A New Probabilistic Approach Based on Markov Process for Matching Problem

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Procedia Technology 22 ( 2016 ) 889 – 896

ScienceDirect

Peer-review under responsibility of the “Petru Maior” University of Tirgu Mures, Faculty of Engineering doi: 10.1016/j.protcy.2016.01.065

9th International Conference Interdisciplinarity in Engineering, INTER-ENG 2015, 8-9 October

2015, Tirgu-Mures, Romania

A New Probabilistic Approach Based on Markov Process for

Matching Problem

Khalid Hati

a,

*, Abdellah El Hajjaji

a

Systems of communications and Detection Laboratory, Faculty of SciencesM’hannech II, Tetuan 93002, Morocco

Abstract

Stereo matching is one of the main topics in computer vision. It consists to find in two images of a same scene, taken from different viewpoints, the pairs of pixels which are the projections of a same scene point. Since the last twenty years, many local and global methods have been proposed to solve this problem. Such stereo configurations reduce the search for correspondences from two-dimensions (the entire image) to one-dimension; the epipolar constraint can be used to facilitate matching by reducing the search. The application area of the matching problem is very wide, including biological sequences, the comparison of different files, speech recognition.

In this paper, we will present a new probabilistic approach method based on Markov process for the matching of two epipolar images; where each scan-line is delimited by two successive points of interest.

Peer-review under responsibility of the “Petru Maior” University of Tirgu-Mures, Faculty of Engineering. Keywords: Probability; Markov process; Matching; 3D reconstruction; Epipolar image.

1. Introduction

In the last two decades, numerous algorithms [1,2,3] for image matching have been proposed. They can roughly be classified into two categories: The template matching, and feature matching. Much good work has been done. Stereo matching is one of the main topics in computer vision. It consists to find in two images of a same scene, taken from different viewpoints, the pairs of pixels which are the projections of a same scene point [4,5].

* Corresponding author. Tel.: +212-610-117-082.

E-mail address: [email protected]

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Fig. 1. Epipolar geometry

Matching is the key for determining the altitude “H” (1) of the observed scene and therefore the visual reconstruction of 3D image. Extracting the depth of a scene point using stereo pairs is based on finding correspondences, i.e. finding for each point in one image the matching point in the other image. Mapping all pixels in one image (the reference image) to disparity values, results in a set of values termed a dense disparity map.

/ /

H% # h (1)

where % and j1 j2

#/ h

is calculated previously, this last is a mathematical entity without any physical sense.

Fig. 2 shows two epipolar images token by satellites (France). Since the last twenty years, many local and global methods have been proposed to solve this problem. Such stereo configurations (fig.1) reduce the search for correspondences from two-dimensions (the entire image) to one-dimension; the epipolar constraint [6,7] can be used to facilitate matching by reducing the search.

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Fig. 3. Left epipolar image

In this presentation and for the sake of clarity, it is assumed that the detection and matching of points of interest was already done (Fig. 4), in this paper we will be focused into the rest of points in each interval delimited by two successive points of interest, as can be seen in Fig. 6, by using a new probabilistic approach method based on Markov process

Fig. 4. 3D reconstruction steps

Acquisition of epipolar images

Detection and matching of point of Interest(P.I)

Matching of scan-lines

Disparity Map Calculation

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Nomenclature

Į correlation coefficients H altitude

M.P makov process P probability P.I points of interests

2. Modeling and preliminaries

2.1. Preliminaries

Stochastic processes [8,9,10] are sequences of random variables generated by probabilistic laws. Markov processes are a class of stochastic processes that are distinguished by the Markov property (2) and have many applications in, for example, operations research, and economics. In this section, we introduce some basic concept of Markov processes

Generally, a stochastic process is a succession of experiments whose result depends on the chance. Let’s be {Xk}

a family of random variables, where k denote the

1 0 0 1 1 1

(

|

,

,...,

)

= (

|

)

n n n n n n

X

y X

x X

x

X

x

X

y X

x

+ +

ΙΡ

=

ΙΡ

=

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Estimating a transition matrix (3) is relatively straight for hard process. This transition matrix allows the looking for the optimal path.

1 ,1 ( , ) i n j n p i j _{b b} _{b b} *3 (3) 11 12 1 21 22 2 1 2 n n n n nn p p p p p p p p p _¬ ® " " # # % # " And 1 1 n ij j p

for each i=1,2,..n (4)

2.2. Example of a classic technique

The matching of all the pixels between each two successive point of interest can be done by the way of the calculation of correlation coefficients (Fig. 4), often denoted Į (5) [11], measuring the degree of correlation, or it can be matched by using the dynamic programming [12]. Those techniques suffer from the calculation times, cause of its complexity. The correlation technique suffer also from the problem of parallax, hidden area and images are taken at different moments.

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i=1 1 1 1 ( )( ) = ( )( ) N M ij ij j N M ij ij i j X X Y Y X X Y Y α = = = − − − −

¦¦

(5) Where 1 1

( )

, * N M i j i j X X N M = = =

¦¦

(6) And

( )

, 1 1 * N M i j i j Y Y N M = = =

¦¦

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Fig. 5. Exemple of matching by correlation technique of two epipolar lines

2.3. Modeling

The need for introducing the concept of stochastic process come from that the values of the pixels can be considered as a random variable.

Markov processes are characterized by a special form of dependence structure that makes them a useful class of stochastic processes in stochastic modeling. A Markov model is defined by the likelihood of transitions.

A Markov chain is described as follows:

Assuming that a line the picture is comparable to a process Markov or stochastic (random field) that checks the properties of Markov, then local maximization of these probabilities well ensure the maximization of total cost of matching.

Let’s be Ai and Ai+1(resp. Bi and Bi+1) two successive points of interest, where Ai and Bi (resp. Ai+1 and Bi+1) are

100% matched without error.

And let’s S (resp. T) a right scanline of pixels, S=S1,S2….SN (resp. T=T1T2…TM) two string, and the length of a

string S is denoted by |S| (resp. |T|). We denote by S[i], for all 1i<|S| , the pixel at index i of S. And S1=Ai and

SN=Ai+1 (resp. S2=Bi and TM=Bi+1).

Generally in mathematics, stochastic describes an approach to anything that is based on probability; a stochastic process is a statistical process obtained from a corresponding sequence of jointly distributed random variables depending on variable parameter (which is usually time).

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Fig. 6. Exemple of the detection and matching of P.P results

3. Results and discussion

Assuming that a line the picture is comparable to a process Markov or stochastic (random field) that checks the properties of Markov, then local maximization of these probabilities well ensure the maximization of total cost of matching and all vertices are connected.

Let I be a countable set,

{

i, j, k, . . . . Each

}

i∈I is called a state and I is called the state-space.

A path is a set of boxes {Xn} travelled successively leading point D

A Bi j

to pointC

Ai1Bi1

. Each Xi

is a random variable, and all vertices are connected.

The figure 6 shows all possible paths from one exact point X1(A1,B1) to another exact point X2(A2,B2). A path is

a set of boxes {X } n travelled successively leading point X1 to point X2. Each Xi is a random variable.

Each matching path has a global cost; it can be calculated as the sum of the local cost along it. The optimal matching path is dened as one minimizing the global cost.

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The table 1 shows an example of results obtained by using our contribution, each move from any stat Xi to other stat Xi+1 has it’s our probability calculated in versus it’s neighborhood. Reducing the neighborhood helps faster to limit the calculation time.

The dimensions difference between the two strings S and T is due to the presence of a set of hidden pixels in a string well-known by the problem of occlusions.

Table 1. Example of result of probability.

S1=Ai S2 S3 S4 S5 S6 S7 S8 S9 S10=Ai+1 T1=Bi 1 0 0 0 0 0 0 0 0 0 T2 0.05 0.6 0.35 0 0 0 0 0 0 0 T3 0 0.2 0.55 0.25 0 0 0 0 0 0 T4 0 0 0.15 0.35 0.5 0 0 0 0 0 T5 0 0 0 0.02 0.4 0.5 0.08 0 0 0 T6 0 0 0 0 0 0.05 0.75 0.2 0 0 T7 0 0 0 0 0.05 0.1 0.6 0.25 0 0 T8 0 0 0 0 0 0.21 0.59 0.2 0 0 T9 0 0 0 0 0 0.3 0.6 0.1 0 0 T10 0 0 0 0 0 0 0.43 0.37 0.2 0 T11 0 0 0 0 0 0 0 0.1 0.56 0.34 T12=Bi+1 0 0 0 0 0 0 0 0 0 1 4. Conclusion

In this paper, we had presented a new probabilistic process. This technique was compared with the classical method, and gives the best result and reduces the computation time.

The application area of the matching problem is very wide, including biological sequences, the comparison of different files, speech recognition. This technique will allow the best real-time result in robotic application as an example [13,14,15,16,17,18].

The application area of the matching problem is very wide, including biological sequences, the comparison of different files, speech recognition.

The aim of the future study is to improve the manner which the path are calculated by using different kind of algorithms. In fact, each path should not take into consideration the case with th probability equal to zero.

Acknowledgements

The authors would like to thank Dr. Peter Benedek from Pulispace for his support. Appreciation is also extended to technical support from colleague, special thanks to Nirmine El Mezouari for her help.

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