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Al-Hanbali, Nedal 
(1995). Full details related to this case-study and how one should deal with refraction effects are given in Al-Hanbali 
(1998). The on-site test proves that the LSS can operate and provide reliable measurements under high temperature and 
vibration effects. Furthermore, the local scaling approach provides non-contact surface deformations is a simple and 
quick approach, and can be used reliably to show deformation trends on a surface of a machine or an object. 
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(b) 
Figure 7: Scaled measured depth distances and their corresponding precision compared to the actual introduced 
movements. 
5 THE LSS MATHEMATICAL MODEL 
4.1 Interpretation of Digital Values 
Each point in the depth-coded image produced by the LSS is characterized by its pixel coordinates (i and j) and its 
registered value p0 (that has a dynamic range between 0 and 32768). In a standard unit system (e.g. metric units), the X, 
Y, and Z coordinates of a point are functions of p. 0 and $, which are functions of the observations p0, i, and j where: 
p=p0.scale value , | 0-0,4-i.60 and = da + j- 00 Equation 1 
where 0 and $ are the X and Y-axes rotational angles, and the angles 0, and dy are the initial angular field of view and 
80 and 3¢ are the step angles of the X and Y-axes of the mirrors. These variables are assumed to be constant for each 
LSS configuration (e.g. angular field of view, lens mode, acquisition strategy, image resolution, and/or other 
configurations). The intensity image is useful and provides better precision for detecting edges and locating target 
centers. Note that the pixel image coordinates are the same for both the depth-coded and the intensity images. For 
detailed explanation, see Al-Hanbali (1998) and Al-Hanbali et. al. (1999). 
The distortion model used is based on a third order polynomial function with five distortion parameters. The polynomial 
functions absorb the distortion effects in the depth values p, the X-axis rotational angle © and the Y-axis rotational angle 
9. 
4.2 The LSS Equations 
The collinearity equations of the LSS are developed by NRC (Beraldin 1993, 1994). The equations are based on: 
X iss = AR. (K,®,@)[X-X, Io; Equation 2 
Yıss - AR, (K,,0)|Y-Y,)], Equation 3 
Zisg 7 AR, (K, o, @) ZZ LL. Equation 4 
Where X,, Y, and Z, are the 3-D coordinates of the LSS centre, and X, Y and Z are the 3-D coordinates of an object 
Space point, both are in the object space coordinates system. The X1ss, YLss and Zi,ss are the 3-D coordinates of the 
object space point in the LSS coordinate system (correspond to LSS collinearity equations). Each is written as a 
function of the scale factor À, the rotation matrix R between the object space coordinate system and the LSS coordinate 
System and the position of an object space point coordinates with respect to the LSS centre . The angle À is the scale 
between the LSS coordinate system and the control field (target field) coordinate system. The value of the scale factor is 
assumed to be equal to one. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 13