Feature Line Detection on Triangulated MeshesA Geological Application

Dimitri KudelskiLSIS, UMR CNRS 6168

Campus de LuminyCase 925

France, 13288 Marseille cedex 9kudelski@univmed.fr

Jean-Luc MariLSIS, UMR CNRS 6168

Campus de LuminyCase 925

France, 13288 Marseille cedex 9mari@univmed.fr

Sophie ViseurGSRC, EA 4234

Université de ProvenceCase 67

France, 13331 Marseille cedex 3sophie.viseur@univ-provence.fr

ABSTRACTWe present in this article an algorithm dedicated to the feature line detection on 3D triangulated outcrop meshes.These lines corresponding to geological elements can be extracted by geometrical properties. Our approach usesdifferential quantities and especially principal curvatures and their derivatives. The roots of these derivativesdescribe particular lines called ridge lines for convex parts and ravine lines for concave parts. Then it is possibleto build a set of polylines matching with ridges and ravines. Finally we apply a directional filtering to keepgeological structures oriented in a particular direction. The proposed algorithm fits in a basis of a tool devoted toassist geologists during the outcrop analysis and interpretation.

Keywordsgeometric modeling, differential geometry, discrete curvatures, crest lines

1. INTRODUCTIONMany works dedicated to the crest line de-tection have been proposed these last years(e.g., [PKS+01, OBS04, YBS05]). Applicationfields of these methods are wide and various: non-photorealistic rendering [JDA07], mesh segmen-tation [SF04], medical imaging [MAM95], andgeology [Nam08].

Since a few years, the LIDAR1 scanning technol-ogy is used to capture cliffs or, more generally,outcrops (i.e., formations of rock strata that crop out).It generates a 3D point cloud which is afterwardstriangulated to obtain a surface corresponding to theoutcrop geometry. Combined with photo mappingtechniques, it is possible to construct 3D models calledDOMs2 [BKJ05]. From this point, we propose asemi-automatic method devoted to the detection ofgeological objects (i.e., fractures and stratigraphiclimits) from outcrop surfaces. This kind of elements ischaracterized by differential properties explained in thefollowing. Therefore, the extraction is a problematicsimilar to the crest line detection. However beforeapplying a method of crest line detection to outcrop sur-faces, several particular constraints must be considered:

Outcrop rugosityThe intrinsic rugosity of observed outcrops makes

1 LIght Detection And Ranging2 Digital Outcrop Model

the generated surfaces highly complex. As the crestlines are characterized by curvature derivatives, thisextraction is noise-sensitive. It is then necessary to usea noise-invariant and triangulation-invariant curvatureestimator.

Results matching with observationsThe presented method aims at detecting geologicalobjects. Nevertheless, when applying traditional algo-rithms of crest line detection, the extracted features donot entirely correspond to elements with a geologicalmeaning. An a priori knowledge is then necessary torealize a filtering to only extract targeted geologicalstructures.

InteractivityAn additional constraint is the computational time dueto the final application. The detection must be per-formed in a few seconds in order to keep interactivitywith a real-time procedure. Moreover, this is partic-ularly crucial as LIDAR scans often generate hugedata sets which are difficult to manipulate. Because ofthis, we take great care to implement process with lowcomputational time.

To understand the crest line detection problem,Section 2 describes the different criteria characterizingthe geological objects. We review in Section 3 therelated work established in the domains of curvatureestimation and crest line detection. Then we detaileach step of our approach in Section 4. Section 5

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finally presents the results obtained with our algorithmapplied on LIDAR data scans.

2. CHARACTERIZATION OF GEO-LOGICAL OBJECTS

Fractures are like crevices more or less opened thataffect a rock mass. Stratigraphic limits of geologicalbodies correspond to a change of rock type. Bothof these geological features are displayed along theoutcrop surface because of the erosion. It leads finallyto step-like or a gutter-like shapes at their location.Figure 1 represents a diagram with the differentpatterns of targeted geological objects. Moreover, thispattern often varies along the same fracture or stratalimit. Given Figure 1, it is indeed possible to see thatthe expected objects (depicted by the thick dashedlines) are located in the highest concave parts of thesurface. These elements have a common geometricalcriterion: they define lines located in areas with highcurvature. Thus crest line algorithms can be applied toachieve the detection of such objects.

rock

rock

rock

rock

Figure 1: Diagram showing the different patterns of ge-ological objects. The thick dashed lines illustrate thehighest concave areas characterizing the expected fea-tures.

Feature lines are then defined by curvature ex-trema and then it corresponds to a zero-crossing ofcurvature derivatives. However, the rough set of crestlines extracted from a DOM does not represent the setof targeted geological objects. This is due to severalfactors, among which: (1) fractures often cut acrossstrata limits. Depending on the erosion effect, anextracted crest line could then encompass features withdifferent geological meaning; (2) the intrinsic rugosityof the rock or the variable direction of the outcropcan generate salient lines which do not represent anyexpected geological feature.

For these reasons, we suggest to add an a prioriknowledge (i.e., a global direction) to guide the extrac-tion and filter feature lines. Our approach is dedicatedto the detection of slightly sinuous structures. It isalways the case for the fractures and very frequentlyfor the strata limits.

The proposed method then relies on the crest lineprinciple. It previously requires a per-vertex estimation

of differential quantities. Before describing eachalgorithm step, the following section gives an overviewof existing crest line detection techniques.

3. RELATED WORK3.1 Curvature EstimationDifferential properties characterize the local geometryof meshes. The notion of curvature describes preciselyhow the surface is locally bent. These geometricaldescriptors are then used since a few years and severalapproaches have already been proposed in this domain.Some of them are presented in the following (foradditional references see [MD02, GG06]).

Continuous methodsThis type of methods tends to fit locally the surface withsimple primitives (e.g., plane, sphere or polynomial)or parametric functions or even implicit functions.These different techniques permit an analytical com-putation of curvatures. For example, in [Ham93], theauthors proposed to approximate locally the surfacewith quadratic polynomials. Alternatively in [GI04],the fitting is performed via bi-cubic polynomials.Bi-quadratic Bézier patches can also be used to fit thesurface such as in [RB05].

Discrete methodsTo reduce the high computational time produced bylocal fitting, differential operators have been proposed.In [MDSB02], the authors suggested to use a curvatureestimation based on cotangent weights and Voronoïareas. In another way, the dihedral angle (i.e., anglebetween the normals of two adjacent faces) can beused as a discriminant property to compute curva-tures [CSM03]. Additionally, the curvature tensorcan be estimated by studying the per-vertex normalvariation such as in [Rus04, BW07].

3.2 Crest Line DetectionThe properties of crest lines are widely used fortheir efficiency as shape descriptors. This domainhas become a field of intensive researches sincethe last decades and several approaches have beenthen proposed. The first family of techniques isbased on extrema searching. It can be performedeither by thresholding [RKS00, SF03], curvaturederivatives [CP04, OBS04, YBS05], focal sur-faces [LA98, WB01, YBYS07], or discretizedoperators [HPW05].

The second kind of methods relies on other dif-ferential properties. The dihedral angle can be usedto detect sharp features such as in [HG01, PSK+02].Then in [GPHW05], the authors proposed to applyactive contour theory stemming from image processing

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domain to detect characteristic lines. In additionin [LVJ05], Lee suggested to use a measure of aregional importance named mesh saliency based oncontextual and visual criteria.

4. GEOLOGICAL FEATURE DETEC-TION

On the one hand, DOMs represent natural surfaces.These objects are characterized by an inherent noisedue to the acquisition technology and a high intrin-sic rugosity because of the surface alteration. On theother hand, due to their definition by high differen-tial quantities, crest lines are very noise-sensitive. Itis then necessary to select a robust curvature estimator.For these reasons, we chose to apply the method pro-posed in [GI04] considering its quality, accuracy andstable results (see [GG06]). Concerning the crest linedetection method, we opted for the criteria expressedin [OBS04]. It relies on curvature derivatives and thusis scale-invariant. It is then possible to extract geologi-cal objects with different sizes.

4.1 Pre-processing StepThe inherent noise and rugosity of the data make thedetection of smooth and continuous lines difficult. Wethen propose to use a pre-processing to increase thesecontinuity and smoothness. Among all existing tech-niques, we chose to integrate a Laplacian smoothing(cf. Equation 1) on surface coordinates:

p′ = p+λ1n

n

∑i=1

(qi− p), (1)

where n is the number of adjacent vertices qi to thevertex p and λ represents a step-size parameter.

Once the smoothing performed, the next step isto compute the differential quantities in order to detectthe surface crest lines.

4.2 Estimation of Curvatures and theirDerivatives

Several techniques of curvature estimation have beenpreviously presented. The approach proposed in [GI04]fits locally the surface with a bi-cubic polynomial in theleast-squares sense. Thus, the surface is expressed foreach vertex thanks to the following equation:

f (x,y) =A2

x2+Bxy+C2

y2+Dx3+Ex2y+Fxy2+Gy3.

(2)The Weingarten matrix (i.e., the matrix of the secondfundamental form) of the surface is therefore composedas:

W =

[A BB C

]. (3)

The curvature values κmax and κmin (with|κmax| > |κmin|) are defined by the eigenvalues ofW and the eigenvectors of W correspond to the prin-cipal curvature directions ~tmax and ~tmin. To obtainthe curvature derivatives, it is possible to use thecoefficients D, E, F and G of Equation 2 as suggestedin [YBS05]:

e =∂κ

∂~t=

[u2

v2

]T [D EF G

][uv

](4)

where~t = (u,v) (5)

can correspond to either~tmin or~tmax. Consequently, twovalues called extremality coefficients (cf. [Thi96]) arethen defined by:

emax =∂κmax

∂~tmaxemin =

∂κmin

∂~tmin. (6)

These coefficients are the support for the crest line de-tection, as described in the next section.

4.3 Crest Line DetectionThe extremality coefficients describe curvature varia-tions and crest lines are located where curvature ex-trema are reached. Thus, the crest lines are character-ized by:

emax =∂κmax

∂~tmax= 0,

∂emax

∂~tmax< 0, κmax > |κmin| (7)

for the ridge lines (convex areas) and:

emin =∂κmin

∂~tmin= 0,

∂emin

∂~tmin> 0, κmin <−|κmax| (8)

for the ravine lines (concave areas).

The curvature sign gives information about the locallyconvexity or concavity of the surface. Ridges andravines are dual notions according to the surface ori-entation: by flipping the surface orientation, convexityand concavity are swapped as for ridge and ravine lines.

As previously mentioned, extremality coefficientsas derivatives, are highly sensitive to noise. For thisreason, the pre-processing of smoothing the surfacegeometry is applied to compute the derivatives.However, original coordinates are restored beforeperforming the detection. In this way, noise impact isreduced and even several artifacts due to the intrinsicsurface rugosity are removed while maintaining the ac-curacy about the locations of the extracted feature lines.

Crest line detection is performed by searchingcrest vertices and curvature extrema (i.e., roots ofcurvature derivatives). Let be ε an edge composed by

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the vertices v1 and v2. A vertex is considered as a crestvertex since a set of conditions described in [OBS04]is satisfied. For the sake of clarity and simplicity, onlythe case of ridge vertices is explained below. As ridgesand ravines are dual notions, explained conditions canbe easily transposed from a ridge to a ravine detectionalgorithm.

First, if the angle between principal directions~tmax(v1) and ~tmax(v2) is obtuse, the vector ~tmax(v2) isflipped as the sign of emax(v2). The second step isto check if there is a zero-crossing of the curvaturederivative on the edge. It appears when the signs ofemax(v1) and emax(v2) are different:

emax(v1) · emax(v2)< 0. (9)

Curvature must also reach a local maxima which can beverified by a derivative test:

emax(v1)[(v2− v1) ·~tmax(v1)

]> 0. (10)

When Equations 9 and 10 are satisfied, the coordinatesof the ridge vertex are found by a linear interpolationbetween v1 and v2:

vridge =|emax(v2)|v1 + |emax(v1)|v2

|emax(v1)|+ |emax(v2)|. (11)

This process is applied on each edge of the mesh toobtain all the crest lines. These lines are defined bypolylines built from crest vertices. Figures 2 and 3summarize the method of crest line extraction andconstruction.

P1

P2

P4

P3

emax > 0

emax < 0

emax < 0

emax < 0Ridge vertexemax = 0

Figure 2: Process of ridge vertex extraction.

The proposed algorithm does not aim at extract-ing all crest lines but only geological feature lines.Thus, particular conditions must be honored during thefeature extraction.

Figure 3: Construction of a feature line. On left, anisolated crest vertex can not define a line. In the middle,two crest vertices generate a straight line. Lastly, 3 crestvertices produce a T-junction between the three verticesand the triangle barycenter.

4.4 Directional FilteringIn order to keep only lines which have a geolog-ical meaning and are roughly oriented in a sameuser-defined direction ~D, an a priori knowledge isintegrated. It corresponds to a filtering process addedto the detection algorithm previously described.

First, as mentioned in Section 2, only concave partscorrespond to fractures or strata limits. Therefore onlyravine lines characterize relevant objects. Secondly,geological structures are generally slightly sinuous.Their detection can be guided via an user-defineddirection ~D, corresponding to the rough direction of afamily of targeted geological structures observed alongthe outcrop.

Let S be a surface of R3 and p a point of S.Principal directions of p are contained in a plane Poriented according to ~Np (i.e., the normal vector of p).As the shape of the geological objects can be locallydescribed as parabolic surfaces, the curvature vec-tor~tmin tends to follow this shape as shown by Figure 4.

tmax

tminPFigure 4: Principal curvature directions along aparabolic shape.

It is thus possible to use the direction of ~tmin tofilter lines oriented in the same direction of ~D. How-ever the direction ~D is set globally by the user on theoutcrop. The outcrop surface is not totally flat andits direction can vary locally. Thus it is not ensuredthat the vector ~D will be contained in the plane P .Therefore, a rotation is applied to transform ~D into ~D′

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and to place this vector into the plane P . This rotationhas the following parameters:

−−→axis = ~D× ~Np

angle = ~D · ~Np. (12)

Once the rotation is applied, ~D′ is contained in theplane P . A projection of ~D onto P could not havebeen considered as it may generate a null vector ~D′ assoon as P is perpendicular to ~D.

Finally, on the edges containing a root of curva-ture derivative, the absolute value s′ of the dot productbetween ~D′ and ~tmin is computed. Therefore whenboth ~tmin vectors of an edge are collinear to ~D′, theline is preserved otherwise it is removed. This step isillustrated by Figure 5.

D’

tmin

s’

Figure 5: Directional filtering according the vectors ~D′and~tmin.

The direction ~D is specified globally by the userand corresponds to the rough direction of the expectedstructures. However the direction of these objects mayvary locally. Thus a threshold T , ranged from 0 to 1 isapplied on s′ as a tolerance factor: if T is equal to 1, thevectors ~tmin and ~D′ must be strictly collinear to keepthe line and inversely if T equals 0, all the ravine linesare kept.

5. RESULTS AND VALIDATIONThe proposed approach devoted to the detection ofgeological objects onto numerical outcrop surfaces iscomposed of four main operations:- a pre-processing smoothing;- an estimation of curvatures and their derivatives;- a crest line extraction;- a directional filtering.

This algorithm is dedicated to the detection ofgeological features (i.e., fractures and strata limits)

from 3D triangulated meshes built from LIDAR datapoints. Figure 6 shows the results obtained withdifferent outcrop models. Figures 7 and 8 display theimpact of the direction ~D and the threshold T onto thedetection of targeted geological features.

These parameters have to be set up manually bythe user. They represent an a priori knowledge aboutthe targeted geological structures to interpret. Thedirection ~D can be determined by the geologiststhrough the observation along the numerical outcrop.It may be noticed that this parameter could be alsoautomatically deduced from a heuristic such as aprincipal component analysis. However the primarygoal of the proposed approach is to assist the geologistsin the outcrop interpretation. Moreover, due to thecomplexity of geological structure spatial organization,the full automatization of the algorithm could easilylead to several mismatch between geological realityand extracted lines which should be in fact removed aposteriori using manual or automated filtering. Then,the tolerance threshold T is used to constraint moreor less the detection to the fixed orientation. It is setup according to the aspect of the observed limits (i.e,straight or slightly sinuous).

The results obtained with the presented approachmatch with geological objects observed on outcropsand manually modeled by geologists. We noticehowever that some lines are incomplete or non-significative. This is due to umbilical points (i.e.,points locally spherical) without principal direction.

The computational time of our algorithm is low: itonly requires less than 5 seconds (in part due to thecomputation of curvature values and their derivatives)to detect geological features of a surface composed byabout 100k triangles (computed on an Intel Core 2 Duo2.8 Ghz).

6. CONCLUSIONSeveral methods of crest line detection have beenproposed in the litterature. However none was directlyapplicable to the context of geological feature extrac-tion from 3D digital outcrop models. By relying onexisting methods, we thus present an algorithm devotedto the feature line detection from LIDAR data scansatisfying new constraints.

The proposed approach is based on the estimationof curvature values and their derivatives. The extremal-ity coefficients are computed from curvature derivativesto obtain ridge and ravine lines. Finally, a directionalfiltering is applied to preserve lines with geologicalmeaning and oriented in a particular direction.

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Feature lines corresponding to fractures and stratalimits are extracted. The proposed tool enables thegeologists to be assisted during the outcrop interpre-tation stage. As mentioned previously, the obtainedresults match with the elements manually modeled bygeologists.

This approach is promising and can be improved.We plan to add post-processing to increase the qualityof results concerning, for instance, the connectivityenhancement and the artifacts removal. In addition, ex-tracted lines are slightly sinuous which concerns mostof the targeted geological objects. Though, some stratalimits are actually sinuous. This requires a pertinentrelaxation of the proposed directional filtering.

The feature extraction corresponding to geologi-cal objects is a first step in the outcrop interpretationworkflow. The next step would be the construction,from the extracted elements, of a graph to reproducethe layout of observed geological structures.

ACKNOWLEDGMENTSThe authors would like to thank ENI S.p.A. to supportthis research.

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Figure 6: Application of our algorithm on LIDAR data scans without any filtering. On left, feature detectionperformed on the Malaval section (' 60000 triangles). On right, extraction of lines of the Pas-Morta section(' 20000 triangles).

Figure 7: Application of two directional filters. The strata limits are extracted with a horizontal direction (leftimage) while the fractures are detected with a vertical direction (right image).

Figure 8: Influence of the tolerance threshold T with values of 0.85 (left image), 0.70 (middle image) and 0.55(right image).

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