Journal of Signal and Information Processing, 2013, 4, 308-313
http://dx.doi.org/10.4236/jsip.2013.43039 Published Online August 2013 (http://www.scirp.org/journal/jsip)
Application of PDE and Mathematical Morphology in the
Extraction Validation of the Roads
Fabricio Leonardi, Viviane Sampaio Santiago, Carolina Dias Chaves, Erivaldo Antônio da Silva
Department of the Cartography, Faculdade de Ciências e Tecnologia-FCT/UNESP, Presidente Prudente, Brasil.
Received June 13th, 2013; revised July 13th, 2013; accepted August 10th, 2013
Copyright © 2013 Fabricio Leonardi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The digital images generated by remote sensors often contain noises that are inherent in the process of imaging and
transmission. The application of digital processing techniques greatly enhan ces the ability to extract information on sur-
face targets from remote sensing data. When digital images are used with high spatial resolution, one of the problems
emerging the high variability of targets presents in such images. From the computational point of view, the use of partial
differential equations is favored by the large number of numerical methods showed in the literature. Many of the models
are considered non-complex both from the mathematical and computational standpoints, due to the characteristics of
explicit equations. This work u ses techniques of the partial differential equations (PDE) and mathematical morphology
to extract cartog raphic featur es in digital images of the remo te sensing. The selected study ar ea correspond s to an image
containing part of the Mário Covas Ring Road, located in the metropolitan reg ion of São Paulo (SP), Brazil. The results
are promising and show the high potential of using mathematical morphology in the field of cartography.
Keywords: Extraction Roads; Partial Differential Equations; Mathematical Morph ology
1. Introduction
With the intense modifications that are occurring in the
cities, highways and the agricultural zone, the carto-
graphic update appears as the basic element for the map-
ping of the perimeters urban and agricultural as well as
of the road mesh.
In this direction, it is essential to search a way for the
accomplishment of these updates, so that the same ones
occur in the objective and satisfactory form. The extrac-
tion of highways using images of remote sensing has
been developed in some works in the area of picture
processing, contributing to the management of traffic and
planning of urban and industrial areas.
According to [1], in the area of cartographic sciences,
the problem of extraction of features has been of basic
importance for over two decades, in the automation of
processes of collection of cartographic features, as build-
ings, rivers, roads, etc.
Being like this, the highways are cartographic features
that need constant update, due to its dynamism in func-
tion of modifications in its form or texture.
In this work a sub-picture, is acquired through the
Quickbird satellite, high resolution spacially.
Due to the high spatial resolution offered by satellite,
several applications can be performed, highlighting the
mapping of urban and agricultural areas such as registra-
tion, transportation, telecommunications and planning.
A problem when working with the high resolution of
images is the complexity of its structure, that is, the di-
versity of targets with different forms, tonalities and tex -
tures, such as houses, shades of buildings, automobiles
and trees [2].
In this direction, the use of PDE in picture processing
of remote sensing, associated with the techniques of
mathematical morphology produces satisfactory results
in the extraction of roads, since that they are used in mod-
els of PDE and by appropriate morphol ogic operators.
The model of PDE used in the process of smoothing of
the image in this work was developed by [3]. This model,
part of an only structure, was formulated from the model
of Malik and Perona. Although simple, the model of
Malik and Perona is of great importance, the main idea
suggests a smoothing selection of the image, where the
diffusion is more intense in homogeneous regions and
less in regions of edges.
Through substitutions and additions of terms, the mo-
del of Barcelos and Chen explores the structures of PDEs

Application of PDE and Mathematical Morphology in the Extraction Validation of the Roads 309
getting superi o r re sul t s.
As the implementations of the model, how much the
morphologic routine they had been carried through in
software Matlab 7.0?
The objective of this study was to perform the
smoothing of the digital image through PDE, and there-
after extracting the highway through suitable morpho-
logical operators, for the purpose of cartographic up-
dating.
2. Partial Differential Equat io ns
The use of partial differential equations in picture pro-
cessing has grown significantly in recent years. The basic
idea is to modify one given image, curve or surface with
a PDE and obtain the expected results as the solution of
this equation.
In other words, the use of EDP in picture processing,
not only makes possible the use of good computational
algorithms as well as, the use of important theoretical re-
sults as existence and unicity of solution.
Thus, some researchers had started to develop models
with the purpose to remove, or at least, to minimize the
noises gifts in digital images. The idea is that th is pro cess
purifies the images and keeps their main characteristics
such as texture and preservation of edges.
The PDE model used in this study was to [3], which
will be described in item 4.
3. Optimal Smoothing Filter
There are two physical considerations to determine the
appropriate smoothing filter. The first is that the purpo se
for filtering the image, is to reduce the ranges of scales in
which changes in intensity occur. The goal of filtering an
image is to reduce the intensity differences. Thus, the
result of filtering should be a smooth image. This condi-
tion characterizes a low-pass filter, ie a filter that allows
passage of only low frequency signals. This condition
can be expressed by imposing that the variance is small.
The second consideration is best expressed as a re-
striction of the spatial domain which we call the restric-
tion of spatial location, each point of the filtered image
should be generated from points near and not random
point of the image, because the factors that influence the
intensity changes are spatially localized. Factors influ-
encing the inten sity changes in an image are:
 lighting, which include shadows, light sources and
illumination;
 orientation or distance from the observer to the visible
surfaces;
 reflective surfaces.
Thus, in their own scales, these factors can be consid-
ered as spatially located. Consequently, each poin t of the
filtered image should be generated from an average score
close to that, rather than any other medium of random
dots.
4. The Model of Barcelos and Chen
The model proposed by Barcelos and Chen, deals with
smoothing in digital images and the removal of noise.
However, there is great concern about the targeting. So it
is a model that acts selectively seeking to maintain the
contour of the clearest picture possible.
The model Barcelos and Chen is obtained through the
equation (see Equation (1)):
with


,,0 ,uxy Ixy

is
0
R
u
n


where

is the function that controls the speed of the
diffusion process and

is a parameter. It is hoped that
the large smoothing homogeneous regions suffer and that
the boundary regions are preserved. A good choice for

is the function g is given by:

2
1
*1* gGu kGu


  
 (2)
Every term of the equation is assigned a specific func-
tion, such as decrease the process of smoothing near the
boundary regions, which produces remarkable effects in
edge detection.
5. Mathematical Morphology
[4] defines as a mathematical morphology theory to the
analysis of spatial structures. It is called morphology be-
cause it is the analysis of form and structure of objects;
and mathematics in the sense that the analysis is based on
set theory, integral geometry and boolean algebra.
[5] cite that mathematical morphology is considered a
powerful tool for image analysis, particularly for those
applications where geometric aspects are relevant. The
authors also mention that the main idea of mathematical
morphology to analyze the shape of objects through a
geometric model called structuring element.
[6] defines structuring element as a set completely de-
 
div
tu
uxxu xuuuI
u
 




 






(1)

Application of PDE and Mathematical Morphology in the Extraction Validation of the Roads 310
fined and known (form and size), that it is compared,
from a transformation to the unknown set of the image.
The result of this transformation allows evaluate the un-
known set.
To make this assessment, the structuring element
moves over the image pixel by pixel, covering the whole
scene. In the process of displacement of the structuring
element of the image, it performs some transformations
in the neighborh ood of pixels analyzed. The result of this
transformation pixel is assigned to the corresponding
new image [7].
5.1. Morphological Operators Used
Hdome: Operator hdome removes random noise in the
input image (ƒ) through the detection of peaks with
higher contrast than the prescribed threshold (h) in the
structuring element (bc) chosen. The Expression (1) de-
monstrates mathematically the operator.
 
cc
hbhfbf hf   (1)
Histeq: Operator histeq enhances the contrast of ima-
ges by transforming the intensity values in an image, or
the values in the colormap of an indexed image, so that
the histogram of the output image appro ximately matches
a specified histogram.
Binary: The function converts a binary image in sha-
des of gray (F) in a binary image by comparing each
pixel to the threshold chosen (A). The values of the pix-
els below the threshold (A) are all associated with zero (0)
and up to a maximum of radiometric resolution of the
image. The expression 3 demonstrates mathematically
the operator.
 
1
1
1 if
0 othsewise
k
f
xk
Xfx





 (2)
Neg: For the morphological operators to act properly
on the image, where torque is a necessary condition that
the trait of interest has the binary value 1. Analyzing the
result of binarizing can see that the presented result is
reversed in relation to the condition mentioned. To corre-
ct this reversal is necessary to apply the operator neg.
 
min max
c
f
xk kfx (3)
Areaopen: Function areaopen removes any compo-
nent connected to the image (F) with an area less than the
threshold (A) limit. The connectivity is given by the
structuring element BC. The Expression (5) demonstrates
mathematically the operator.

 
,
,: is connected,Area
cBa
c
c
BB
Ba C
fa fB
5.2. Otsu Method
For a threshold t we think needs to be done an analysis of
the histogram of the image to be ninarized. This process
consists in mapping the gray tones of the image to the set
{0,1}, where 0 (zero) denotes the black color and one (1)
white color.
The method treats the image histogram as a probability
distribution given as follows:
pknk N

wherein nk is the number of pixels with
gray tone k, N is the total number of pixels, so pk is the
probability of a present pixel tone k.
Grouping the pixels into two classes C0 and C1 prob-
abilities of each class are given respectively by:
00
max
11
t
k
k
k
k
kt
p
p








wherein k max is the largest gray level and, by definition,
10
1



.
The mean (01 ,

 ) and variance

22
01
,


 classes
C0 and C1 calculated as follows:


00
0
max
11
1
2
2
00
0
2 max
2
11
1
t
k
k
k
k
kt
t
k
k
k
k
kt
kp
kp
kp
kp


0
1
















Then we find the following medias separability be-
tween classes:

222
00 11
2
201 10
W
B


 


where is the variance 2
W

intraclass and interclass 2
B

the variances.
Thus we find a T that maximizes the interclass varian-
ce or minimize the intraclass variance. This method tests
all possible thresholds and calculates the average sca-
ttering for the pixels in each class C0 and C1.
The method is applied to histogram separated into two
peaks, allowing it to be applied to binarizing with a
threshold T which separates the two peaks. In the case of
a histogram that shows three peaks applying a single
threshold may cause problems, and its mapping to be
done as follows: The values of t1 (first threshold separat-
ing the second peak) and t2 (threshold separates the sec-
ond peak of the third) take the value 1 (one) denoted by
white, otherwise get the value 0 (zero), black color.

X
EX BXa


 



∨
(4)
6. Technique for Feature Extraction
To carry out this work we used a two panchromatic sub-
picture containing an excerpt from Mario Covas Ring

Application of PDE and Mathematical Morphology in the Extraction Validation of the Roads 311
Road, located in region of São Paulo (SP), Brazil, which
was acquired in October 2010, through the Quickbird
satellite, high resolution spatial. This image is shown in
Figure 1.
According [8] the use of linear filters, although widely
used, is not always the most appropriate method of
smoothing images, a time that distort the information of
borders affecting the identification and distinction of
structures of interest in the process of detecting features.
In order to reduce losses in regions of edges and smooth
homogeneous areas, an alternative is the use of filters
nonlinear base d o n PDEs .
To conduct the study we used Matlab, both for the im-
plementation of the PDE model as for routine morphol-
ogy. The procedure for the extraction of road is described
in Figure 2.
7. Presentation and Analysis of Results
The first step was to convert the original image to gray-
scale, for after that carrying through the smoothing
through the PDE model. The result of smoothing by the
PDE model of Barcelos and Chen can be seen in Figure
3. The smoothing was performed with a time t equal to
12.
(a) (b)
Figure 1. Original image.
Result
Removal of
unwanted areas
Application of
the binary
operator.
Original
image Smoothing trough the BC
model.
Quality
Analysis
Extraction the
feature of
interest
Figure 2. Diagram of algorithm for detecting the feature of
interest.
Each of the three terms in the equation has a specific
function. The objective of the first term is to smooth both
sides of a border region, with conducting a minimum of
smoothing the contour. The second term serves to reduce
the smoothing process near the boundary and the last
term is the approximation of the solution to the original
image.
After smoothing the image, it was binarized by Otsu
method with threshold 110 and 95, i.e., was transformed
into shades of black and white. The values of pixel that
they are below of the threshold assume the value 0 (black)
and the values that are above assume the value 1 (white).
In order to remove the remaining of area around the
highway were applied morphologycal operators. These
operators, aims to remove any component with area
smaller than a specified value in a binary image.
The Figure 4 shows the results obtained after removal
of all targets present in the image that do not correspond
to the target of interest highway.
The analysis may be performed visually by overlap-
(a) (b)
Figure 3. Result of the application of the Barcelos and Chen
model (a) e (b).
(a) (b)
Figure 4. Result of the extraction (a) e (b).

Application of PDE and Mathematical Morphology in the Extraction Validation of the Roads 312
ping images, and also numerically, through specific soft-
ware.
In this study we used the System Analysis of Cartogra-
phic Features Extracted for numerical analysis of the
results. The results are shown in Figure 5.
To analyze the results are necessary two files, the re-
ference image, in which case is the original image and
the end of the extraction process by morphological. The
user must select samples of the feature of interest in the
o rigin al image to be carried out analysis of correspondence
between the two images.
It can be considered as a match (C) between the ima-
ges the value of the ratio between the sum of TB and TB
TA and TV as a percentage. The percentage of match can
be obtained by the following formula:
100
TB
CTBTA TV






(3)
where,
 TB represents the total number of white pixels in the
image end of the extraction process, which corre-
spond to pixels correctly extracted;
 TV total red pixels, which correspond to pixels not
detected in the extraction;
(a)
(b)
Figure 5. Result of numerical analysis (a) e (b).
 TA total blue pixels in the image end of the extraction
process, which correspond to pixels that are not ex-
tracted part of the feature of interest;
By correspondence analysis it was verified that the
method is shown suitable for the detection of the feature
of interest, with a correspondence of 82.53%. It is note-
worthy that the technique of EDP combined with mathe-
matical morphology is effective in extracting of features
with features curvilinear.
8. Conclusions
From the result it is possible to conclude that the objec-
tives to smooth the image through a PDE model and to
carry through the extraction of the features of in- terest
with morphologic operators are reached.
The model proposed by Barcelos and Chen had a good
result for image smoothing. You can see the edges of the
region of interest (in case the roads) are preserved while
the image is smoothed. From the computational point of
view, the use of partial differential equations is favored
due to the characteristic of explicit equations, in addition
to the extensive amount of numerical methods in the lit-
erature.
The extraction of the feature of interest by routine
morphology was also quite satisfactory, resulting in good
detection of edges and axes of the tracks.
The choices of thresholds used in the functions must
be done carefully, because if they are not suitable, they
can cause loss of objects.
The structuring element to be used in functions is the
main factor for obtaining good results in detection tech-
niques using Mathematical Morphology, considering that
it is the parameter which identifies the shape of the ob-
ject to be detected.
As the result is satisfactory, the work can be used as a
stage of daily preprocessing for processes of automatic
extraction of the road mesh, beyond contributing as an
alternative method for cartographic update.
REFERENCES
[1] A. P. Dal Poz, R. B. Zanin and G. M. Vale, “Extração
Automática de Feições Rodoviárias em Imagens
Digitais,” Revista Controle & Automação, Campinas, Vol.
18, No. 1, 2007, pp. 44-54.
[2] Y. Chen, B. C. Vemuri and L. Wang, “Image Denoising
and Segmentation via Nonlinear Diffusion,” Computers
Mathematics with Applications, Vol. 39, No. 5-6, 2000,
pp. 131-149. doi:10.1016/S0898-1221(00)00050-X
[3] C. A. Z. Barcelos and Y. Chen, “Heat Flow and Related
Minimization Problem in Image Restoration,” Computers
Mathematics with Applications, Vol. 39, No. 5-6, 2000,
pp. 81-97. doi:10.1016/S0898-1221(00)00048-1
[4] P. Soille, “Morphological Image Analysis: Principles and
applications,” Springer-Verlag, Berlin, 2003.

Application of PDE and Mathematical Morphology in the Extraction Validation of the Roads 313
[5] J. Goutsias and H. J. A. M. Heijmans, “Mathematical
Morphology,” IOS Press, Amsterdan, 2000.
[6] J. Facon, “Morfologia Matemática: Teorias e Exemplos,”
Editora Universitária Champagnat da Pontifícia Univer-
sidade Católica do Paraná, Curitiba, 1996.
[7] J. I. Miranda and J. Camargo, “Detecção de Bordas com o
Modelo de Difusão Anisotrópica,” Anais XIII Simpósio
Brasileiro de Sensoriamento Remoto, Florianópolis, 2007,
pp. 5957-5964.
[8] C. A. Z. Barcelos, “Restauração e Análise de Imagens via
Equações Diferenciais Parciais,” Tendências em Mate-
mática Aplicada e Computacional, Vol. 3, No. 2, 2002,
pp. 1-13.