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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol XXXV, Part B7. Istanbul 2004 
Actually these numeric derivative values {g, , gy, 8, are x-y-z 
components of the local surface normal vector at that point. 
  
|2e" Og" ag" 
Ta | ax y. Oz je li n c] 
n [Ve | r | Vg | co rash (23) 
  
In the case of representation of search surface elements as 
parametric bi-linear surface patches, which are constituted by 
fitting the bi-linear surface to 4 neighboring knot points P;;: 
G(u, w) : Po (L- u)(1— w)-* Pj, (1 -u)w P, ou(1 7 w) * P,,uw 
(24) 
2 ~ 3 . . 
where u,w € [0,1] and G, Pi € JV. Again the numeric 
derivative terms {g, , g, , g,} are calculated from components of 
the local surface normal vector on the parametric bi-linear 
surface patch: 
__  AG(u.w) 3G(u,w) 
PT LEO D (25) 
; [va Iva] 
  
  
  
  
With this approach a better a posteriori sigma value could be 
obtained due to a smoothing effect. In the case of insufficient 
initial approximations, the numeric derivatives 1g: gy. 2, can 
be calculated on the template surface patch f(x,y,z) instead of on 
the search surface g(x,y,z) in order to speed-up the 
convergence. 
2.2 Precision and Reliability Issues 
The standard deviations of the estimated transformation 
parameters and the correlations between themselves may give 
useful information concerning the stability of the system and 
quality of the data content (Gruen, 19852). 
^ 
0, 2 0,4 doo: vi EAL ty fm eq Ki, de Q6) 
As pointed out in (Maas, 2000), the estimated standard 
deviations of the translation parameters are too optimistic due to 
stochastic properties of the search surface. 
Because of the high level redundancy of a typical data 
arrangement, a certain amount of occlusions and/or outliers do 
not have significant effect on the estimated parameters. 
Baarda’s data-snooping method can be favourably used to 
localize the occluded or gross erroneous measurements. 
2.3 Computational Aspects 
The computational complexity is of order O(N?), where N is the 
number of employed points in the matching process. The actual 
problem is to search the correspondent element of the template 
surface on the search surface patch, whereas the adjustment part 
is a small system, and can quickly be solved using back- 
substitution followed by Cholesky decomposition. Searching 
the correspondence is an algorithmic problem, and needs 
professional software optimization techniques and programming 
skills, which are not within the scope of this paper. 
Since the method needs initial approximations of the unknowns 
due to the non-linear functional model, one of the methods for 
pre-alignment in the literature (Habib and Schenk, 1999, 
Murino et al., 2001, Lucchese et al., 2002, Vanden Wyngaerd 
and Van Gool, 2002) should be utilized. 
963 
Two 1* degree C" continuous surface representations are 
implemented, and explained in detail. In the case of multi- 
resolution data sets, in which point densities are significantly 
different on the template and search surface patches, higher 
degree C' continuous composite surface representations, e.g. bi- 
cubic Hermit surface (Peters, 1974), should give better results, 
of course increasing the computational expenses. 
2.4 Convergence of Solution Vector 
In a standard LS adjustment calculus in geodesy and 
photogrammetry, the function of the unknowns is unique, 
exactly known, and analytically continuous everywhere, e.g. the 
collinearity equations in the bundle adjustment. Here the 
function g(x,y,z) is discretized by using a definite sampling rate, 
which leads to slow convergence, oscillations, even divergence 
in some cases with respect to the standard adjustment. The 
convergence behaviour of the proposed method basically 
depends on the quality of the initial approximations and quality 
of the data content, and it usually achieves the solution after 4^ 
or 5" iterations (Figure 1), as typically in LSM. 
  
  
dp;/c; 
omg S omg 
Z0 
100 S 
kap~ à kap 
X Z0 -. 
0 ent mE an: a 
X0 
-100+ 
| phi/ phi 
| YO xo/ lvo iterations 
L- ii M 
HR A a 4 13 2559 do 5i, quiis 
(a) (b) 
Figure 1: Typical examples for fast convergence (a) and slow 
convergence (b). Note that scale factor is fixed to unity. 
3. THE EXPERIMENTAL RESULTS 
Two practical examples are given to show the capabilities of the 
method. All experiments were carried out using own self- 
developed C/C++ software that runs on Microsoft Windows® 
OS. Processing times given in Table 1 were counted on such a 
PC, whose configuration is Intel® P4 2.53 GHz CPU, 1 GB 
RAM. The first example is the registration of three surface 
patches, which were photogrammetrically measured 3D point 
clouds of a human face from multi-images (Figure 2). For the 
mathematical and implementation details of this surface 
measurement method the author refers to (D'Apuzzo, 2002). 
Left and right search surface patches (Figure 2-a and 2-c) were 
matched to the centre template surface patch (Figure 2-b) by use 
of LS3D. Since the data set already came in a common 
coordinate system, the rotation angles (w,p,k) of the search 
surfaces were deteriorated by ~10% in the first iteration. 
Numerical results of the matching of the left surface and the 
right surface patches are given at parts I-L and I-R of Table | 
respectively. Relatively high standard deviations for the 
estimated t, and ¢ (note that high physical correlation between x 
and © due to a conventional axes configuration) exhibit the 
narrow overlapping areca along the x-axis, nevertheless the 
matching result is successful. The estimated o, values prove the 
accuracy potential of the surface measurement method, given as 
0.2 mm by D'Apuzzo (2002).