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ALGORITHM FOR CONSTRUCTION OF THICKNESS RASTER
A.E.Hmelnov1
1 Institute
for System Dynamics and Control Theory SB RAS, 664033 Irkutsk, Lermontov St.
134, (3952)-453-102, [email protected]
This report introduces the concept of thickness raster and presents an algorithm for its
construction. The thickness raster is constructed using the Euclidean distance raster of a
source image. Using the thickness raster the skeletons of uniform areas of the image
could also be found.
Introduction
The distance raster (distance map) is an image,
which contains in its pixels the distances from
the corresponding points to the nearest object
from a given set of objects [1]. A distance
transformation is an algorithm for construction
of the distance raster from a binary image
describing the set of objects.
As a result of distance map construction the
information about source image is assigned to
each pixel, which wasn’t locally available
before, i.e. each pixel receives information,
which didn’t contain in its fixed neighborhood.
By analyzing this information it is possible to
implement effectively different image
processing algorithms (see [1] for review of
applications of distance transformation).
This report introduces the concept of thickness
raster and presents an algorithm for its
construction. As a result of thickness raster
construction it becomes known for each pixel,
to how significant uniform image part it
belongs.
By analyzing the thickness raster it is possible
to detect the skeleton of the image areas [2],
and, in contrast to traditionally used for this
purpose morphological analysis [3], the most
significant skeleton points are naturally
detected here, as a points, which have some (or
enough) support in the thickness raster.
Problem statement
Let us consider the raster image I of the size
MxN (width M and height N). The expression
I[i,j] denotes the color of j-th pixel in the line i.
The distance raster Dt(I) is an image of the
same size as I, with the element Dt[i,j]
containing the distance to the nearest to the
point (i,j) pixel of I, having the color different
from I[i,j].
Fig. 1. An example of distance raster
It is possible to use various functions as a
distance when computing the distance raster.
Among them the Euclidean distance is
preferable for most part of problems, because
this distance is isotropic. Other distances are
used mostly to simplify computation of the
distance transform.
Let us call the thickness raster Wt(I) of a
binary image I the image, such that the value
of its pixel is the maximal radius of the circle
of constant color in I, which covers the pixel.
We can consider Dt(I) as an image, such that
the value of its pixel is the maximal radius of
the circle of constant color in I, centered in the
pixel. When defining Wt(I) we check not only
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the uniform circles centered in the point, but
also all the uniform circles, which cover the
image point by any their part. So the thickness
raster contains information about thickness of
the area of constant color in each point of the
image I.
A simple algorithm for construction of
thickness raster
Let us denote WI=Wt(I), DI=Dt(I). It is
possible to use very simple but inefficient
algorithm for construction of WI from DI,
which, meanwhile, allows to better understand
the definition of Wt(I):
1.
2.
3.
4.
Algorithm 1. Construction of thickness
raster by circle fill
Initialize WI[i,j] := 0;
For each image point p=(i,j) do:
For each image point q=(k,l), such that
D(p,q)<DI[i,j] do:
WI[k,l] := max(WI[k,l], DI[i,j]);
Thus the algorithm fills the circle of radius
DI[i,j] in each image point (i,j), using the
raster operation of the maximum of two values
(the value of the pixel and the value of the fill
color).
It should be noted that this algorithm computes
an approximation to the true thickness raster,
in which the best distance values found could
be less than the maximal circle radii, but at
most by 0.51/2 pixel, because the algorithm
considers only those uniform circles, which
are centered in some pixel (and could miss the
very best circles, with fractional centers). The
improved thickness transform algorithm,
which is discussed below, also constructs just
the same approximation to the true thickness
raster.
The time complexity of the simple algorithm is
O(M*N*(D2)m) , where (D2)m denotes the
mean squared circle radius (the mean squared
value of DI elements).
An improved algorithm for construction of
thickness raster
To construct a more effective algorithm we’ll
use the data structure, which represents the
borders of the circles around each pixel of DI the limit table. The limit table (LT) is an array,
which contains for each raster column the list
of active circle borders for already visited
pixels of DI.
The proposed algorithm works in two passes:
the first is top-down and from the left to right,
the second is bottom-up and from the right to
left. The same approach is used in the majority
of sequential algorithms for distance
transform.
Fig. 2. The order of raster processing
The two-pass mode allows to check the pixels
of a DI pixel influence area after visiting its
center. (Рис. 2), thus substantially simplifying
the algorithm structure. This way each circle
will be completely considered after completing
both passes.
Algorithm 2. A general structure of each
pass
1. For each image row i do:
2. For each image column j do:
3.
Add to LT the information about the
circle with radius DI[i,j] and center
(i,j).
4.
D := the first radius in the list of
borders of the column j in LT;
5.
WI[i,j] := max(WI[i,j], D);
6.
If the first border in the column j is
limited by i, then remove the border
from LT.
The information about circle borders is
represented by records TLimit, which contain
the following fields:
 Next – the pointer to the next border in the
list for the raster column;
 D – the radius of the circle;
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
L – the raster line, after which the border
goes.
Each list of column borders is ordered by
ascending of L (and, hence, by descending of D).
A
a
A
r
Fig. 3. Circle overlapping
To justify the last statement (that ordering the
list by ascending of L means its ordering by
descending of D) let us consider the
interaction of two circles, which form the
influence areas of the corresponding pixels of
DI (Fig. 3). The area of intersection of the
circles belongs to the influence area of the
largest of them. That’s why the influence area
of the smaller circle is the part of the circle
which emerges from the larger one. Because
the algorithm starts to consider each circle just
after visiting its center (Fig. 2), the interval,
which corresponds to any of the records in the
limit table (from the current raster line to the
circle border) is completely contained in the
circle. The intervals for all the previous
records in the list are also contained in the
circle for the limit considered, because they
have smaller values of L. So, once the
previous borders remain in the list, they should
have larger values of D, or else they would be
covered by the current limit.
Let us consider the most complex part of the
proposed algorithm – Step 3, which adds
circles to the limit table. Let us denote
R=DI[i,j].
Algorithm 3. Adding circle information to
the limit table
1. Find the range of columns j0…j1 influenced
by the circle;
2. For each k from j0 to j1 do:
3
Find the limit line l of the circle in the
column k
l := i+max {t| D((j,0),(k,t))<R};
In the list LT[j] skip all the borders,
having D>R (skip the stronger limits);
5
If there exists the stronger border with
Ll then Exit (the added circle is
covered by the existing ones in the
column);
6
Remove from LT[k] the weaker (D<R)
limits having Ll;
7
Add to LT[k] the record for the new
border of the level R after the line L or
merge with the existing record of the
same R and smaller L;
Let us make several remarks about effective of
implementation of the algorithm steps.
The simplest way to execute step 1 is to assign
the maximal possible values to the range
limits: j0 := j-R; j1 := j+R;
But it is possible to substantially reduce the
range by taking into account the values of
neighbor pixels.
Let DI[i,j] = R, and DI[i-1,j] = Q, Q>R, i.e. the
immediate neighbor pixel in the column has
larger distance to the objects. Let us move the
origin of coordinate system to the point (i,j).
The points (x,y) of intersection of the circles
should satisfy the equations:
x2+y2=R2 , x2+(y+1)2=Q2/
Hence
y=(Q2-R2-1)/2, x2= R2-((Q2-R2-1)2)/4.
The expression for x allows to contract the
range of column indices to consider.
Let us consider now the case when DI[i,j-1] =
Q, Q>R, i.e. the immediate neighbor in the
row has larger radius. By replicating the
previous considerations after swapping axes
we get: x=(Q2-R2-1)/2 . When Q>R and both
values are integer, the value of x0, and,
hence, j0 = j. In a similar manner we can
justify that when the right pixel (i,j+1) has
larger radius j1 = j.
The numbex of times required to execute all
the steps #3 of the loop is estimated by O(R),
because it is possible to compute the value of t
starting from the value found for the previous
step, and the limit value passes the range from
0 to R and back to 0 (at most 2R-1 values to
check).
The drawback of the algorithm considered is
that it requires to check from the very
4
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beginning for each loop step all the elements
of the lists LT[k]. To clear this drawback, we
should add some complexity to the algorithm,
by adding links between the elements of
neighbor lists, which immediately refer to the
items of the corresponding level. The record
TLimit is extended by adding the following
two fields:
 NextR – the limit in the next column of the
same or higher level;
 NextL – the back (to NextR) link, the limit
in the previous column of the same or
lower level.
After changing the data structure the step 4 of
the algorithm will be executed in constant time
for most steps of the loop by k.
The total number of borders removed from LT
on the step 6 is not higher than the number of
added borders, which is, in turn, limited by the
sum by all circles of the length of the column
range influenced by the circle (j1-j0+1)2R+1.
Skel(A) = {x | |Arg min yA D(x,y)|>1}.
Fig. 5. The skeleton points detected using thickness
raster
Each center of the circle, which forms the
thickness
raster
(has
some
pixels,
corresponding to them in WI), is a part of the
skeleton of the corresponding image area,
because the growth of each circle is limited by
at least three border points. To detect the
skeleton pixels it is required to collect all the
pixels of DI, which form WI.
Conclusion
This report introduces the concept of thickness
raster of binary image and presents an
algorithm for its construction. Computation of
thickness raster can be used as a constituent
part of complex image analysis and processing
algorithms.
Fig. 4. An example of thickness raster (for the distance
raster from Fig. 1).
Thus, by considering the total time required to
execute all the steps of the loop, it may be
inferred that the time complexity of the
proposed algorithm is estimated by
O(M*N*(D)m) , where (D)m denotes the mean
value of circle radius (mean value of the
element of DI).
The skeletons of the image areas
The skeleton of an area A is the geometric
locus of points, which have more than one
projection to the border of A.
References
1. O. Cuisenaire. Distance transformations: fast
algorithms and applications to medical image
processing // Ph.D. Thesis, UCL, Louvain-la-Neuve,
Belgium October 1999.
2. M. van Eede, D. Macrini, A. Telea, C.
Sminchisescu, S. Dickinson. Canonical Skeletons
for Shape Matching // Proceedings, International
Conference on Pattern Recognition (ICPR), Hong
Kong, August, 2006.
3. Zhanrong Li, Jianqing Zhang. Image Segmentation
Based on Inscribed circle // 18th International
Conference on Pattern Recognition (ICPR'06) pp.
247-250.