ul 2004 International Archives of the Photogrammetry, Remote Sensing
curve that best-fits them is sought. Then, by equations 2
and 3, the components D; must satisfy
zc BO,
3, B'(f;)QV,
for certain values of t;, where à = 0,...,k— 1, and Np <
k.
This linear system is more compactly written in matrix
formas D — K(Q* QV), where the k x Np elements
of the real matrix Æ are given by K;; = Bat). with: =
In the most general case Np < k and, therefore, K 1s
not a square matrix. In this case, the pseudo-inverse ma-
trix form (Q^ QV) — K* D is used to find the B-spline
fitting curve. A useful set of values for the parameters
(fo, ..., fy1] is given by
gp - Dol pr
Se) ID; im Dii | =
iz]
to = 0, : Le m
The knot set to build the B-spline basis functions is arbi-
th three trarily chosen.
nulating
tion un-
ion. 4 BOUNDARY DETECTION
eo. =
e 5 In this section we describe the algorithms developed for
EN boundary detection once the noise was removed.
3 >
cach 4.1 Radial lines algorithms for boundary detection
tructed
deel, Let E be a scene made up by the background D and a set
of regions {R1, Ri», .... Ry} with their respective bound-
aries TOF, .... OR4}. For each region R;, we want to find
the curve C; that fits boundary OR; in the image. In the
first step a box-counting fractal dimension estimation is ap-
plied in order to remove the noise. We define an initial
(2) search area, which are specified by polygons, the vertexes
of which are control points that generate a B-spline curve,
(3) as shown on Figure 2 with a thin line. Once the initial
search zones are determined the centroid of each of them
onents
is calculated.
e weight
ents of
33 0f
s a fit-
Joints,
e now
and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
0... k—1,j 2 0,..., Npg—1, and D — (Do, Di,.. ny.
‚Di Dau-1}
ine
Figure 2: Initial areas of interest determinated by polygons, the
vertexes of which are control points that generate a B-spline curve
1161
If a point belongs to the object boundary, then a sample
taken from the neighborhood of that point exhibits a change
in the intensity level of the box-counting image and it is
considered to be a transition point. Then N segments s),
i € {1,..., N) with the form s(0 — CP, are considered.
Here C' is the centroid of the initial region, the extreme P;
is a point outside of the region and 0 — ang (se) se)
Vi is the angle between two consecutive segments, as shown
on Figure 3.
Figure 3: Radial straight lines s‘) ; = 1,., N projected from
the centroid C of an initial curve to the external part of the region.
Q9 is the angle between s? and seit} à = dat 7 À
The segment s(? is an array of m elements coming from a
discretization of the straight line on the image. The border
point b; is found convolving the data of the segment with a
mask given by [-2, —1, 0, 1, 2]. After find {b,,… b V, the
algorithm build the B-spline curve interpolating the border
points.
4.2 Possible Problems
This algorithm gives very good results when it is applied
to convex objects, but there are some situations where this
method could fail. For example, it is possible that some
point on the object boundary be obstructed by other point
of the same boundary, so the radial straight lines will not
reach this point, making it impossible to find it, as shown
on Figure 4.
Figure 4: A non convex object. The procedure fails on the
straight line segment shown on this figure. It will not detect the
border points marked with a circle.
In order to solve these problems we have modified the al-
gorithm, using an estimation of the curves’s derivative at
step n to predict the center of the segment sg (n) at step
n + 1. This modified version calculates the velocity vec-
tor between two given boundary points b; and b;4, in the
