New

Approach for Geological Faults Detection

A.O. Cepero D´ıaz

1,2

, V. Di Ges`u

, and C. Valenti

2,?

Departamento de Autom´atica y Computaci´on

Universidad Tecnol´ogica de La Habana, Cuba

Dipartimen

to di Matematica e Applicazioni

Universit`a degli Studi di Palermo, Italy

Abstract. This paper will introduce a new approach, mainly based on

wavelets transform and mathematical morphology, for the detection of

geological faults in satellite and aerial photos. This can be useful for ter-

ritorial analysis, digital cartography and the use of geographical infor-

mation systems. The proposed method is fast enough and unsupervised.

Diﬀerent techniques will be compared and some examples on real data

will be presented to indicate the applicability of these methods.

Keywords: geological faults, mathematical morphology, fuzzy morphol-

ogy, wavelet image analysis.

1 Introduction

Geological systems for the analysis of multispectral images taken from satellites

and airplanes usually require diﬀerent experts and a lot of time. Actually, it is

possible to spend months or even years in order to create an accurate map of a

wide region. This information is embedded in a database which can be used, for

instance, to create thematic maps and temporal reconstructions of the events

(e.g. the evolution of the pollution, the type of the vegetation, the establishment

of settlements, ...). Recently, a speciﬁc problem has been arisen by the research

group of Dipartimento di Scienze della Terra in Siena, Italy, and it consists of the

recognition of the outlines due to faults, usually perceived as darker zones. This is

of great interest in several tasks such as cartography, underground or superﬁcial

characterization, the location of spring of water, the estimation of the stability

of the slopes, and the driving in the extraction of stones. Traditionally, this duty

is based on the identiﬁcation of certain features (e.g. geometrical relations of the

faults such as size, position and orientation) through campaigns and handmade

analysis of the photos by using expensive hardware and software.

Although we have tested diﬀerent techniques, such as the Hough transform,

we will refer to three approaches which have given the most promising results.

Here the suitability for high resolution image analysis of a well known wavelet

algorithm is examined and its results are compared with those obtained by other

methods already present in the literature. Tests on real images show that it is

possible to analyze complex structures in a very fast way.

This

ork has been partially supp orted by ASI - Italian Space Agency - under

grant I/R/117/00: “Tecniche ottimizzate di elaborazione, memorizzazione ed indi-

cizzazione di immagini telerilevate”.

O. Cepero Díaz A., Di Gesù V. and Valenti C. (2004).

A New Approach for Geological Faults Detection.

In Proceedings of the 4th International Workshop on Pattern Recognition in Information Systems, pages 216-221

DOI: 10.5220/0002686702160221

 SciTePress

2 Standard methods

Mathematical morphology is a branch of digital image analysis which uses con-

cepts of algebra and geometry [1, 2]. It has been successfully applied in the ﬁeld

of remote sensing to discriminate tree species [3], to segment soil maps [4], and

to extract roads, rivers, villages [5].

2.1 Standard mathematical morphology

A fast algorithm for directional morphological ﬁltering to extract long thin ob-

jects in satellite images has been introduced in [6]. In our ﬁrst technique, we

have implemented both the erosion ε and the dilation δ of the input image I:

(I)(p) = min

q ∈X

I(p + q) and δ

(I)(p) = max

q ∈X

I(p + q)

where X represents, along a variety angles, a binary bar with generic length and

width. In practice, the erosion (dilation) is deﬁned on the grey level image I

by computing the local minimum (maximum) around each point p. The exact

shape of this neighborhood is easily deﬁned by that of the structuring element

X. Assuming a fault as the union of consecutive segments, darker than the

background, we have enhanced it by applying the opening operator γ = δε. We

must do it a lot of times along diﬀerent orientations to highlight all the faults

and the ﬁnal map will contain the local maxima among all opened images. An

example for 16 angles and bars of 15 × 3 pixels is shown in Figures 1a,b.

2.2 Fuzzy mathematical morphology

In order to apply fuzzy mathematical morphology [7, 8], we transform both the

image I and the non-ﬂat structuring element X in fuzzy sets and reinterpret their

intensity values as a grade of membership. All usual mathematical morphology

operators can be generalized and it is possible to extend to grey level images some

inequalities that hold for binary images in the case of standard mathematical

morphology. Here, we will recall only the deﬁnition of correlation:

1 +

q ∈X

(X(q) − µ

) · (I(p + q) − µ

)

2|σ

where µ

(σ

) and µ

(σ

) are the mean (standard deviation) values of X

and the sub-image of I, with the same shape and size of X and centered in p.

Best correlation (i.e. ρ = 1) is obtained when the considered sub-image is

exactly like X. This deﬁnition of correlation exalts both dark and bright outlines

and therefore we have to select the former ones by computing their average

luminosity. As for the morphological opening, we then calculate the image of

local maxima. Figure 1c shows the map of matching degree obtained by rotating

along 16 directions a Gaussian structuring element of 21 × 9 pixels.

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Fig. 1. a: aerial photo (4096 × 4096) of the Caprera Isle. The black zones corresponds

to military sites. b: mathematical morphology erosion. c: fuzzy correlation. The struc-

turing elements, not on scale, are superimposed on (b,c). d–h, i: wavelet and residual

planes by the `a trous algorithm. All histograms have been stretched for clarity.

3 Analysis based on wavelets transform

Wavelets provide an alternative approach to signal processing and constitute a

link between mathematics, physics and electrical engineering. The so called `a

trous procedure [9] maps the input signal to its coeﬃcients with respect to a

basis of wavelet functions, constructed by dilation and translation of a mother

wavelet function [10]. The underlying idea is simple and the process, sketched as

follows, is very fast. At step i, get a smoother version c

of I by convolving with

a high-pass ﬁlter and compute the wavelet coeﬃcients w

by a low-pass ﬁlter:

(p) = I(p), c

(p) =



i−1

(p + 2

i−1

q), h(q)

, and w

(p) = c

i−1

(p) − c

(p).

In practice, both c

and w

have the same size of I. Figures 1d–i show the ap-

plication of the `a trous algorithm. The planes w

and w

contain the highest and

lowest frequencies, that is the ﬁnest and coarsest details of I, respectively. When

the desired resolution j has been reached, the original image can be expressed

as the sum of all w

and the ﬁnal smoothed approximation c

I(p) =

i=1

(p) + c

(p).

218

The kernel has been obtained via a B

-spline function:

h =





1 2 1

2 4 2

1 2 1





This function has been selected thanks to the fact that it is compact (reducing

the boundary artifacts at block edges), isotropic (not privileging any direction),

and limited to 9 elements, whatever the level i of the scale is (an average personal

computer can calculate each plane w

of a high resolution photo in about 5

seconds). Moreover, other classical scaling functions such as B

-spline, M eyer

or Daubechies

[11, 12] seem to return too smoothed wavelet images.

4 Experiments and comparison of the methods

After applying the above methods we binarize their results by just setting a

threshold value equal to the average luminosity of each single image. This is

valid for all three approaches but, of course, we select the right wavelet plane

according to a priori knowledge of the faults in input image. To compare the

output of each method we have considered only the plane w

In order to put better in evidence the faults we calculate the skeleton of the

selected binary images [13]. Usually the corresponding graph is too dense and

we have to prune it. In particular, many of its cycles are due to small bright

spots inside the faults of the input image. We have veriﬁed that, before of the

skeletonization, it is suﬃcient to eliminate in the binary images all black 4-

connected components which are too small (area less than 35 pixels in the case

of morphology or area less then 150 pixels in the case of the wavelet transform)

and/or too dark in the original photo (average luminosity less than 25%). Please

note that a white zone in the binary images correspond to a possible fault in the

original photo. Finally, we remove all arcs (considered as 8-connected objects)

that are isolated and too small (length less than 20 pixels). Figures 2 illustrates

the main steps of the pruning procedure. We want to stress that we have carried

out an extensive simulation to determine a suitable set of parameters needed to

threshold the auxiliary images or to prune the graphs.

We have analyzed high resolution photos, provided by Dipartimento di Scienze

della Terra in Siena, Italy, [14] or freely downloaded from the web [15, 16]. The

results obtained by the pipeline of methods above describ ed have been compared

with those carried out by photo-interpreters. See Figure 3 for the output rela-

tive to Figure 2a. To verify the robustness of the whole procedure we have also

introduced a range of uniform salt and pepper noise. A successive convolution

with a Gaussian ﬁlter was enough to disperse the inﬂuence of the wrong pixels.

Morphological opening is simple and fast, thanks to the use of temporary

look-up tables. In practice, ﬁltering along each orientation can be applied in

linear time, so allowing to reach a high angular resolution. We have observed

that pairs of close and parallel faults are often misinterpreted as one bigger line.

Fuzzy correlation makes use of ﬂoating point operations and has a time com-

plexity proportional to the size of the structuring element. As for mathematical

morphology, the choice of the structuring element is very important and we

have deﬁned many of them to correctly detect faults with diﬀerent shapes and

dimensions. Nevertheless, it has correctly detected most of the faults.

219

A faster execution and a higher accuracy have been achieved by the wavelet

transform. Its multiresolution support natively puts in evidence structures with

diﬀerent sizes and, despite the other two previous approaches, the skeleton is

smooth. Moreover, no custom kernel is needed to analyze faults with particular

slopes, which can be extracted during the pruning phase.

Fig. 2. a: a detail (400 × 400) of Figure 1a. b,c: raw and cleaned skeletons of w

5 Conclusions and perspectives

A new unsupervised detector of geological faults in high resolution photos taken

from satellite and airplane has been presented. This can be useful in solving

several tasks such as digital cartography, underground or superﬁcial character-

ization, geographical information interpretation. At present, geologists have to

analyze almost by hand a huge amount data, to be integrated with information

obtained through campaigns. The whole procedure is expensive and very slow.

We have compared three diﬀerent methods based on mathematical morphol-

ogy, fuzzy morphology and wavelet analysis. The ﬁrst two methods process the

input image through structuring elements with a variety of sizes and orienta-

tions and have returned similar outputs. Their accuracy strictly depends on the

choice of the shape of the structuring element and are sensitive to the change of

luminosity. For example, they usually tend to cross the streets, when partially

occluded by trees. The latter approach has shown to be fast and accurate when

putting in evidence both ﬁne and coarse outlines. It is based on a sequence of

convolutions with a compact kernel. Moreover, it does not privilege faults with a

speciﬁc direction and therefore there is no need to consider a set of orientations.

A skeletonization let us reduce the size of the binarized regions of interest,

obtained by any one of the above methods. The new image can be considered

as a graph and its arcs represent pieces of faults. We have demonstrated that

pruning the graph enhances the ﬁnal result though more eﬀorts can still be made

in order to make the shape of the graph smoother by computing, for example,

the minimum spanning tree and regularization formulas.

The eﬃciency of the detector has been veriﬁed with a database of real images

and we have introduced salt and pepper noise to test the robustness of the meth-

ods. We have roughly estimated the running time, showing that these algorithms

are fast enough. Moreover, it should be relatively easy to embed them on the

actual software, such as Imagine and ArcView, already employed by geologists.

Acknowledgements

We wish to thank the group of Dipartimento di Scienze della Terra in Siena, led

by Prof. Carmignani, and esp ecially Dr. Disperati for providing the input data.

220

Fig. 3. a: morphological opening. b: fuzzy correlation. c : w

of the `a trous algorithm.

g–i: cleaned skeletons of (a–c) superimposed on Figure 2a.

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