fine (w = 10) and a coarse (w = 30) LoG operator respec- 
tively. In addition, the traces of edges in various resolutions 
offer a complete representation of the original signal, thus 
allowing its reconstruction [Yuille & Poggio, 1983]. 
3. DIFFERENTIAL SCALE VARIATIONS 
When representing the scale space family of a digital image 
as a pyramid, we create a number of discrete representations 
of the original image with each representation correspond- 
ing to a specific scale level. However, unless the image- 
generating projection is parallel, the exposure vertical and 
the object surface planar, features within the same image 
pyramid level will not have the same geometric scale, ex- 
pressed as ; 
A 
E (11) 
with A' the image of a feature A of the object space. For 
the projective transformation governing the image formation 
process, the scale factor 5* at a point (25, y*) of the image, 
corresponding to a point (.X*, Y*, Z*) in the object space will 
be given through the formula 
54 
z! : x —X 
y | =SR| Y+—-Y (12) 
—c 2 
where R is the rotation matrix and (X,, Y,, Z;) the exposure 
station coordinates of the photo. It is apparent that different 
features in the same image will have different scale factors. 
In addition, the images of the same object space feature in 
two or more different images will have different scales, par- 
ticularly when the exposure conditions (rotations, exposure 
stations) differ significantly (e.g., converging photography) 
or the object space surface displays high variations. In the 
extreme case, the scale becomes 0 and occlusions occur. 
  
Assuming each image pyramid level i to correspond to an 
average scale S;, features within this image will thus appear 
in scales 
0Z OZ 
"ax oy) (13) 
which in general will not coincide with any of the discrete 
scales represented by the image pyramid. Image pyramids 
though are discrete representations of the scale space which 
itself is continuous. While the discrete representation is ob- 
tained using only a number of values of the scale parameter c 
of the Gaussian kernel used to convolve the image, a contin- 
uous representation is the outcome of the same convolution 
allowing o to receive any allowable real value. 
  
S; + dS;, dS; = f(X,Y, Z,w, ‚KK 
(w = 10) 
Scale variations between members of stereopairs become ap- 
parent in digital photogrammetric operations, with match- 
ing serving as a good example. In least squares matching, 
we attempt to match windows of pixels by minimizing their 
radiometric differences. This is achieved by forming one ob- 
servation equation for every pair of conjugate pixels within 
a pair of approximately conjugate image windows gi(zr, VL) 
and gn(zn, yn) in the left and right image respectively 
  
gi(z1,yL) — gn(zn. yn) — e(v,v) (14) 
Figure 5: Edges detected with a coarse LoG operator The solution is obtained by allowing one of the two windows 
(w — 30) to be geometrically reshaped according to an affine transfor- 
mation and by resampling gray values for this newly defined 
588