THE UNIFORM DECOMPOSITION
Properties and Applications
Franklin C´esar Flores and Airton Marco Polidorio
Department of Informatics, State University of Maring´a, Av. Colombo, 5790, Jd. Universit´ario, Maring´a, PR, Brazil
Keywords:
Uniform Decomposition, Morphological Segmentation, Hierarchical Segmentation.
Abstract:
The uniform decomposition is an hierarchical segmentation method which main property is the preservation of
a given characteristic for each subset of the partition provided by the decomposition. Its computation is incre-
mental and, in each iteration, a candidate region is split only if it is found a valid split where each new subset
respects an input criterion, for instance size or shape. This paper discusses some properties of the uniform
decomposition and proposes several applications for this decomposition such as hierarchical segmentation,
stereo vision pre-processing and interactive segmentation.
1 INTRODUCTION
Let E Z × Z be a rectangular finite subset of points.
A partition X of E is a family of subsets X
i
E, such
that X
i
X
j
= , i 6= j, and
S
i
X
i
= E. Let X = (X
i
)
be a family of partitions such that X
1
= E and
X
j
X
i
X
k
X
i1
: X
j
X
k
, (1)
i > 1. The family of partitions X defines an hierar-
chy (F. Meyer, 2001) of nested partitions: the union
of adjacent regions in finer partitions forms a region
in a coarser partition. And the contour of a region in
a coarser partition will also belong to the finer parti-
tions.
Hierarchical segmentation is the segmentation of
an image in function of an hierarchy of nested par-
titions. Given a family of such partitions X = (X
i
),
assigned to an image f, the hierarchical segmentation
of f may be given by a partition X X . Such segmen-
tation makes possible to represent the objects from an
image in several levels of details and has been applied
to solve problems such as segmentation of MR im-
ages (M. A. G. Carvalho, R. A. Lotufo and M. Cou-
prie, 2003b), cells segmentation (M. A. G. Carvalho,
R. A. Lotufo and M. Couprie, 2003a) and video cod-
ing (P. Salembier; A. Oliveras and L. Garrido, 1998).
One way to provide hierarchical segmentation is
by application of the watershed from markers tech-
nique (Beucher and Meyer, 1992; Vincent and Soille,
1991). The idea is to compute nested partitions by
discarding regional minima that are not relevant and
using the important minima as markers to the appli-
cation of the watershed operator. An hierarchy may
be defined by the way the regional minima from an
image are qualified and selected, usually according to
structural features such as area, volume or contrast (C.
Vachier and F. Meyer, 1995; C. Vachier, 1995).
The method described above is classical under the
morphological hierarchical segmentation framework.
However, it does not provide a way to control the
properties of each subset in the output partition and,
thus, nothing can be globally stated about any sub-
set of the partition. It could be interesting to have an
hierarchical morphological segmentation where each
subset of the partition should respect a given criterion,
such as size or shape. It is useful when the subsets of
the partitions should stand after an area filtering or a
erosion by a given structuring element (s.e.).
The uniform decomposition is a proposal to com-
pute partitions with all of its subsets respecting a
given criterion. Such computation is incremental and,
in each iteration, a candidate region is split in two new
ones only if both new regions are approved by a cri-
terion function. In that way, the method guarantees
that all regions in the decomposed image respect the
input criterion. More, the decomposition is done in-
side a domain of application, what makes possible to
decompose only a specific region of the image.
Besides the presentation of the uniform decompo-
sition (Section 2), the goal of this paper is to point
some of decomposition properties (Section 3) and to
show three applications of the decomposition (Sec-
tion 4). That is the way the paper, concluded in Sec-
tion 5, is organized.
498
César Flores F. and Marco Polidorio A..
THE UNIFORM DECOMPOSITION - Properties and Applications.
DOI: 10.5220/0003870304980504
In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 498-504
ISBN: 978-989-8565-03-7
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
2 THE UNIFORM
DECOMPOSITION
Let K = [0,k] be a totally ordered set. Denote by
Fun[E,K] the set of all functions f : E K. An im-
age is one of these functions (called graylevel func-
tions). Particularly, if K = [0,1], f is a binary im-
age. An image operator (operator, for simplicity) is
a mapping ψ : Fun[E,K] Fun[E,K].
Let N(x) be the set containing the neighbourhood
of x, x E. We define a path from x to y, x,y E
as a sequence P(x,y) = (p
0
, p
1
,..., p
n
) from E, where
p
0
= x, p
n
= y and i [0,n 1], p
i
N(p
i+1
).
A connected subset of E is a subset X E such
that, x,y X, there is a path C entirely inside X.
The criterion function is a function that will be ap-
plied to assess connected subsets according to a given
criterion crit and a parameter p. Its general formula
is given by
C
crit
(C, p) =
1,&if C satisfies the criterion crit,
0,&otherwise,
(2)
where C E and p is a numerical parameter.
This paper will use two criterion functions : the
area criterion and the disc one. The area criterion
function outputs 1 if the area of the connected sub-
set C is greater or equal to p. It is given by,
C
area
(C, p) =
1, if #(C) p,
0, otherwise,
(3)
where #(C) : E Z
+
returns the cardinality ofC E.
The disc criterion function outputs 1 if C E
stands when erode by a disc s.e. with radius p. It
is given by,
C
disc
(C, p) =
1, if ε
B
p
(C) 6= ,
0, otherwise,
(4)
where ε is the morphological erosion (Serra, 1982) B
p
is the disc s.e. with radius p.
Let A, B E be two disjoint connected subsets.
Let W
A,B
(g) be the watershed from markers operator,
which computes the partition of E in function of g,
applying A and B as markers. This partition is given
by two subsets E
A
and E
B
, such that i) E
A
S
E
B
= E,
ii) E
A
T
E
B
= , iii) A E
A
and iv) B E
B
.
Let V E be a domain of validation. Given a cri-
terion function C
crit
and a parameter p, the disjoint
sets A,B V define a valid pair of markers if,
C
crit
(E
A
V, p) = C
crit
(E
B
V, p) = 1, (5)
where E
A
,E
B
E are the two subsets that denote a
partition of E computed by W
A,B
. The validation con-
cept introduced above will be applied in the decom-
position, in order to check if a subset of the hierarchi-
cal segmentation may be split in function of a pair of
while doEmptyQueue( )NOT Q
Isthereavalidpairofmarkers
and inC?M1 M2
C xDw x Elem{ : ( )= };
Î
Elem QRemoveQueue( );
M Min( )g C ;
Pick,ifexists,tworegionalminima , ,
suchthat and defineavalidpairofmarkers;
M1M2M
M1 M2
Î
lbl lbl +1;
wi Watershed gM1M2( , , );
Wi { : ( )=2};xCwi x
w(x) ;lbl:xWi
InsertQueue( , );QElem
InsertQueue( , );Qlbl
Î
Î
wE: ;Z
+
w x( ) 1: ;x DÎ
w x( ) 0 : ;x DÏ
g img( );Ñ
g x g y y D x D( ) max{ ( ): }: ; Î Ï
lbl 1;
InitializeQueue( );Q
InsertQueue( ,1);Q
return ;w
InputData:
Inputimage ;
Domainofapplication ;
Criterionfunction;
Criterionparameter ;
img
D
p
YES
NO
YES
YES
NO
NO
Does
Drespectthecriteriumfunction
andparameter ?p
Figure 1: The uniform decomposition.
markers. The splitting must only occur if both new
subsets satisfies the decomposition criterion.
2.1 The Algorithm
The uniform decomposition algorithm has the follow-
ing input data:
the input image img Fun[E,K];
the subset D E that defines the domain of appli-
cation of the decomposition;
the criterion C
crit
, applied to the iterative split-
ting of regions; and
the parameter p Z
+
used when the criterion
above is applied to.
The algorithm outputs an image w Fun[E,Z
+
] con-
taining the image decomposed in one or more subsets.
Each subset will receive a distinct label lbl Z
+
in
order to be uniquely identified.
The fluxogram in Figure 1 shows how the decom-
position is computed. The first block of the fluxogram
initializes the output w and other auxiliary structures,
such as the queue Q that will store the label of the re-
gions being processed. The morphological gradient is
computed and the minima located outside the domain
of application are removed. Image w receives, in all
points x D, the label 1. The remaining points are
not labeled and set to 0.
The domain of application is evaluated according
to the criterion function and its parameter. If it satis-
fies the criterion, its label is inserted in Q. Otherwise,
the process stops and returns D labeled with just one
label. In the following, the iterative section runs until
Q is empty.
THE UNIFORM DECOMPOSITION - Properties and Applications
499
Figure 2: Hierarchical segmentation. First line : 13 regions. Second line : 34 regions. Third line : 101 regions. First column :
Area criterion. Second column : Disc criterion. Third column : Classic method (extinction value threshold).
In each iteration, a label Elem is removed from
Q and the current subset C D labeled by Elem is
selected. The set M of all regional minima located
inside C are selected as candidate markers to define a
valid pair of markers.
The evaluation of a pair of markers is done by
Equation 5. If a valid pair of markers is found, the
label lbl (initialized as 1) is updated, and the water-
shed operator is applied to the image gradient g using
M1 and M2 as markers. The resulting image w
i
is la-
beled as 1 in the region that contains M1 and labeled
as 2 in the region related to M2.
The set W
i
C is selected by taken all points val-
ued 2 in W
i
(i.e., the subregion of C related to M2).
The set W
i
defines the points in the output image w
that will be updated with the new label lbl.
Finally, label Elem returns to Q and, in the follow-
ing, label lbl is inserted in the queue.
A more detailed and formal presentation of this
algorithm is found in (F. C. Flores and R. A. Lotufo,
2010).
3 PROPERTIES OF THE
UNIFORM DECOMPOSITION
Besides the hierarchical property, the uniform decom-
position also has other important characteristics:
Given f Fun[E,K] and a set of domains of ap-
plication D
i
E : D
i
D
j
= , i 6= j. The decom-
position may be applied independently to each do-
main D
i
and different criterions crit
i
and param-
eter p
i
may be assigned to each D
i
.
All partition subsets given by the decomposition
respect the criterion and parameter applied to the
domain of application. I.e., all subsets measure-
ments are greater than p according to criterion
crit. It provides a more balanced (or uniform)
distribution of subsets.
The most important objects in the image tend to
have a more robust representation.
The application of criteria in the decomposition
computation allows the design of image opera-
tors that preserves the structures represented by
the subsets of the decomposition. For instance, if
the area criterion is applied to, all subsets stand to
an area opening application (with parameter p),
According to disc criterion, all subsets stand to
erosions which s.e’s are discs with diameter lower
or equal to p.
4 APPLICATIONS
The support for a new version of the watershed from
propagated markers (F. C. Flores and R. A. Lotufo,
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
500
Table 1: Segmentation error assessed by function F
(I) of Borsotti et al.
Method Regions 13 34 101
Classic 57.3692 30.8142 23.0695
Area 57.4575 49.3614 23.1594
Disc 55.3578 49.4325 51.5475
Table 2: Segmentation error assessed by function Q(I) of Borsotti et al.
Method Regions 13 34 101
Classic 581.4834 229.1602 110.9979
Area 535.1354 319.6443 103.0404
Disc 524.6157 325.5783 270.1857
2010) was the first reported application of the uni-
form decomposition. This Section presents and dis-
cusses other applications of the uniform decomposi-
tion, such as hierarchical segmentation, stereo vision
pre-processing and interactive segmentation.
4.1 Hierarchical Segmentation
The first application presented in this paper is the
use of the uniform decomposition to do hierarchical
segmentation. Since it is a straightforward applica-
tion of the decomposition, it is preferable to present
the application along the comparison with the classi-
cal hierarchical segmentation by selection of regional
minima via an extinction value threshold. It will be
demonstrated here that the uniform decomposition is
comparable to the classical hierarchical segmentation
besides retaining the respect the decomposition crite-
rion.
The comparison will be done with three decom-
positions: the uniform decomposition under the area
criterion, the uniform decomposition under the disc
criterion and the classical one. The three decompo-
sitions require different kind of parameters, so we
decided to do the comparison according to the num-
ber of regions generated by the segmentation meth-
ods. The three methods will be applied to in or-
der to produce the same number of regions and then
the segmentation results will be assessed subjectively
and quantitatively by application of two functions de-
signed to evaluate the quality of a segmentation.
Classical decomposition is the easiest way to seg-
ment an image in n regions: after compute the extinc-
tion value for each regional minima, it is taken the n-
th greatest extinction value to be the threshold value to
select the n regional minima with the highest extinc-
tion values
1
. There is no straightforward way to seg-
ment an image in n regions by applying the uniform
1
Let us assume, for simplification, that each regional
minimum has a distinct extinction value.
decomposition and it is a drawback of the method. So,
in order to compare the uniform decomposition to the
classical one using the chosen criterion, we searched
for area and disc parameters that generated segmen-
tations with the same number of regions and, then,
applied this number to segment the image using the
classical method.
Three experiments were accomplished. In the first
one, two uniform decompositions were done by ap-
plying an area parameter equals to 5000 and a disc
parameter equals to 33. Both decompositions gener-
ated 13 regions. To complete the first experiment, it
was applied the classical method to achievea segmen-
tation with 13 regions. The second experiment ap-
plied area parameter equals to 1600 and disc parame-
ter equals to 20. The two segmentations provided 34
regions and were compared to the classical method.
Finally, the third experiment compares three decom-
positions that generated 101 regions. The applied area
and disc parameters to this experiment were, respec-
tively, 500 and 10. Figure 2 shows all images gener-
ated to the three experiments.
Following an subjective criteria, we can see that
the segmentation provided by the uniform decompo-
sition is comparable to the provided by the classi-
cal one. In some case, the uniform decomposition
presents a better representation of some objects. Be-
sides the uniform decomposition presents similar re-
sults to the classical one, the subsets of the uniform
results respect the criteria applied to generated them.
For instance, in Fig. 2 (second line-left), no region has
its area lower than 1600 pixels and, in Fig. 2 (second
line-center), all regions stands an erosion done using
a s.e. with radius equals to 10. Such structures preser-
vation does not occur in the classical decomposition.
In order to demonstrate the similarity among both
uniform decompositions and the classical one, it was
done and evaluation experiment. The segmentation
evaluations were done by applying two functions pro-
posed by Borsotti et al. (M. Borsotti, P. Campadelli
THE UNIFORM DECOMPOSITION - Properties and Applications
501
(a) (c) (e)
(b) (d) (f)
Figure 3: Support to image registration. (a) Image displacement. (b) Markers applied to segment Image Two. (c) Uniform
Decomposition of Image One. (d) Segmentation of Image Two. (e) Image One (with overlaid segmentation result). (f) Image
Two (with overlaid segmentation result).
and R. Schettini, 1998) to assess segmentation of
color images according to heuristic criteria such as
homogeneity and simplicity. When comparing seg-
mentation results, the lower results are provided by
the best segmentation according to the cited criteria.
Tables 1 and 2 show, respectively, the evaluation
results given by the application of functions F
(I) and
Q(I) proposed by Borsotti et al. Note that the val-
ues are very similar among themselves and each de-
composition wins in at least one situation. However,
no one victory condition was significative enough to
highlight a decomposition in particular. Quantita-
tively, the three segmentation were equivalent. And,
as a reminder, the uniform decomposition holds the
criteria advantages.
4.2 Pre-processing for Stereo Vision
The second application presented in this paper is the
matching of a pair of stereo images, in order to pre-
process them before the displacement estimation for
each object in the image registration framework. Ba-
sically, the proposed pre-processing method aims to
segment the objects contained in one image and use
them as markers to segment the same objects in the
other image. It is not added any semantics to the pre-
processing method: the segmentation of the objects
is given by the uniform decomposition of Image One,
using as criterion the disc with radius p. The subsets
are then erode separately by a disc s.e. with radius
no greater than p/2. This eroded subsets will be the
markers applied to segment Image Two via watershed
from markers.
This matching framework proposed here was de-
signed under the assumption that the objects con-
tained in Image One are also contained in Image Two,
considering a reasonable displacement. Given an ob-
ject represented by a subset in the partition of Image
One, it is expected that the erosion of the subset, by
a s.e. which radius is chosen in function of the image
displacement, provides a marker located on the same
object in Image Two.
In the following, it is presented the re-
sult of the pre-processing described in the pre-
vious paragraphs. It is applied to a pair of
images extracted from the dataset available at
http://vision.middlebury.edu/stereo/. Figure 3 (a)
shows the displacement of two Images One and Two
(this displacement was computed by the symmetrical
difference between the graylevel version of both im-
ages). Figure 3 (c) shows the uniform decomposition
of Image One, using as criterion a disc with radius
20. All regions are eroded separately by a disc with
radius 10 and used as a set of markers (Fig. 3 (b)) to
segment Image Two (Fig. 3 (d)). This segmentation is
quite good.
4.3 Multiscale Segmentation by
Interactive Splitting of Markers
In this application, the uniform decomposition is used
VISAPP 2012 - International Conference on Computer Vision Theory and Applications
502
(a) (b)
(c) (d)
Figure 4: Interactive segmentation. Decomposition and switching of (a) upper-left region. (b) upper-right region. (c) down-
right region. (d) down-left region (composed with the original image).
to achieve interactive segmentation of static images.
The idea in this approach is not to add or to remove
markers to segment objects via watershed from mark-
ers application. The regions of the image are seg-
mented by the local hierarchy control in each region.
The user points the mouse cursor to a given region
and selects with the mouse buttons the desired oper-
ations. The activation of such operations depends on
the current mode the interface is switched. There are
two interface modes: in the decomposition mode, the
user segments hierarchically the region pointed by the
mouse cursor using the mouse buttons to control the
hierarchy. Left button lowers the hierarchy in a region
while right button raises the hierarchy. In the edition
mode, the user may choose two operations: to split a
region in two ones using the same decomposition or
to define if the region is a foreground or a background
one. Except in the decomposition mode where the
user goes up and down in the hierarchy of a given
region, there is no merging of regions in this segmen-
tation framework. Once a decomposition operation is
done, it is not possible to join the split regions in the
further steps.
Two images are provided in the method inter-
face: one of them shows the current hierarchical seg-
mentation; all tasks are done according to the par-
tition shown in this image. The second image just
show which subsets of the partition are classified as
foreground (yellow) and which ones are classified as
background (blue).
Figures 4 shows some samples of such images,
taken from a sequence of operations over the Fore-
man input image. Each sample shows a significative
instant where part of the foreman is segmented. Fig-
ure 4 (a)) shows the decomposition and switches of
a region in the upper-left location in the image, after
a global initial decomposition. Figures 4 (b-d) show
the decomposition and switches of other regions in
the image. Figure 4 (d) shows the composition of fi-
nal result with the input image.
5 CONCLUSIONS
The uniform decomposition is a hierarchical segmen-
tation method which incremental process guarantees
that all regions in the decomposed image respect an
input criterion, such as area or diameter of a contained
disc. In each iteration, it is investigated is a candidate
region may be split in two new ones, and this splitting
is done according to the input criterion. It provides
a more balanced distribution of subsets in the output
image partition.
Several applications supported by the uniform de-
composition were proposed and the results are quite
satisfactory. It was also done a experimental compar-
ison with the classical hierarchical segmentation pro-
vided by the threshold of the regional minima extinc-
tion values. The comparison was done subjectively
and quantitatively and the results provided by the uni-
form and classical decomposition were comparable.
More, the uniform decomposition still gives the ad-
vantage of retain the structure criterion for each sub-
set of the partition.
Future works include the design of new criterion
functions, the proposal of methods to find valid pair
of markers and the proposal of new applications of the
uniform decomposition.
THE UNIFORM DECOMPOSITION - Properties and Applications
503
ACKNOWLEDGEMENTS
Authors would like to thank Fundac¸˜ao Arauc´aria for
financial support.
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