of 3D data to cylinder-centred system. plane 
farther from O than its diametrical, O lies on the cylinder, ^3, Space resection using appropriate control points. 7. The 
O « C etc. - can be confronted by the program using the — 4, Manual tracing of image vector detail (x, y). is est 
control information available (Theodoropoulou, 1996). 5. Space intersection of projective rays with the analyti- 8. AS 
cal surface resulting in 3D vector data. a dh 
Another aspect of particular interest regards the propaga- ^ 6. Merging of data from all images for the whole object. i f 
tion of image measuring errors oxy to space coordinates. — 7, Development of 3D vector data in suitable projection. 9, From 
Important is here the angle £ under which a projective ray — 8. Representation in 3D and 2D. 0 Imag 
H intersects the surface (Figure 1). This angle decreases ^ |n the adopted cylinder-centred XYZ system developing is M hd. 
rapidly with p tending towards the tangent. Small angles € simplest in a 2D system Xp, Yo with Xpi - aR, Ypi — Y;, in T. 
cause high uncertainty, especially regarding depth. When 
planning photography of cylindrical objects one, therefore, 
needs to fix an accepted sun value and then decide upon 
imaging distance and total number of images. These pa- 
rameters also determine the limits of mapping. For given 
mean photographic scale, narrow or normal angle lenses 
allow mapping of larger areas than wide angle lenses. 
  
  
  
  
  
Figure 2 The effect of departures r from the cylinder. 
Finally, a last point to be made regards errors introduced 
which a; (0 < ai < 2rt) denotes the angle formed by each 
individual radius CQ; and the positive Z-axis. 
4. DIGITAL UNWRAPPING OF ORIGINAL IMAGES 
The second approach introduced here provides the final 
product not in vector but in raster form. It is based on the 
fact that a known analytic surface is practically equivalent 
to a surface DEM. Thus, production of digital orthoimages 
of the solid is now conveniently possible. Generally, how- 
ever, the conventional orthoimaging of curved surfaces of 
architectural or archaeological interest does not fully meet 
the requirements of the user. In these cases, and for de- 
velopable surfaces in particular, it is an unwrapping of the 
surface in question which is basically desirable. Evidently, 
such photographic (raster) object presentations cannot be 
directly based on conventional DEMs but rather on 'DDMs' 
(digital development models), namely planar (Xo, Yo) grids 
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
into monoplotting due to radial departures r of the actual uniquely referenced to the actual surface (Vozikis, 1979). 5.1 Tes 
A late 1 
X X: linder of 
Zi Zr was fully 
Noe ur cer pear onan taken wi 
Y were en 
Ywx |... P DA MxN are in Fi 
pr ek: ETT object space 
. pixelsize De 
in object space E [t 
S» Pepsi Dur rer a Ag, es 
Y y ipi 
area of size of | 
E development 7 unwrapped image i 
1S 
DLT collinearity equations 
  
  
  
  
  
resampling 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
E 
  
  
  
  
ET 
  
  
  
  
  
  
  
  
  
  
  
  
m | affine — 
FREE transformation 
AI Tee 
  
  
  
  
  
  
  
  
  
  
digital image 
analogue image 
  
Figure 3 The basic phases of digital unwrapping of right circular cylinders. 
292 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996