rater om 
PR RE 
ERAI IS RAMS 
  
ON THE APPLICATION OF SCALE SPACE TECHNIQUES 
IN DIGITAL PHOTOGRAMMETRY 
Anthony Stefanidis 
Toni Schenk 
Department of Geodetic Science and Surveying 
The Ohio State University, Columbus, Ohio 43210-1247 
USA 
Commission III 
ABSTRACT 
Scale space techniques are widely used in digital photogrammetry. Typical implementations use the scale 
space as a discrete representation, thus inherently assuming that all features represented in images of similar 
resolutions belong to the same scale space level. However, this approach ignores differential scale variations that 
exist between conjugate features in multiple images, or even between different features in a single image. The 
subject of this paper is an investigation into theoretical and practical aspects associated with the use of scale 
space techniques in both the image and object space domains. The interrelationship between the scale space 
representations of these two domains and the effects of differential scale variations in digital photogrammetric 
operations, such as matching, object space reconstruction, and orthophoto production are also addressed. 
1. INTRODUCTION 
Physical phenomena in object space occur over a wide va- 
riety of spatial extents. Macro-variations of a surface ex- 
press its major trend, while micro-variations correspond to 
trends of smaller extent. The concept of macro- and micro- 
variations is relative and depends on the specific applica- 
tion. What is considered a macro-variation in one applica- 
tion might very well be viewed as a micro-variation in an- 
other. In digital images, changes in gray values correspond 
to object space phenomena, which can also be perceived 
within areas of different sizes, ranging from few pixels to 
large regions. However, even region-wise changes occur over 
an extensive array of region sizes, ranging from as little as 
a few pixels to as much as a large part of the image. The 
identification of these changes is essential in decoding the 
information which inherently exists in an image. 
The scale space representation of signals in general, or dig- 
ital images in particular, is widely used to successfully pro- 
duce several versions of the same image in which the infor- 
mation content is changing in a systematic and, therefore, 
easy to exploit fashion [Lindeberg, 1990], [Yuille & Poggio, 
1983]. Physical phenomena of various extents can be easily 
identified through the behavior of their images in different 
levels of scale space [Lu & Jain, 1989], [Witkin, 1983]. 
In our paper, we present the basic axioms of scale space, 
and we analyze the corresponding mathematical aspects, 
together with the proper selection of scale-generating func- 
tions. The effect of differential scale variations on pho- 
togrammetric procedures is discussed, and we report how 
a continuous scale space can be used to bypass the short- 
comings of this effect. Finally, the scale space representation 
of object space and its potential use in photogrammetry are 
explored. 
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2. SCALE SPACE 
The scale space representation of a signal f(z, y) is a set of 
signals ( f?(z, y; n)), representing the original one in various 
scale levels as function of a scale parameter n. The set 
of signals (f7(z,y;n)) is called the scale space family of 
f(z.v). 
The objective of the scale space representation of any signal 
is to create a scale space family in a way that information 
conveyed by this signal will become more explicit. In order 
for this goal to be met, the generation of scale space family 
has to follow some basic guidelines [Lindeberg, 1990]: 
e The scale space family has to be generated by the 
convolution of the original signal with a single scale- 
generating function s(z, y; n) 
fz(z,y;n) 5 s(z,y;n) * f(z. v) (1) 
e The scale-generating function should be selected in a 
proper manner, such that larger values of n would cre- 
ate coarser versions of the original signal through elim- 
ination of the finer details which correspond to higher 
frequency phenomena. We want to be able to identify 
large trends in lower resolutions and include spatially 
limited details in finer levels. For n = 0, at the finest 
resolution of scale space, we have the original signal 
itself 
fz(z,y0) — f(=,9) (2) 
which is obviously the upper limit as far as fine reso- 
lution is concerned. 
A Gaussian filter is mathematically expressed as a function 
g(z,y) ke ^l (3) 
e- Oo e <