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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
points with fixed interval. Each sample point would be treated 
as an observation in the LSMDF to solve the plane parameters 
as optimal fit. Fig. 4 depicts the flowchart of the plane fitting. 
Data 
Process 
Example 
Figure 4. The flowchart of plane fitting. 
the distance djj represents a discrepancy between a sample point 
Tjj and its corresponding edge v m v„, which is expected to be 
zero. Therefore, the objective of the fitting function is to mini 
mize the squares sum of d^. Suppose an edge is composed of 
the vertices v m (x m , y„) and v„(x„, y n ), and there is an edge pixel 
Tfxy, y-j) located inside the buffer. The distance d l} from the 
point Tjj to the edge v m v n can be formulated as the following 
equation: 
__ \(y m -y n ) x ij + ( x n -xJy» + (y n x m ~I 
J( x m ~ x J 2 +(y m -yj 2 (1) 
The coordinates of vertices v m (x m , y m ) and v„(x„, y„) are func 
tions of the unknown plan parameters. Therefore, d t j will be a 
function of the plan parameters. Taking a box model for in 
stance, d^ will be a function of w, /, a, dX, and dT, with the hy 
pothesis that a normal building rarely has a tilt angle (t) or 
swing angle (s). The least-squares solution for the unknown pa 
rameters can be expressed as: 
'Ldif = E[Fy ( w, /, a, dX, dF)] 2 -* min. (2) 
Eq.(2) is a nonlinear function with regard to the unknowns, so 
that the Newton’s method is applied to solve for the unknowns. 
The nonlinear function is differentiated with respect to the un 
knowns and becomes a linear function with regard to the in 
crements of the unknowns as follows: 
OF. dF„ dF, 
(—^) 0 A dY + (—Mo A dZ + (-i-) 0 Aa 
odY odZ da 
Since the model has been manually fit, the bottom edges of the 
wireframe model should be close to the building’s boundary on 
the map. Benefited from the approximate fitting, the LSMDF 
iteratively pulls the model to the optimal fit instead of blindly 
searching for the solution. To avoid the disturbance of irrelevant 
sample points, only those points distributed within a specified 
buffer zone are adopted for fitting calculation. Figure 5 depicts 
the sample point 7^ and a w bu ff er wide buffer determined by an 
edge v m v„ of the model. The suffix i is the index of edge line L, 
and j is the index of sample points. Filtering edge pixels with 
buffer is reasonable, because the discrepancies between the bot 
tom edges and the corresponding sample points should be small, 
as the model parameters have been fit approximately. Ffowever, 
the buffer size has to be carefully chosen because it will di 
rectly affect the convergence of the computation. 
in which, Fyo is the approximation of the function Fy calculated 
with given approximations of the unknown parameters. Given a 
set of unknown approximations, the least-squares solution for 
the unknown increments can be solved, and the approximations 
are updated by the increments. Repeating this calculation, the 
unknown parameters can be solved iteratively. 
The linearized equations can be expressed as a matrix form: 
V=AX-L, where A is the matrix of partial derivatives; X is the 
vector of the increments; L is the vector of approximations; and 
V is the vector of residuals. The objective function actually can 
be expressed as q=V T V^-min. For each iteration, X can be 
solved by the matrix operation: X=(A T A)' , A T L. The standard 
deviation of each increment can also be calculated as the accu 
racy index of the LSMDF. 
Discard Sample Points 
■ V Adopted Sample Points 
Figure 5. Buffer zone for fitting. 
The fitting condition we are looking for is that the model edge 
exactly falls on the building boundary on the map. In Eq.(l), 
The objective of the height fitting is the building’s roof in the 
LiDAR data. As the distance from sample point to the edge is 
the observation function in the plane fitting, the observation 
function should be the distance from LiDAR point to the roof 
plane in the height fitting. The roof plane equation is composed 
of model parameters. However, the calculations of 3D fitting 
would be much more complicated than 2D. And it will also in 
crease the iteration number and the chance to divergence. Con 
sidering the efficiency and the practicality, we adopt an easier 
method for the height fitting in this paper. Since the plane pa 
rameters have been fit optimally, LiDAR points within the 
plane range are supposed to belong to the model. These points 
are then projected to a local 2D coordinate system which is de 
fined on the façade of the model. Fig. 6 illustrates the transfor 
mation of a ridge-roof building. Thus the observation function 
is simplified as the distance from 2D point to edge, similar to 
the plan fitting.