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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
3. LEAST-SQUARES MODEL-IMAGE FITTING 
The principle of model-image fitting algorithm is to adjust the 
shape and pose parameters of a primitive model, so it can fit the 
corresponding features extracted from the images. Since the 
floating model can be taken as a wire-frame model, the features 
for fitting are edge pixels. The optimal fit is achieved by 
minimizing the sum of the perpendicular distances from the 
edge pixels to the corresponding projected line of the wire- 
frame model. Figure 6 depicts the optimal fitting procedure. A 
model base which is a collection of various floating models has 
been pre-established. The selected primitive model is projected 
onto the image and fit the extracted edge pixels. 
  
  
  
Optimal Fitting 
    
  
Edge 
Detection [^ 
  
  
  
  
  
  
     
   
Model Base 
&e[mce 
Projection 
  
  
  
Figure 6. The procedures of optimal fitting. 
3.1 Coordinate Systems 
The proposed LSMIF algorithm performs the fitting in the 
photo coordinate system. A primitive model, however, is 
defined in the model space. It is necessary to transform a 
primitive model from model space to object space by 
introducing a set of shape and pose parameters. Then, it has to 
be projected onto the photo coordinate system with the known 
exterior orientation parameters. On the other hand, edge pixels 
extracted from the images should be transformed to the photo 
coordinate system for matching. Figure 7 shows the 
transformation of a box from model to photo coordinate system 
and the edge pixels from image to photo coordinate system. 
Shape & E.O. 1.0. 
Pose Para. Parameters Parameters 
Zi Y3 Zi Y 3 
» 
Es Model =) —=)| Exterior [T= fu Interior m) 
to Orienta- Orienta- c 
X Object X tion X tion 
Model Coord. | Trans. Object Coord. Photo Coord. 
System System System System 
Figure 7. The coordinate systems involved with LSMIF. 
  
  
  
  
  
  
  
  
  
  
  
  
  
The primitive model defined in the model space is a simple unit 
solid, e.g. a box is a unit cube of which width, length, and 
height are all equal to 1. The shape parameters will elongate or 
shorten the box to the correct size, and the pose parameters will 
rotate and move the box to the correct altitude and position in 
the object space. Table 1 lists the eight vertices coordinates 
transforming from model space to object space. Each vertex in 
the object space is then projected onto the images by the 
Image Coord. 
collinearity condition equations with the known exterior 
orientation elements 
3.2 Approximate Fitting and Buffer 
An approximate fitting is required before applying the LSMIF 
algorithm. An interactive program is developed for model 
selection, approximate fitting, and visualization. To obtain as 
close as to the right fitting, this program provides a user 
interface that allows the operator to resize, rotate, and move a 
model to fit the corresponding building images approximately. 
Benefited from the approximate fitting, the LSMIF iteratively 
pulls the model to the optimal fit instead of blindly searching 
for the solution. To avoid the disturbance of irrelevant edge 
pixels, only those edge pixels distributed within the specified 
buffer zones will be used in the calculation of the fitting 
algorithm. Figure 8 depicts the extracted edge pixels 7; and the 
buffer determined by a projected edge v;;v;> of the model. The 
suffix i represents the index of edge line, j represents the index 
of overlapped image, and k/ represents the index of the edge 
pixel. Filtering edge pixels with buffer is reasonable, because 
the discrepancies between the projected edges and the 
corresponding edge pixels should be small, as the model 
parameters are known approximately. However, the buffer size 
has to be carefully chosen because it will directly affect the 
convergence of the computation, i.e., the pull-in range. 
   
  
  
A > 
(x " Extracted Pixels 
iN in Yi * En Jn) 
E d. ? ©. e e 
$0 jj °° 
eo ^ e e 
e Bhd 
f 
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Yo 7.) 
» x 
Figure 8. The extracted edge pixels and the buffer. 
3.3 Objective Function and Least-squares Adjustment 
That the fitting condition we are looking for is the projected 
model edge line exactly falls on the building edges in the 
images. In Eq.(1), the distance dj represents a discrepancy 
between an edge pixel 7;; and its corresponding edge line vj;v;», 
which is expected to be zero. Therefore, the objective of the 
fitting function is to minimize the squares sum of dj. Suppose a 
projected edge line is composed of the projected vertices v;;(x;;, 
vi) and vix(x;2, yi). and there is an edge pixel Z;,(xy yj) 
located inside the buffer. The distance dj; from the point 7;, to 
the edge v;;v;; can be formulated as the following equation: 
  
  
Table 1. Vertices coordinates from model space to object space. 
  
After Translation 
  
  
  
  
  
  
  
  
  
  
  
Mertex No Model Space Multiply After Rotation ; 
: Coordinate Shape Parameters (Object Space Coordinate) 
Vi (0, 0, 0) (0, 0, 0) (0, 0, 0) (dX, dY, dZ) 
Vo (1, 0, 0) (w, 0, 0) (wcosa, wsina, 0) (wcosa--dX, wsinac-dY, dZ) 
V3 (1,0) (w, 1,0) (weosa-/sina, wsina+icosa, 0) (wcosa-Isina+dX, wsina+/cosa+dY, dZ) 
V4 (0, 1, 0) (0, 7, 0) (-Isina, /cosa, 0) (-[sina+dX, /cosa+dY, dZ) 
Vs (0, 0, 1) (0, 0, h) (0,0, h) (dX, dY, h+dZ) 
V6 (1,0, 1) (w, 0, 4) (wcosa, wsina, h) ((vcosa+dX, wsina+dY, h+dZ) 
V7 (1,11) (w, 1, h) (wcosa-Isina, wsina+/cosa, h) (rcosa-/sinatdX, wsina+/cosa+dY, h+dZ) 
Vg (0.1, 1) (0, /, h) (-[sina, /cosa, h) (-Isina+dX, lcosa+dY, h+dZ) 
  
  
  
  
443