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Binary Image Skeleton - Continuous Approach.

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In this paper we propose a building technique of a correct model of continuous skeleton for discrete binary image. Our approach is based on approximation of each connected object in an image by a polygonal figure. Figure boundary consists of closed paths of minimal perimeter which separate points of foreground and background. Figure skeleton is constructed as a locus of centers of maximal inscribed circles. In order to build a so-called skeletal base from figure skeleton, we cut unnecessary noise from it. It is shown, that the constructed continuous skeleton exists and is unique for each binary image. This continuous skeleton has the following advantages: it has a strict mathematical description, it is stable to noise, and it also has broad capabilities of form transformations and shape comparison of objects. The proposed approach gives a substantial advantage in the speed of skeleton construction in comparison with various discrete methods, including those in which parallel calculations are used. This advantage is demonstrated on real images of big size.
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BINARY IMAGE SKELETON
Continuous Approach
Leonid Mestetskiy
Department of Mathematical Methods of Forecasting, Moscow State University, Moscow, Russia
l.mest@ru.net
Andrey Semenov
Department of Information Technologies, Tver State University, Tver, Russia
semenov@tversu.ru
Keywords: Binary image, continuous skeleton, discrete skeleton, polygonal figure, pruning, skeletal base.
Abstract: In this paper we propose a building technique of a correct model of continuous skeleton for discrete binary
image. Our approach is based on approximation of each connected object in an image by a polygonal figure.
Figure boundary consists of closed paths of minimal perimeter which separate points of foreground and
background. Figure skeleton is constructed as a locus of centers of maximal inscribed circles. In order to
build a so-called skeletal base from figure skeleton, we cut unnecessary noise from it. It is shown, that the
constructed continuous skeleton exists and is unique for each binary image. This continuous skeleton has
the following advantages: it has a strict mathematical description, it is stable to noise, and it also has broad
capabilities of form transformations and shape comparison of objects. The proposed approach gives a
substantial advantage in the speed of skeleton construction in comparison with various discrete methods,
including those in which parallel calculations are used. This advantage is demonstrated on real images of
big size.
1 INTRODUCTION
Mathematical concept of a skeleton has been
formulated initially only for continuous objects
(Blum, 1967). A skeleton of a closed region on
Euclidean plane is defined as a set of centers of
maximal empty disks. A disk is empty if each
internal point of it is also internal point of the region.
In order to use the concept of a skeleton as a
research tool of image shape in digital images, one
needs to extend this concept to discrete space.
However, in spite of seeming simplicity, it is not
possible to extend this definition to discrete images
immediately (Smith, 1987; Ogniewicz and Kubler,
1995; Bai et al., 2007). Efficient algorithms of
continuous skeleton construction are known only for
polygonal regions (Lee, 1982; Fortune, 1987; Yap,
1987; Klein and Lingas, 1995). However, for exact
polygonal approximation of discrete form
boundaries, one needs to use many small rectilinear
segments. This leads to an increase in the number of
vertices of approximating polygons. But the more
vertices there are in polygons, the more noisy
branches of skeleton are generated. And these
branches are not important for an analysis of image
shape.
Since it is impossible to use continuous
skeleton for image analysis, «discrete skeleton», an
analogue of continuous skeleton, is constructed for
these purposes. A discrete skeleton is usually
defined as a binary image derived by a certain
transformation of the initial image. The skeleton
consists of pixel-wide lines and all of these lines are
approximately equidistance from the boundary of
the initial object. There exist several approaches of
construction of such transformation: topological
thinning, morphological erosion and allocation from
a distance map (Costa and Cesar, 2001). However,
discrete skeletons obtained by these methods have
essential disadvantages in comparison with their
continuous analogues. In methods of topological
thinning and morphological erosion the Euclidian
metric is lost. Skeletonization methods by distance
map cause loss of skeleton connection. In addition
presentation of skeletons as binary images
complicates their comparison. It is also impossible
251
to transform image shape on the basis of a discrete
skeleton.
Another approach, proposed in (Ogniewicz and
Kubler, 1995), uses a subgraph of the Voronoi
diagram of object boundary points as a skeleton of a
discrete object (Fig.1(a-d)). This subgraph is
extracted from the Voronoi diagram on basis of
regularization procedure. Therefore, the resulting
continuous skeleton is a planar linear graph. Since it
is continuous and not discrete, it suits much better
for image shape transformation and comparison. The
disadvantage of the resulting skeleton is that its
branches are often zigzag. This disadvantage
becomes especially pronounced for images of low
resolution (Fig.1(d)). In addition, when this method
is applied to a complex image of high resolution,
which also has regular elements (for example, to a
drawing with rectilinear fragments), a big number of
"redundant" boundary pixels becomes involved in
processing, which leads to an unnecessary increase
in the dimension of Voronoi diagram and the total
amount of calculations.
Figure 1: (a) – the binary image, (b) – the boundary points,
(c) – the Voronoi diagram of boundary points (only finite
edges), (d) – regularization of the Voronoi diagram, (e) –
the approximating polygonal figure, (f) – the figure
skeleton, (g) – the skeleton regularization, (h) – the radius
function of skeleton.
Not only the quality of a skeleton constructed
by a specific algorithm is important. The speed with
which this algorithm works is very important in
computer vision systems. Currently speed
enhancement is usually achieved by the
development algorithms of parallel discrete
skeletonization (Manzanera et al., 1999; Deng et al.,
2000; Strzodka and Telea, 2004). However this
acceleration has its limits since there remain
sequential steps in discrete skeleton construction
algorithms and the number of these steps increases
with the growth of image size. Image size, in turn,
increases steadily as resolution of cameras and
scanners increases.
In reality, the time necessary for skeletonization
of big images even on modern computers is still too
big for many applications.
Therefore, the issue of extension of the concept
of continuous skeletons on discrete images seems far
from being resolved. The purpose of this paper is to
describe a continuous approach to skeletonization of
binary images (Fig.1(e-h)) developed by the authors
and its application to real-world problems
(Mestetskiy, 1998, 2000, 2006). The advantages of
the proposed method are also demonstrted in the
paper. The main advantages include superiority in
computer efficiency.
2 DISCRETE FIGURE AND ITS
SKELETON
We will define a skeleton of a discrete image on the
basis of the following concepts:
- а discrete figure;
- аn approximating minimal perimeter polygonal
figure;
- а continuous skeleton of a polygonal figure;
- а skeletal base of a polygonal figure.
2.1 Discrete Figures in Binary Image
A binary image is a two-colored picture where one
or several objects of one color are located on a
background, which has another color. Without loss
of generality, we will consider a binary image as a
black-and-white image: object is black, and
background is white. Such image is represented in a
computer as a matrix of black and white pixels.
Let us define an adjacency structure on a set of
pixels as follows. For a black pair of pixels we will
define neighborhood as 8-adjacency, and for a white
pair and a two-colored pair – as 4-adjacency.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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A set of one-colored pixels is called connected
if for each pair of pixels in it there is a path from one
pixel to another, consisting of sequentially
neighboring pixels of the same color. Maximal
connected set of pixels of one color is called a
connected component. If all pixels of a component
lie on the same straight line, such a component is
called degenerated. Let us define discrete figures as
connected black-colored components. There are 5
connected components in the image in Fig.2(a), two
of them are discrete figures.
2.2 The Continuous Approximation of
Discrete Figure
Let us regard pixels as points with integer co-
ordinates on Euclidean plane.
We will call a pair of 4-adjacent two-colored
points a boundary pair, a segment connecting these
points – a boundary segment. Two components to
which points of a boundary pair belong are called
adjacent, and the boundary pair is called dividing for
these components. The set of all dividing boundary
pairs for two adjacent components we will name a
boundary corridor (Fig.2(b)). Each discrete figure
defines one or more boundary corridors.
Let us say, that a closed path lies in a boundary
corridor if it crosses all boundary segments of this
corridor. We will consider that a path crosses a
segment if it has a common point with it and lies on
different sides from this segment in some
neighborhood of the intersection point. There will
exist a minimal length path in the set of paths lying
in a boundary corridor. This path will be a closed
polyline and we will call it a separating minimal
perimeter polygon (MPP). If a discrete figure and all
its holes are not degenerated then all its MPP are
simple polygons (Fig.2(c)). For a degenerated figure
or a degenerated hole MPP degenerates in a line
segment. The set of all MPP of a discrete figure
defines a polygonal figure – «a polygon with
polygonal holes».
Thus, we have defined minimal perimeter
polygonal figures that approximate discrete figures
of binary image. It is important to note that the set of
approximating polygonal figures always exists and
is unique for a given binary image.
2.3 Polygonal Figure Skeleton
Degenerated disks of zero radiuses centered in the
convex vertices of a polygonal figure are empty as
they have no internal points and, therefore, don’t
contain boundary points of a figure. Besides, they
are the maximal empty disks since they don’t
contain other empty disks. Therefore points, which
coincide with convex vertices of a polygonal figure,
belong to a polygonal figure skeleton.
(a)
(b)
(c)
Figure 2: (a) – the binary image with 5 components and 2
discrete regions, (b) – boundary corridors, (c) – minimal
perimeter polygons.
A polygonal figure skeleton is a planar graph
with edges consisting of line segments and parabolas
(Lee, 1982). The vertices of this skeleton are
comprised from the convex vertices of a polygonal
figure (one degree vertices) and from the points,
which are centres of the inscribed circles, tangent to
figure boundary in three or more points (three and
more degree vertices). The radial function is defined
in each skeleton point as the radius of an inscribed
circle centered in this point.
It is important to underline that a polygonal
figure skeleton always exists and is unique.
2.4 Polygonal Figure Skeletal Base
The problem of “noise” branches exists for both
continuous and discrete skeletons. Small
BINARY IMAGE SKELETON - Continuous Approach
253
irregularities in figure boundary lead to occurrence
of skeleton branches, unessential for analysis of
image form. The task of skeleton regularization is to
remove these branches and leave only fundamental
part of the skeleton which at the same time
characterizes properties of the shape. This
fundamental part looks like a skeleton subgraph. We
will name it a skeletal base. Since the transformation
of a skeleton into a skeletal base is achieved by the
removal of unessential vertices and edges, this
process is called pruning.
Let C be a polygonal figure. Let us call its
boundary C, its skeleton – S and its skeleton radial
function –
ρ
(s), sS. The skeleton will be a planar
graph ),( EPS = with the set of vertices P and
edges E. We will call a skeleton vertex with one
incident edge terminal, and with two or more edges
– internal. An edge incident to a terminal vertex is
also called terminal; an edge incident to two internal
vertices is called linking. Linking edges can enter in
one or more cycles and in this case they are called
cyclic.
Pruning is a consecutive removal of some
terminal vertices and skeleton edges incidental to
them. In the process of pruning, degree of some
vertices changes. In particular, internal vertex can
become terminal or its degree can become 2.
Pruning guarantees preservation of skeleton
connectivity and also preservation of all cycles in a
skeleton as it doesn’t touch cyclic edges.
Let us consider an assessment criterion of
“essentiality” of a terminal edge. Essential edges
remain in a skeletel base, and unessential edges are
cut.
Let ),( EPS
=
be some adjacent subgraph
of a skeleton ),( EPS =, such that PP
,
EE
and also such that in the set EE
\ there
are no cyclic edges of skeleton. This means, that
graph S can be obtained from skeleton S by the
removal (“pruning”) of some vertices and edges of
skeleton, and such removal doesn’t destroy cycles
and doesn’t break connectivity of the graph. This
graph S we will call truncated subgraph of S. We
will consider the set of points formed by union of all
inscribed circles, centered in points of the truncated
subgraph S, whose radiuses are equal
ρ
(s), sS
.
This set of points forms a closed region which we
will call a silhouette of subgraph S. The important
property of a truncated subgraph silhouette is its
topological equivalence to figure C. In particular, a
silhouette is a connected set.
Let a skeletal base of figure C be the minimal
truncated subgraph S
of its skeleton S with
silhouette S
V satisfying a condition
ε
),( S
VCH , where 0>
ε
is a regularizing
parameter, and ),( S
VCH – Hausdorff distance
between figure C and silhouette S
V.
It is necessary to note, that for each value of
parameter
ε
the skeletal base always exists and is
unique.
We will call the derived skeletal base a
continuous skeleton of a discrete figure.
3 ALGORITHMS
3.1 Boundary Corridor
The construction of a boundary corridor consists of
two stages: the corridor search and its tracing.
Corridor search is understood as a problem of
finding one boundary pair of points (Fig.3(a)).
Search of such pair can be executed by row scanning
of the binary image. After finding the boundary pair,
the boundary tracing algorithm will work. This
algorithm reveals all other boundary pairs of a
corridor. After corridor tracing is finished, the
algorithm starts the search of next corridor from that
location where the first boundary pair of the
previous corridor has been found. The process ends,
when the single line scanning ends.
Figure 3: Corridor tracing: (a) the initial position of tracer
pair, (b) the consecutive positions of tracer pair, (c) the
sequence of test points (tracing track).
The algorithm starts contour tracing from the
first boundary pair and finds sequentially other
boundary pairs of a corridor. A boundary pair of
points currently found by the algorithm we will call
a tracer pair. Tracing process corresponds to the
consecutive movement of the white end of the tracer
pair in a positive direction relative to the black end
(Fig.4). The derived point is called a test point. All
possible variants of test point choice at different
positions of the tracer pair are presented on Fig.4.
(a) (b) (c)
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Current position of the tracer pair is shown by a
solid line, and new possible positions depending on
color of a test point – by a dotted line.
A new position of a tracer pair is determined
from the color of a test point by the following rule.
The test point replaces in the tracer pair a point of
the same color as it is.
Consecutive moving of a tracer pair allows to
single out all boundary points corresponding to one
boundary contour (Fig.3). Tracing process ends
when tracer pair will return to its initial position.
3.2 Minimal Perimeter Polygon
The sequence of test points forms an ordered list
called a tracing track (Fig.3(c)). We will attain
"walls" of a boundary corridor by sequentially
connecting all black points of this list among
themselves and all white points. The left wall
consists of black points, and the right one – of white
points. The minimal perimeter polygon lies between
the corridor walls. All vertices of MPP are points of
a tracing track. We will call such points a corner.
The task of MPP construction is to choose corner
points from a tracing track.
The first corner point is defined from the initial
position of the tracer pair (Fig.3(a)). It is obvious,
that the right point of this pair is always corner. Let
us note, that the two consecutive vertices in MPP
should be connected by line segment lying between
corridor walls completely. It means, that if another
(in particular, the first) corner point is found, it is
necessary to search for the next corner point as for a
point lying from it «in the line of sight» inside a
corridor.
Let us define a concept of a «coverage sector»
for a corner point. At the initial moment (for the
current found corner point) it equals 360 ° and isn’t
limited by anything. As the algorithm proceeds, the
points of the track after this corner point are
sequentially considered and the coverage sector is
modified by following rules (Fig.5):
1. If the test point is located inside the coverage
sector, the sector changes (Fig.5(a,b)). If the test
point is black (Fig.5(a)), it is declared as the left side
of the sector, if it is white (Fig.5(b)) – as the right
one.
2. If the white point is located outside the
coverage sector to the left of its left side (Fig.5(c)),
the left black point of the sector is declared the new
corner point. Similarly, if the black point is located
outside the sector to the right of its right side
(Fig.5(d)), the right white point is declared the new
corner point.
3. In all other cases (Fig.5(e,f)) the coverage
sector doesn’t change.
As these rules are followed, all corner points
are sequentially found. In the process, the corridor
track is regarded as a circular list of points. The
process ends when the initial corner point is chosen
as a new corner point (but not as a test point!) once
again.
3.3 The Construction of Skeletons
Fast algorithms for skeleton construction of simple
polygons with n vertices have computational
complexity O (n log n) (Lee, 1982) and O (n) (Klein
and Lingas, 1995). Known generalizations to the
case of a polygonal figure with holes (Srinivasan et
al., 1992, Lagno and Sobolev, 2001) have
computational complexity O (kn + n log n), where k
is the number of polygonal holes and n is the general
number of vertices. For some problems such
computational complexity takes too much. For
example, in the task of construction of an external
skeleton for segmentation of the text document
(c) (d)
(a) (b)
(
e
)
(
f
)
Figure 4: Choice of the next test point (labeled as square)
for different positions of tracer pair (solid line) during
tracing process of boundary corridor.
Figure 5: (a,b) correction of coverage sector, (c,d) new
angular point forming, (e,f) coverage sector doesn’
change.
BINARY IMAGE SKELETON - Continuous Approach
255
image (Mestetskiy, 2006) values k and n have an
order 103 and 105 accordingly. At the same time,
efficient algorithms for Voronoi diagram
construction of linear segment set (Fortune, 1987;
Yap, 1987) don’t use specific features of segment set
of polygonal figure boundary because of their
universality. In particular, these algorithms build
Voronoi partitioning not only inside, but also outside
of a polygonal figure and this is superfluous work.
Our solution is based on the concept of
adjacency of polygonal figure boundary contours
and on the construction of so-called adjacency tree
of these contours.
Figure 6: Figure boundary adjacency tree construction: (a)
the polygonal figure and intercontour circles, (b) the
boundary adjacency graph, (c) the boundary adjacency
tree, (d) transforming of the figure to the polygon.
Two boundary polygons are adjacent if the
circle inscribed into a figure, which contacts both of
these polygons exist. The given relation of contour
adjacency defines a graph of contour adjacency. It is
obvious, that this graph is connected. Each spanning
set of it (the minimal connected spanning subgraph)
is a tree. Such tree we will call a polygonal figure
boundary adjacency tree. In figure 6a the image with
12 boundary contours is presented. Inscribed circles,
contacting pairs of contours, show the adjacency
relation. In Fig.6(b) the polygonal figure boundary
adjacency is shown, and in Fig.6(c) one of the
boundary adjacency trees is presented.
The boundary adjacency tree gives the chance
to reduce a problem of a polygonal figure
skeletonization to a problem of a simple polygon
skeletonization. For this purpose let us transform
chains of polygon sides into one chain by «cutting-
in» them into one another. As a result the polygonal
figure conditionally transforms to "polygon"
(Fig.6(d)). An O (n log n) sweepline algorithm for
boundary adjacency tree finding and figure skeleton
construction is described in (Mestetskiy, 2006).
3.4 Skeletal Base
It is possible to present the process of a skeletal base
construction as a construction of a sequence of
truncated subgraphs of skeleton
{}
m
S. Here
SS
=
0, mm SS
+1, m=0,…,M, and for all m
S
the following condition is satisfied:
ε
),( m
S
VCH . The last element of this sequence
M
S
is the required skeletal base. According to our
definition of a skeletal base, for each truncated
subgraph M
SS
condition
ε
>
),( S
VCH
takes place or there are no terminal edges in M
S
.
The described process is illustrated by an example in
Fig.7. Here 2
=
ε
.
Figure 7: Skeletal base construction: (a) the initial image,
(b) the polygonal figure and its skeleton, (c,d,e) the
skeleton subgraphs and their silhouettes.
Computational complexity of this algorithm
depends on the number of skeleton vertices linearly,
i.e. it is at worst O(n), where n is the number of
polygonal figure vertices.
4 EXPERIMENTS
The described method of continuous skeleton
construction of a binary image has been
implemented and has passed multiple checks in
different applications.
Theoretical estimates of computational complexity
of algorithms, with all their importance, don’t give
full conception about the possible application of
algorithms in computer vision systems. Therefore
there is a necessity to perform
(
a
)
(
b
)
(
c
)
(
d
)
(
e
)
(c) (d)
(a)
(b)
0
1
2
3 4
5
6
7
8 9
10
11
12
1
2
11
10
9
8
7
4
3
0
12 6 5
6
5
10
1
2
11
9
7
3
0
12
8
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256
Figure 8: Test examples: (a) Billygoat, (b) leaf1, (c) room,
(d) neuron, (e) roots.
experimental estimates based on real working
algorithms and on practical examples. There is not
many publications describing such experiments.
Usually there is no information about software
implementation and algorithm running time at all
(Manzanera et al., 1999) or there are only results of
computing experiments with "toy" examples of very
simple images (Deng et al., 2000).
The most difficult examples (Fig.8) of images
and real time expenses for their skeletonization are
presented in works (Ogniewicz and Kubler, 1995;
Strzodka and Telea, 2004).
Table 1: Comparison of our algorithm CS and algorithm
OK (Ogniewicz and Kubler, 1995).
OK CS OK/CS
sites 11104 1874 5.92
edges 31381 3721 8.43
vertices 20303 3730 5.44
time 9.82 0.05 196.4
Results of comparison of our algorithm with the
algorithms described in these works, are given in
tables 1 and 2. Quality of the derived continuous
skeletons is shown on examples in Fig. 9, 10.
The running time of our algorithm was
estimated using Intel processor 1.6 GHertz with 512
Mb of memory. Time in tables is specified in
seconds.
Comparison with algorithm (Ogniewicz,
Kubler, 1995) shows, that using MPP for image
boundary approximation allows to reduce dimension
of the problem substantially: number of elements in
a polygonal figure skeleton is about 6-8 times less
than in a corresponding Voronoi diagram of image
boundary points. The reduction in computation time
(in 196 times) is partially due to this dimension
reduction, and partially due to processors capacity
increase as compared with SPARCstation-2.
Table 2: Comparison of our algorithm CS and algorithm
ST (Strzodka and Telea, 2004).
Size ST CS ST/CS
Leaf1 410×440=182040 0.14 0.02 70
Room 413×506=208978 0.64 0.03 21
Neuron 839×731=613309 2.5 0.1 25
Roots 1800×1810=
3258000
3.79 0.41 9.22
A new parallel discrete skeletonization algorithm
is described in Strzodka and Telea (2004). Authors
show that the running time of this algorithm is a
record for discrete algorithms so far. The table
shows the results attained by the authors on GPU
GeForce FX 5800 Ultra chip, containing tens
independent computers working in a parallel mode.
However, it is apparent from table 2, the running
time of our algorithm is less than of that algorithm
by 1-2 orders. It is necessary to note, that our
algorithm can be parallelized too and its operation
speed on multicore processors will grow.
5 CONCLUSIONS
The continuous approach to image skeleton
construction exceeds in many criteria traditionally
applied discrete methods.
1. The continuous skeleton is described by the
strict mathematical model. The discrete skeleton
doesn’t have such strict description; it is validated
only as an analogue of a continuous skeleton.
2. Regularization of continuous skeletons,
directed on noise overcoming, can be performed by
strict mathematical methods; and as for discrete
skeletons, it is done on the basis of heuristic devises.
3. The continuous skeleton with the radius
function gives more ample opportunities on shape
transformations of an object. Comparison of
continuous skeletons is reduced to a problem of
planar graphs comparison by topological and metric
criteria.
(a)
(e)
(b)
(c)
(d)
BINARY IMAGE SKELETON - Continuous Approach
257
Figure 9: Continuous skeletons: (a) leaf1, (b) room,
(c) Billygoat (external), (d) Billygoat (internal).
Figure 10: The fragment of the skeleton for “neuron”.
4. Running time of continuous skeletonization
algorithm is less by at least an order than that of the
best samples of discrete skeletonization algorithms.
The downside of application of continuous
skeleton construction algorithm is the complexity of
its software implementation which demands rather
refined programming technique.
ACKNOWLEDGEMENTS
The authors thank Dr. R.Strzodka who has granted
us image samples for experiments. Also authors are
grateful to the Russian Foundation of Basic
Research, which has supported this work (grant 05-
01-00542).
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... To improve the algorithm, the coefficient of asymmetry was determined between the left and right regions of the worm's body, into which the medial axis divides the shapes. Note that since we are essentially dealing with the search for an extended curvilinear axis of symmetry of the worm, it is natural to use the model of continuous skeleton (or, synonymously, medial axis) of binary image (Mestetskiy and Semenov, 2008). The skeleton consists of lines equidistant from two or more sections of the boundary (all solid lines in Fig. 3a) and contains the required axis as a subgraph. ...
... Hence, we employ the fact that GVDs can be obtained based on the medial axis transform [46]. Accordingly, we use the morphological operation of "skeletonization" as in [47], [48] to determine an approximation of the VB that consists of the image pixels on the medial axis with an equal minimum distance to obstacle pixels (see (7)). Then, BPs are determined as image pixels that are connected to at least three segments of the VB. ...
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The subject of this paper is the computation of paths for mobile robots that navigate from a start position to a goal position in environments with static obstacles. Specifically, we focus on paths that are represented by straight lines. Such paths can for example directly be followed by omni-directional robots or can be used as an initial solution for path smoothing. In this context, the most common performance metrics are the path length, the obstacle clearance and the computation time. In this paper, we develop a new path planning algorithm that addresses all the stated performance metrics. Our method first determines all possible connections between the start position and goal position along the edges of the generalized Voronoi diagram (GVD) of a given obstacle map. The shortest connections are then refined using a balanced method for creating shortcuts along existing waypoints and introducing new waypoints in order to cut corners. As an important feature, our method reduces the number of required waypoints by iteratively adding new waypoints and then removing unnecessary waypoints along solution paths. Moreover, our method takes into account multiple start-goal connections, since the shortest start-goal connection along the edges of the GVD might not lead to the shortest solution path. A comprehensive computational evaluation for a large number of maps with different properties shows that the proposed method outperforms sampling-based algorithms such as Probabilistic Roadmaps (PRM) and exact methods such as Visibility Graphs (VG) by computing close-to-optimal solution paths with a specified minimum obstacle clearance in less time.
... This method is used by some authors as in the following for transport phenomena in porous media [39][40][41]. However, disadvantages are the error rate for image artefacts and over-segmentation [65], since, as with the inscribed sphere approach, the voxel-based data are used directly. For this, the data must have a good signal-to-noise ratio. ...
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In recent years, non-destructive X-ray microscopy (XRM) has become a common method to characterize particle systems in various scientific fields: Besides the size and shape of particles in bulk powders, the insight into filter cake structures provides additional information about micro processes during filtration and dewatering. Distributed particle properties mainly influence the porous network build-up with possible local deviation in vertical and horizontal alignment. This article focusses on the model-based correlation between the distributed particle properties and characteristic network parameters like tortuosity, pore radii and preferred capillaries for dewatering, using tomography data as model input. Therefore, cake-forming filtration experiments were carried out with a down-scaled, self-constructed in-situ pressure nutsch. The entire tomographic dataset consists of seven individual scans at certain desaturation steps at different pressure levels. For the experiments, a lognormal distributed particle system (crushed Al2O3) in the range of 55 to 200 µm inside an aqueous suspension was used, containing additives for contrast enhancement. Image data processing based on reconstructed 360° projections allows the identification of the background, solid particles and liquid phase by a two-step segmentation. The subsequent modelling uses experimentally verified particle size distributions from laser diffraction measurements (integral value), 2D- (limited number of particles) as well as tomographic analysis, based on calculated single-particle volumes given by the voxel-dataset (all particles within the scanned volume). To characterize the porous network, a developed tetrahedron model is first applied to follow the shortest way through the porous matrix, then again to calculate the widest capillary related to the pore entrance. Furthermore, with information about the pore throat distribution and the wetting line from the tetrahedron side faces, the force balance is evaluated. This results in an entrance pressure distribution, the capillary pressure curve. Experimental data according to VDI 2762 built filter cakes and mercury intrusion tests are taken as reference for validation.
... Then, photos of NFA and NCA human ears was taken under the same condition. According to the existing studies, the profiles of scaffolds were extracted from the binary images [37][38][39]; so the profiles of the blueprint and 3D printed human ear with NFA and NCA inks were extracted. Eventually, the image similarity of the binary images of the blueprint (used as the reference) and NFA and NCA human ears for analyzing shape fidelity [40,41]. ...
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Cell-laden printing is the most commonly used approach in 3D bioprinting. One of the major drawbacks of cell-laden printing is that cell viability is highly affected by the extrusion pressure and shear force in the printing process. We present a new cell-deposition method by using the superabsorbent capability of 3D printed scaffolds with four ink formations: 20:10 nanocrystal/alginate (NCA 20/10), 20:10 nanofiber/alginate (NFA 20/10), 20:02 nanocrystal/alginate (NCA 20/02) and 20:02 nanofiber/alginate (NFA 20/02). Limited pores were observed from the surface of inherent NCA and NFA scaffolds, which may limit the numbers of cells to enter into the scaffolds. Therefore, we designed a dual-porous (DP) structure to connect the inherent pores (IPs) to the scaffold surface. Due to these porous structures, NCA and NFA scaffolds exhibit an excellent capability to absorb cell suspension, which may be used for depositing cells to 3D-printed scaffolds, namely self-absorbent (SA) deposition. Compared to the conventional top-loading (TL) method, the SA method had more uniform cell distributions in the entire 3D-printed scaffolds and higher efficiency of cell deposition. For the TL method, DP scaffold exhibited a more uniform cell distribution, which may provide a better microenvironment for the cells in comparison to the IP scaffold. For both cell loading methods, a rapid increase of cell number was observed in the first 4 days of culture in the 3D-printed NCA and NFA structures. NFA 20/02 exhibits the best cell viability compared to the other three inks. In conclusion, the SA method may serve as a new approach for loading cells in cell-free 3D-bioprinting, and DP design could improve the efficiency of the cell deposition.
... For this purpose each symbol is drawn as a binary raster image at such a scale that the height H of the capital letter is 1000 pixels. Continuous skeletons are constructed from these images using the method described in (Mestetskiy and Semenov, 2008). Morphological moments up to the order of 3 with radius step 0.5 of the pixel value are calculated based on continuous skeletons. ...
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The concept of morphological moments of binary images is introduced. Morphological moments can be used as a shape descriptor combining an integral width description of an object with a description of its spatial distribution. The relationship between the proposed descriptor and the disc cover of the figure is discussed and an exact analytical method for descriptor calculation is proposed within the continuous morphology framework. The approach is based on the approximation of the shape by a polygonal figure and the extraction of its medial representation in the form of the continuous skeleton and the radial function. The proposed method for calculation of morphological moments achieves high accuracy and it is computationally efficient. Experimentations have been conducted. Obtained results indicate that the morphological moments are a more informative and rich shape descriptor than the area of the disc cover. Application of morphological moments to the font recognition task improves the recognition quality.
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LetX be a given set ofn circular and straight line segments in the plane where two segments may interest only at their endpoints. We introduce a new technique that computes the Voronoi diagram ofX inO(n logn) time. This result improves on several previous algorithms for special cases of the problem. The new algorithm is relatively simple, an important factor for the numerous practical applications of the Voronoi diagram.
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This paper investigates the skeletonization problem using parallel thinning techniques and proposes a new one-pass parallel asymmetric thinning algorithm (OPATA 8). Wu and Tsai presented a one-pass parallel asymmetric thinning algorithm (OPATA 4) that implemented 4-distance, or city block distance, skeletonization. However, city block dis-tance is not a good approximation of Euclidean distance. By applying 8-distance, or chessboard distance, this new algorithm improves not only the quality of the resulting skeletons but also the efficiency of the computation. This algorithm uses 18 patterns. The algorithm has been imple-mented, and has been compared to both algorithm OPATA 4 and Zhang and Suen's two-pass parallel thinning algorithm. The results show that the proposed OPATA 8 has good noise resistance, perfectly 8-connected skeleton output, and a faster speed without serious erosion.
Book
Features Serves as both an introduction to and a reference for computer-based analysis and recognition of shapes Includes a comprehensive review of the basic mathematical concepts involved Examines various techniques for shape characterization and analysis, including shape contour analysis and extraction of different shape measures for statistical classification Explains several multiscale techniques, such as wavelets and multiscale skeletonization Focuses on two-dimensional shapes but includes concepts and techniques that can be generalized for 3-D shapes Identifies future trends and developments Includes numerous illustrations and real-world examples Summary Advances in shape analysis impact a wide range of disciplines, from mathematics and engineering to medicine, archeology, and art. Anyone just entering the field, however, may find the few existing books on shape analysis too specific or advanced, and for students interested in the specific problem of shape recognition and characterization, traditional books on computer vision are too general. Shape Analysis and Classification: Theory and Practice offers an integrated and conceptual introduction to this dynamic field and its myriad applications. Beginning with the basic mathematical concepts, it deals with shape analysis, from image capture to pattern classification, and presents many of the most advanced and powerful techniques used in practice. The authors explore the relevant aspects of both shape characterization and recognition, and give special attention to practical issues, such as guidelines for implementation, validation, and assessment. Shape Analysis and Classification provides a rich resource for the computational characterization and classification of general shapes, from characters to biological entities. Both students and researchers can directly use its state-of-the-art concepts and techniques to solve their own problems involving the characterization and classification of visual shapes.
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We introduce a geometric transformation that allows Voronoi diagrams to be computed using a sweepline technique. The transformation is used to obtain simple algorithms for computing the Voronoi diagram of point sites, of line segment sites, and of weighted point sites. All algorithms haveO(n logn) worst-case running time and useO(n) space.
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This paper investigates the state of the art in computer analysis of digitized line images, with emphasis on the requirements of character recognition. The main discussion concerns the requirement of converting a bit mapped, grey-level or binary image to a polygonal approximation of the “skeleton” or “medial axis” of the character.A wide range of thinning algorithms are outlined, followed by some methods of polygonal approximation. The paper concludes that without cheap highly parallel processing power, the common iterative methods are of no use in a cost effective OCR system.
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