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where c is the associated standard deviation. In applica- 
tions, the multiplicative factor k may receive various values, 
creating a large array of Gaussian filters which are essen- 
tially scaled variations of the core function, e.g., 
1 z24 2 
ze (4) 
G(z9,959) — ré 
attempts to preserve the output within a prespecified range 
[Agouris et al., 1989]. The use of a Gaussian filter, with 
standard deviation c as the associated scale parameter, as 
a scale-generating function satisfies the above set criteria 
[Babaud et al., 1986]. Therefore, the scale space family of a 
signal f(z,y) can be created as 
f; (z,9;0) = G(æ,, 99; 9) * f(z, y) (5) 
A digital image is a two-dimensional discrete signal I(x,y). 
Its convolution with the Gaussian kernel 
= eo oo 1 5 
Czas Yo) soe 17 M2 — 25,4 — yo) (6) 
Tg=-—00 Yg=—00 
can be used to construct its scale space family. Members of 
the scale space family may have the same dimensions as the 
original image, or, more commonly, their dimensions may 
decline in coarser resolutions. Assuming the original image 
I(z,y) to have dimensions 4096 x 4096 pixels, we can form 
its scale space family by creating m versions of the image (all 
of dimensions 4096 x 4096 pixels), each one by convolving 
I(z,y) with a Gaussian kernel of different scale parameter 
c. 
  
Figure 1: An image pyramid as a representation of discrete 
scale space 
However, in most applications coarser levels of scale space 
are represented by images of smaller dimensions. By con- 
volving the image with a Gaussian kernel and resampling 
every n^ pixel we can create a lower resolution copy of size 
4096 /n x 4096/n. A scale space family in which lower resolu- 
tion members are represented by smaller size images is called 
an image pyramid [Fig. 1]. Various members of the image 
pyramid can be perceived as images of the same object scene 
in various geometric scales. For practical reasons the dimen- 
sions of the members of the scale space family are integer 
powers of two. Typically, the image pyramid of an original 
image of 4096 x 4096 pixels includes versions of the image in 
dimensions of 2048 x 2048, 1024 x 1024 and 512 x 512 pixels. 
Fig. 2 shows two windows of equal dimensions, one from the 
912 x 512 pixel member of an image pyramid and the other 
587 
from the 2048 x 2048 pixel version to demonstrate the asso- 
ciated differences in resolution. Both images were obtained 
by the convolution of the original 4096 x 4096 image with a 
Gaussian kernel, and by proper resampling. 
  
Figure 2: Two windows of equal size in pixels, one in 512 x 
512 resolution (left) and the other in 2048 x 2048 resolution 
(right). 
The use of a Gaussian kernel as a scale-generating func- 
tion offers certain advantages, most notably exploited when 
combining smoothing with edge detection. Edges are iden- 
tified as discontinuities in the image function, and therefore 
correspond to zero-crossings of the twice-differentiated im- 
age. The orientation independent second derivative of a 
two-dimensional function is obtained through a Laplacian 
operator 
0 0 
Vie — tI 7 
Oz? t Oy? (7) 
The associative property of convolution allows the combi- 
nation of scale space generation with a Gaussian function 
G(z,, yg) and differentiation with a Laplacian operator, thus 
substituting two convolutions by a single one 
V'[G(z,, 5:0) * I(z,y)] 7 [V^G(zs,y5:0)] * I(z,y) (8) 
Instead of scaling the image with G(z,,y,) and then looking 
for edges in the smoothed image, we simultaneously smooth 
the image and extract its orientation-independent second 
derivative in a single convolution by the Laplacian of Gaus- 
sian (LoG) function 
z 2 + V. 2 Cw 2.y 2 
V2iG(z,,y,0) = [2 — aid 207 (9) 
The size of the LoG operator is determined by the value of 
c or altenatively, by the diameter w of its positive central 
region, which is related to o through the equation 
w= 2v20 (10) 
Scale space family generation and edge detection can thus 
be succesfully combined. By using the Gaussian kernel for 
scaling we ensure that in any scale level fewer edges occur 
than in finer resolutions and more than in coarser ones, thus 
performing proper scale space generation. This property has 
a qualitative aspect in addition to its obvious quantitative 
meaning. Edges detected in coarser levels using large c (or 
w) values will also appear in finer levels. The same edge can 
be traced through various resolutions, since its images dis- 
play a certain degree of geometric similarity, with the degree 
of localization (closeness to the true edge) increasing with 
resolution [Lu & Jain, 1989], [Witkin, 1983]. This is demon- 
strated in Fig. 4 and Fig. 5 which show edges of the original 
image (shown in Fig. 3) produced by its convolution with a