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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B2. Istanbul 2004
APPENDIX A: DETERMINATION OF THE MAXIMUM
CURVATURE
The computation of the maximum value of the curvature at a
certain position with the help of a 2.5D hybrid raster/grid
surface model (Kraus, 2000) can be performed by differential
geometry.
The 2.5D hybrid raster/grid surface can be described as a
discrete function z(x,y) which provides height information
along the stored vector data and on each raster/grid position. In
the first step, based on this hybrid surface description, the first
(z z,, ) have to be
x» Z,) and second derivatives ( Z,, , Z
Xy >
computed taking into account the surface discontinuity on
breaklines. Then the maximum value X44, of the principle
curvature can be determined with the help of the coefficients of
the first (E, F, G) and second (L, M, N) fundamental form by
solving the following quadratic equation:
EL >. 2
ag EN 20M CI LN — M 20 (A-1)
EG -F? EG-F?
with:
E4rz.?
F2,
G=1+2Z,?
and
Ks eR
Jingle,
lk.
M=kz,,
N =kz,
APPENDIX B: ELEVATION ACCURACY OF A PLANE
SURFACE ELEMENT DERIVED OF 3D POINTS
To fit a plane to 3D points by least squares, the following error
equations are used:
V; * 8g * aX; F a, Y, — Z, (B-1)
Xi X Zi three dimensional coordinates of the reference
points
ay, aj, ay parameters of the plane
V; residuals (corrections) to occur when the
number n of terrain points exceeds 3
Normal equations:
n IX] Iv] (a. (R21) 2
Ble] Bye, mil TS A
Mixed be) a) (DZ)
118
This system of equations becomes much simpler when the
origin of the XY coordinates is moved to the points’ center of
gravity. In this case parameter a, becomes the elevation in that
center; this can be readily seen in analyzing equation (B-1). The
accuracy of this elevation. 1s representative for DTM
interpolation. To estimate this accuracy, one has to re-write the
system N of normal equations (B-2) with the coordinates
reduced to the weight center — denoted here by x and y:
(n ie 0
N= 0 |x*| [xy] B-
0 xy] vl m
This is an invertible hyper-diagonal matrix; this means, that by
inverting the individual matrices N, the corresponding cofactor
matrices Q carrying the weight coefficients can be computed.
We are interested in inverting the element [1,1] only:
dde (B-4)
This result is not surprising: in case when the plane is
horizontal, the result of the modeling becomes the arithmetic
mean. The accuracy of the arithmetic mean will be computed of
the accuracy of the individual observations divided by the
square root of their number.
ACKNOWLEDGEMENT
This research has been supported by the Austrian Science
Foundation (FWF) under project no. P15789.
REFERENCES
Borgefors, G., 1986. Distance Transformations in Digital
Images. Computer Vision, Graphics and Image Processing,
CVGIP 34 (3), pp. 344-371.
Kraus, K., 2004. Photogrammetrie, Band 1, Geometrische
Informationen aus Photographien und Laserscanner-
aufnahmen, 7. Auflage, Walter de Gruyter.
Kraus, K., Pfeifer, N., 1998. Determination of terrain models in
wooded areas with airborne laser scanner data. ISPRS-
Journal, Vol. 53.
Briese, C., Kraus, K., 2003. Datenreduktion dichter Laser-
Gelándemodelle. zfv, Heft 5, 128. Jahrgang, S. 312-317.
Kraus, K., Attwenger, M., Briese, C., Mandlburger G., 2004.
QualititsmaBe für digitale Gelándemodelle (DGMe) am
Beispiel eines Photogrammetrie- und eines Laserscanner-
Projektes. Wissenschaftlich-
Technischen Jahrestagung der DGPF. Halle, im Druck.
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