Fault and gradient fault detection and reconstruction from scattered data

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Highlights

  • Fault and gradient fault detection and reconstruction from scattered data.

  • Detection algorithm based on minimal differentiation formulas.

  • Applications to edge detection and geology.

Abstract

We suggest an algorithm for fault detection from scattered data that relies on new fault indicators to detect clouds of points enclosing the faults, and a reconstruction phase that includes a narrowing step to thin the detected point clouds, the construction of ordered sequences of points representing all fault curves, and localization of possible areas of intersection among different faults. The fault indicators are based on recently introduced minimal numerical differentiation formulas for gradient or Laplacian on irregular centers, which bypasses any intermediate gridding of the data. We show that our indicators provide lower bounds for local Hölder norms of any function the data may have been sampled from, and investigate their asymptotic behavior when the spacing between the data sites goes to zero. An application to edge detection in 3D surfaces is also proposed and a selection of numerical examples illustrates the performances of the method, including the identification and reconstruction of multi-branch faults and applications to terrain investigation and edge detection.

Introduction

Discontinuity curves in surfaces approximating geographical data are usually indicated as faults or gradient faults for ordinary and first order derivative discontinuity, respectively. The detection of discontinuities is a common problem arising also in different application settings, like 2D and 3D edge detection in image processing and computer graphics. Note that discontinuities in derivatives are also sometimes called creases.

Fault detection and reconstruction algorithms have usually to be combined with scattered data approximation techniques to properly deal with irregular real data configurations. On the other hand, schemes based on computational grids that exploit their regular structure are more common. For example, the approximation of scattered data with (ordinary) discontinuities was addressed in Arge and Floater (1994) by using two-dimensional grid functions and assuming that the fault curve was provided in advance. The approximation of unknown fault lines instead was addressed, e.g., in Allasia et al. (2009); Crampton and Mason (2005); Gutzmer and Iske (1997). Examples of both ordinary and derivative discontinuity detection in one and two dimensions based on central differences of various orders can be found in Cates and Gelb (2007) and Bozzini and Rossini (2013), respectively. Alternative solutions that avoid intermediate grid approximations include the non-smooth surface reconstruction scheme based on moving least squares investigated in Amir and Levin (2018), and methods based on radial basis functions, see e.g., Romani et al. (2019) and references therein. Discontinuity detection in one and two dimensions was also addressed in Archibald et al. (2005) by exploiting local Taylor expansions and polynomial annihilation.

Edge detection algorithms that enable the automatic identification of sharp features characterizing digital models defined in terms of 3D point clouds are of significant interest in different application, ranging from computer aided design geometries of industrial complexity to modeling and processing techniques in graphics algorithms. The data sets are often obtained through 3D scanning devices which generate point clouds of different nature. Recent results in this context include for example Weber et al. (2010); Tran et al. (2015); Cao et al. (2017). Methods addressing the problem of detecting not only edges but also other kinds of features (e.g. local extrema of the curvature) can be found for example in Dey and Wang (2013); Hildebrandt et al. (2005); Pauly et al. (2003).

To directly deal with the non-uniform nature of scattered data sets, we rely on the minimal numerical differentiation formulas (MNDFs) defined on irregular centers proposed by Davydov and Schaback (2018). For this class of formulas that minimize a given absolute seminorm, error bounds expressed in terms of a growth function that encapsulates the information on the geometry of the centers can be derived.

In this paper, we develop a fault detection algorithm by introducing suitable fault indicators that also allow us to distinguish ordinary and gradient discontinuities. We show that our indicators provide lower bounds for appropriate local Hölder norms of any function from which the data may have originated, and hence the high values of the indicators strongly suggest that the local sample is crossed by a fault. Moreover, we show that the indicators are unbounded for points located directly on or near the faults when the spacing between the data sites goes to zero. Consequently, they can be suitably exploited for detecting the faults when the data are sufficiently dense. The problem of fault curve reconstruction is also addressed with special attention to the treatment of intersections and multi-branch configurations. Finally, we also present the extension of our method to 3D edge detection. A selection of numerical examples illustrates the performance of the method, including the identification and reconstruction of multi-branch and closed faults, the detection of faults in geophysical data, and the application of our 3D edge detection algorithm on two well known benchmarks. Note that preliminary results on the application of MNDFs to the detection of ordinary faults were presented in Bracco et al. (2018).

The structure of the paper is as follows. Section 2 introduces minimal numerical differentiation formulas, while Section 3 presents the new fault detection indicators directly acting on the given set of scattered data. A related algorithm is given, together with a rigorous theoretical analysis of the behavior of the indicators. Section 4 explains the curve reconstruction algorithm adopted here, that covers the case of intersecting faults. An application of our indicators to edge detection in 3D surfaces is also given in Section 5, while Section 6 offers a selection of numerical examples aimed to cover all geometrically distinct cases (faults of different kind to be classified, intersecting or tangent faults). These include an example of detection of ordinary faults on a terrain sampled at scattered locations with a scanner technology and two examples of application to 3D edge detection. A conclusion is given in Section 7.

Section snippets

Minimal differentiation formulas

Minimal numerical differentiation formulas introduced in Davydov and Schaback (2018) are aimed to approximate the value assumed at a given point by any linear differential operator applied to an admissible multivariate function known only on a set of scattered points. Thus, they may be considered an extension to scattered data of well-known finite difference formulas that require function information on a grid. In this section we give a brief introduction to MNDFs, since this is the

Fault detection

Given a set of scattered data XΩ with associated function values f(x), xX, our goal is to use MNDFs to detect and reconstruct curves across which a piecewise smooth real valued function f either has a finite jump (ordinary fault) itself or its gradient has a finite jump (gradient fault) but f is continuous. Note that ordinary and gradient faults are evident on the surface obtained as a graph of f, since they correspond to curves along which the surface respectively has a jump or a crease.

This

Fault curve reconstruction

Even if there are important real-world applications for which the results from detection are already the expected output – see Section 6 for two significant cases – in other situations it can be desirable to have in output a set of curves approximating all the present faults (for example for the formulation of a surface approximation scheme which requires in input such kind of shape information). Thus in this section we introduce the approach we have used for this aim which is able to handle

Application to edge detection

The approach we have presented essentially allows to detect discontinuities on a surface S when the given data are pairs (x,f(x)), xR2, that is, we are assuming that S is the graph of f. This suggests that we can extend our approach to any surface S which can be considered, at least locally, as the graph of a bivariate function. In particular, we will focus on the detection of gradient faults, as this naturally allows to perform edge detection, a very relevant application in the context of 3D

Numerical examples

In this section we present several examples to highlight the features of our method. In 6.1 Non-intersecting fault and gradient fault, 6.5 Intersecting faults forming closed curves the employed data are obtained by sampling a test function at the points of a given set. In these cases we assume that the type of faults is unknown, so we apply the full detection algorithm of Section 3 to detect and classify the faults, which are then reconstructed too with the method presented in Section 4. The

Conclusion

In this paper we introduce and investigate theoretically and numerically a new fault detection method based on indicators defined in terms of minimal numerical differentiation formulas with irregular centers. As a consequence, our scheme naturally applies to arbitrary scattered data configurations, and is able to distinguish between ordinary and gradient faults. The presented reconstruction algorithm extends a technique based on the computation of local linear regression lines to the case of

Declaration of Competing Interest

None.

Acknowledgements

This work has been partially supported by INdAM through Finanziamenti Premiali SUNRISE and Progetto di Ricerca GNCS 2019 “Metodi di approssimazione locale con applicazioni all'analisi isogeometrica e alle equazioni integrali di contorno”. Cesare Bracco, Carlotta Giannelli and Alessandra Sestini are members of GNCS-INdAM.

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