\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.3 - Estimation of Local Differential Properties of Point-Sampled Surfaces

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User Manual

Authors
  Marc Pouget and Frédéric Cazals

Figure 55.1 Principal directions associated with \(k_{max}\) scaled by \( k_{min}\).

This package allows the estimation of local differential quantities of a surface from a point sample, given either as a mesh or as point cloud.

Note that this package needs the third party library Eigen to be installed to compile the example code..

# Introduction

## Overview

Consider a sampled smooth surface, and assume we are given a collection of points \( P\) about a given sample \( p\). We aim at estimating the differential properties up to any fixed order of the surface at point \( p\) from the point set \( P^+ = P\cup \{ p\}\) - we denote \( N=\mid P^+\mid\). More precisely, first order properties correspond to the normal or the tangent plane; second order properties provide the principal curvatures and directions, third order properties provide the directional derivatives of the principal curvatures along the curvature lines, etc. Most of the time, estimating first and second order differential quantities is sufficient. However, some applications involving shape analysis require estimating third and fourth order differential quantities. Many different estimators have been proposed in the vast literature of applied geometry [4] (section 3, page 7), and all of them need to define a neighborhood around the point at which the estimation is computed. Our method relies on smooth differential geometry calculations, carried out on smooth objects fitted from the sample points. Datasets amenable to such a processing are naturally unstructured point clouds, as well as meshes - whose topological information may be discarded.

Estimating differential properties from discrete date always raises a philosophical issue. On one hand, estimating differential quantities subsumes a smooth surface does exist. In this spirit one wishes to recover its differential properties, so that any estimation method must come with an asymptotic convergence analysis of the results returned. For the method developed in this CGAL package, the interested will find such an analysis in [1], (Theorem 3) - it should be stressed the error bounds proved therein are optimal.

On the other hand, any estimation method may be applied to arbitrarily data - surface unknown, surface piecewise smooth etc. In such a case, no analysis can be carried out, and it is up to the users to check the results match their needs.

Unlike most of the CGAL packages, this package uses approximation methods and is not intended to provide an exact canonical result in any sense. This is why internal computations are performed with a number type possibly different from that of the input data, even if for convenience the results are returned with this original number type. A reasonable choice for this internal number type is for example the `double` type.

## Smooth Surfaces, d-Jets and the Monge Form

To present the method, we shall need the following notions. Consider a smooth surface. About one of its points, consider a coordinate system whose \( z\)-axis does not belong to the tangent space. In such a frame, the surface can locally be written as the graph of a bivariate function. Letting \( h.o.t.\) stand for higher order terms, one has :

\[ \begin{equation} z(x,y)=J_{B,d}(x,y) + h.o.t. \ ; \quad J_{B,d}(x,y)=\ccSum{k=0}{d}{(\ccSum{i=0}{i}{ \frac{B_{k-i,i}x^{k-i}y^{i}}{i!(k-i)!}})}. \end{equation} \]

The degree \( d\) polynomial \( J_{B,d}\) is the Taylor expansion of the function \( z\), and is called its \( d\)-jet. Notice that a \( d\)-jet contains \( N_d=(d+1)(d+2)/2\) coefficients.

Recall that an umbilical point of a surface - or umbilic for short, is a point where both principal curvatures are identical. At any point of the surface which is not an umbilic, principal directions \( d_1, d_2\) are well defined, and these (non oriented) directions together with the normal vector \( n\) define two direct orthonormal frames. If \( v_1\) is a unit vector of direction \( d_1\), there exists a unique unit vector \( v_2\) so that \( (v_1,v_2,n)\) is direct; and the other possible frame is \( (-v_1,-v_2,n)\). Both these coordinate systems are known as the Monge coordinate systems. In both these systems, the surface is said to be given in the Monge form and its jet has the following canonical form :

\begin{eqnarray} z(x,y) = & \frac{1}{2}(k_1x^2 + k_2y^2)+ \frac{1}{6}(b_0x^3+3b_1x^2y+3b_2xy^2+b_3y^3) \\ & +\frac{1}{24}(c_0x^4+4c_1x^3y+6c_2x^2y^2+4c_3xy^3+c_4y^4) + h.o.t. \end{eqnarray}

The coefficients \( k_1, k_2\) are the principal curvatures, \( b_0,b_3\) are the directional derivatives of \( k_1,k_2\) along their respective curvature line, while \( b_1,b_2\) are the directional derivatives of \( k_1,k_2\) along the other curvature lines.

The Monge coordinate system can be computed from any \( d\)-jet ( \( d\geq 2\)), and so are the Monge coefficients. These informations characterize the local geometry of the surface in a canonical way, and are the quantities returned by our algorithm.

## Algorithm

Based on the above concepts, the algorithm consists of 4 steps.

1. We perform a Principal Component Analysis (PCA) on \( P^+\). This analysis outputs three orthonormal eigenvectors and the associated eigenvalues. The fitting basis consists of these three vectors so that the vector associated to the smallest eigenvalue is the last vector of the basis. (Indeed, if the surface is well sampled, one expects the PCA to provide one small and two large eigenvalues, the eigenvector associated to the small one approximating the normal vector.)
2. We perform a change of coordinates to move the samples into the coordinate system of the fitting basis and with origin the point \( p\) at which the estimation is sought. We then resort to polynomial fitting, so as to either interpolate or approximate the \( d\)-jet of the surface in this coordinate system. This bivariate polynomial approximation reduces to linear algebra operations.
3. From the fitted \( d\)-jet, we compute the Monge basis \( (d_1,d_2,n)\).
4. Finally, we compute the Monge coefficients : \( k_i, b_i, c_i\).

Further details can be found in section Mathematical and Algorithmic Details and in [1] (section 6).

## Degenerate Cases

As usual, the fitting procedure may run into (almost) degenerate cases:

- Due to poor sampling, the PCA used to determine a rough normal vector may not be good. The nearer this direction to the tangent plane the worse the estimation.

- As observed in [1] (section 3.1), the interpolating problem is not well posed if the points project, into the fitting frame, onto an algebraic curve of degree \( d\). More generally, the problem is ill posed if the condition number is too large.

In these cases, even if a result is provided, the estimation may not be relevant. To inform the user of these issues, we provide the PCA results and the condition number of the fitting. In any case, it is up to the user to judge if the result meets its need.

# Software Design

## Options and Interface Specifications

The fitting strategy performed by the class `Monge_via_jet_fitting` requires the following parameters:

- the degree \( d\) of the fitted polynomial ( \( d \geq 1\)),
- the degree \( d'\) of the Monge coefficients sought, with \( 1 \leq d' \leq \min(d,4) \),
- a range of \( N\) input points on the surface, with the precondition that \( N \geq N_d = (d+1)(d+2)/2\). Note that if \( N=N_d\), interpolation is performed; and if \( N >N_d\), approximation is used.

## Output

As explained in Section Introduction , the output consists of a coordinate system, the Monge basis, together with the Monge coefficients which are stored in the `Monge_via_jet_fitting::Monge_form` class. In addition, more information on the computational issues are stored in the `Monge_via_jet_fitting` class.

The `Monge_via_jet_fitting::Monge_form` class provides the following information.

- Origin. This is the point on the fitted polynomial surface where the differential quantities have been computed. In the approximation case, it differs from the input point \( p\) : it is the projection of \( p\) onto the fitted surface following the \( z\)-direction of the fitting basis.

- Monge Basis. The Monge basis \( (d_1,d_2,n)\) is orthonormal direct, and the maximal, minimal curvatures are defined wrt this basis. If the user has a predefined normal \( n_0\) (e.g. the sample points come from an oriented mesh) then if \( n_0 . n >0\) then max-min is correct; if not, i.e. \( n_0 . n <0\), the user should switch to the orthonormal direct basis \( (d_1',d_2',n')=(d_2,d_1,-n)\) with the maximal curvature \( k_1'=-k_2\) and the minimal curvature \( k_2'=-k_1\). If \( n_0 . n =0\) or is small, the orientation of the surface is clearly ill-defined, and the user may proof-check the samples used to comply with its predefined normal.

- Monge Coefficients. The coefficient of the Monge form is \( (k_1, k_2 (\leq k_1), b_0, b_1, b_2, b_3, c_0, c_1, c_2, c_3, c_4)\) for \( d' = 4\).

In addition, the class `Monge_via_jet_fitting` stores

- the condition number of the fitting system,
- the coordinate system of the PCA (in which the fitting is performed).

## Template Parameters

### Template parameter DataKernel

This concept provides the types for the input sample points, together with \( 3d\) vectors and a number type. It is used as template for the class `Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>`. Typically, one can use `Cartesian<double>`.

### Template parameter LocalKernel

This is a parameter of the class `Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>`. This concept defines the vector and number types used for local computations and to store the PCA basis data.

Input points of type `DataKernel::Point_3` are converted to `LocalKernel::Point_3`. For output of the `Monge_via_jet_fitting::Monge_form` class, these types are converted back to `DataKernel` ones. Typically, one can use `Cartesian<double>` which is the default.

### Template parameter SvdTraits

This concept provides the number, vector and matrix types for algebra operations required by the fitting method in `Monge_via_jet_fitting<DataKernel, LocalKernel, SvdTraits>` . The main method is a linear solver using a singular value decomposition.

### Compatibility Requirements

To solve the fitting problem, the sample points are first converted from the `DataKernel` to the `LocalKernel` (this is done using the `Cartesian_converter`). Then change of coordinate systems and linear algebra operations are performed with this kernel. This implies that the number types `LocalKernel::FT` and `SvdTraits::FT` must be identical. Second the Monge basis and coefficients, computed with the `LocalKernel`, are converted back to the `DataKernel` (this is done using the `Cartesian_converter` and the `NT_converter`).

# Examples

## Single Estimation about a Point of a Point Cloud

The first example illustrates the computation of the local differential quantities from a set of points given in a text file as input. The first point of the list is the one at which the computation is performed. The user has to specify a file for the input points and the degrees \( d\) and \( d'\).
File Jet_fitting_3/Single_estimation.cpp

#include <CGAL/Simple_cartesian.h>
#include <CGAL/Monge_via_jet_fitting.h>
#include <fstream>
#include <vector>
typedef double DFT;
typedef CGAL::Simple_cartesian<DFT> Data_Kernel;
typedef Data_Kernel::Point_3 DPoint;
typedef CGAL::Monge_via_jet_fitting<Data_Kernel> My_Monge_via_jet_fitting;
typedef My_Monge_via_jet_fitting::Monge_form My_Monge_form;
int main(int argc, char *argv[])
{
size_t d_fitting = 4;
size_t d_monge = 4;
const char* name_file_in = "data/in_points_d4.txt";
//check command line
if (argc<4)
{
std::cout << " Usage : " << argv[0]
<< " <inputPoints.txt> <d_fitting> <d_monge>" << std::endl
<< "test with default arguments" << std::endl;
}
else {
name_file_in = argv[1];
d_fitting = std::atoi(argv[2]);
d_monge = std::atoi(argv[3]);
}
//open the input file
std::ifstream inFile(name_file_in);
if ( !inFile )
{
std::cerr << "cannot open file for input\n";
exit(-1);
}
//initalize the in_points container
double x, y, z;
std::vector<DPoint> in_points;
while (inFile >> x) {
inFile >> y >> z;
DPoint p(x,y,z);
in_points.push_back(p);
}
inFile.close();
// fct parameters
My_Monge_form monge_form;
My_Monge_via_jet_fitting monge_fit;
monge_form = monge_fit(in_points.begin(), in_points.end(), d_fitting, d_monge);
//OUTPUT on std::cout
CGAL::set_pretty_mode(std::cout);
std::cout << "vertex : " << in_points[0] << std::endl
<< "number of points used : " << in_points.size() << std::endl
<< monge_form;
std::cout << "condition_number : " << monge_fit.condition_number() << std::endl
<< "pca_eigen_vals and associated pca_eigen_vecs :" << std::endl;
for (int i=0; i<3; i++)
std::cout << monge_fit.pca_basis(i).first << std::endl
<< monge_fit.pca_basis(i).second << std::endl;
return 0;
}

## On a Mesh

The second example (cf Mesh_estimation.cpp in the example directory) illustrates the computation of local differential quantities for all vertices of a given mesh. The neighborhood of a given vertex is computed using rings on the triangulation. Results are twofold:

- a human readable text file featuring the `Monge_via_jet_fitting::Monge_form` and numerical informations on the computation: condition number and the PCA basis;
- another text file that records raw data (better for a vizualization post-processing).

Figure 55.1 and Figure 55.2 provide illustrations of principal directions of curvature.

Figure 55.2 Principal directions of curvature and normals at vertices of a mesh of the graph of the function \( f(x,y)=2x^2+y^2\).

# Mathematical and Algorithmic Details

In this Section, we detail the mathematics involved, in order to justify the design choices made. To begin with, observe the fitting problem involves three relevant direct orthonormal basis: the world-basis \( (w_x,w_y,w_z)\), the fitting-basis \( (f_x,f_y,f_z)\), the Monge basis \( (d_1,d_2,n)\).

Figure 55.3 The three bases involved in the estimation.

## Computing a Basis for the Fitting

Input: samples Output: fitting-basis

Performing a PCA requires diagonalizing a symmetric matrix. This analysis gives an orthonormal basis whose \( z\)-axis is provided by the eigenvector associated to the smallest eigenvalue.Another possibility is to choose as z-axis the axis of the world-basis with the least angle with the axis determined with the PCA. Then the change of basis reduces to a permutation of axis. Note one may have to swap the orientation of a vector to get a direct basis.

Let us denote \( P_{W\rightarrow F}\) the matrix that changes coordinates from the world-basis \( (w_x,w_y,w_z)\) to the fitting-basis \( (f_x,f_y,f_z)\). The rows of \( P_{W\rightarrow F}\) are the coordinates of the vectors \( (f_x,f_y,f_z)\) in the world-basis. This matrix represents a orthogonal transformation hence its inverse is its transpose. To obtain the coordinates of a point in the fitting-basis from the coordinates in the world-basis, one has to multiply by \( P_{W\rightarrow F}\).

As mentioned above, the eigenvalues are returned, from which the sampling quality can be assessed. For a good sampling, the eigenvector associated to the smallest eigenvalue should roughly give the normal direction.

## Solving the Interpolation / Approximation Problem

Input: samples, fitting-basis Output: coefficients \( A_{i,j}\) of the bivariate fitted polynomial in the fitting-basis

Computations are done in the fitting-basis and the origin is the point \( p\). First, one has to transform coordinates of sample points with a translation ( \( -p\)) and multiplication by \( P_{W\rightarrow F}\).

The fitting process consists of finding the coefficients \( A_{i,j}\) of the degree \( d\) polynomial

\[ \begin{equation} J_{A,d}= \ccSum{k=0}{d}{(\ccSum{i=0}{k}{ \frac{A_{k-i,i}x^{k-i}y^{i}}{i!(k-i)!}})}. \end{equation} \]

Denote \( p_i=(x_i,y_i,z_i), \ i=1,\ldots , N\) the coordinates of the sample points of \( P^+\). For interpolation the linear equations to solve are \( A(x_i,y_i)=z_i \ i=1,\ldots,N\), and for approximation one has to minimize \( \sum_{i=1}^N (A(x_i,y_i)-z_i)^2\). The linear algebra formulation of the problem is given by

\begin{eqnarray} A = & (A_{0,0}, A_{1,0},A_{0,1}, \ldots , A_{0,d})^T \\ Z= &(z_1, z_2,\ldots , z_N)^T \\ M= &(1,x_i,\ y_i,\ \frac{x_i^2}{2},\ldots , \ \frac{x_iy_i^{d-1}}{(d-1)!},\ \frac{y_i^d}{d!})_{i=1,...,N}\\ \end{eqnarray}

The equations for interpolation become \( MA=Z\). For approximation, the system \( MA=Z\) is solved in the least square sense, i.e. one seeks the vector \( A\) such that \( A = \arg \min_A ||MA-Z||_2\).

In any case, there is a preconditioning of the matrix \( M\) so as to improve the condition number. Assuming the \( \{x_i\}\), \( \{y_i\}\) are of order \( h\), the pre-conditioning consists of performing a column scaling by dividing each monomial \( x_i^ky_i^l\) by \( h^{k+l}\) - refer to Eq. ( eqfitlinalg ). Practically, the parameter \( h\) is chosen as the mean value of the \( \{x_i\}\) and \( \{y_i\}\). In other words, the new system is \( M'Y=(MD^{-1})(DA)=Z\) with \( D\) the diagonal matrix \( D=(1,h,h,h^2,\ldots,h^d,h^d)\), so that the solution \( A\) of the original system is \( A=D^{-1}Y\).

There is always a single solution since for under constrained systems we also minimize \( ||A||_2\). The method uses a singular value decomposition of the \( N\times N_d\) matrix \( M= U S V^T\), where \( U\) is a \( N \times N\) orthogonal matrix, \( V\) is a \( N_d \times N_d\) orthogonal matrix and \( S\) is a \( N\times N_d\) matrix with the singular values on its diagonal. Denote \( r\) the rank of \( M\), we can decompose \( S= \left( \begin{array}{cc} D_r & 0_{r,\ N_d-r}\\ 0_{N-r,\ r} & 0_{N-r,\ N_d-r} \end{array} \right). \) The number \( r\), which is the number of non zero singular values, is strictly lower than \( N_d\) if the system is under constrained. In any case, the unique solution which minimize \( ||A||_2\) is given by:

\[ \begin{equation} A= V \left( \begin{array}{cc} D_r^{-1} & 0_{N_d-r,\ r}\\ 0_{r,\ N-r} & 0_{N_d-r,\ N-r} \end{array} \right) U^TZ. \end{equation} \]

One can provide the condition number of the matrix \( M\) (after preconditioning) which is the ratio of the maximal and the minimal singular values. It is infinite if the system is under constrained, that is the smallest singular value is zero.

Implementation details.

We assume a `solve` function is provided by the traits `SvdTraits`. This function solves the system MX=B (in the least square sense if M is not square) using a Singular Value Decomposition and gives the condition number of M.

Remark: as an alternative, other methods may be used to solve the system. A \( QR\) decomposition can be substituted to the \( SVD\). One can also use the normal equation \( M^TMX=MTB\) and apply methods for square systems such as \( LU\), \( QR\) or Cholesky since \( M^TM\) is symmetric definite positive when \( M\) has full rank. The advantages of the \( SVD\) is that it works directly on the rectangular system and gives the condition number of the system. For more on these alternatives, see [3] (Chap. 5).

## Principal Curvature / Directions

Input: coefficients of the fit \( A_{i,j}\), fitting-basis

Output: Monge basis wrt fitting-basis and world-basis

In the fitting basis, we have determined a height function expressed by Eq. ( eqanswer ). Computations are done in the fitting-basis. The partial derivatives, evaluated at \( (x,y)=(0,0)\), of the fitted polynomial \( J_{A,d}(x,y)\) are \( A_{i,j}=\frac{\partial^{i+j}J_{A,d}}{\partial^ix \partial^jy}\) Expanding Eq. ( eqanswer ) yields:

\begin{eqnarray} J_{A,d}(x,y)&= A_{0,0}+A_{1,0}x+A_{0,1}y+\frac{1}{2}(A_{2,0}x^2+2A_{1,1}xy+A_{0,2}y^2) + \frac{1}{6}(A_{3,0}x^3+3A_{2,1}x^2y+\ldots )+ \ldots \end{eqnarray}

- The origin, that is the point of the fitted surface where the estimation is performed, is \( (0,0,A_{0,0})\).
- The normal is \( n=(-A_{1,0},-A_{0,1},1)/\sqrt{A_{1,0}^2+A_{0,1}^2+1}\).
- Curvature related properties are retrieved resorting to standard differential calculus [2] (Chap. 3). More precisely, the Weingarten operator \( W=-I^{-1}II\) is first computed in the basis of the tangent plane \( \{ (1,0,A_{1,0}), (0,1,A_{0,1}) \}\). We compute an orthonormal basis of the tangent plane using the Gram-Schmidt algorithm, and then we compute Weingarten in this basis (applying a change of basis with the matrix \( W'=P^{-1}WP\)). In this orthonormal basis, the matrix of the Weingarten map is symmetric and we diagonalize it. One finally gets the principal curvatures which are the eigenvalues of \( W\), and the associated principal directions. This gives an orthonormal direct basis \( (d_1,d_2,n)\). Let us denote \( P_{F \rightarrow M}\) the matrix to change coordinates from the fitting-basis to the Monge basis. Its rows are the coordinates of the vectors \( (d_1,d_2,n)\) in the fitting-basis. It is an orthogonal matrix \( P_{F \rightarrow M}^{-1}=P_{F \rightarrow M}^T\). The Monge basis expressed in the world-basis is obtained by multiplying the coordinates of \( (d_1,d_2,n)\) in the fitting-basis by \( P_{W\rightarrow F}^{-1}\), (the same holds for the origin point which has in addition to be translated by \( p\), i.e. the coordinates of the origin point are \( P_{W\rightarrow F}^{-1} (0,0,A_{0,0}) +p\).

## Computing Higher Order Monge Coefficients

Input: coefficients of the fit, Monge basis wrt fitting-basis ( \( P_{F \rightarrow M}\))

Output: third and fourth order coefficients of Monge

We use explicit formula. The implicit equation of the fitted polynomial surface in the fitting-basis with origin the point \( (0,0,A_{0,0})\) is \( Q=0\) with

\[ \begin{equation} Q=-w-A_{0,0} +\ccSum{i,j}{}{\frac{A_{i,j}u^iv^j}{i!j!}}. \end{equation} \]

The equation in the Monge basis is obtained by substituting \( (u,v,w)\) by \( P^T_{F\rightarrow M}(x,y,z)\). Denote \( f(x,y,z)=0\) this implicit equation. By definition of the Monge basis, we have locally (at \( (0,0,0)\))

\[ \begin{equation} f(x,y,z)=0 \Leftrightarrow z=g(x,y) \end{equation} \]

and the Taylor expansion of \( g\) at \( (0,0)\) are the Monge coefficients sought. Let us denote the partial derivatives evaluated at the origin of \( f\) and \( g\) by \( f_{i,j,k}=\frac{\partial^{i+j+k}f}{\partial^ix \partial^jy \partial^kz}\) and \( g_{i,j}=\frac{\partial^{i+j}g}{\partial^ix \partial^jy}\). One has \( f_{1,0,0}=f_{0,1,0}=f_{1,1,0}=0\), \( g_{0,0}=g_{1,0}=g_{0,1}=g_{1,1}=0\) and \( g_{2,0}=k_1\), \( g_{0,2}=k_2\). The partial derivative of order \( n\) of \( f\) depends on the matrix \( P_{F\rightarrow M}\) and the partial derivatives of order at most \( n\) of \( J_{A,d}\). The third and fourth order coefficients of are computed with the implicit function theorem. For instance:

\begin{eqnarray*} &b_0=g_{3,0}=-{\frac { f_{3,0,0} f_{0,0,1} -3\, f_{1,0,1} f_{2,0,0} }{ f_{0,0,1} ^{2}}} \\ &b_1=g_{2,1}=-{\frac {- f_{0,1,1} f_{2,0,0} + f_{2,1,0} f_{0,0,1} }{ f_{0,0,1} ^{2}}} \\ & .... \\ \end{eqnarray*}