A differential approach for fissures detection on road surface images Bertrand AUGEREAU(1), Benoît TREMBLAIS(1), Majdi KHOUDEIR(1), Vincent LEGEAY(2) (1)

Laboratoire IRCOM SIC, UMR 6615 CNRS Université de Poitiers, Bâtiment SP2MI, BP 179 86960 Futuroscope Chasseneuil Cedex France phone : (33) 549 496 567, fax : (33) 549 496 570 email : [email protected] poitiers.fr (2)




Laboratoire Central des Ponts et Chaussées 44340 Bouguenais France

Topics : Defect detection Methods : P.D.E. Preference : None

Extended Abstract

1. Introduction The present work is part of a more general study concerning the characterisation of road surface from images. Here, we deal with the search for some particular deteriorations, the surface's fissures. The encountered difficulties come from the nature of these images, with an example in Figure 1. On one hand, we have a grey levels distribution which is quite perfectly gaussian. On the other hand, the road's texture induces possible confusions between fissures and usual stone's separation. Most authors involved with the general problem of crack's detection base their researches on the following principle : a crack may be considered like a break of a given model. Our approach is absolutely different in that we don't directly use any homogeneity's hypothesis concerning the cracked object. In this paper, we present how some pure differential observations about the image surface introduce an original fissures detection process. We use a new and dedicated PDE's scheme to generate an appropriated scale space from which it's rather easy to extract fissures edges. Finally, we show some results from real roads images.




Figure 1 : example image of road surface with fissure ( extract from original image, zoom 100% ) 2. A differential characterisation of fissures





In this first part we point out some useful differential geometry. First of all, we assume that a 3 grey level image is a two dimensional differential manifold locally embedded in , and we adopt the hessian as a metric on this manifold. The associated map, namely the surface, is denoted by I x , y and if we use the Monge notations



r



2

the hessian matrix will be



I x, y 2 x

,

  

s



H x, y



r s

2

s t

I x, y x y

t

,





2



I x, y 2 y

,

.

So the matrix has two eigenvalues

  1

1 r 2

 t

r



t

2



4 s

2



and



1 r 2

2

 t

r



t

2



4 s

2



which are the expression of the two main surface curvatures according to the directions given by the associated eigenvectors. More precisely, is called minimal curvature, maximal 1 2 2 is the total curvature, while the curvature and H x , y H x, y rt s 1 2 ,



I x, y  r

laplacian operator

t ,



I x, y



 2

1

2

is the mean curvature. So the

curvatures can also be



1



x, y 2

x, y

 

1 2 1 2

I x, y H x, y  I x, y   I x, y  H x, y I x, y

2

2

2

and 2

.

Numerically we use usual discrete approximations of the partial derivatives and denote these approximations with symbol tilde. As example, in Figure 2, we can see the images of the curvatures computed from the synthetic surface visualised on first image. However, it's well known that the computation of partial derivatives is an extremely noise sensitive problem, as shown in Figure 3 where the previous surface has been perturbed with a gaussian noise.

Figure 2 : images of respectively original map, minimal, maximal and total curvature.

Figure 3 : images of respectively noisy original map, minimal, maximal and total curvature. For concluding this preliminary, we simply remark that the curvatures remain a characteristic of the fissures, even if they are significant for all surface's bending.

3. A PDE's scheme preserving significant bending



We begin this second part with an observation. In our first example, the non noisy surface, the total curvature is quite always null ( the grey pixels on the image ). And obviously, that is not the case for our second example. So we decide to transform the noisy surface into a map with a quasi null total curvature. In other words, we have to start from the given surface and try to eliminate some perturbations. A classical approach for performing smoothing restoration is to convolve the surface with a gaussian kernel. In this prospect one have to fix, sometimes empirically, a standard deviation of the kernel, and that's why we prefer to implement an equivalent iterative process which generates a scale space of iterated gaussian convolutions.





So and from now, we consider that I x , y , t is a map obviously associated to a three maps are build from the original two dimensional map dimensional manifold. The I x , y , t I x , y , in a way such as I x , y , 0 I x, y and we can refer to parameter t as an evolution scale for the two dimensional maps. In this scope, gaussian smoothing arise from the initial PDE







 It  

I

(1).

The scale space comes from the discrete expression of (1), that gives the relations



I0 x , y

,

I x, y

I

n



1

x, y





I

n

x, y

 

I

n

x, y

with n denoting the process' epochs, being the laplacian discrete approximation and a gathering the various discretization's steps, h x , h y and h t . This process, parameter implementing a numerical resolution of the PDE, can be seen as a fix point search method converging towards the map I such as I x , y , i. e. such as I x, y I x, y I x, y 0 .



$" 





!#$ " 

 



Anyway, the gaussian convolution or its iterative equivalence doesn't produce the expected map, the isotropic smoothing erasing every bending without any discrimination. To avoid this drawback while reaching ours aim, their are different possibilities. The more natural is to preserve the smoothing efficiency of previous process while trying to enhance the surface according to curvatures. So we choose to implement the PDE

%

% It # &

(2).

1

'

This choice's justification lies in the iterative discrete scheme generating the scale space. The scheme is

2

&"

( *,- +

I n 1

x, y

n

1

I

n



-+ x, y .

x, y

I

n

x, y I

n

)! &" x, y

n 1 2

x, y

. H+

n

, x, y

2

(3),

and its analysis points out two fundamental facts. First of all,

/+

n 1

x, y

can be separated in two terms : a pure smoothing part

10

2

10

2

0

1 1 n n 2 n 2 I x, y I x, y H x, y and an enhancing part which combines 2 2 the elementary opposite laplacian enhancement with a weighting term depending on the total I such as curvature. Next, the process converges towards an I x, y I x, y 0 . Now, if we develop the , i. e. such as 1 x, y 1 x, y 2 2 , that is last equation, we obtain I x, y I x, y H x, y H x, y 0 . So and theoretically, this process converges towards a map whose total curvature discrete approximation is identically null. In practice, we can stop the process when the modification between the maps resulting of two consecutive iterations becomes lower than a given ε. Furthermore this convergence criterion allows to obtain a compact scale space where latest maps conserve some non null total curvature.

3

4 3

:3

4

57698 3

10 3

83 ; 4 : 3

4

< : 3

3

=

=

4. Application to fissures detection on road surfaces images Finally, we apply the above process to road surfaces images. The format of these images is 480 over 760 pixels. Here, we only present the results obtained with one of them. In this case, the ε convergence criterion is 10 5 and the process runs over 37 iterations. To point out the efficiency of our process we choose to show, in Figure 4, conjointly with the original and last maps, the results of a very simple segmentation, a gradient hysteresis. In Figure 5 we can see the results of an existing method, at LCPC, which combines a first segmentation based upon textures and fissures models, a second phase of pixels agglomeration, a third phase using a markovian model and at end the fissure extraction.

>

Figure 4 : images of respectively 1st map, segmented 1st map, 37st map, segmented 37st map ( whole images, zoom 15% ).

Figure 5 : same initial map segmented with the reference method. 5. Conclusion In this paper we demonstrate how the use of the differential properties of cracks on maps, especially the curvatures, enables to produce an original process for the fissures detection from road surface images. So, we propose to build a compact and dedicated scale space which can be the first but quite efficient step towards a complete treatment procedure. There are several possibilities for our future works, the insertion of our differential method as a pre process for the markovian detection appears to us being the most promising of them.

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