RECONSTRUCTING SMALL SURFACE PATCHES FROM MULTIPLE IMAGES
Toni Schenk
Charles K. Toth
Department of Geodetic Science and Surveying
The Ohio State University, Columbus, Ohio 43210-1247
USA
Commission III
ABSTRACT
Matching multiple images simultaneously greatly increases the reliability of determining conjugate points automatically. It
offers significant advantages for aerotriangulation because tie points cannot only be determined automatically but also more
reliably. In this paper we develop models where the matching is combined with reconstructing a surface patch geometrically
and radiometrically from multiple image patches.
KEY WORDS: Image Matching, Aerotriangulation, Machine Vision.
1. INTRODUCTION
Determining conjugate points is a fundamental task that
occurs in almost any photogrammetric application. In dig-
ital photogrammetry it has become customary to call this
process ?mage matching. It is fascinating to observe the de-
velopment of image matching during the last decade. Much
progress has been made since Ackermann (1984) and Fórs-
tner (1984) presented the first rigorous mathematical models
for the image matching process. Apart from extending the
basic mathematical model by introducing geometrical con-
straints (see, e.g., Grün & Baltsavias, 1988), a decisive step
was to combine matching with reconstructing the surface
(see e.g. Wrobel, 1987; Helava, 1987; Ebner et al., 1987).
Ebner and Heipke (1988) propose a new approach where
matching several images and surface reconstruction is treated
as a simultaneous adjustment problem. Matching multi-
ple image patches is of great practical importance, most
notably in aerotriangulation where as many as nine pho-
tographs may partially overlap. Identifying and measuring
tie points, particularly between strips, is a notorious prob-
lem, since only two photographs can be viewed stereoscop-
ically at the same time. The reliability and accuracy of tie
points is expected to significantly increase when multiple
image matching methods are employed.
The purpose of this paper is to formulate suitable math-
ematical models for rigorously solving the multiple image
matching problem. We describe a general solution by in-
troducing a geometric and radiometric relation between a
surface patch and the corresponding image patches. We
then investigate several geometric transformation models
between object and image space, including linearized ob-
servation equations. We conclude with describing two ap-
proaches of using multiple image matching in aerotriangu-
lation. The first approach is the most general one where
the exterior orientation, the surface patches with elevations
(DEM) and gray levels are all determined simultaneously.
In the second approach conjugate points are determined in-
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dependently from one another and without the exterior ori-
entation parameters. This solution corresponds to the tra-
ditional method where all points are individually measured
and then entered into a block adjustment.
2. GENERAL APPROACH
Fig. 1 depicts four images I,,..., I, with image patches
Di,-..,P4 Covering the surface patch S. To generalize let p
be the number of image patches, size n xn pixels. Associated
with every image patch p; is a gray level function g;(z, y). We
may consider the gray levels as observed.
The surface patch S is represented as a DEM with a res-
olution of m x m grid points, m « m. The task is now
to reconstruct the surface patch S from the observed gray
levels of the image patches. This involves both, geometric
and radiometric reconstruction. Let Z(X,Y) be the geo-
metric function and G(X,Y) the radiometric function for
representing the surface patch S. Capital letters are used
to better differentiate object space functions and variables
from their counterparts in image space. Obviously, the im-
age functions gi(z, y) correspond to G(X,Y), the gray level
distribution of the surface patch S. The discrete represen-
tation of Z(X,Y) can be considered the DEM of S. The
reconstruction of 5 involves 2m? parameters (m? elevations,
m? gray levels).
Since S is small we may approximate it by a Lambertian sur-
face. The gray levels of the image patches are then directly
related to the gray levels of the surface patch. Suppose the
surface is flat and parallel to all image patches. In this (un-
realistic) case, the gray levels of S would simply be the mean
of the gray levels of all image patches.
Next, we need to define the geometrical relationship between
the image patches and the surface patch. This is accom-
plished by a geometrical transformation T€. Combining the
geometric and radiometric relationship leads to the following
non linear observation equations:
