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.