# Augmented Semantic Signatures of Airborne LiDAR Point Clouds for Comparison

Jaya Sreevalsan-Nair Pragyan Mohapatra
Graphics-Visualization-Computing Lab,
International Institute of Information Technology Bangalore, Karnataka 560100,
India
http://www.iiitb.ac.in/gvcl<EMAIL_ADDRESS>
###### Abstract

LiDAR point clouds provide rich geometric information, which is particularly
useful for the analysis of complex scenes of urban regions. Finding structural
and semantic differences between two different three-dimensional point clouds,
say, of the same region but acquired at different time instances is an
important problem. Comparison of point clouds involves computationally
expensive registration and segmentation. We are interested in capturing the
relative differences in the geometric uncertainty and semantic content of the
point cloud without the registration process. Hence, we propose an
orientation-invariant geometric signature of the point cloud, which integrates
its probabilistic geometric and semantic classifications. We study different
properties of the geometric signature, which are image-based encoding of
geometric uncertainty and semantic content. We explore different metrics to
determine differences between these signatures, which in turn compare point
clouds without performing point-to-point registration. Our results show that
the differences in the signatures corroborate with the geometric and semantic
differences of the point clouds.

_K_ eywords Covariance tensor, Tensor voting, Airborne LiDAR point clouds,
Local geometric descriptors, Barycentric coordinates, Visualization, Image
processing, Shannon entropy, Signature, Comparison, Uncertainty analysis

##  1 Introduction

Topographic Light Detection and Ranging (LiDAR) technology provides three-
dimensional (3D) point clouds, which are used for reconstructing regions.
While there has been an active interest in investigating what analyses the
point clouds can exclusively provide [1], it has limitations in its data
acquisition process, amongst others. Airborne LiDAR technology suffers from
the known problem of reproducibility of logistics of acquisition of point
clouds to study temporal changes [2]. Thus, the point clouds of the same
region recorded in two different time instances may have significantly
different point densities and poses/orientations, thus leading to
misregistration errors [2]. Such differences lead to the strict requirement of
a registration process of point-to-point matching between point clouds for
further analysis. There also exist other gaps in comparison of different point
clouds. There are several methods to compare either their geometric
properties, such as Fisher modified vector [3], or semantic compositions, such
as Hamming distance of normalized distributions or symmetric Kullback Leibler
(KL) divergence. Measures that integrate both geometric and semantic
differences are rare.

Figure 1: Overview of our proposed method of using probabilistic geometric
classification for generating our proposed geometric signature of an airborne
LiDAR point cloud. We generate the augmented semantic signature by adding
semantic classification. The data is of Area-3 of the Vaihingen data [4], the
signatures are generated using $T_{\text{3DCM}}$, with Multiscale approach,
and the visualizations are point renderings. (Top row) Top view of the 3D
point cloud.

To alleviate the need for registration, the point cloud could be projected to
a different coordinate system by preserving the relationships between points,
but not their absolute position coordinates. Hence, we propose the use of
coordinate space transformations, which enable us to create a rotation-
invariant global signature of the point cloud, as shown in the overview of our
work (§Figure 1). This signature is the image of the two-dimensional (2D)
barycentric representation of the entire point cloud, constructed using
multiscale local geometric descriptors [5]. We call it the geometric signature
$\mathcal{S}_{Gm}$ of the point cloud, which encodes probabilistic geometric
classification into geometric classes, namely, line-, surface-, and (critical)
point-type features. A variant of the signature incorporates semantic
classification, which is referred to as the augmented semantic signature
$\mathcal{S}_{AgSm}$. It corresponds to a variant of the augmented semantic
classification, which labels each point with a 2-tuple comprising of geometric
and semantic class labels [6]. Semantic classes are dataset-specific, and
include buildings, trees, low-vegetation, and roads, in most urban datasets,
e.g., Vaihingen site [4]. $\mathcal{S}_{Gm}$ must satisfy the requirements of
metric-invariance, orientation-invariance, and, additionally, class-dependence
in the case of $\mathcal{S}_{AgSm}$. These requirements ensure that the
signature contains the geometric classification of the point cloud, and
optionally, its semantic classification necessary for its contextual
understanding [7].

One of the motivating applications of $\mathcal{S}_{Gm}$ of the point clouds
is their comparison. Comparing $\mathcal{S}_{Gm}$ of point clouds implies
finding the differences between their geometric classifications, arising from
the differences in data acquisition (e.g., change in sampling density,
orientation, etc.), or processing (e.g., use of different local geometric
descriptors (LGDs) for geometric classification, multiscale approach, etc.).
We are interested in point cloud comparison, which integrate both geometric
and semantic information to indicate an event, e.g., deforestation in urban
regions over time.

Apart from visualizing the change in signatures, we propose metrics to
quantify their differences using approaches based on root mean square
deviation, image processing, and information theory. We propose a new
definition for Shannon entropy computed using probabilistic geometric
classification to serve two purposes. Firstly, this measure referred to as
saliency map-based Shannon entropy is used in the multiscale approach for
computing local geometric descriptors. Secondly, it is used to determine the
information-theoretic difference between $\mathcal{S}_{Gm}$ of point clouds.
Overall, $\mathcal{S}_{Gm}$ of a point cloud reveals the uncertainty in its
geometric classification.

Our contributions are:

  1. 1.

We propose a novel geometric signature $\mathcal{S}_{Gm}$ of an airborne LiDAR
point cloud using probabilistic geometric classification, which is a global
feature descriptor.

     * •

Its variant, the augmented semantic signature, incorporates its semantic
classification.

  2. 2.

We propose a saliency map-based Shannon entropy for determining the optimal
scale of local geometric descriptors and for point cloud comparison.

  3. 3.

We propose quantitative metrics computed from the geometric signature for
point cloud comparison.

##  2 Related Work

In this section, we discuss the relevant literature on signatures of point
clouds and on point cloud comparisons.

Geometric Signatures of Point Clouds: Histograms have been widely used as
signatures of point clouds, e.g., persistent feature histograms [8],
histograms of the distribution of geometric distances [9]. Spin image is a
rotation-invariant signature of point clouds [10], which is the distribution
of surface normals in connected components. These have been used for shape
recognition, classification, and reconstruction.

Similar to our method, a barycentric coordinate representation of pointwise
local geometric descriptor has been used for the class-based analysis of point
clouds in geomorphology [11]. This signature, referred to as the
dimensionality density diagram (DDD) uses normalized eigenvalues, and collates
the signatures across multiple scales in a feature vector for semantic
classification. In our work, the multiscale integration occurs before
computation of eigenvalue-based saliencies [5], used as barycentric
coordinates, and the signatures are used for determining structural and
semantic changes in point clouds.

Aggregated Local Feature Descriptors: Given that 3D LiDAR point clouds
implicitly encode a surface, they are 2.5D data characterized by surface
feature descriptors. 3D surface feature descriptors can be categorized into
the global and the local feature descriptors. The local feature descriptors
encode the characteristics of the local neighborhoods around specific
keypoints. They are, therefore, more suitable for recognizing partially
visible or incomplete objects in a cluttered scene with occlusions. In
contrast, the global feature descriptors ignore shape details and require a
priori segmentation of the object from the scene. Thus, they have difficulty
recognizing partially visible or incomplete objects from cluttered scenes.
Hence, global features, that aggregate local features, also embed the
geometric information.

The local geometric descriptors used for LiDAR point clouds include covariance
tensor [12], $T_{\text{3DCM}}$, and anisotropically diffused tensor
$T_{\text{3DVT}}$, obtained from tensor voting [5]. Eigenvalue decomposition
of the covariance tensor [13] provides point-wise features used for tangent
plane estimation [13], geometric reconstruction [12, 14], and semantic
classification [7]. Kernel heat signatures [15], or local surfaces at each
point (multiscale features [16], kernel correlation [17]) are other widely use
local descriptors for general point clouds. Rotational Projection Statistics
(RoPS) [18] and, Tri-Spin-Image (TriSI) [19] are robust to disturbances such
as, noise, varying point density, clutter, and occlusion. RoPS is a local
surface descriptor computed from multiple viewpoints, and TriSI is computed
with respect to three orthogonal axes. However, they require the triangulation
of the unordered point clouds, which is an overhead in real applications.
Binary shape context descriptor is a robust descriptor computed on a scattered
point cloud that encodes both the projection point density and distances from
the local surface [20].

Fisher vectors (FV) [21] and Vector of Locally Aggregated Descriptors (VLAD)
[22] are information-based aggregates used as global feature descriptors. FVs
have been computed as global descriptors of images using the local descriptors
and the Fisher kernel [23]. The FV is the normalized gradient vector of the
local descriptors. A Gaussian Mixture Model (GMM) is used as the probability
density function used for modeling the local descriptors. The gradient vector
is first computed from the log-likelihood of the sample with respect to the
model parameters (i.e., weight, mean and, covariance). The gradient vector is
normalized using the Cholesky factor of the Fisher information matrix [23].
The Fisher kernel of two images is a similarity measure computed as the dot
product of FVs of the two images. The kernel could also be computed with a
selected generative model [24]. The VLAD, which is a global feature descriptor
of an image, is computed using a codebook of clusters of SIFT (scale-invariant
feature transform) descriptors in images, and accumulation of residuals
between descriptors and corresponding cluster center. The VLAD has been
further extended to point clouds to a deep learning network for place
recognition [25]. Similar to FV used for point cloud analysis [3] and VLAD,
our proposed $\mathcal{S}_{Gm}$ is aggregated from local geometric descriptors
explicitly, but using coordinate space transformation. Our $\mathcal{S}_{Gm}$
encodes shapes of local neighborhoods, which implies the information of the
geometric classification. Thus, $\mathcal{S}_{Gm}$ is used for comparisons,
different from the uses of the state-of-the-art for object detection and
localization.

Point Cloud Comparison: A common technique for performing change detection in
a point cloud is to compute its distance to a 3D reference model. The
reference model can either be theoretical or simulated from real data. Point-
to-mesh and mesh-to-mesh distances have been very well studied and
demonstrated by Metro [26], Mesh [27], and other tools. However, the learning
curve for using these software applications is steep. Also, 3D shape
reconstruction of a point cloud is computationally intensive and hence, is
neither efficient nor scalable. An out-of-core method has been used for change
detection in massive point clouds, without reducing the raw data [28]. This
involves computing the distance of each point to its closest point, which
quantifies the change in geometry, using a memory-efficient multi-data octree.
But, constructing such an octree for every comparison is inefficient.

Direct comparison of point clouds can be made with one-to-one point mapping by
using robust distance measures, such as the Gromov-Hausdorff (GH) distance
[29]. GH distance generalizes the Hausdorff distance between two compact
metric spaces, including probability measures and all isometric embeddings. It
can be used for comparing LiDAR point clouds, which may have rotational
transformations in follow-up data acquisitions, but after isometric
transformations. However, computing a discrete approximation of GH distances
using pair-wise geodesic distances is inefficient in large-scale point clouds,
such as in airborne LiDAR. Recently, a feature set derived from local
geometric descriptors, namely the covariance tensor, has been used for change
detection in airborne LiDAR point clouds [30]. The feature set used in random
forest (RF) classifier stores the information of the change in each point
between two epochs and, thus, is further used to classify a specific pre-
determined change in buildings or trees in the dataset. This method, however,
requires registration between epochs. Our novel method quantifies integrated
changes in semantic composition and geometric information, as in these
methods.

Multiscale Approaches: There are two classes of approaches of integrating
multiple scales in LiDAR point cloud processing, namely, determining an
optimal scale and aggregating specific features computed from multiple scales.
Generally, scales are chosen from a specific range, which can be fixed or
adaptive across points. The use of optimal scale in semantic classification
enables the use of adaptive scales. The optimal scale is where Shannon entropy
of geometric features in the local neighborhood is minimum for the point [31].
The feature vector for semantic classification has been widely generated at
the optimal scale for supervised learning, e.g., using random forest
classifier [7]. Aggregating specific features in a fixed range of scales is
done by averaging across scales for saliency maps for geometric classification
or reconstruction [32, 33, 5]. Here, we experiment with aggregation as well as
optimal scale method for computing local descriptors needed for
$\mathcal{S}_{Gm}$.

Other recent multiscale methods include a scale-invariant method for
generating hierarchical point clusters by aggregating scales in a multiscale
and hierarchical point classification method [34]. Latent Dirichlet Allocation
(LDA) and AdaBoost classifiers are then used for feature extraction from the
clusters and classification, respectively. A variant of this method uses
natural exponential function thresholds for point-cluster extraction, and
joint LDA and sparse coding (SCLDA) for feature extraction [35] for airborne
LiDAR point clouds of complex scenes. To add contextual information, e.g.,
topology, in classification results, an RF classifier has been integrated into
a Conditional Random Field (CRF) framework for urban objects classification
[36]. Multiscale features accumulated in the CRF as a long feature vector have
been used in combination with RF classifier [37] and Markov RFs [38].

Barycentric Coordinate Representation: Barycentric coordinate system has been
widely used in point cloud processing, e.g., for point-in-triangle test in
various applications [39], and for curvature estimation [40]. Barycentric
coordinate representation has been used in triangle-occlusion tests for
texture mapping of triangular meshes of LiDAR point clouds [41]. Generalized
barycentric coordinate representation has been used in geometric modeling, for
instance, in applications of interpolation and deformation [42].

##  3 Methodology

Our objective is to generate a signature of the entire point cloud, which is
invariant to the orientation of the point cloud, sampling density, and the
metric system used for data capture. The motivating application is to use our
proposed geometric signature $\mathcal{S}_{Gm}$ for finding the differences in
geometric uncertainty and semantic content of two different LiDAR point
clouds. The rationale for finding new measures of difference between point
clouds is to find an integrated one, which captures the changes in point
clouds holistically.

We propose the use of probabilistic geometric classification [5] to project
the point cloud to the 2D barycentric coordinate space. The barycentric
coordinate space, thus, shows the information of geometric classification and
is referred to as geometric signature $\mathcal{S}_{Gm}$. Upon adding the
semantic classification information to the geometric signature, we create its
variant, called as augmented semantic signature $\mathcal{S}_{AgSm}$.

######  Definition 3.1.

A geometric signature $\mathcal{S}_{Gm}$ of a point cloud $\mathcal{P}$ is an
image of the points projected in the barycentric coordinate space, using the
probabilistic geometric classification.

######  Definition 3.2.

An augmented semantic signature $\mathcal{S}_{AgSm}$ of a point cloud
$\mathcal{P}$ is its geometric signature which is colored as per its semantic
classification.

$\mathcal{S}_{Gm}$ represents the probabilistic geometric classification, and
not the widely used deterministic geometric classification [6]. Thus, our
proposed geometric signature of a point cloud is a global feature descriptor,
which aggregates local geometric descriptors. The point rendering in the
barycentric coordinate space gives us the scope to use visual encodings such
as color, size, and shape of the point to display its attributes111However,
the use of visual encodings of size and shape of the point (glyph) is subject
to the constraint of permissible visual clutter in the image., similar to that
in the 3D Cartesian coordinate space. When the color of the point corresponds
to its semantic class, we augment the geometric signature with the semantic
classification, thus creating $\mathcal{S}_{AgSm}$, which corresponds to
augmented semantic classification [6]. Unlike the existing signatures,
$\mathcal{S}_{Gm}$ does not require intermediate 3D representation, point
registration between point clouds, or point-wise comparisons. This is because
we measure only the overall differences between the point clouds in geometric
and semantic classifications. Even though $\mathcal{S}_{AgSm}$ uses label
information, our method does not explicitly perform semantic classification,
unlike similar methods [30]. Our work is novel in using 2D image comparisons,
which is traditionally used in 2D imagery in remote sensing [30], for 3D data.

###  3.1 Probabilistic Geometric Classification

The local geometric descriptors are predominantly used as features for
semantic classification, which internally uses geometric classification [6,
32]. The geometric classification of a point is based on the cylindrical,
disc, or spherical shape of points in the local neighborhood of the point. The
shape of the neighborhood thus gives line-, surface- or (critical) point-type
features, respectively. The shape is determined using the eigenvalue analysis
of the local geometric descriptors. The saliency map is a 3-tuple that gives
the likelihood of a point belonging to each of the three geometric classes,
namely, line-, surface-, and point-type features. The saliency map is
$\\{C_{l},C_{s},C_{p}\\}(x)$, such that $C_{l}(x)+C_{s}(x)+C_{p}(x)=1.0$.
Thus, probabilistic geometric classification is represented by the saliency
map. Geometric classification is inherently probabilistic [5], but it can be
rounded off to make it deterministic [6] by using the maximum saliency values.
Augmented semantic classification involves tuple of labels from semantic and
deterministic geometric classifications [6]. Here, we use augmented semantic
classification as a tuple of the semantic label and the saliency map of the
probabilistic geometric classification.

Augmented semantic classification is a terminology introduced by Kumari and
Sreevalsan-Nair [6], where each point in the point cloud has a tuple of
labels, namely the semantic and geometric classes. The uniqueness of the
combination of semantic and geometric classes in the tuples ensures clear
separability of augmented semantic classes. For instance, augmented semantic
classes include “line-type feature on a building,” “point-type feature on a
tree,” etc. Thus, the augmented semantic classification is of finer-
granularity compared to its constituent classifications.

Geometric classification of LiDAR point clouds is determined by the local
geometric descriptors [5, 14]. Other than covariance tensor, tensor voting
based on the perceptual organization using Gestalt principles [43] also is
used to construct a positive semidefinite second-order tensor for unoriented
points [44]. Such a tensor is not the same as the covariance tensor, as the
former is generated in normal space and the latter, in tangent space [5].
Hence, anisotropic diffusion is applied to get the geometrical features
similar to those of covariance tensor [5, 45]. After applying anisotropic
diffusion, $T_{\text{3DVT}}$ is further modified using gradient energy tensor
obtained from 2D tensors in cylindrical neighborhood [14, 33]. This tensor,
referred to as $T_{\text{3DVT-GET}}$, has a higher percentage of point-type
features, especially for the tree class.

The local neighborhood used for descriptor computation is conventionally
determined either as nearest k neighbors for $k\in\mathbb{Z}^{+}$, or as a set
of points within the spherical neighborhood of radius $s\in\mathbb{R}^{+}$. In
each case, the size of the neighborhood, $k$ or $s$, is considered as the
scale. It has been widely understood that the outdoor environmental scenes
captured by airborne LiDAR point clouds have larger uncertainties involved in
scene understanding. Hence, a single scale analysis of the local neighborhood
of the points is insufficient [5, 37, 38]. Therefore, multiscale approaches
are used to aggregate the information across scales. In this work, we use
scales from both $k$ nearest and spherical neighborhoods. We then integrate
the scales in each case using two different approaches, namely, the optimal
and aggregated methods. An optimal scale can be computed using minimum Shannon
entropy values across different scales [7, 31]. An aggregated method can be
used by averaging saliency maps across different scales.

###  3.2 Geometric Signatures

As the saliency map at a point is a set of probabilities that sum up to one,
it is equivalent to barycentric coordinate representation of a point in the
reference triangle222The reference triangle in 2D barycentric coordinate space
is equivalent to the reference axes in the 3D cartesian coordinate system. in
the 2D barycentric coordinate space (§Appendix A). Thus, our proposed
$\mathcal{S}_{Gm}$ maps points in 3D cartesian to 2D barycentric coordinate
spaces. $\mathcal{S}_{Gm}$ is influenced by factors such as choice of LGDs,
choice of multiscale approaches, and the characteristics of the function/map
between two coordinate spaces. $\mathcal{S}_{AgSm}$ is influenced by the
semantic composition.

Figure 2:  Patterns observed in the geometric signatures in tensor voting and
optimal scale are shown in red dotted lines and ellipses, respectively, in the
reference triangle in barycentric coordinate space. (Top left) The
partitioning line between (0,1,0) and (0.25,0,0.75) occurs due to the
computations of tensor voting as ball tensors. (Bottom left) This line
transforms to a curve between (1,0,0) and (0,0.93,0.07) upon applying
anisotropic diffusion to get $T_{\text{3DVT}}$. Upon using optimal scale in
$T_{\text{3DVT}}$, additionally (top right) a “dent” transforms to (bottom
right) a “hole” in $T_{\text{3DVT}}$ after applying anisotropic diffusion. The
points located away from the centroid ($\frac{1}{3},\frac{1}{3},\frac{1}{3}$)
and closer to the vertices have lower Shannon entropy, used for optimal scale
determination. These images are generated for LiDAR point cloud of Area-3 of
Vaihingen dataset [4] (323,896 points) for $T_{\text{3DVT-GET}}$, a variant of
$T_{\text{3DVT}}$ after integrating GET, with same patterns in
$\mathcal{S}_{Gm}$ seen for $T_{\text{3DVT}}$.

####  3.2.1 Choice of Local Geometric Descriptors

Covariance tensor $T_{\text{3DCM}}$ and anisotropically diffused tensor using
tensor voting $T_{\text{3DVT}}$ [5] are LGDs of our interest. In the latter
case, we use the tensor integrated with 2D gradient energy tensor [33],
$T_{\text{3DVT-GET}}$333We have used the computations for $T_{\text{3DCM}}$,
$T_{\text{3DVT}}$, and anisotropic diffusion as given by Sreevalsan-Nair and
Kumari [33], and for $T_{\text{3DVT-GET}}$ as given by Sreevalsan-Nair et al.
[14].. The choice of LGD gives characteristic geometric signatures, as the
distribution of saliency maps in the LiDAR point cloud varies across different
LGDs, e.g., $T_{\text{3DCM}}$ gives more point-type features than
$T_{\text{3DVT}}$ [14]. $T_{\text{3DVT}}$ is computed using ball tensors, as
initialization for unoriented points. Hence, $T_{\text{3DVT}}$ is
characterized by $(C_{l}\leq C_{p}+C_{s})$ in $T_{\text{3DVT}}$, since the
ball tensor is computed in the normal space and not in the tangent space, as
it is for $T_{\text{3DCM}}$ [5]. The computation in normal space causes the
vote from each neighbor to be a curvel or a plate tensor where the minor
eigenvalue is 0.0 [46]. Thus, when these votes add up, a constraint on $C_{l}$
builds up, which manifests as $(C_{l}\leq C_{p}+C_{s})$. Using Equation B
(§Appendix B), we find the upper bound, $\frac{C_{s}}{C_{l}}\leq 3.0$,
manifesting as the partitioning line in the barycentric space. The
partitioning line is between $(0,1,0)$ and $(0.25,0,0.75)$ (§red dotted line
in Figure 2). Anisotropic diffusion is performed on tensor voting to
strengthen weak line-type features [45] and to make the tensor voting-based
LGD substitutable for $T_{\text{3DCM}}$ [5]. Anisotropic diffusion transforms
the partitioning line in the signature of $T_{\text{3DVT}}$ to the curved
boundary between $(1,0,0)$ and $(0,0.93,0.07)$ (§Figure 2). This pattern in
$\mathcal{S}_{Gm}$ demonstrates two known characteristics of $T_{\text{3DVT}}$
[5, 14], which are that applying anisotropic diffusion increases the number of
line-type features and reduces the number of point-type features in
$T_{\text{3DVT}}$. Thus, the $\mathcal{S}_{Gm}$ gives visual validation of the
characteristics of the tensor voting based LGD, $T_{\text{3DVT}}$, especially
the influence of anisotropic diffusion on it.

####  3.2.2 Multiscale Approaches & Novel Entropy Metric

Here, we use both multiscale aggregation of saliency maps as well as optimal
scale determination to integrate multiple scales from a fixed range of scales.
The multiscale aggregation refers to averaging of the saliency maps calculated
at a range of scales [5, 33] (§Equation B in Appendix B gives saliency map
computation for a single scale). Computing an optimal scale at each point is
done by finding the local minimum of Shannon entropy computed of the local
neighborhood of each point, across scales. Shannon entropy is computed based
on either dimensionality [31] or eigen-entropy [47]. The dimensionality method
uses linearity, planarity, and scattering of $T_{\text{3DCM}}$ [31], whereas,
the eigen-entropy method computes the same from normalized eigenvalues of
$T_{\text{3DCM}}$ [47]. Thus, eigenvalue decomposition (EVD) of LGD is
significant in multiscale approaches. In fact, saliency maps capture
information of the EVD. Since the saliency maps at each point obey the
partition to unity rule and is a variant of normalized eigenvalues, it can be
used computing Shannon entropy444For interested readers, Weinmann et al. have
explained in detail the use of eigenvalue-based features for use in Shannon
entropy [7].. We propose a novel variant of Shannon entropy $E_{\text{geom}}$
using saliency maps, which gives uncertainty in probabilistic geometric
classification. Our saliency map-based Shannon entropy at a point x is given
as555As an operator,
$E_{\text{geom}}=-C_{l}.ln(C_{l})-C_{s}.ln(C_{s})-C_{p}.ln(C_{p})$.,

$E_{\text{geom}}(x)=\sum\limits_{k\in\\{l,s,p\\}}-C_{k}(x).ln(C_{k}(x)).$

Thus, we use the minimum value of $E_{\text{geom}}$ to select the optimal
scale $s_{\text{opt}}$, and then use the saliency maps at $s_{\text{opt}}$.
$E_{\text{geom}}$ is the measure of the uncertainty in probabilistic geometric
classification, which we refer to as geometric uncertainty.

We observe that the $\mathcal{S}_{Gm}$ using optimal scale based on
$E_{\text{geom}}$, have a distinct “hole” or empty region in the neighborhood
of the centroid, where $E_{\text{geom}}$ is the maximum (§Figure 2). This
pattern occurs because the optimal scale is at local minimum of
$E_{\text{geom}}$, corresponding to the saliency maps closer to the vertices
of the reference triangle, i.e., where either $C_{l}$, $C_{s}$, or $C_{p}$ is
1. We observe the same pattern for $T_{\text{3DCM}}$ (§Section 5). In the case
of optimal scale in tensor voting, we observe an additional hemispherical
“dent” in the partitioning line in $T_{\text{3DVT}}$, which transforms to the
“hole”, upon applying anisotropic diffusion (§Figure 2). Thus, the pattern in
the $\mathcal{S}_{Gm}$ with $T_{\text{3DVT}}$ shows the combined effects of
using the optimal scale and anisotropic diffusion.

####  3.2.3 Influence of Semantic Class in Signatures

The semantic classification has conventionally been implemented using feature
vectors derived from LGDs. If these features enable class separability in
different classification methods, we can hypothesize that the different
semantic classes, such as trees, low-vegetation, buildings, etc. seen in
airborne LiDAR point clouds, in turn, have characteristic $\mathcal{S}_{Gm}$.
Our hypothesis stems from the observed differences in the composition of
line-, surface-, and point-type features in each class [31]. For instance, the
tree and low- vegetation classes consist of predominantly point-type features
[31, 33]. The building class consists of a relatively higher proportion of
surface-type features along with a lower proportion of line-type features
[31], which is the characteristic of the roof surface and contour [14]. Our
hypothesis is the basis of our motivating application of using
$\mathcal{S}_{Gm}$ for determining differences between two point clouds,
including their semantic composition.

####  3.2.4 Non-injective Surjective Function Map

The probabilistic geometric classification is a many-to-one mapping, which
implies that several points in the point cloud could have the same saliency
map. This, in turn, causes those points to have the same barycentric
coordinate representation. At the same time, all points in the point cloud
will have a saliency map, and hence a barycentric coordinate representation.
Thus, the mapping for generating geometric signatures is a non-injective
surjective function. One of the implications of the non-injective function is
that several points in the point cloud overlap in the $\mathcal{S}_{Gm}$. This
nature of the function has implications on the applicability of the
signatures. A point to note with respect to the non-injective mapping in the
$\mathcal{S}_{Gm}$ is that, though the signature encodes spatial locality, it
does not linearly transform spatial locality. Thus, our proposed signature
loses the spatial context and hence, can be used only for applications or
analytical tasks to visualize the composition of the scene using the geometric
properties of different semantic classes whose objects are present in the
scene.

The sampling density of the point cloud influences the overlap of points in
the signature. Thus, sparser point clouds tend to give even sparser
$\mathcal{S}_{Gm}$. However, we can say that our $\mathcal{S}_{Gm}$ are
resilient to sampling density, i.e., small differences in sampling density for
a point cloud do not manifest as perceptual differences in the signature (more
in Section 5).

###  3.3 Comparing LiDAR Point Clouds

We visualize the $\mathcal{S}_{Gm}$ of two different LiDAR point clouds to
compare them qualitatively. For quantitative comparison, conventional methods
use injective mapping after performing registration. Injective mapping refers
to the one-to-one mapping of points between two point clouds, where the more
accurate distances, such as, point-wise root mean square error (or difference)
(RMSE), can be computed. Distribution-based metrics are apt for non-injective
mapping, such as in the case of absence of registration. For determining
differences in semantic composition, the difference between count distribution
across different classes can be computed using distance metrics, such as,
Hamming distance of normalized distributions and symmetric Kullback Leibler
(KL) distance. In the same vein, we propose to use distributions of
$E_{\text{geom}}$ in point clouds to quantify differences in geometric
uncertainty. For integrating semantic composition and geometric uncertainty,
we propose image-based metrics for both grayscale and colored images, for
$\mathcal{S}_{Gm}$ and $\mathcal{S}_{AgSm}$, respectively.

####  3.3.1 Difference in Uncertainty in Probabilistic Geometric
Classification

The distribution of $E_{\text{geom}}$ of the point cloud reflects the
uncertainty in its probablistic geometric distribution. The plots of the
probability distribution function (pdfs) of $E_{\text{geom}}$ of a point cloud
give the uncertainties in geometric classification across different semantic
classes when using different local geometric descriptors. Highest uncertainty
in probabilistic geometric classification occurs at the $\bar{G}$ of the
reference triangle of the barycentric coordinate space in $\mathcal{S}_{Gm}$.
When uncertainty is high, $E_{\text{geom}}$ is high, and points tend to be
closer to $\bar{G}$. We compare two point clouds $\mathcal{P}$ and
$\mathcal{Q}$ using the Earth Mover’s Distance (EMD) [48] between the pdfs of
$E_{\text{geom}}$ of each point cloud. EMD determines the minimum cost of
transforming a histogram into another, where higher EMD implies higher
transformation cost. Thus, our information-theoretic distance measure is given
as:

$d_{\text{EMD-
Info}}=d_{\text{EMD}}(\text{pdf}(E_{\text{geom}}(\mathcal{P})),\text{pdf}(E_{\text{geom}}(\mathcal{Q})))$.

####  3.3.2 Image-based Metric

The signatures generated in congruent reference triangles can be compared
using the difference in the point distribution in the interior of the
triangles. We store the graphical rendering of the point cloud in the interior
of the triangle as a pixellated image. Thus, the difference between
$\mathcal{S}_{Gm}$ of two point clouds gives the difference between geometric
uncertainties, and between $\mathcal{S}_{AgSm}$ of the point clouds gives the
difference in semantic composition.

For grayscale images of $\mathcal{S}_{Gm}$: Structural Similarity Index (SSIM)
is used to measure the similarity between geometric signatures. SSIM measures
the similarity score $\mathbf{s}_{\text{SSIM}}\in[-1,1]$, rather than
differences. A score of $+1$ indicates no difference between the two images.
SSIM is conventionally computed using the luma of the image, i.e., the
achromatic portion. Thus, we use SSIM to compute the similarity between the
geometric uncertainty of two point clouds.

For color images of $\mathcal{S}_{AgSm}$: The distance between the discretized
histograms of the probability density functions (pdfs) of the color channels
give the difference in semantic classification in the point clouds. There are
two groups of distance measures for a histogram, namely, bin-to-bin and cross-
bin. The bin-to-bin computes the difference in the content of the bin with the
corresponding bin of the second histogram, while the latter additionally
considers the neighborhood values of the concerned bin. Conventionally used
Bhattacharyya distance [49] $d_{\text{BD-Img}}$ is a bin-to-bin method which
gives values in the range $[0,1]$, where 0 and 1 mean a perfect match and
maximum difference between the images, respectively. In addition to the
symmetric variant of Bhattacharyya distance, we determine EMD $d_{\text{EMD-
Img}}$ between the images, which is a cross-bin distance measure. We use the
normalized color histogram to determine the color distribution of the interior
of the reference triangle in the image. Using a mask with size same as that of
the triangle exclusively ignores the image background, thus reducing the
discrepancies in the distances. These image metrics are sensitive to point
sampling density, just as in the case of perceptual differences in the
visualization of $\mathcal{S}_{Gm}$ or $\mathcal{S}_{AgSm}$.

##  4 Experiments

We have used the airborne LiDAR dataset for Area-2 and Area-3 in the Vaihingen
site, from the ISPRS benchmark [4]. We have used the labeled dataset, with
Area-2 (N = 266,675 points) having high-rise buildings (B = 69,097 points)
with trees (T = 107,485 points), and Area-3 (N = 323,896 points) having
independent small houses (B = 69,563 points), and trees (T = 135,499 points).
As is done with all classification studies, we use the height values of all
points above ground, thus assuming a flat terrain. We compare Area-2 and
Area-3 by using $\mathcal{S}_{Gm}$ to capture the integrated differences in
geometry and semantic composition.

Choice of Multiscale Approach and Local Neighborhoods: Our experiments include
geometric signatures computed using the covariance tensor, $T_{\text{3DCM}}$,
and $T_{\text{3DVT-GET}}$. $T_{\text{3DVT-GET}}$ is computed using tensor
voting, and modified using anisotropic diffusion and 2D gradient energy tensor
of $T_{\text{3DVT}}$ [5]. The LGDs have been computed using the following
combination of local neighborhood types and multiscale approaches666The
rationale behind choosing the specific combinations for our chosen LGDs is to
compare the influence of the choice of the descriptor for computing the
signatures, where the descriptor is already studied in the state-of-the-art.:
(a) k-nearest neighbors for optimal scale computation [7], and (b) spherical
neighborhood for averaging saliency maps over multiple scales [5]. Hereafter,
we refer to our chosen combination of multiscale approaches and local
neighborhoods, as:

(a) “Multiscale” for saliency map in spherical neighborhoods average across
multiple scales, and

(b) “Optimal-scale” for saliency map from the local geometric descriptor at
optimal scale determined specifically using $E_{\text{geom}}$777Eigen-entropy
[7] and $E_{\text{geom}}$ give comparable results, as both use normalized
eigenvalues., at multiple scales of k-nearest neighborhoods.

For (a), the scales we have used are $k_{min}=10$, $k_{max}=100$, with step
size888This is different from the step size used in Weinmann et al. [7],
$\Delta k=1$, for semantic classification. Even with larger step size, $\Delta
k=10$, we have gotten good results for geometric classification for the same
$k$ range as for semantic classification. We get the benefit of a lesser
number of computations here. of $\Delta k=10$. For (b), in the case999We have
fitted the 3D (volumetric) data in a (canonical) bounding box of
$[-1,1]\times[-1,1]\times[-1,1]$ using uniform scaling transformation, as done
in computer graphics. In this normalized point cloud, we have used scales of
$r_{min}=0.009$, $r_{max}=0.011$, with $\Delta r=0.001$. Though we have
methodically taken the same range of scales in this transformed space for all
datasets, the scales in metric units vary from dataset to dataset. Hence, the
different values for Area-2 and Area-3. of Area-2, we have used the scales
$r_{min}=1.89m$, $r_{max}=2.31m$, with step size of $\Delta r=0.21m$, and in
the case of Area-3, we have used the scales $rscale_{min}=1.92m$ ,
$r_{max}=2.34m$, and $\Delta r=0.21m$

Choice of Local Geometric Descriptors: The outcomes of probabilistic geometric
classification of the descriptor tensors, $T_{\text{3DCM}}$,
$T_{\text{3DVT}}$, and $T_{\text{3DVT-GET}}$, in combination with each of the
multiscale approaches, namely, the Multiscale and the Optimal-scale approaches
show anomalies in the case of Optimal-scale $T_{\text{3DVT}}$ (§Figure 3).
Hence, we use $T_{\text{3DCM}}$ and $T_{\text{3DVT-GET}}$ in our experiments.
Since Multiscale $T_{\text{3DCM}}$, Optimal-scale $T_{\text{3DCM}}$, and
Multiscale $T_{\text{3DVT-GET}}$ show similar geometric classification results
(§Figure 3), we use these tensors for further discussion in Section 5.

Figure 3:  Top-down views of point rendering of Area-2 and Area-3 demonstrate
the variations in geometric classification for different local geometric
descriptors with different multiscale approaches. Each point is colored with
saturation in red, green, and blue channels corresponding to $C_{l}$, $C_{s}$,
and $C_{p}$ of its saliency map, respectively. $T_{\text{3DCM}}$ and
$T_{\text{3DVT-GET}}$ are the local geometric descriptors used for generating
signatures in our experiments, owing to anomalies in Optimal-scale
$T_{\text{3DVT}}$. The checks indicate the specific tensors used for further
discussion of our results, owing to similarity in saliency maps.

(i) Area-2

(ii) Area-3

Figure 4:  Semantic class-wise probability density functions (pdfs) of saliency maps ($C_{l}$, $C_{s}$, $C_{p}$) for (i) Area-2 and (ii) Area-3 of Vaihingen site. The dotted circles show the distinctive behavior of the pdfs in our selected local geometric descriptors with appropriate multiscale approach, for the tree class. Table 1: The count distribution of 3D points in each semantic class in point clouds of Vaihingen site from the ISPRS benchmark data [4]. Dataset | Tree | Building | Low-vegetation | Road
---|---|---|---|---
| (#points) | (#points) | (#points) | (#points)
Area-2 with 266,675 points for area of $225m\times 256m$ | 107,485 | 69,097 | 38,866 | 51,227
(point sampling density of 2.76 points/$m^{2}$) |  |  |  |
Area-3 with 323,896 points for area of $193m\times 277m$ | 135,449 | 69,563 | 35,512 | 83,372
(point sampling density of 2.54 points/$m^{2}$) |  |  |  |

###  4.1 Visualizations and Comparative Analysis

All $\mathcal{S}_{Gm}$ use rendering of points in the barycentric coordinate
space, using fixed position coordinates for the reference triangle in the
barycentric coordinate space. For point rendering, the position coordinates of
the points in the $\mathcal{S}_{Gm}$ are computed as the weighted sum of the
vertices reference triangle, where the saliency map is used as weights
(§Appendix A). Fixing the position coordinates of the reference triangle is
important for computing image-based differences between signatures.

##  5 Results and Discussion

(i) Area-2

(ii) Area-3

Figure 5:  Augmented semantic signature with geometric signature inset, for
(i) Area-2 and (ii) Area-3 of Vaihingen site show the augmented semantic
signatures using different local geometric descriptors and multiscale
approaches.

The pdfs of semantic class-wise saliency maps (§Figure 4) explain the patterns
observed in signatures. The pdfs of saliency maps of building and tree classes
show multimodal distributions, which manifest as highly dispersed points in
the signatures. The saliency map using Multiscale $T_{\text{3DVT-GET}}$ gives
more (critical) point-type features in the tree class [14] and thus the
$\mathcal{S}_{Gm}$ and $\mathcal{S}_{AgSm}$ of the LGD show more points in the
vertex corresponding to $C_{s}=1$ (lower-right corner of the reference
triangle, §Figure 5).

Figure 6:  Effects of sampling density in point cloud in the augmented
semantic signatures of Area-3 of the Vaihingen site, using uniform spatial
sampling.

The observable differences in the signatures are caused by the difference in
the uncertainty in geometric classification. The key aspect of our work is to
use our proposed signatures with point clouds with different sampling
densities. Hence, we study the resilience of our signatures with respect to
change in sampling density (§Figure 6). We observe that the $\mathcal{S}_{Gm}$
and $\mathcal{S}_{AgSm}$ exhibit sparsification only after the sampling
density falls below 60% of the original point cloud, when using uniform
sampling in the 3D cloud, demonstrating the resilience of signatures to
sampling density reduction. Thus, our proposed signatures can be effectively
used for comparing two point clouds with $\lesssim$40% relative difference in
point count.

Distance Metrics: Our work is generalized for comparing signatures and finding
integrated distance between any two point clouds. For instance, $d_{\text{BD-
Img}}$ values between Area-3 and Area-2, which is at 82% point sampling
density, are 0.2269, 0.2432, and 0.228 using Multiscale $T_{\text{3DCM}}$,
Optimal-scale $T_{\text{3DCM}}$, and Multiscale $T_{\text{3DVT-GET}}$,
respectively. Also, $\delta_{KL}$, $L_{1}$, and the symmetric $\delta_{FV}$
(using Multiscale $T_{\text{3DCM}}$) between Area-2 and Area-3 are 0.0382,
0.1609, and 0.0835, respectively. Since $d_{\text{BD-Img}}$ is bounded in
[0,1], we can say that the two point clouds have relatively small integrated
distance. Further work in making our proposed measures bounded or
dimensionless will improve the usability of our distance measures in the
generalized application.

Discussions: The state-of-the-art Fisher vector representation is high-
dimensional, owing to which the representation cannot have an easy visual
representation as our proposed geometric signatures. Overall, the
Bhattacharyya distance between signatures $d_{\text{BD-Img}}$ gives an
effective integrated measure of change in semantic composition and geometric
uncertainty.

However, there are three inherent limitations in our signatures. Firstly, it
does not preserve the spatial contextual information which is a disadvantage
for spatial inferential tasks. Secondly, the geometric signature depends on
the local geometric descriptor. The effectiveness of the signature in
demonstrating differences relies on the clear separability of the classes in
the pdfs of the saliency maps (§Figure 4). Thirdly, our $\mathcal{S}_{Gm}$
suffer from information loss owing to the non-injective mapping of Cartesian
coordinate space to barycentric coordinate space. This causes overlap of
points in barycentric space, and hence, occlusion of points in the
visualization of the signature. Also, the occlusion of signatures is
determined by the order of point rendering, which indirectly affects
$\mathcal{S}_{AgSm}$. However, this shortcoming can be alleviated by using
appropriate colormaps, which exploit transparencies for $\mathcal{S}_{AgSm}$
to reduce occlusion.

##  6 Conclusions

Three-dimensional point clouds acquired using LiDAR technology are useful for
topographical study, given the speed and accuracy of data acquisition LiDAR
can provide. By the same token, LiDAR point clouds also come with challenges
in its analysis given high unpredictability and variability in the type of
objects encountered in outdoor environments, especially. In this work, we have
proposed a geometric signature $\mathcal{S}_{Gm}$ of an airborne LiDAR point
cloud, which can be visualized and used for comparing datasets.
$\mathcal{S}_{Gm}$ encapsulates the probabilistic geometric classification of
the point cloud into line-, surface-, and (critical) point-type features.
Augmented semantic signatures $\mathcal{S}_{AgSm}$ are a variant of
$\mathcal{S}_{Gm}$ that use color corresponding to the semantic class.
$\mathcal{S}_{Gm}$ is generated in image format after converting 3D points to
2D barycentric coordinate representation using the saliency map from the
probabilistic geometric classification.

Our proposed signatures are global descriptors of the point cloud, and we have
demonstrated its usability in comparing point clouds of two regions in the
same (Vaihingen) site. We propose integrated measures of differences in both
semantic composition and geometric uncertainty. We have proposed an entropy
measure based on probabilistic geometric classification, namely, the saliency
map-based Shannon entropy, $E_{\text{geom}}$. We have found that our
signatures are resilient for relative differences in sample size to be
$\lesssim$40%.

Overall, we have shown that our novel augmented semantic signatures are
effective global descriptors of point clouds for measuring differences in
geometric uncertainty and semantic composition between two datasets. This work
can be effectively used as generalized integrated measures of distances
between any two point clouds.

## Acknowledgments

The authors are grateful to Satendra Singh for the code refactor that preceded
the proposed work and Intel India Research Fellowship. This work has been
supported by the Early Career Research Award of the first author, by the
Science and Engineering Research Board (SERB), Government of India. This work
has used third-party software libraries, namely, PCL [50], CGAL [51], OpenCV
[52], and Python libraries [53] for implementations of point cloud processing,
geometric processing, and image processing algorithms, respectively. The
Vaihingen data set has been published by the German Society for
Photogrammetry, Remote Sensing and Geoinformation (DGPF) [54]101010Vaihingen
dataset acquisition project: http://www.ifp.uni-stuttgart.de/dgpf/DKEP-
Allg.html

##  Appendix A Computations in Barycentric Coordinate System

The barycentric coordinate system is an alternative to an affine or cartesian
coordinate system, where there is a notion of a barycenter or a center of
mass. It is a coordinate system described on a simplex, such as a triangle,
tetrahedron, etc. In the barycentric coordinate system, the simplex
corresponds to its dimensionality $d$ of the cartesian coordinate system,
which means, triangle corresponds to a 2D space, tetrahedron corresponds to a
3D space, and so on.

There are four key features of the barycentric coordinate space. Firstly, the
representation of a point is a $(d+1)$-tuple for a $d$-dimensional space,
unlike a $d$-tuple in an equivalent cartesian coordinate system. That said, in
both cases, only $d$ dimensions can be strictly linearly independent, which
means in the barycentric coordinate system, there is one dependent dimension.
As an example, in a 2D barycentric coordinate, a point $A$ is described as
$(x_{0}(A),x_{1}(A),x_{2}(A))$, with $\sum\limits_{i=0}^{2}x_{i}=1.0$.

Thus, the second feature is that the coordinates of a point in a barycentric
coordinate system have the property of partition to unity. The position
coordinates of $A$ in the barycentric coordinate system can be rewritten as
$(x_{0}(A),x_{1}(A),1-(x_{0}(A)+x_{1}(A)))$, and similarly, any of the three
dimensions can be chosen to be linearly dependent one. Thirdly, for a point in
the interior of the simplex, i.e., inside the triangle in 2D case, all
coordinates of all points in the simplex lie in the interval $[0,1]$. The
fourth feature of barycentric coordinate system is that each coordinate
$x_{i}$ corresponds to a vertex $V_{i}$ of the $d$-dimensional simplex, where
$0\leq i\leq(d-1)$ and $i\in\mathbb{Z}^{+}$. Summing up all four features, any
property $q(A)$ at a point $A$ with barycentric coordinates
$(x_{0},x_{1},\ldots,x_{d-1})$, will be a convex combination of the property
at the vertices of the simplex, e.g. $q$ can be the 3D coordinates of $A$, the
color at $A$, etc. Thus,

$q(A)=\sum\limits_{i=0}^{d-1}x_{i}.q(V_{i})$

.

We can see that the property of partition to unity is analogous to the
probabilities in the saliency map. Owing to the analogy of barycentric
coordinates with the sum of likelihoods in probabilistic geometric
classification, i.e., $C_{l}(A)+C_{s}(A)+C_{p}(A)=1.0$, at point A, we can map
the saliency values as barycentric coordinates. Thus, we propose the
transformation of point cloud from 3D cartesian coordinate system to 2D
barycentric space to generate the geometric signature of the point cloud.

##  Appendix B Multiscale Computations for Local Geometric Descriptors

Here, we explore the use of multiple scales for both averaging saliency maps
across scales, as well as computation of saliency map using an optimal scale
based on a novel variant of Shannon entropy using saliency maps themselves
(§Section 3.2).

Multiscale Aggregation of Saliency Maps: We compute saliency maps for line-,
surface-, and (critical) point-type features in a point cloud using local
geometric descriptors (LGDs). The saliency map is computed as a 3-tuple, using
the likelihood of a point being a line-type $C_{l}$, surface-type $C_{s}$, and
point-type $C_{p}$ feature. Since these three are the only possibilities of
geometric classes in 3D space, we get $C_{l}+C_{s}+C_{p}=1.0$ at each point.
The saliency map at each point $x$ is computed using the eigenvalue
decomposition of its LGD [5], for each scale $s$ at each point $x$, as given
below, where the eigenvalues of the LGD are sorted as
$\lambda_{0}\geq\lambda_{1}\geq\lambda_{2}$:

$\displaystyle C_{l}(x,s)$ $\displaystyle=$
$\displaystyle\mathbf{S}.(\lambda_{0}(x,s)-\lambda_{1}(x,s)),$ $\displaystyle
C_{s}(x,s)$ $\displaystyle=$
$\displaystyle\mathbf{S}.(2(\lambda_{1}(x,s)-\lambda_{2}(x,s)),\text{ and }$
$\displaystyle C_{p}(x,s)$ $\displaystyle=$
$\displaystyle\mathbf{S}.(3(\lambda_{2}(x,s))),\text{ where }$
$\displaystyle\mathbf{S}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda_{0}(x,s)+\lambda_{1}(x,s)+\lambda_{2}(x,s)}.$
(1)

There are two strategies for aggregating the saliency maps across scales. The
saliency maps are averaged across scales, as they are likelihoods of points
belonging to a geometric class, and the different scales are mutually
exclusive events [5], in which case, for $N_{s}$ scales in $[s_{min},s_{max}]$

$\\{C_{l},C_{s},C_{p}\\}(x)=\frac{1}{N_{s}}\sum\limits_{s=s_{min}}^{s=s_{max}}\\{C_{l}(x,s),C_{s}(x,s),C_{p}(x,s)\\}$.

The second strategy is to use optimal scale, which is determined based on
minimum value of chosen Shannon entropy. In the case of optimal scale,

$\\{C_{l},C_{s},C_{p}\\}(x)=\\{C_{l}(x,s_{\text{opt}}),C_{s}(x,s_{\text{opt}}),C_{p}(x,s_{\text{opt}})\\}$.

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