Geosphere
Robust best-fit planes from geospatial data
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Full Title:
Robust best-fit planes from geospatial data
Short Title:
Robust best-fit planes from geospatial data
Article Type:
Research Note
Keywords:
orthogonal regression; total least squares; plane-fitting; eigenvalues
Corresponding Author:
Richard R. Jones
Geospatial Research Ltd / University of Durham
Durham, UNITED KINGDOM
Corresponding Author Secondary
Information:
Corresponding Author's Institution:
Geospatial Research Ltd / University of Durham
Corresponding Author's Secondary
Institution:
First Author:
Richard R. Jones
First Author Secondary Information:
Order of Authors:
Richard R. Jones
Mark A. Pearce
Carl Jacquemyn
Francesca E. Watson
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UNITED KINGDOM
Abstract:
Total least squares regression is a reliable and efficient way to analyse the geometry of
a best-fit plane through georeferenced data points. The suitability of the input data, and
the goodness of fit of the data points to the best-fit plane are considered in terms of
their dimensionality, and quantified using two parameters involving the minimum and
intermediate eigenvalues from the regression, as well as the spatial precision of the
data.
Suggested Reviewers:
Thomas Seers, awaiting PhD viva
University of Manchester
[email protected]
He has recently worked on similar matters.
Nigel Woodcock
Reader, University of Cambridge
[email protected]
He has a comprehensive understanding of the use of eigenvalues to solve 3D
problems.
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Robust best-fit planes from geospatial data
Richard R. Jones1, Mark A. Pearce2, Carl Jacquemyn3*, Francesca E. Watson1
1 Geospatial
Research Ltd., Dept. of Earth Sciences, University of Durham, DH1 3LE, UK.
Mineral Resources, Australian Resources Research Centre, 26 Dick Perry Avenue, Kensington, WA
6151, Australia
3 Dept. of Earth and Environmental Sciences, KU Leuven, Celestijnenlaan 200 E, 3001 Heverlee, Belgium.
2 CSIRO
Corresponding author: [email protected]
* Current address: Dept. of Earth Science & Engineering, Imperial College London, SW7 2AZ, London, UK.
ABSTRACT
Total least squares regression is a reliable and efficient way to analyse the geometry of a bestfit plane through georeferenced data points. The suitability of the input data, and the goodness
of fit of the data points to the best-fit plane are considered in terms of their dimensionality,
and quantified using two parameters involving the minimum and intermediate eigenvalues
from the regression, as well as the spatial precision of the data.
Keywords: orthogonal regression; total least squares; plane-fitting; eigenvalues.
INTRODUCTION
Geological and geomorphological surfaces observed in nature are rarely perfectly planar.
Nevertheless, in some situations it can be useful to approximate surfaces as planes, for
example to improve computational efficiency (e.g. in Discrete Fracture Network modelling),
or when analysing the spatial distribution of deviation from planarity (e.g. Jones et al. 2009).
To define the orientation of a plane requires three non-collinear points; this is the basis of
the “three-point problem” used widely in geological mapping (e.g. Threet 1956; Berger et al.
1992) and taught in undergraduate classes and textbooks (e.g. Lisle 1988; Boulter 1989).
Most geospatial methods typically allow many more than three points (101-107 points or
more) to be rapidly measured on an exposed surface. Relevant technologies include
reflectorless total stations (e.g. Feng et al. 2001), laser range finders (e.g. Xu et al. 2000),
terrestrial and airborne lidar (e.g. Ahlgren et al. 2002; Hasbargen 2012; Cracknell et al. 2013),
digital photogrammetry (e.g. Pringle et al. 2004; Jacquemyn et al. 2015), Global Positioning
System data (GPS; e.g. McCaffrey et al. 2005), and satellite imagery (e.g. Banerjee & Mitra
2004).
The purpose of this paper is to review different regression methods used to derive a best-fit
plane from geospatial data, and to assess the criteria used to determine the quality of the
regression. We propose new criteria to replace those described by Fernández (2005).
PLANAR REGRESSION METHODS
One common method of plane-fitting is based on Ordinary (or “Standard”) Least Squares
regression, initially developed by Legendre (1806) and Gauss (1809). The approach is easiest
to illustrate in terms of two variables (Fig. 1A), where x is the independent variable and y is
dependant (or “measured”), though it is equally applicable in three (or more) dimensions.
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This method is the basis for the planar regression method used in Section 2 of Fernández
(2005).
An important limitation of the ordinary least squares method is that it assumes that all error
lies in the dependant dimension, and therefore the regression will minimise the residual errors
in that variable (Fig.1A). For most types of geospatial data this assumption is invalid, since
there are measurement errors in x, y, and z. Consequently, in many situations a more
appropriate strategy is to use “Total Least Squares” (or “orthogonal”, or “errors-in-variables”)
regression in which the residual errors are minimised in a direction perpendicular to the bestfit plane (Fig. 1B). The total least squares approach is not new (Adcock 1877, 1878; Pearson
1901), and according to Anderson (1984) has been re-discovered many times (e.g. Sturgul &
Aiken 1970, and subsequent discussion including Gower 1973 and references therein). For
many types of 3D geospatial data, using total least squares regression, in which the likelihood
of error is equal in x, y, and z, will be a valid approach to plane-fitting. For example, in virtual
outcrops derived from terrestrial lidar and/or digital photogrammetry data, we usually
consider there to be no systematically greater error in any one dimension relative to the others.
For types of data where errors are unequal (such as GPS data, where measurement precision is
poorer in z than in x and y), a weighted total least squares approach can be used (e.g.
Markovsky & Van Huffel 2007). Other modifications to the total least squares approach
include ways to identify and disregard outlying points (e.g. Li & De Moor 2002).
A significant advantage of using orthogonal regression is that the magnitudes of the
residual errors are independent of the orientation of the best-fit plane with respect to any of
the reference coordinate axes. In contrast, with ordinary least squares the magnitude of the
residuals varies with the orientation of the plane relative to the chosen coordinate axes (see
section 4.1 of Fernández 2005).
Implementation
The orientation of the best-fit plane through a set of geospatial data-points using
orthogonal regression can be calculated using singular value decomposition (Golub & Van
Loan 1980), or by deriving the eigenvectors of the covariance matrix. In programming
languages such as Octave or Matlab these can each be achieved with a single line of code:
[U, S, W] = svd(xyzPoints)
(where xyzPoints is the input matrix of geospatial data-points, S contains the singular values,
and U & W contain the left and right singular vectors), or
[V,D] = eig(cov(xyzPoints))
(where xyzPoints is the input matrix of geospatial data-points, and V & D are the
eigenvectors and eigenvalues of the covariance matrix of the input data). The best-fit plane
passes through the centroid of the input data. The maximum and intermediate eigenvectors lie
within the best-fit plane, and the minimum eigenvector defines the pole to the plane.
QUALITY OF BEST-FIT SOLUTION
As discussed by Fernández (2005) there are two aspects determining the quality of a bestfit solution: the suitability of the sampled data-points to define a plane, and the goodness of fit
between the input data and the resultant regression plane. The suitability of the input points
depends upon their spatial distribution in 3D, while the goodness of fit is related to the
magnitude of the residual errors. The eigenvalues of the covariance matrix (1 > 2 > 3)
derived from total least squares regression are useful in analysing both these aspects of the
best-fit results.
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Dimensionality
One way to conceptualise the robustness of a best-fit plane is to consider the
dimensionality of the sample data (c.f. Jones et al. 2008). Since points within a plane are
intrinsically 2D in extent, best-fit solutions for data points that sample a space that tends
towards 1D or 3D will have poorer quality (Fig. 2). If the data points are nearly collinear (Fig.
2A), such that 2 and 3 are small, then the sampled points are unsuitable for defining a bestfit plane, irrespective of the magnitude of the residual errors (in this situation, the covariance
matrix is close to a rank-one matrix, and the plane is degenerate). In contrast, if the data are
non-collinear (3 << 2), and the total residual errors are low (3 is small; the covariance
matrix is close to rank-two), then the input data are nearly coincident with a 2D plane (Fig.
2B). Finally, where the residual errors are larger (3 >> 0) then the data points are markedly
non-coplanar (i.e. the dimensionality of the data is > 2) and the goodness of fit is poor (Fig.
2C).
As with all regression methods, for orthogonal regression there is no universally applicable
measure of what constitutes an acceptable fit, so the quality of the best-fit solution is
arbitrary, and depends on the nature of the input data and the purpose to which the interpreted
best-fit planes will be used. We illustrate this with analysis of fractured outcrops measured
using terrestrial laser-scanning (lidar).
Case study: best-fit fracture planes from lidar data
Here we describe an approach that we typically use to assess best-fit planes derived from
fracture surfaces that are measured using terrestrial lidar, as part of a workflow to produce
deterministic discrete fracture network (DFN) models from outcrop analogues. For
illustrations of the lidar equipment used and further details of the method of field acquisition
see Labourdette & Jones (2007), White & Jones (2008), and Jones et al. (2009).
Following field acquisition, lidar data from multiple scanner positions are co-registered,
and georeferenced using differential GPS control points, to give a point cloud that represents
a digital copy of the outcrop surface (a “virtual outcrop model”). Representative points from
each exposed fracture surface are picked, and form the input data for the calculation of the
best-fit plane for that fracture. For further examples of fractures picked from virtual outcrop
models and analysed using the methods described here, see Jones et al. (2009), Jacquemyn et
al. (2012), and Pless et al. (2015).
A critical step is to decide two appropriate thresholds for the quality of the plane-fitting.
Firstly, to ensure that the sampled data points are sufficiently non-collinear (c.f. Fig. 2A), the
key requirement is that the extent of the data in a direction parallel to the intermediate
eigenvalue (2) is large compared with the precision of the input data. Therefore it is
important to be able to quantify the typical measurement precision of the specific type of lidar
equipment used (which varies with distance). For example, high precision medium range
scanners such as the Leica ScanStation C10 or Leica HDS 3000 (see Figure 1A of White &
Jones 2008) have low noise levels, and according to the manufacturer have an accuracy of ca.
2-6 mm at 50m range (see also Lindenbergh et al. 2005). With these types of scanners, our
empirical tests have shown that a low 2 threshold can be used; e.g. input data with 2 > 0.001
typically give well defined planes throughout the scanners’ measurement range of ca. 200m.
In comparison, when using a scanner prone to slightly more measurement noise, such as the
Riegl LMS-Z420i (Figure 3 of Jones et al. 2009), our tests indicate that a 2 threshold of
0.005 or 0.01 is often more appropriate. According to Riegl, under test conditions this scanner
can achieve an accuracy of ca. 10mm at 50m (see also Renard et al. 2006). In situations where
merged lidar data are smoothed (decimated), higher 2 threshold values may need to be used.
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A second threshold is needed to determine whether the goodness of fit of the regression
plane is adequate (c.f. Fig. 2C). A measure based on the mean square of the total residuals can
provide a useful indication of goodness of fit, independent of the size of the fracture. For
example, the maximum likelihood estimate of the noise variance is given by:
2 = (sum of squared residuals) / N
where N is the number of sampled data points. If we are examining a single surface in detail,
it can be useful to map the spatial variation in the residual errors across the surface (Jones et
al. 2009).
However, for the statistical analysis of hundreds or thousands of fractures in relation to
DFN modelling, we generally find it more suitable to allow larger fracture planes to have
higher variance, and therefore we choose to define the threshold in terms of the ratio of 2 to
3 (e.g. data points with 2/3 < 10 are rejected).
DISCUSSION
In our experience, the criteria outlined above to assess the quality of best-fit planes
consistently give robust results. This is based on the analysis of many virtual outcrop datasets
derived from lidar and/or digital photogrammetry, in a range of lithologies including
carbonates, sandstones, mudstones, and crystalline rocks. Other criteria might also be useful
and may be equally valid for different purposes.
There are important differences between the criteria we use and those described by
Fernández (2005), in which the quality of the best-fit solution is assessed based on a method
developed to quantify the shape of tectonic fabrics (Woodcock 1977). Fernández uses two
criteria, M and K, to assess quality of fit, both involving eigenvalue ratios:
M = ln(1/3)
K = ln(1/2) / ln(2/3),
where higher M and lower K values are defined to be higher quality. Critically, both M and K
are functions of 1, and we have found empirically that they do not represent good criteria for
assessing the quality of a planar regression. This is also evident from theoretical
considerations, as both M and K are scale independent ratios, and hence are not influenced by
the spatial precision of the measured data-points. This routinely causes small best-fit planes to
be erroneously classified as reliable. By way of example, in Fig. 3 if the units of scale are
millimetres, and if the input points were picked from lidar data acquired with a relatively
‘noisy’ scanner such as the Riegl Z420i, then the precision of the data is too poor for a best-fit
plane for these fractures to have statistical significance (see Trinks et al. 2005 for an example
of a high-precision short-range scanner capable of resolving features of this size). However, in
contrast, if the units of scale in Fig. 3 are decimetres (or larger) then lidar data acquired from
any scanner we have tested will have sufficient spatial precision to allow fractures on this
scale to be clearly defined.
Fig. 3 also illustrates how the K parameter is particularly inappropriate, because planes
with high 1 are likely to be rejected as unreliable because the resultant K values will be high,
although these planes are actually often very well defined in virtual outcrop data. It is
common for outcrops to contain fractures (or other types of plane) that can be reliably
delineated by picking along the trace of the intersection of the plane with the irregular outcrop
surface. These intersection traces are often very long relative to their width, but as long as the
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outcrop surface has sufficient rugosity along the trace, then the definition of the plane is
robust. For example, the intersection of the fracture in Fig. 3A with the outcrop surface is
very elongate. This results in a high 1 value, and therefore a K value that is high enough to
be deemed unreliable. In reality, however, the only critical factors are that 3 should be small,
and that there is sufficient relief on the outcrop surface, such that 2 is large in absolute terms
in comparison with the spatial precision of the measurements; the relative size of 2 in
comparison with 1 is irrelevant. This is further demonstrated in Fig. 3B&C. The fracture
shown in Fig. 3B is defined by an equant distribution of points so that its K value is low
(provided that 3 is small). Assume that we are able to extend the analysis by picking an
additional point situated on the same planar fracture (beyond the area of fracture initially
used), depicted by the red square in Fig. 3C. (Assume also that 3 remains small). Counterintuitively, the addition of an extra point, which should help to improve the definition of the
fracture, actually increases its K value significantly, such that the best-fit plane might become
assessed as unreliable. Adding further additional points (magenta stars) continues to cause the
K value to change, but it always remains higher (i.e. less reliable) than in Fig. 3B.
Consequently, our opinion is that the M and K criteria of Fernández (2005) are invalid, and
to assess the robustness of best-fit fracture planes in relation to DFN modelling, we use the
following two parameters:
Parameter 1: reject the best-fit plane if 2 < 2, (where  ∝ precision)
Parameter 2: reject the best-fit plane if 2 / 3, < C (where lower values of C allow more
curvature or irregularity of the fracture surface).
These parameters are applicable to many plane fitting problems involving 3D data, including
analysis of digital outcrop models. The specific threshold values to be used for both
parameters depend on the scale of analysis, the precision of the geospatial data, and the
desired stringency when assessing goodness of fit.
CONCLUSIONS
Total least squares regression, in which errors are minimised in the direction orthogonal to
the regression plane, provides a robust way to define best-fit planes from many types of
geospatial data. Eigenvectors and eigenvalues from the regression analysis are used to define
the orientation of the best-fit plane and to assess the quality of the results.
Criteria proposed by Fernández (2005), based on adaptation of a method to quantify
clustering in orientation data, are inappropriate and are unable to identify reliable best-fit
planes in a consistent way. Fundamentally, the method of Woodcock (1977) is a widely used,
robust, and elegant way to characterise shape fabrics from a set of orientation measurements
(i.e. vectors with no defined spatial position), but is poorly suited to analyse the spatial
relationship of georeferenced point data.
Instead, we propose a pair of criteria that in combination allow robust best-fit planes to be
derived. Firstly, the intermediate eigenvalue from the orthogonal regression must be
sufficiently large, in comparison with the spatial precision of the input data, to ensure that the
best-fit plane has sufficient breadth to be statistically meaningful (i.e. to test that the input
data are non-collinear). Secondly, the minimum eigenvalue needs to be sufficiently small to
ensure that the input points are satisfactorily close to being co-planar. This can be assessed
using the standard deviation of the residual errors from the regression, although for some
purposes a less stringent approach based on the ratio of the intermediate and minimum
eigenvalues gives appropriate results.
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Acknowledgements
We are very grateful to Sabine Van Huffel for her input on total least squares regression. ENI
funded fieldwork including lidar acquisition and data analysis during CJ’s PhD, supervised by
Rudy Swennen.
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Figure Captions
Fig. 1. Comparison between ordinary least squares regression, in which residual errors are minimised in y; and
total least squares in which errors are minimised orthogonally to the best-fit curve. Graphical depiction is easiest
in 2D, however the principle is equally applicable to plane-fitting in 3D. Data are modified from Pearson 1901.
Fig. 2. Influence of the input data (small grey spheres) on the robustness of the best-fit plane. (A) data points are
nearly collinear (sample space ~ 1D) so are unsuitable for defining a plane. (B) data points are nearly coplanar,
and the quality of the best-fit plane is good. (C) The residual errors from orthogonal regression are high (sample
space is a 3D volume), so the quality of the best-fit plane is poor. Asterisks are the centroids of the input data.
Eigenvalues are derived from total least squares regression, assuming x, y & z units are decimetres. The edges of
the bounding boxes containing the data points are parallel to the eigenvectors of the covariance matrix. Data in
the xz plane are based on values from Pearson 1901 (see input data table for each graph).
Fig. 3. Examples from Fig.2 of Fernández (2005) to illustrate how using the M and K parameters to determine
the quality of the best-fit plane gives erroneous and counter-intuitive results (see text for further explanation). In
each case the fracture lies in the plane of the diagram, such that 1 and 2 are in-plane, and 3 is parallel to the
view direction. Assume 3 to be small. The dotted grey trace represents the intersection of the fracture with the
outcrop surface. Circles, stars and square symbols are the points picked as input to define the best-fit plane, and
‘x’ depicts the centroid of these data points.
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Figure 1
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Figure 2
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Figure 3
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