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window. Differences in scale are accommodated by the two
scale factors assumed in the six-parameter affine transfor-
mation
Er = 41 + 4281 t asy (15)
and
yR = bi + baæz d bsyr (16)
Updating the above affine transformation parameters by the
solution of the linearized observation equations
gi(zn,yL) — e(z,y) = 9r(=R,YR) + JR.da1 + gn. 21da;
+ gr.yrdas + gr, db; + 9r,e14b2
+ gn.yrdbs (17)
we define a new window in the right image within which we
resample the gray values.
Scale variations will affect this procedure in various stages.
When two image patches are represented in two different
scale levels in a stereopair, their scale difference will be
both geometric and radiometric. When resampling the gray
values gr(Zr,Yr) We use the original image, spreading or
shrinking its gray values over a new area, according to the
updated affine transformation parameters. As a result we
produce a new window in the right image which might be-
long to the same geometric level of scale space as its conju-
gate left image template gr(zr, yr) but will still differ from
it in the radiometric scale space. This will have obvious ef-
fects on the observation equations, since we use gray level
differences as observations. The same problem occurs dur-
ing digital image warping or rectification for orthophoto pro-
duction [Doorn, 1991],[Novak, 1992]. Conjugate patches in
two overlapping orthophotos are brought to the same scale
level geometrically, using as a reference a digital elevation
model of the object space. Radiometrically though, these
patches remain unequal to the same degree that the corre-
sponding windows in the original stereopair were unequal.
This causes conjugate patches in overlapping orthophotos to
differ radiometrically, even when their gray level histograms
are adjusted for average and standard deviation differences.
To accommodate for the problem of different scales, the scale
concept has to be introduced into the matching process it-
self. This will be conceptually performed by the alteration
of the observation equations to accommodate for scale as
gi(zp, yr; SL) — gr(ZR, YR; 5R) = e(v, y) (18)
which would correspond to a matching process adapting it-
self into various scales. The above equation may be lin-
earized with respect to x, y and s, essentially adding to the
previously mentioned (eq. 17) linearized observation equa-
tions one term
gi(zr,yp,sr)-— e(z,vy) — gm(*m Vm 5R)- gn.dzn *
+9r,dyr + gnsdsn (19)
The added term gr, expresses how gray levels change at a
point whenever the scale level of the window within which
this point is located changes within the continuous scale
space. The term s has conceptual meaning and may be sub-
stituted by the o of the Gaussian filter or any other quantity
sufficiently describing scale.
The introduction of a scale parameter in least squares match-
ing may introduce linear dependency. The terms gg, and
gr, also express gray level gradients, but are different than
589
the term gg, in that they are highly localized and obviously
orientation dependent. Even in the case that high depen-
dency exists, matching may be implemented in two distinct
sets, properly constraining some of the parameters to real-
istic estimated values. To assure succesful implementation,
matching has to be performed in the highest possible com-
mon resolution of the two conjugate patches. That will ob-
viously be the resolution of the coarser patch, and therefore
the finer patch has to be transferred into another scale level
using a Gaussian filter.
4. SCALE SPACE REPRESENTATION OF
OBJECT SPACE
Object space can be described by the combination of two
two-dimensional continuous signals, one (Z(X,Y)) express-
ing its geometric and another (R(X,Y)) expressing its ra-
diometric properties. Discretized, these signals are repre-
sented by a Digital Elevation Model and a Digital Radiom-
etry Model which can be together referred to as DERM.
Each of the signals can be individually expressed in a scale
space representation using the Gaussian kernel, thus pre-
serving the scale space family properties that we presented
in section 2. The scale space family of the DEM will consist
of DEM of lower resolutions, with each lower resolution level
representing a smoothed version of the original signal. Tak-
ing advantage of the self-reciprocity of the Gaussian func-
tion which states that the Fourier transform of a Gausian is
another Gaussian
F(G(=)] = G(w) (20)
we see that convolution with a Gaussian function in the
space domain is equivalent to a filtering with a filter of the
same shape in the frequency domain [Weaver, 1983]. There-
fore, Gaussian convolution can be perceived as filtering with
a low-pass filter, the cut-off frequency of which is determined
by the scale parameter c. Coarse scale representations of
the DEM preserve the major geometric trends of the sur-
face, corresponding to the lower frequencies of its frequency
domain equivalent. In finer resolutions, frequencies of higher
order are introduced. Edge detection, with the application
of an LoG function to the DEM signal, will locate break-
lines [Chakreyavanich, 1991]. Breakline detection can be
applied hierarchically, similarly to edge detection in images.
In coarse levels of scale space (large w parameter) we detect
major breaklines in the topographic surface, while moving
to finer resolutions we not only improve the spatial accuracy
of these breaklines, but we also identify breaklines of smaller
spatial extent.
In a similar fashion, the Digital Radiometry Model (DRM)
of the surface can be processed with a Gaussian filter for the
generation of its scale space family. Edges in the DRM will
correspond to positions where the radiometric properties of
the surface present discontinuities.
The recorded image gray values represent the DRM as al-
tered due to the geometric properties of the object space.
In the scale space family of DRM there will exist a member
which most closely corresponds to the image depicting this
DRM. For a DERM with no geometric variations, the edges
detected in the image function would correspond to discon-
tinuities in DRM. In realistic situations though, DEM is not
flat and the image edges reflect the combined effect of geo-
metric and radiometric discontinuities. Taking advantage of
