BRANCHES FILTERING APPROACH FOR MAX-TREE
Ketut E. Purnama
1
, Michael. H. F. Wilkinson
2
, Albert G. Veldhuizen, Peter. M. A. van Ooijen
Jaap Lubbers, Tri A. Sardjono and Gijbertus J. Verkerke
3
1,3
Department of Biomedical Engineering, University Medical Center Groningen, University of Groningen
P.O. Box 196, 9700 AD, Groningen, The Netherlands
1
Department of Electrical Engineering, ITS, Surabaya, Indonesia
2
Institute for Mathematics and Computing Science, University of Groningen
P.O. Box 800, 9700 AV, Groningen, The Netherlands
Keywords: Branches Filtering, Max-Tree.
Abstract: A new filtering approach called branches filtering is presented. The filtering approach is applied to the
Max-Tree representation of an image. Instead of applying filtering criteria to all nodes of the tree, this
approach only evaluate the leaf nodes. The expected objects can be found by collecting a number of parent
nodes of the selected leaf nodes. The more parent nodes involve the wider the area of the expected objects.
The maximum value of the number of parents (PL
max
) can be determined by inspecting the output image
before having unexpected image. Different images have found have different PL
max
values. The branches
filtering approach is suitable to extract objects in a noisy image as long as these objects can be recognised
from its prominent information such as intensity, shape, or other scalar or vector values. Furthermore, the
optimum result can be achieved if the areas which have the prominent information are present in the leaf
nodes. The experiments to extract bacteria from noisy image, localizing bony parts in a speckled ultrasound
image, and acquiring certain features from a natural image appeared to be feasible give the expected results.
The application of the branches filtering approach to a 3D MRA image of human brain to extract the blood
vessels gave also the expected image. The results show that the branches filtering can be used as an
alternative filtering approach to the original filtering approach of Max-Tree.
1 INTRODUCTION
Separating objects from an image is a main issue in
many applications of computer vision. Many
methods have been proposed including the methods
of mathematical morphology. A family of
mathematical morphology called connected
operators has been introduced and there is a great
deal of development going on (Breen et al., 1996;
Salembier et al., 1995) especially by the introduction
of Max-Tree for image representation (Salembier et
al., 1998). Connected operators, especially the ones
that have anti-extensive property, are used to filter
the expected objects based on one or more criteria.
Objects extraction is not done to the original image.
Instead, it is done to the Max-Tree, and the filtering
criteria are applied to each node of the tree. The
criteria can be shapes (Ouzounis et al., 2006; Urbach
et al., 2002; Wilkinson et al., 2001), vector attributes
(Urbach et al., 2005) or other types of information.
In this paper, we proposed a new filtering
approach called branches filtering which applies the
filtering criteria only to the leaf nodes of Max-Tree.
The expected objects can be found by collecting a
number of parent nodes of the selected leaf nodes.
The next sections are organized as follows.
Section 2 discusses the theory of connected
operators for binary and grey-level image. The Max-
Tree creation is discussed in Section 3. The
description of the proposed branches filtering
approach and its application to four different types
of images are discussed in section 4. The discussion
of our work is presented in section 5.
328
E. Purnama K., H. F. Wilkinson M., G. Veldhuizen A., M. A. van Ooijen P., Lubbers J., A. Sardjono T. and J. Verkerke G. (2007).
BRANCHES FILTERING APPROACH FOR MAX-TREE.
In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 328-332
Copyright
c
 SciTePress

2 CONNECTED OPERATORS
AND ANTI-EXTENSIVE
CONNECTED OPERATORS
Connected operators in mathematical morphology
are well known operators that preserve or remove
connected components based on one or more
filtering criteria. They never introduce new
components. In detecting edges, for example, these
operators can preserve the edge while removing
noise or unwanted components. These operators can
be applied to either binary or grey-level image. In
the following paragraphs, the description of these
operators is presented briefly. A more extensive
description can be found elsewhere (Salembier et al.,
1995; Serra et al., 1993).
A binary connected operator
ψ
is connected if
and only if, for any binary image
X
, the associated
partition of
X
is finer than the associated partition
of
)X(
ψ
. Let
E
be the universe of a set and
partition
}A{
i
is a set of connected components
in
E
. A partition }A{
i
is said finer than partition
}B{
i
if any pair of points of the same partition
class
i
A also belongs to a unique partition class
i
B .
The extension to grey-level image is applicable by
associating a partition to a function. For each grey-
level function
f we obtain flat zones and the set of
these flat zones is called partition of flat zones.
Hence, the grey-level connected operator
ψ
is
connected if and only if, for any grey-level image,
the partition of flat zones of
f is finer than the
partition of flat zones of
)f(
ψ
.
In the rest of this paper, we inspect only
connected operators that have anti-extensive
property (
XXx ⊆∀ )(,
ψ
). The term connected
component is used to refer to flat zone.
3 MAX-TREE FOR IMAGE
REPRESENTATION
Max-Tree was first proposed by Salembier et al.
(1998) as a tree data structure to represent a grey-
level image. In the tree representation, the root node
stores the pixels belonging to the background (pixels
with the lowest grey-level). The nodes at the higher
levels store pixels of each connected component
found at higher grey-level. The leaf nodes represent
the regional maxima of the image. The authors also
proposed a filtering scheme consists of three steps. It
is started by creating a Max-Tree representation of
an image, then applying filtering criteria to all nodes
of the tree, and followed by creating the output
image from the filtered tree (image restitution).
The Max-Tree creation can be described as an
iterative process. In the first iteration, the lowest
grey-level is used as the threshold value h. Then,
using this threshold value the pixels belong to the
background are found, and a set of connected
components is obtained from the pixels with grey-
level higher than h. The background pixels are
assigned to the root node, while the pixels of each
connected component are assigned to a temporary
node.
The next iterations are done as follows. The
threshold value h is increased by one. For each
temporary node, pixels with grey-level h are
assigned to a new node. These nodes are then added
to the tree. Again, a set of connected components is
obtained from the pixels with grey-level higher than
h. The pixels assigned by each connected component
are assigned to a new temporary node. Figure 1
illustrates the Max-Tree creation process.
We use the following notations:
k
h
C
to refer to a
kth node at level h,
k
h
P to refer to a set of pixels that
is assigned to
k
h
C , and
k
h
TC to refer to a kth
temporary nodes at grey-level h.
The image in Figure 1a is composed of ten
connected components (A, B, C, D, E, F, G, H, I,
and J) of four grey-level (0, 1, 2, and 3). The lowest
grey-level of this image is 0. In the first iteration, we
use this value as a threshold value (h). We find that
a. an image
b. iteration 1 c. iteration 2
d. iteration 3 e. empty nodes are
romoved
Figure 1: Max-Tree creation process. (a) input image, the
first, the second, and the third iteration are illustrated in
(b)-(d) respectively, and the empty nodes are excluded (e).
BRANCHES FILTERING APPROACH FOR MAX-TREE
329

pixels in A have grey-level 0. The pixels of A are
assigned to the root node
1
0
C . We obtain three
connected components (E, DJH, BCGFI) from the
pixels with grey-level higher than h. The pixels of
these connected components are assigned to
temporary nodes:
1
1
TC ={E},
2
1
TC ={DJH}, and
3
1
TC ={BCGFI}. The result of the first iteration is
displayed in Figure 1b. In the second iteration, we
increase h with one. Temporary node
1
1
TC has no
pixel with grey-level h. Hence, we assign no pixel to
a newly created node, and we still add this node to
the tree. Temporary node
2
1
TC has one connected
component with grey-level h (D) and one connected
component with grey-level higher than h (JH). We
assign the pixels of D to a new node and add this
node to the tree. The pixels of JH are assigned to a
temporary node. Shortly, for
3
1
TC a new node that
is added to the tree refers to pixels in BC, while
pixels in G and FI are assigned to two temporary
nodes. The result of the second iteration is displayed
in Figure 1c. The same process is applied to each
temporary node in the third iteration, and we obtain
the result in Figure 1d. Finally, the empty nodes are
removed from the tree (Figure 1e).
In the filtering process, (Salembier et al., 1998)
describes the filtering process using increasing and
non-increasing criteria. Each node of Max-Tree is
examined using a specific criterion. The result is
whether the inspected node will be removed or
preserved. Classical criteria were reported having
increasing property. It includes: opening by
reconstruction that preserves node
k
h
C if the binary
erotion of
k
h
P is not the empty set; grey-level area
opening that preserve
k
h
C if the number of pixel of
k
h
P is larger than a limit
λ
; max−
λ
that
preserve
k
h
C if there is at least one non-empty set of
its descendant nodes at level
λ
+h , and min
−
λ
that can be defined by the duality. For non-
increasing criteria three rules are reported: “Direct”,
“Min” and “Max” decision. In the “Direct” decision
k
h
C will be preserved if and only if λκ ≥)C(
k
h
.
In this case,
(.)
κ
is a criterion. The
k
h
P will be
merged with the nearest ancestor. In “Min” decision,
k
h
C will be preserved if λκ ≥)C(
k
h
and all of its
ancestors are also preserved. “Max” decision is dual
of the “Min” decision where
k
h
C will be removed if
λκ <)C(
k
h
and all of its descendants are also
removed.
4 BRANCHES FILTERING
In the original Max-Tree, the filtering criteria are
applied to all nodes of the tree. The branches
filtering approach acts differently; it applies the
filtering criteria only to the leaf nodes. Based on the
selected leaf nodes, a number of their parent nodes
at the higher level is selected and preserved, while
the other nodes are removed resulting in selected
branches of the tree which represent the expected
objects. By increasing the number of selected
parents while inspecting the resulted image, the
maximum value of the number of parents (PL
max
)
can be determined.
The idea of branches filtering approach was
motivated by the fact that in some applications the
expected objects are difficult to differentiate from
unwanted neighbouring objects, or they are in the
noisy image. This filtering approach is suitable if the
expected objects can be recognized although by only
a little information, and these information are
present in the leaf nodes of Max-Tree.
Considering only the leaf nodes in the Max-Tree
is comparable to extract the regional maxima of the
image; (Vincent, 1993) use the grey scale
reconstruction to extract all of the regional maxima
(h-domes). However, branches filtering approach do
not select all of the maxima, but just the maxima
which fulfil the filtering criteria.
Figure 2 shows the application of this filtering
approach on four different types of images. The
images are grey-level images, and the values are
between 0 and 255. We used grey-level as a criterion
in the filtering process, and in the Max-Tree creation
process this information was stored in each node.
The first image is the inverted image of bacteria, and
our objective is to extract the bacteria which has
long rounded shape and typically chained with each
other. Although it has unique shape, we do not use
the shape as a criterion in the filtering process.
Instead, we chose its grey-level. The second image
is an axial view of the cross-section ultrasound
image of human back. The objective is to extract the
bony parts. Although bony structures will give
strong reflection (high grey-level), the parts which
have high grey-level are too small, and the
neighbouring unwanted parts have grey-level
VISAPP 2007 - International Conference on Computer Vision Theory and Applications
330

a. bacteria b. human body c. MRA of human brain d. Natural image
T=88, PL= 0 T=66, PL= 0 T=94, PL= 0 T=85, PL= 0
T=88, PL= 5 T=66, PL= 25 T=94, PL= 8 T=85, PL= 3
T=88, PL= 12 T=66, PL= 50 T=94, PL= 17 T=85, PL= 5
Figure 2: Branches filtering approach for different types of images (the first row) with a threshold value (T) and different
parent level value (PL) in the second until fourth row. In the second row no parent node (PL=0) is selected; the parents
until the PL
max
th (fourth row) and around the half of PL
max
th (third row) are selected.
slightly the same. We need to enlarge the captured
area without getting the unwanted parts. The third
image is a 3D MRA image of human brain and the
objective is to extract the blood vessels ignoring the
cloudy parts around it. The last image is a natural
image of a boy. The objective is to extract his image
and the mask on his face.
The second, third and fourth rows display the
influence of three different PL values to the resulted
images. In our application the PL value was
parameterized. The second row displays the results
with PL is set to the minimum value (0); no parent
node was included in the resulted image. The images
in the fourth row resulted from the PL value equal to
the PL
max
. Different images have different PL
max
value. The values around the half of PL
max
value
were used to obtain the images in the third row.
BRANCHES FILTERING APPROACH FOR MAX-TREE
331

5 DISCUSSION
We have shown a new filtering approach to the
Max-Tree representation of an image called
branches filtering. Instead of inspecting all nodes of
the tree, the branches filtering applies the filtering
criteria only to the leaf nodes. From the selected leaf
nodes, the expected objects can be found by
successively collecting a number of their parent
nodes at the higher level. The maximum value of the
number of parents (PL
max
) can be determined by
inspecting the resulted images before having
unexpected result. Different images have different
PL
max
value. In case the grey-level is used as a
criterion, the grey-levels stored in the leaf nodes can
be sorted, then an automated process to determine
the intensity threshold can be applied; we are
working on it.
The results show that the branches filtering can
be used as an alternative filtering approach to the
original filtering approach of Max-Tree.
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