International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
1983). The underlying idea is that random brightness dis- 
tributions posses self-similar characteristics that are anal- 
ogous to those of classical fractal sets (Cantor set, Sier- 
pinsky gasket, and so on). However, self-similarity mea- 
surements cannot be applied in a strict sense since a ran- 
dom distribution is by definition not algorithmic. How- 
ever, the relation between random distributions and frac- 
tals is very significant (Peitgen and Saupe, 1986, Mandel- 
brot and van Ness, 1968). The box-counting dimension is 
perhaps the best trade off between accuracy and computa- 
tional complexity. In most cases this fractal dimension es- 
timator coincides with self-similarity measurements, and 
even though an absolute measurement may be inaccurate, 
our interest in feature segmentation by relative comparison 
of fractal dimension is not compromised. 
Box-counting dimension is based on counting the pixels 
visited by the set under measurement in a grid of varying 
resolution and position. (This assumes a previously bina- 
rized image under an adequate criterion.) Let s be the size 
ofthe side ofa square cell in the grid, and N (s) the average 
of the cells visited by the set under measurement under dif- 
ferent translations. Then it is expected that increasing the 
resolution of the grid (which in fact decreases the side s of 
the cells of the grid in a fixed size image), N (is) should also 
increase. The slope of this relation in logarithmic space is 
the box-counting dimension of the set: 
(N(s 
»—0  log(l) e 
Accurate estimations should show a value of D that is sta- 
ble and steady along several orders of magnitude. In exper- 
imental settings this is obviously impossible to grant, and 
therefore the values of D are taken as provisory. 
Starting with this schema, we can find local estimators 
that are adequate for image segmentation. The idea is to 
adapt the box-counting dimension to a local context, in 
which only a sub-image centered around a given pixel is 
considered. Larger sub-images produce better measure- 
ments, but also with higher computational cost. In this 
work we used a grid of 5 x 5 pixels centered around the 
pixel under estimation, which showed experimentally to 
be a good compromise. Consider for instance the syn- 
thetic image, under a statistical model, with three different 
areas and a background which was generated simulating 
speckle noise in Figure 1(a) and its brightness histogram 
(Figure 1(b)). Any segmentation produced by simple bi- 
narization will be of little use. In Figure l(c) we show a 
binarization under bayesian classification (marked with a 
circle in Figure 1(b)). However, the box-counting dimen- 
sion performed on the image is remarkably good for con- 
tour extraction (see Figure 1(d)). 
3 B-SPLINE REPRESENTATION 
A brief theoretical review of B-spline representation of con- 
tours is presented; for more details see (Blake and Isard, A set of k data points in the image plane is given by[Do. Di, Dii 
where D; = (x:.9y:)®, í — 0,..., k — 1, and the spline 
1998, Rogers and Adams, 1990). 
1160 
  
  
  
(a) (b) 
  
(9 (d) 
Figure 1: (a) Synthetic image under a statistical model with three 
different areas and a background which was generated simulating 
speckle noise. (b) Its brightness histogram. (c) Binarization un- 
der bayesian classification. (d) The box-counting dimension. 
Let {Qo, ..., Qn, —1} be a set of control points, where Q,, = 
(Tn, Yn)’ = R?, 0 € n € Ng — 1, and let {So «Si € 
s9 < +++ < 81-1} C R be a set of L knots. A B-spline 
curve of order d is defined as a weighted sum of N p poly- 
nomial basis functions B,, 4(5) of degree d — 1, within each 
interval [5;, 5;,1] with 0 € i € L — 1. The constructed 
spline function is r(s) = (x(s),y(s))’, 0 < s < L — 1, 
being 
Np-1 
re) =i SOBs, 
n=0 
and 
fs} BE (2) 
y(s) = B'(s)G” (3) 
where the basis functions vector B(s) of N 3 components 
is given by B(s) — (Boa(s), ..., By, 14(s))*. The weight 
vectors Q* and Q" give the first and second components of 
the Q,,, respectively. 
The curves used in this work are closed of order d = 3 or 
d = 4 specified by periodic B-spline basis functions. 
3.1 B-spline curve fit 
The problem of determining a polygon that generates a fit- 
ting B-spline curve with known number of control points, 
Np, was studied by (Rogers and Adams, 1990). We now 
present a brief review of this subject. 
   
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