GENERAL MATHEMATICAL MODEL OF LEAST SQUARES 3D SURFACE
MATCHING AND ITS APPLICATION OF STRIP ADJUSTMENT
Z. Q. Zuo* *, Z. J. Liu, L. Zhang?, S. Tuermer?
* Chinese Academy of Surveying and Mapping, 28, Lianhuachixi Road, Haidian District, Beijing, P.R. China 100830-
(zqzuo, zjliu, zhangl)(g)casm.ac.cn
? Remote Sensing Technology Institute (IMF), German Aerospace Center (DLR), Oberpfaffenhofen, Germany-
sebastian.tuermer@dlr.de
Commission III, WG III/2
KEY WORDS: 3D surface matching, 3D similarity transformation, strip adjustment, laser altimetry
ABSTRACT:
Systematic errors in point clouds acquired by airborne laser scanners, photogrammetric methods or other 3D measurement
techniques need to be estimated and removed by adjustment procedures. The proposed method estimates the transformation
parameters between reference surface and registration surface using a mathematical adjustment model. 3D surface matching is an
extension of 2D least squares image matching. The estimation model is a typical Gauss-Markoff model and the goal is minimizing
the sum of squares of the Euclidean distances between the contiguous surfaces. Besides the generic mathematical model, we also
propose a concept of conjugate points rules which are suitable for special registering applications, and compare it to three typical
conjugate points rules. Finally, we explain how this method can be used for the co-registration of real 3D point sets and show co-
registration results based on airborne laser scanner data. Concluding results of our experiment suggest that the proposed method has
a good performance of 3D surface matching, and the least normal distance rule returns the best result for the strip adjustment of
airborne laser altimetry data.
1. INTRODUCTION
A laser scanner system consists of three main components:
GPS, IMU and Laser unit. Data collection is carried out in a
strip-wise form and the object coordinates of the laser
footprints are determined using the direct geo-referencing from
the GPS/IMU. Due to the systematic errors in the laser scanner
components or in the alignment, adjacent strips usually have
discrepancies. Such discrepancies are serious to the terrain
modeling and object reconstruction. So the most emphasis of
our approach is to find a general solution for the registration
problem in 3D modeling.
In the field of Computer Vision, one of the most famous
methods is the Iterative Closest Point (ICP) algorithm proposed
by Besl and McKay (1992), Chen and Medioni (1992), and
Zhang (1994). And in the field of LIDAR and photogrammetry,
people use data driven methods to do strip adjustment. The
purpose of data driven methods is establishing a 3D
transformation 7' between two point sets, which represent an
irregular spatial sampling of the same surface (Shan, 2008).
The earliest data driven methods, only consider the difference
on elevation between strips, using linear system parameters
with conjugate points of adjacent strips (Crombaghs, 2000;
Kornus and Ruiz, 2003). And then Kilian (1996) uses a 12
parameters model to replace the linear parameters. In order to
eliminate the strong correlation of the 12 parameters model,
Vosselman and Maas (2001) use a 9 parameters model to
estimate 3D transformation. Morin and El-Sheimy (2001) just
considered the global translation and rotation transformation to
establish a 6 parameters model. And if the scale factor can be
involved, the 7 parameters of a space similarity transformation
model may return an effective solution for the strip adjustment
(Robert, 2004; Gruen and Akca, 2005).
Least squares 3D surface matching (LS3D) is a typical data
driven method, where the sum of squares of the Euclidean
distances between the neighboring surfaces is minimized.
LS3D has many advantages compared with the ICP method,
* Corresponding author
and the significant one is that conjugate points of LS3D can be
obtained using interpolation on 3D surface but ICP needs real
points. Hence, LS3D can achieve higher accuracy in many
cases, especially in the co-registration routine of different
resolution point clouds.
In this approach, we propose a mathematical model for LS3D.
It is a general model for estimating orthomorphic
transformation parameters for conjugate surfaces. Three
different searching rules of conjugate points on adjacent
surfaces are defined in this adjustment system and each rule
can be well used in the new estimate model. Also the
differences are compared to existing methods and are shown in
detail. At last, two groups of real airborne laser scanner data
are used to show the capabilities of our method.
2. 3D SURFACE MATCHING
2.1 New Estimation Model
The major task of 3D surface matching is finding the
transformation parameters between template surface
Flx,y, zZ) and searching surface G(x, X, zZ) The
overlapping area between two surfaces is O(x 3 y.2 y, which
can be defined as O — F' (1G . The goal of least squares
estimation of the orthomorphic transformation parameters is as
follows:
G(x,y,z) =T{F(x,y,z)} (D
To express the geometric relationship between the conjugate
surfaces, a seven parameters similarity transformation is used:
xt f. X
y' -|f, - mR(g,o,K)«| y Q)
24 t, z
