Zhu Xu 
  
stereo models while the LZD algorithm is generally applicable for DEM matching. 
Let Z = F (X, Y) and z z f (x, y) be the two DEMs to be matched and P = [ X Yzj] and P! xy y are a pair of 
corresponding points. The following equation holds: 
Ex Y zZl-RLE v :]e (1) 
where R (rotation matrix) and # (the translation vector) define the transformation from z = f (x, y) to Z = F (X, Y). For a 
point P' , [X, Y, Zo]. is assigned to be the approximate corresponding point, of which Xp = x, Y; — y and Z, is 
interpolated from Z=F(X,Y) . Let X = X,+ 4X, Y = Yo+ À l'and Z =Zo+ AZ. Supposing the DEM Z - F (XY) is 
continuous, we have approximately 
F 
A = SF AX + 9F AY (2) 
0X oY 
Let €, ® and * denote rotations around X, Y and Z axis respectively and f£ - [ t, t, t. ]". Linearization of equation (1) at 
om da KEz0,t,= h=t= 0 gives: 
Xo * AX - x t dt, — ydK * zdó 
Yo t AY - y t dt, * xdK — zd) (3) 
Zo t AZ 2 z tdt. — xdó * ydo 
Since Xp = x and X, - y, the first two portions of (3) become: 
AX - dt, — ydK * zdó 
AY - dt, * xdK — zd (4) 
From equation (2), (3) and (4) we obtain the observation equation for a pair of corresponding points: 
ózZ,-zt(dt,-ydk- ado) ar, +xdk— dn Sa, +xd¢ — ydw (5) 
where Ó is the matching residual. The following objective function is minimized to yield estimates of transformation 
parameters that bring the two DEMS closer: 
FR.D=Y pb; (5.1) 
j=! 
The weight p; is introduced to incorporating robust estimators as will be seen later. The partial derivatives needed in 
equation (5) are approximated by slopes over two DEM grids. 
3. DETECTION OF LOCAL DEFORMATION 
As analyzed in the introduction, the essential task of detecting surface difference without control points is to match the 
two DEMs precisely. The LZD algorithm is able to do so only if the two DEMs are identical except for random errors. 
In the cases that the two DEMS suffer from local deformation, if they are precisely matched according to their identical 
parts, then the matching residuals over all the surface (i.e. 9 in (5) for the LZD algorithm after the matching procedure 
converges) can be divided into two categories: those coming from the identical parts, denoted by R;, and those coming 
from non-identical parts, denoted by Rp. In general, elements in R; can be assumed to obey the same normal 
distribution N(0, 9 ).The mean of elements in Rp would usually differ significantly from zero and the standard 
deviation would be large to some extent. When R; is mixed with Rp, elements in Rp could be therefore considered as 
outliers, or gross errors, in Ry. This brings to us the idea that local deformation could be detected as outliers in the 
observations of matching process. 
3.1 Handling Outliers 
There are two major approaches for dealing with outliers: outlier detection and robust estimation. In the outlier 
detection approach, one first tries to detect (and remove) outliers and then perform estimation with the "clean" 
observations. Many statistics based on residuals (usually resulting from least squares adjustment) are designed to 
measure the "outlyingness" of observations. The most popular statistics are the normalized residual developed by 
Baarda (1968), which is applicable in the case that reference variance (variance of observation of unit weight) is known 
a priori, and the studentized residual proposed by Pope (1976), which is applicable in situations when reference 
variance is estimated from residuals. Statistical testing procedure using either of these two statistics is known as data 
snooping. Given a desired level of significance, a constant is determined and the observation, of which the normalized 
residual in absolute value is larger than this constant, is detected as an outlier. Lower bound of gross error detectable 
with a given probability can be estimated [Forstner 1986]. However, data snooping is only applicable under the 
  
1002 International Archives of Photogrammetry and Remote Sensing. Vol, XXXIII, Part B3. Amsterdam 2000.