Index: sparse_sample/damkjer_sparse_sample.tex
===================================================================
--- sparse_sample/damkjer_sparse_sample.tex	(revision 2)
+++ sparse_sample/damkjer_sparse_sample.tex	(revision 3)
@@ -14,4 +14,6 @@
 %%    2013-APR-19  K. Damkjer
 %%       Expanded on Introduction and Local Statistic Attribution sections.
+%%    2013-JUN-02  K. Damkjer
+%%       General edits. Expanded Structure Features section.
 %%=============================================================================
 
@@ -31,4 +33,31 @@
 \usepackage{graphicx}
 \usepackage{latexsym}
+\usepackage{xcolor}
+\usepackage[pagebackref,colorlinks]{hyperref}
+
+%%***
+%% Set up packages
+%%***
+\definecolor{darkgreen}{rgb}{0 0.5 0}
+\definecolor{darkblue}{rgb}{0 0 0.7}
+
+\hypersetup{%
+   linkcolor=darkblue,%
+   citecolor=darkgreen,%
+   urlcolor=blue,%
+}
+
+\renewcommand*{\backref}[1]{}
+\renewcommand*{\backrefalt}[4]{%
+\ifcase #1 %
+   % case: not cited
+   (not~cited).
+\or
+   % case: cited on exactly one page
+   (see~p.~#2).%
+\else
+   % case: cited on multiple pages
+   (see~pp.~#2).
+\fi}
 
 %%***
@@ -40,5 +69,5 @@
 %% Participating authors
 %%***
-\author{Kristian L. Damkjer\thanks{e-mail: kdamkjer@knights.ucf.edu} \ and Hassan Foroosh\thanks{e-mail: foroosh@eecs.ucf.edu}\\
+\author{Kristian L. Damkjer\thanks{e-mail: \href{mailto:kdamkjer@knights.ucf.edu}{kdamkjer@knights.ucf.edu}} \ and Hassan Foroosh\thanks{e-mail: \href{mailto:foroosh@eecs.ucf.edu}{foroosh@eecs.ucf.edu}}\\
 University of Central Florida}
 
@@ -63,17 +92,17 @@
 \section{Introduction}
 
-Light detection and ranging (LiDAR) systems have developed to support mapping and surveying produce a staggering amount of information-rich true three-dimensional (3D) data. Modern systems sample several thousand to over a million points per second resulting in several million to billions of point samples per product to be stored, processed, analyzed and distributed.\cite{Parrish:2012,Smith:2012,Young:2012}
-
-Managing such large data sets through a production pipeline consisting collection, processing, analysis, storage and dissemination presents a host of challenges to content providers. Limiting data sets to small areas of interest (AOIs) can mitigate data management issues inherent in processing and storing individual LiDAR point clouds. However, user demands for simultaneous wide area coverage and precise scene content keep data sizing considerations at the forefront of content provider concerns. Further, AOIs do not address the ultimate storage constraints imposed on processing and archival systems that must maintain large libraries of raw and processed data sets.
-
-Raw LiDAR data taken directly from the collection system is usually in a compact, vendor-specific, proprietary binary format that must be converted into a common public format to facilitate analysis and information exchange. The conversion to an exchange format has traditionally inflated the size of the raw LiDAR data holdings. A significant cause of inflation is the conversion of 1D range data into 3D spatial data. Further inflation and loss of precision may result depending on the data types and structures used to represent the 3D point clouds.
-
-An industry standard binary exchange format---LASER File Format (LAS)---was introduced by the American Society for Photogrammetry and Remote Sensing (ASPRS) to facilitate data exchange and minimize overhead.\cite{ASPRS:2012} Recent development of a compressed version of LAS---LASzip (LAZ)---has gained wide-spread adoption and offers lossless, non-progressive, streaming, order-preserving compression, random access, and typical compression ratios between 10 and 20 percent of original file size.\cite{Isenburg:2011} 
-
-While the compression achieved by the LAZ format can be significant, achieving optimal results requires that the uncompressed input data stream be aliased to a regular point spacing. This operation may not be an appropriate modification of the data stream in all scenarios. Even with an effective compression strategy, additional data reduction may be necessary to support users in bandwidth-limited and mobile device environments or to support efficient querying and comparison of data holdings in processing and archival systems.
-
-An ideal data reduction algorithm should remove elements from a data set in an information-preserving manner. Several approaches have been developed to identify salient elements in dense scenes. These salience metrics can then be applied to guide the reduction process. West et al introduced structure-tensor eigenvalue based feature classifiers for LiDAR point sets.\cite{West:2004} These basic classifiers were enhanced by Gross and Thoennessen to extract strong linear features to support scene modeling from laser data.\cite{Gross:2006} Demantk\'{e} et al  also extend this basic point classification to direct optimal neighborhood scale selection for feature attribution.\cite{Demantke:2011}
-
-In this paper, we present a method for performing unsupervised sparse sampling of LiDAR point data while preserving scene information content.
+Mapping and surveying Light Detection and Ranging (LiDAR) systems produce a staggering amount of information-rich true three-dimensional (3D) data. Modern systems sample several thousand to over a million points per second resulting in several million to billions of point samples per product to be stored, processed, analyzed and distributed.\cite{Parrish:2012,Smith:2012,Young:2012}
+
+Managing such large data sets through a production pipeline consisting of collection, processing, analysis, storage and dissemination presents a host of challenges to content providers. Limiting data sets to small areas of interest (AOIs) can mitigate data management issues inherent in processing and storing individual LiDAR point clouds. However, user demands for simultaneous wide area coverage and precise scene content keep data sizing considerations at the forefront of content provider concerns. Further, AOIs do not address the ultimate storage constraints imposed on processing and archival systems that must maintain large libraries of raw and processed data sets.
+
+Raw LiDAR data taken directly from collection systems is usually in a compact, vendor-specific, proprietary binary format that must be converted into a common public format to facilitate analysis and information exchange. The conversion to an exchange format has traditionally inflated the size of the raw LiDAR data holdings. A significant cause of inflation is the conversion of 1D range data into 3D spatial data. Further inflation and loss of precision may result depending on the data types and structures used to represent the 3D point clouds.
+
+An industry standard binary exchange format---LASER File Format (LAS)---was introduced by the American Society for Photogrammetry and Remote Sensing (ASPRS) to facilitate data exchange and minimize overhead.\cite{ASPRS:2012} Recent development of a compressed version of LAS---LASzip---has gained wide-spread adoption and offers lossless, non-progressive, streaming, order-preserving compression, random access, and typical compression ratios between 10 and 20 percent of original file size.\cite{Isenburg:2011} 
+
+The compression achieved by the LASzip format can be significant, however achieving optimal results requires that the uncompressed input data be aliased to a regular point spacing. This operation may not be an appropriate modification of the data in all scenarios. Even with an effective compression strategy, data reduction may be necessary to support users in bandwidth-limited and mobile device environments or to support efficient querying and comparison of data holdings in processing and archival systems.
+
+An ideal data reduction algorithm should remove elements from a data set in an information-preserving manner. Several approaches have been developed to identify salient elements in dense scenes. Basic features were proposed based on structure-tensor--eigenvalue analysis of local point neighborhoods.\cite{West:2004} These feature sets have been enhanced to extract strong spatially linear features to support scene modeling applications.\cite{Gross:2006} Methods have also been developed to direct optimal neighborhood scale selection for feature attribution.\cite{Demantke:2011}
+
+In this paper, we present a method for extending the previously mentioned metrics and methods to higher dimensional spaces support unsupervised sparse sampling of LiDAR point data while preserving scene information content.
 
 %TODO Provide overview of paper structure.
@@ -81,7 +110,7 @@
 \section{Local Statistic Attribution}
 
-Modern LiDAR sensors produce data sets in low-dimensional spaces typically consisting of 3D position and, depending on the operational mode of the detector array, a measure of intensity proportional to the incident signal strength. Avalanche photodiode (APD) detectors operated in Geiger mode are single-photon sensitive and report only time of flight information, while APD detectors operated in linear mode produce an output signal proportional to the detected optical intensity that can be used to augment peak returns or full waveform data.
-
-To guide the sparse sampling process, we propose extending the attribution of the raw feature set using structure tensor principal component based local statistic attribution. These metrics were first developed by West to guide automated target recognition \cite{West:2004} and later extended by Gross and Thoennessen to extract linear features from LiDAR to support automated generation of 3D models.\cite{Gross:2006}
+Modern LiDAR sensors produce data sets in low-dimensional spaces typically consisting of 3D return locations and, depending on the operational mode of the detector array, a measure of intensity proportional to the incident signal strength. Avalanche photodiode (APD) detectors operated in Geiger mode are single-photon sensitive and capture only ranging information, while APD detectors operated in linear mode produce an integrated output signal proportional to the detected optical intensity that can be used to augment peak returns or may directly represent full-waveform return data.
+
+To guide our sparse sampling process, we propose extending the attribution of the raw feature set using structure tensor principal component based local statistic attribution. These metrics were first developed by West \textit{et al.} to guide automated target recognition \cite{West:2004} and later extended by Gross and Thoennessen to extract linear features from LiDAR to support automated generation of 3D models.\cite{Gross:2006}
 
 In this section we describe the computation of the principal component based attributes and generalize the attributes to arbitrary dimensional spaces.
@@ -89,7 +118,5 @@
 \subsection{Locality}
 
-This paper considers the analysis of real-valued multidimensional points, $\bm{x}$. Each point is assumed to be of the following form where $N$ is the set of attribute dimensions for the native point, including spatial dimensions, and $\lvert N\rvert = n$ is the dimensionality of the native point space.
-
-%TODO add footnote on converting boolean and enumerated values to real space for analysis
+This paper considers the analysis of real-valued\footnote{Boolean and finite-class attributes may be simply represented by an appropriate integer enumeration.} multidimensional points, $\bm{x}$. Each point is assumed to be of the following form where $N$ is the set of native attributes for the point, including spatial coordinates, and $\lvert N\rvert = n$ is the dimensionality of the native feature space.
 
 \begin{align}
@@ -101,5 +128,5 @@
 \end{align}
 
-The standard concept of a point cloud database, $\mathcal{D}$, then is simply a set of real-valued multidimensional points of consistent dimension and description.
+The standard concept of a point cloud, $\mathcal{D}$, then is simply a database of real-valued multidimensional points with consistent feature space definition.
 
 \begin{align}
@@ -107,5 +134,5 @@
 \end{align}
 
-We establish a database of query points, $\mathcal{Q}$, where $M\subseteq N$ is the search space of attributes for the determination of locality and $\lvert M\rvert = m$ is the dimensionality of the search space. In some cases, it may be desirable to restrict queries to a subset of the available native dimensions (\textit{e.g.}, spatial dimensions).
+In some cases, it may be desirable to restrict queries against the point cloud to a subset of the available native feature space (\textit{e.g.}, spatial coordinates). To support this capability, we establish a database of query points, $\mathcal{Q}$, where $M\subseteq N$ is the search space of attributes for the determination of locality and $\lvert M\rvert = m$ is the dimensionality of the search space.
 
 \begin{align}
@@ -113,11 +140,7 @@
 \end{align}
 
-Our analysis is performed on neighborhoods of points from the point cloud database about the query points, $\mathcal{V}_{\bm{q}}$. Note that there is no restriction on the query points, $\bm{q}$,  to be in the point cloud itself. Though, in practice, we typically perform the analysis treating each  $\bm{x}\in\mathcal{D}$ as a query location. This approach effectively results in the need for a reasonable all nearest-neighbor (ANN) search.
-
-\begin{align}
-\mathcal{V}_{\bm{q}}\subseteq\mathcal{D}
-\end{align}
-
-We investigated two neighborhood definitions and each presents merits. The $k$-nearest neighborhood, $\mathcal{V}^{k}_{\bm{q}}$, consists of the $k$ closest points to $\bm{q}$ in $\mathcal{D}$ whereas the fixed-radius neighborhood, $\mathcal{V}^{r}_{\bm{q}}$, consists of all points within the ball of radius $r$ centered at $\bm{q}$ in $\mathcal{D}$.
+Our analysis is performed on neighborhoods of points from the point cloud  about the query points, $\mathcal{V}_{\bm{q}}\subseteq\mathcal{D}$. The neighborhoods are defined by an $m$-dimensional distance metric $d$ between the query points, $\bm{q}\in\mathcal{Q}$, and the data points, $\bm{x}\in\mathcal{D}$. Note that there is no restriction that the query points, $\bm{q}$, must be a member of the point cloud itself. Though, in practice, we typically perform our analysis treating each  $\bm{x}\in\mathcal{D}$ as a query location (\textit{i.e.}, $\mathcal{Q}=\mathcal{D}$). This approach requires a reasonable all nearest-neighbor search.
+
+We investigated two neighborhood definitions that each present merits. The $k$-nearest neighborhood, $\mathcal{V}^{k}_{\bm{q}}$, consists of the $k$ closest points to $\bm{q}$ in $\mathcal{D}$ whereas the fixed-radius neighborhood, $\mathcal{V}^{r}_{\bm{q}}$, consists of all points within the ball of radius $r$ centered at $\bm{q}$ in $\mathcal{D}$.
 
 \begin{equation}
@@ -129,32 +152,67 @@
 \end{equation}
 
-\begin{align}
-\mathcal{V}_{\bm{q}}^{r}=\left\{\bm{q}\in\mathcal{Q},\bm{x}\in\mathcal{D}:d\left(\bm{x},\bm{q}\right)\leq r\right\}
-\end{align}
-
- We discuss the relative merits and demerits of each neighborhood selection strategy later in the paper.
+\begin{equation}
+\begin{aligned}
+\mathcal{V}_{\bm{q}}^{r}=\left\{\mathcal{A}\subseteq\mathcal{D}\right.:&\left.
+\bm{q}\in\mathcal{Q},r\in\mathbb{R},\right.\\
+&\left.\left(\forall\bm{x}\in\mathcal{A},\bm{x}^{\prime}\in\mathcal{D}\setminus\mathcal{A}\right)\right.\\
+&\left.\left(d\left(\bm{x},\bm{q}\right)\leq r<d\left(\bm{x}^{\prime},\bm{q}\right)\right)\right\}
+\end{aligned}
+\end{equation}
+
+We discuss the relative merits and demerits of each neighborhood selection strategy later in the paper. %KLD - Where?
  
- \subsection{Density}
-
-Description of density metric and interpretations.
+\subsection{Density}
+
+One of the simplest local attributes that can be defined for the neighborhood of points is an estimate of the concentration of observations in the neighborhood. This attribute is especially useful for visualizing Geiger-mode LiDAR data since the native point data contains no intensity information. The point density with respect to spatial dimensions is proportional to the reflectivity of the scene at the imaging wavelength and acts as a suitable estimate of "intensity" for many applications.
+
+The definition of density can be generalized to arbitrarily large dimensional spaces, but is most informative in low-dimensional spaces. This is due to the fact that "volume" tends to zero at high dimensions.
+
+The radius used to define the neighborhood volume is based on the maximal distance between the query point and the neighborhood points, $R_{\bm{q}}$. For a fixed-radius neighborhood, this value is simply the query radius, $r$.
+
+\begin{align}
+R_{\bm{q}}&=\max\left\{d\left(\bm{x},\bm{q}\right):\bm{q}\in\mathcal{Q},\forall\bm{x}\in\mathcal{V}_{\bm{q}}\right\}
+\end{align}
+
+The volume of the neighborhood is then estimated as the volume of an $n$-ball in the metric space defined by $d$. When using a Euclidean distance metric, the volume of the neighborhood is estimated as follows. However, similar volume definitions can be derived for other metric spaces.
+
+\begin{align}
+V_{m}\left(R_{\bm{q}}\right)&=\frac{\pi^{\frac{m}{2}}}{\Gamma\left(\frac{m}{2}+1\right)}R_{\bm{q}}^{m}
+\end{align}
+
+The density of the neighborhood is calculated by by the number of observations in the neighborhood divided by the $n$D-volume of the neighborhood. For a $k$-nearest neighborhood, the number of observations is simply the query set size, $k$.
+
+\begin{align}
+\rho_{\bm{q}}=\frac{\lvert\mathcal{V}_{\bm{q}}\rvert}{V_{m}\left(R_{\bm{q}}\right)}
+\end{align}
+
+Given the sensitivity of this attribute to the volume estimate, a useful alternative definition may be to use the low-dimensional embedding dimensionality to determine the dimensionality of the $n$-ball used for the volume estimate instead of the query or native attribute dimensionality. This alternative definition was not examined in depth for this paper.
 
 \subsection{Principal Component Analysis}
 
-Data matrix:
-
-\begin{align}
-\bm{X}=\left[\begin{array}{ccc}\bm{x}_{1}&\cdots&\bm{x}_{n}\end{array}\right],\forall\bm{x}\in\mathcal{V}_{\bm{q}}
-\end{align}
-
-Covariance matrix:
-
-\begin{align}
-\bm{C}=\frac{1}{k-1}\bm{X}\left(\bm{\mathrm{I}}_{k}-\frac{1}{k}\bm{\mathrm{J}}_{k}\right)\bm{X^{\mathsf{T}}}
-\end{align}
-
-Alternatively (more computationally efficient):
-
-\begin{align}
-\bar{\bm{x}} = \frac{1}{k}\sum_{p=1}^{k}{\bm{x}_{p}}
+The remainder of the attributes explored in this paper are derived from eigenanalysis of the query neighborhoods. Eigenanalysis is performed via eigenvalue decomposition of the empirical covarance matrix for each query neighborhood. For simplicity, we will refer to the cardinality of the neighborhood as $k$, even when considering the fixed-radius neighborhood definition.
+
+\begin{align}
+k=\lvert\mathcal{V}_{\bm{q}}\rvert
+\end{align}
+
+The process begins by organizing the point data for each query neighborhood into a data matrix.
+
+\begin{align}
+\bm{X}=\left[\begin{array}{ccc}\bm{x}_{1}&\cdots&\bm{x}_{k}\end{array}\right],\forall\bm{x}\in\mathcal{V}_{\bm{q}}
+\end{align}
+
+
+
+The empirical covariance matrix for the neighborhood is computed by re-centering the data matrix about its mean and computing the outer product of the resulting matrix with itself. This can be represented succinctly as follows:
+
+\begin{align}
+\bm{C}=\frac{1}{k-1}\bm{X}\left(\bm{\mathrm{I}}_{k}-\frac{1}{k}\bm{\mathrm{J}}_{k}\right)\bm{X}^{\mathsf{T}}
+\end{align}
+
+However, data volume generally prohibits computing the empirical covariance matrix in this manner. Instead, it is more computationally efficient to compute the mean of the data matrix and the matrix of mean deviations directly.
+
+\begin{align}
+\bar{\bm{x}} = \frac{1}{k}\sum_{p=1}^{k}{\bm{x}_{p}},\forall\bm{x}\in\mathcal{V}_{\bm{q}}
 \end{align}
 
@@ -169,12 +227,14 @@
 
 \begin{align}
-\bm{C}=\frac{1}{k-1}\bm{M}\bm{M^{\mathsf{T}}}
-\end{align}
-
-Perform Eigendecomposition:
+\bm{C}=\frac{1}{k-1}\bm{M}\bm{M}^{\mathsf{T}}
+\end{align}
+
+The construction of the empirical covariance matrix guarantees that it is a square non-negative definite matrix and thus is guaranteed to be diagonalizable. We can therefore perform the eigendecomposition by factorizing the empirical covariance matrix as follows:
 
 \begin{align}
 \bm{C}=\bm{U}\bm{\Lambda}\bm{U^{\mathsf{T}}}
 \end{align}
+
+The factorization results in a matrix of eigenvectors of the system, $\bm{U}$, and a diagonal matrix, $\bm{\Lambda}$, whose main diagonal elements are the corresponding eigenvalues of the system.
 
 \begin{align}
@@ -193,7 +253,13 @@
 \end{align}
 
+By convention, the eigenvalues and corresponding eigenvectors are provided in decreasing order of eigenvalue magnitude. That is, $\lambda_{1}\geq\lambda_{2}\geq\hdots\geq\lambda_{n}$, which implies that the most significant components are listed first in the $\bm{U}$ and $\bm{\Lambda}$ matrices. We assume that the eigenvectors have been normalized.
+
+It is also worth noting that the eigenvalues matrix is simply the square of the singular values matrix.
+
 \begin{align}
 {\bm{\Lambda}}=\bm{\Sigma}\bm{\Sigma^{\mathsf{T}}}
 \end{align}
+
+This means that it is trivial to define two feature vectors for the neighborhood, $\mathcal{V}_{\bm{q}}$. The first is derived from the diagonal elements of the eigenvalue matrix:
 
 \begin{align}
@@ -207,4 +273,6 @@
 \end{align}
 
+The second is derived from the diagonal elements of the singular value matrix, which are simply the square roots of the diagonal elements of the eigenvalue matrix:
+
 \begin{align}
 \bm{\sigma}=\left(\begin{array}{c}\bm{\Sigma}_{1,1}\\
@@ -214,7 +282,78 @@
              \sigma_{2}\\
              \vdots\\
-             \sigma_{n}\end{array}\right)
-\end{align}
-
+             \sigma_{n}\end{array}\right)=\left(\begin{array}{c}\sqrt{\lambda_{1}}\\
+             \sqrt{\lambda_{2}}\\
+             \vdots\\
+             \sqrt{\lambda_{n}}\end{array}\right)
+\end{align}
+
+These two feature vectors provide the basis for all subsequent analysis in this paper.
+
+\subsection{Structure Features}
+
+The Defense Advanced Research Projects Agency (DARPA) Exploitation of 3-D Data (E3D) Program  identified several structure tensor features to facilitate automated target recognition. West \textit{et al.} present six features that proved most applicable to their work in segmentation and object recognition: omnivariance, anisotropy, linearity, planarity, sphericity, and entropy.\cite{West:2004} They define each of the features with respect to the three spatial dimensions since their analysis was limited to structure tensors of points and normals. However, each of these features can be generalized to higher order dimensions.
+
+Linearity, planarity, and sphericity are closely related features that each represent the concept of dimensional participation. That is, how much does the local neighborhood "spread" into the dimension under consideration. We generalize this concept as "dimensionality".
+
+The highest order dimensionality considered by West \textit{et al.} is 3-dimensionality, which they referred to as "sphericity". We feel that it is worth considering the highest order dimensionality for the data set as a unique feature as well, and generalize the concept to "isotropy". The complement of this value, "anisotropy", is thus easily understood and maintains a definition consistent with West's.
+
+In addition to the 
+
+\subsubsection{Isotropy}
+Further discussion of information provided by this classifier.
+
+\begin{align}
+Iso\left(\mathcal{V}_{\bm{q}}\right)&=\frac{\lambda_{n}}{\lambda_{1}}
+\end{align}
+
+\subsubsection{Anisotropy}
+Further discussion of information provided by this classifier. Estimate of degree of structure present in local area feature. Complement of isotropy.
+
+\begin{align}
+Ani\left(\mathcal{V}_{\bm{q}}\right)
+&=\frac{\lambda_{1} - \lambda_{n}}{\lambda_{1}}
+\end{align}
+
+\subsubsection{Dimensionality}
+Further discussion of information provided by this classifier. Estimates the embedded dimension of local area feature.
+
+\begin{align}
+Dim_{D}\left(\mathcal{V}_{\bm{q}}\right)&=\left\{\begin{array}{ll}
+\frac{\lambda_{D} - \lambda_{D+1}}{\lambda_{1}}&,\,D<n\\
+Iso\left(\mathcal{V}_{\bm{q}}\right)&,\,D=n
+\end{array}\right.
+\end{align}
+
+\subsubsection{Low-Dimensional Embedding}
+
+\begin{align}
+Emb\left(\mathcal{V}_{\bm{q}}\right)&=\operatorname*{\arg\!\max}_{D\in[1,n]}\left(Dim_{D}\left(\mathcal{V}_{\bm{q}}\right)\right)
+\end{align}
+
+\subsubsection{Entropy}
+Further discussion of information provided by this classifier. Estimates the number of dimensions needed to encode information content of local area feature.
+
+\begin{align}
+H\left(\mathcal{V}_{\bm{q}}\right)&=-\sum_{D=1}^{n}{\hat{\lambda}_{D}\log_{n}\left(\hat{\lambda}_{D}\right)}
+\end{align}
+
+\subsubsection{Omnivariance}
+Further discussion of information provided by this classifier. Geometric mean of structure-tensor eigenvalues. Estimates data variance across dimensions of local area feature.
+
+\begin{align}
+Omnivar\left(\mathcal{V}_{\bm{q}}\right)&=
+\left(\prod_{D=1}^{n}{\lambda_{D}}\right)^{\frac{1}{n}}
+\end{align}
+
+\subsubsection{Fractional Anisotropy}
+
+Further discussion of information provided by this classifier. 
+
+\begin{align}
+FA\left(\mathcal{V}_{\bm{q}}\right)&=
+\left(\frac{n\sum\limits_{D=1}^{n}
+{\left(\lambda_{D}-\bar{\lambda}\right)^{2}}}
+{\left(n-1\right)\sum\limits_{D=1}^{n}{\lambda_{D}^{2}}}\right)^{\frac{1}{2}}
+\end{align}
 
 \subsection{Normal Estimation}
@@ -234,65 +373,4 @@
 
 Curvature\cite{Kalogerakis:2009}
-
-\subsection{Structure Features}
-
-Structure Tensor features introduced by West\cite{West:2004}:
-
-\subsubsection{Isotropy}
-Further discussion of information provided by this classifier.
-
-\begin{align}
-Iso\left(\mathcal{V}_{\bm{q}}\right)&=\frac{\lambda_{n}}{\lambda_{1}}
-\end{align}
-
-\subsubsection{Anisotropy}
-Further discussion of information provided by this classifier. Estimate of degree of structure present in local area feature. Complement of isotropy.
-
-\begin{align}
-Ani\left(\mathcal{V}_{\bm{q}}\right)
-&=\frac{\lambda_{1} - \lambda_{n}}{\lambda_{1}}
-\end{align}
-
-\subsubsection{Dimensionality}
-Further discussion of information provided by this classifier. Estimates the embedded dimension of local area feature.
-
-\begin{align}
-Dim_{d}\left(\mathcal{V}_{\bm{q}}\right)&=\left\{\begin{array}{ll}
-\frac{\lambda_{d} - \lambda_{d+1}}{\lambda_{1}}&,\,d<n\\
-Iso\left(\mathcal{V}_{\bm{q}}\right)&,\,d=n
-\end{array}\right.
-\end{align}
-
-\subsubsection{Low-Dimensional Embedding}
-
-\begin{align}
-Emb\left(\mathcal{V}_{\bm{q}}\right)&=\operatorname*{\arg\!\max}_{d\in[1,n]}\left(Dim_{d}\left(\mathcal{V}_{\bm{q}}\right)\right)
-\end{align}
-
-\subsubsection{Entropy}
-Further discussion of information provided by this classifier. Estimates the number of dimensions needed to encode information content of local area feature.
-
-\begin{align}
-H\left(\mathcal{V}_{\bm{q}}\right)&=-\sum_{d=1}^{n}{\hat{\lambda}_{d}\log_{n}\left(\hat{\lambda}_{d}\right)}
-\end{align}
-
-\subsubsection{Omnivariance}
-Further discussion of information provided by this classifier. Geometric mean of structure-tensor eigenvalues. Estimates data variance across dimensions of local area feature.
-
-\begin{align}
-Omnivar\left(\mathcal{V}_{\bm{q}}\right)&=
-\left(\prod_{d=1}^{n}{\lambda_{d}}\right)^{\frac{1}{n}}
-\end{align}
-
-\subsubsection{Fractional Anisotropy}
-
-Further discussion of information provided by this classifier. 
-
-\begin{align}
-FA\left(\mathcal{V}_{\bm{q}}\right)&=
-\left(\frac{n\sum\limits_{d=1}^{n}
-{\left(\lambda_{d}-\bar{\lambda}\right)^{2}}}
-{\left(n-1\right)\sum\limits_{d=1}^{n}{\lambda_{d}^{2}}}\right)^{\frac{1}{2}}
-\end{align}
 
 \section{Salience Weighting}