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Even though on the finer scale more edges are detected which 
disappear at coarse levels, it is to observe that in tracking from 
coarse to fine contour elements merge together. 
In the next sections we want to deepen insight mainly from an 
experimental point of view. 
2 EDGE DETECTION IN MULTISCALE IMAGES 
The surface obtained by convolving the image with a Gaussian 
kernel with varying scale parameter is called the scale space 
image. In practice only a series of a finite number of multiscale 
representations can be generated. For our experiments we there- 
fore approximate the Gaussian kernel by binomial kernels which 
are recursively obtained by convolution with dá 1). 
The interest operator proposed by Förstner for point detection is 
described in detail in (Förstner and Gülch 1987). In the two di- 
mensional formulation the operator is designed to detect, classi- 
fy and precisely locate corners, circular features and other iso- 
trop textures. The 2x2 matrix (e.q. 1), written in terms of con- 
volutions and matrix multiplication, is the so-called normal 
equation matrix used within the point location step. In the one- 
dimensional case this matrix reduces to a scalar quantity and the 
point operator is just an edge operator. Denoting the signal with 
f(x) and the scale space image generated from the signal by 
convolution with a Gaussian kernel by according to 
g(x,0)= fix) * G,, 
the formalism for edge location with the 1D-interest operator 
reads as 
v A 2 
@ * g,) X= px * 8, (2) 
As outlined above p can also be a Gaussian, i.e. p- Gu The 
scale o, stands for the width of the convolution window and by 
this also for the window size. Varying 0, as investigated by 
Heikkilä (see above) produces scale space of the squared gra- 
dient image. In this investigation we keep o, and chooseo, 
throughout all the experiments. In the discrete approximation of 
the Gaussian kernel by a binomial kernel this fits to a window 
size of 5 pixels. The second operator in (2) denoted by px can 
be computed according to px(i)- p(i) + x(i) for all pixels i 
within the window. For convenience the coordinates x are cen- 
tered to the mean of the window, so that px is symmetric and 
shift invariant. Thus £ gives a field of edge positions, in which 
for each pixel the corresponding edge location is estimated. The 
quantity w- p * gl also found for each pixel is sometimes 
called interest value or weight or response of the window opera- 
tor. It measures the roughness of sale space image. In the expe- 
riments presented below the roughness w and edge locations 
x*£ are computed for all pixels x of the multiscale image in a 
series of discrete scale levels. For calculation of the gradients 
the image is convolved with (-1 1). 
The edge detection scheme is based on the roughness image. 
The local maxima in w are considered to indicate edges. The 
window size for extracting local maxima (or suppressing non- 
297 
maxima) is chosen in accordance with size of the convolving 
operator p, i.e. in the experiments also 5 pixel are used for the 
non-maxima window size. Though a larger non-maxima window 
may suggests that the remaining edges are more distinct, in our 
opinion this is not recommendable. A quite not rare observation 
in scale space is that with increasing scale parameter the weight 
of a local maxima may decrease considerably faster than that of 
a neighboring edge. One common aspect in the strategies behind 
edge focusing (Bergholm, 1987) or RESS (Lu and Jain, 1992), 
but also the work of Lindeberg and Eklundh (1990) on scale 
space blobs is that those features are of interest which achieve 
high weights on all levels or on the coarser scales. 
The rest of the paper is devoted to experiments. Even the first 
experiences with simulated and real image data have shown that 
for example the very nice rules for reasoning in the linear LoG 
scale space presented by Lu and Jain can not simply be applied 
to the nonlinear edge detector (eq. 2). The characteristics of the 
zero crossings in the LoG scale space we have outlined above. 
Moreover a further property we have to include which occurs 
just in the case of nonlinear operator: new edges may be gene- 
rated as the scale parameter increases (Yuille and Poggio, 
1986). 
3 TRACKING IN A SYNTHETIC IMAGE 
In this section we would like to illustrate the tracking problem 
at the example of an idealized synthetic image. The signal 
shown together with the multiscale representation in fig. 3.1 has 
just two intensity levels. It consists of a series of pulses whose 
width is chosen randomly. The number of pixels plotted is 
approximate 200. 
  
  
1 1 ' 
0.0 50.0 100.0 150.0 200.0 
pixel 
Figure 3.1 Multiscale image with 23 levels and step size ao= 1 
The following three figures show the roughness of the signal in 
scale space. The logarithm In w(x,0) is plotted. The partitioning 
of the scale range (o= 0-23) in three parts mainly intents to im- 
prove the visual impression from details of the plots. The global