Full text: The 3rd ISPRS Workshop on Dynamic and Multi-Dimensional GIS & the 10th Annual Conference of CPGIS on Geoinformatics

1SPRS, Vol-34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
Taravudh TIPDECHO & Xiaoyong CHEN * 
Space Technology Applications and Research (STAR) Program 
School of Advanced Technologies 
Asian Institute of Technology, THAILAND 
This paper addressed ways to assess accuracy of integrated data, photograph and laser data. Polynomial function was employed for 
this experiment. It was applied to be a relational model between buildings represented in photograph and laser data. The experiment 
was undertaken on one group of data with twenty-four control points, which were randomly selected from photograph with their 
corresponding laser data. Least square method was applied for finding out the best fit of control points on existing data. Expectation 
values were calculated and compared between calculated and referenced data, then ranked from the lowest to highest one. The result 
showed expectation value substantially increased, which was closing to one when increasing number ofsample points. And also few of 
control points resulted an acceptable accuracy. According to this experiment, it implied that the use of polynomial functions could offer 
an effective method to reconstruct 3D building objects from 2D building objects, based on single photograph. Even the accuracy was 
not high, it could be adjusted by integrating it with others instruments such as GPS, Laser and etc. 
Graphical technologies are increasingly becoming important 
topic for computer applications. Those abilities have obviously 
been applied in several kinds of application particularly in the 
filed of landscape. Creating the graphical 3D representation of 
the environment and paint is a common approach now [3]. The 
automated reconstruction of 3D models from real environments 
is the latest technology with respect to 3D spatial object 
reconstruction [5] using the combination between spatial data 
and visual data. Modeling from pictures is another way now 
particularly in the field of computer graphic based on shading of 
objects [2]. They were several available ways to achieve 3D 
reconstruction, most of the techniques aim to use 
photogrammetric-method based, in terms of image geometry 
model, to achieve 3D object reconstruction. Even this technique 
rather gave high accurate results when compared with other 
techniques; it consumed high complex technology, high picture 
resolution, highly accurate algorithms, lots of relative 
parameters, etc. There were basic requirements for system that 
expert person must provide them. Nowadays the technology has 
been changed positively. Some new tools or methodologies 
could be applied for this environment. This study aimed to 
experiment on polynomial function being applied for 3D-building- 
objects reconstruction based on integrated data, a photograph 
and its laser data. The accuracy assessment was the main part 
of this experiment, in order to be known how much of accuracy 
and possibility to apply for 3D reconstruction. Finally the level of 
accuracy was released in terms of expectation value (E). 
For this studying, polynomial equation was applied for serving 
3D reconstruction from single image. Basically the simple 
polynomial equation for one variable can be defined as [1] 
f(x) = a0 + a1x + a2x 2 + ... + anx n (1) 
For two variables, they can be defined as 
f(x,y) = aO + a1x + a2y + a3x 2 + a4y 2 +a5xy (2) 
, in case of three variable they can be defined as 
f(x,y,z) = aO + a1x + a2y + a3z + a4x 2 + a5y 2 +a6z 2 + a7xy+ a8yz 
+a9xz+a10x 3 + ally 3 + a12z 3 + a13x 2 y + a14x 2 z + 
a15/x + a16/z + a17z 2 x+ a18z 2 y + a19xyz (3) 
, from (3) it can be normalized, according to result of testing 
shown (in Figure. 1) 
f(x,y,z) ~ aO + a1x + a2y + a3z + a4x 2 + 35/+ a6z\4) 
Figure 1. Result of testing of normalizing the polynomial 
equation from 6 to 20 constant values [tested with 24 sample 
points, accuracy measured by expectation value, E]. 
With the number of control points, the reduction of matrix based 
on Least Square theory was applied for this study, in order to 
result value of constants and solve polynomial function. [4] as 
square matrix. Many given square matrices of matched control 
points were selected, then only one matrix being qualified as 
group representative with optimum value under the expectation 
value (E). 
.when U| is original referenced data. 
u is mean value of referenced data. 
u, A is calculated value of referenced data by 
polynomial function. 
ST = Z (Uj - u) 2 (5) 
SSE = Z (u, - Uj A ) 2 -> Z e 2 (6) 
SU = ST - SSE 
E (expectation) = SU/ST -> 1 
The way generating 3D objects from single photograph was how 
to reconstruct them with easily and possibly mathematic 
function. Polynomial function was a choice for this study. The 
following processes were entire details of working. 
Find out the best fit of polynomial function on building data 
based on least square method by considering whole 
building in a picture, that is presented as 3 dimension 
* Supported by Visiting Scholar Foundation of Keb Lab. In Wuhan University, P. R. China

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