The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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2. PRESENTING A NOVEL APPROACH TO DEVELOP 
LSM IN ORDER TO BUILDING CHANGE 
DETECTION 
2.1 Object decomposition 
For the purpose of our approach, a building is expressed 
through a wireframe representation based on prior vector 
information (Habib, and Kelley, 2001). For decomposition of 
the lines, a new element is introduced, the Minimum Spatial 
Element (MSE). The MSE describes the resolution of spatial 
change that the user is interested in. Using this information, we 
perform a segmentation of outlines, and lines are essentially 
substituted the corresponding points along the outline (Figure. 
1). 
Figure 1: object decomposition 
f(x,y)-g(x,y)=e(x,y) (2) 
With e(x,y) being the error vector. 
In a typical least squares matching method, observation 
equations can be formed relating the gray values of 
corresponding pixels, and they are linearized as(Grien, and 
Akca, 2005): 
fix, y) - e(x, y) = g° (x, y) + 
dg°jx,y) 
dx 
dx + 
dy 
(3) 
The derivatives of the image function in this equation express 
the rate of change of gray values along the x and y directions, 
evaluated at the pixels of the patch. Depending on the type of 
edge, the geometric relationship describing the two windows 
may be as complex as an affine transformation, or as simple as a 
simple shift and/or rotation. To facilitate solutions we can 
resample template and/or actual image to have edges lying 
along one of the major coordinate axes. For example, we can 
have resampled edges oriented along the y axis of the 
corresponding windows. In this case, the above equation may be 
reduced to: 
f(x, y) - e(x, y) = g° (x, y) + dg ^ dx ( 4 ) 
ox 
Where the two patches are geometrically related through a shift 
along the x direction: 
2.2 Providing template 
In order to provide template, first the image coordinate of the 
point along the building edges is computed for each image by 
colinearity equation. The image coordinates is considered as the 
centre of the template. Determining the size of the templates is 
the second step. This allows us to optimize computational 
performance without compromising accuracy potential. From an 
accuracy and reliability point of view, small image patches, e.g. 
3x3 pel 2 are not suitable for LSM due to small redundancy 
number. An optimal size of matched window also differs with a 
pixel size and scale of the original images. A size of 15 x 15 
pel 2 for good quality images with a lot of details and at least 
21x21 pel 2 for noisy images or when significant differences 
in brightness between a template and a search area occur are 
recommended in (Kraus, 1997). So, the appropriate template 
size for our images used in this research is 21 x 21 pel 2 . 
2.3 Development of least square template matching 
A window depicting an edge pattern is introduced as a reference 
template that is subsequently matched to digital image patches 
in the vicinity of actual edge segments. The concept behind the 
method is simple yet effective: by matching the edge template 
window to an image window, we can identify edge locations in 
the image as conjugate to the a priori known template edge 
positions. Assuming f(x,y) to be the reference edge template and 
g(x,y) to be the actual image patch, a matching correspondence 
is established between them when 
f(x,y)=g(x,y) (1) 
However, considering the effects of noise in the actual image, 
the above equation becomes 
XNEW ~ XOLD + dX (5) 
The dX parameter is the unknown that allows the repositioning 
of the image window to a location that displays better 
radiometric resemblance to the reference template. Regardless 
of the choice of geometric transformation, the resulting 
observation equations are grouped in matrix form as 
-e = Ax - l; P (6) 
Where, / is the observation vector, containing gray value 
differences of conjugate pixels. The vector of unknowns x 
comprises the shift at the x direction, while A is the 
corresponding design matrix containing the derivatives of the 
observation equations with respect to the parameters, and P is 
the weight matrix. A least squares solution allows the 
determination of the unknown parameters as 
x = (A T PAy l A T Pl (7) 
Based on the formulation of the above model, we have to face 
the major challenge about formulation of the weight matrix 
described in the next section. 
In order to better handle noise through the analysis of local edge 
information within the matching windows, we have to allow 
edge pixels to influence the solution more than the rest of the 
template. This can be performed by manipulating the 
corresponding weight matrix P. Indeed, by assigning high 
weight values to certain pixels, we allow them to influence the 
solution more than pixels with low weight values and the effects 
of undesired variations on our solution are kept at minimum.