Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 582870, 11 pages

http://dx.doi.org/10.1155/2015/582870

## Spherical Harmonics for Surface Parametrisation and Remeshing

School of Computer Science, The University of Nottingham, Jubilee Campus, Wollaton Road, Nottingham NG8 1BB, UK

Received 5 August 2015; Revised 2 November 2015; Accepted 9 November 2015

Academic Editor: Masoud Hajarian

Copyright © 2015 Caitlin R. Nortje et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a novel method for parametrisation and remeshing incomplete and irregular polygonal meshes. Spherical harmonics basis functions are used for parametrisation. This involves least squares fitting of spherical harmonics basis functions to the surface mesh. Tikhonov regularisation is then used to improve the parametrisation before remeshing the surface. Experiments show that the proposed techniques are effective for parametrising and remeshing polygonal meshes.

#### 1. Introduction

Polygonal meshes are often used to represent the surfaces of graphical models. Often a mesh is irregular or unsatisfactory due to over- or undersampling or the triangularisation technique used. There is thus the need to improve the quality of the mesh by remeshing.

There are two main categories of remeshing techniques: parametrisation methods [1–5] and mesh adaptation strategies [6–8]. Parametrisation methods indirectly improve a given mesh by parametrising it, often through bijective mapping from the original mesh to a 2D planar domain. New sampling can then be performed on this planar domain and, using the inverse of the original mapping function, is mapped back to the surface to form a new mesh. Unlike parametrisation techniques, mesh adaption strategies directly improve meshes using relaxation or modification processes to iteratively make local improvements to the mesh [9], until either some size or quality criterion is reached.

Parametrisation and adaption approaches can also be combined. An example of this is a method that assumes a mesh is formed of parametric patches bounded by parametric curves [10] and iteratively improves the mesh by calculating the quality of a patch. Vertices and therefore faces are inserted into some areas and deleted from others until the global quality criterion is achieved. The refinement of a mesh region is determined by a local error computed at each vertex in a patch, which is the difference between the surface curvatures of the original patch and the new patch.

Another example of the combined approaches is the isometric parametrisations (MIPs) method [11]. Here, the remeshing is an adaption strategy on a mesh parametrised to a planar domain and the final result is mapped back to the domain of the original mesh. The remeshing process involves an umbrella operator that selects a vertex to be improved while fixing all neighbouring vertices and treats edges connecting the centre vertex and the neighbouring vertices as “springs.” Each spring is assigned an energy, and the minimisation of the combined energy of all springs is used to determine the new position of the centre vertex. Essentially, this minimisation can be seen as a push and pull to reach an equilibrium of energy. The process is repeated for all vertices until the mesh reaches a certain specified quality or the difference in movement in the vertex positions between iterations is below a certain threshold.

Such local improvement methods tend to be greedy in their approach, and therefore the global quality of the resulting mesh can suffer [12]. These meshes may also be inefficient when carrying out numerous sampling operations in 3D space [3]. Furthermore, methods that parametrise surfaces to a single domain are often constrained to meshes that are topologically similar to that domain.

One group of methods exploits the fact that some meshes can be transformed into a single topologically planar mesh through a series of cuts [13]. These cuts are stored in a graph and are applied to the original mesh prior to parametrisation. The cuts allow the mesh to be simplified for parametrisation using a planar domain. Creating the cut graph for these meshes is the major challenge of such methods. Meshes of genus-1 or above will require multiple cuts and the particular location of these can be difficult to compute automatically. The particular location of each cut needs to be considered carefully to strike a balance between the goal of low distortion and discontinuities introduced with each cut. Even though methods such as these overcome the topological limitations of mesh parametrisation, they often suffer from visible discontinuities along the seams of the cuts. Where the discontinuities are not tolerated by some applications, changing the base domain is often worthwhile [12].

Another approach uses harmonic mappings to parametrise individual segments, before each segment is remeshed [14]. Although this method can produce high quality meshes, with very little to no discontinuities, and is computationally robust, the partitioning of the mesh can result in unnecessary segmentation. For example, it may segment a straight tube into two segments [14]. This is due to the planar domain used, limiting the topology of the mesh. There is a considerable advantage to using spherical domain for parametrisation, as many meshes can be parametrised to a sphere without the need to cut the mesh [15]. Planar meshes that can be parametrised by a square domain can be mapped to a spherical domain using the process of UV mapping [12].

Spherical parametrisation is not without its problems. Many methods face the issue of not being able to guarantee a bijective (one-to-one) mapping between the sphere and the mesh. Some attempts to overcome this create a geometry image that approximates the mesh [16]. First, the mesh is mapped to the spherical domain, and then an arbitrary polyhedron is spherically parametrised. The 2D geometry image is then created by unfolding the polyhedron and then remeshing it by mapping back to the original surface domain. However, the method demonstrates an inherent limitation of spherical parametrisation. A trade-off is required between stretching of the mesh and conformality, as both cannot be attained simultaneously for highly deformed shapes.

This paper proposes a technique for parametrising 3D meshes for remeshing which uses the theory of spherical harmonics to approximate a continuous surface. Spherical harmonics are a natural basis for representing functions defined over spherical and hemispherical domains. They have been used for many applications in 3D modelling, including face recognition [17], lighting and systems [18, 19], and diffusion imaging [20–22]. Spherical harmonics have been suggested as a parametrisation technique for surfaces [13]. To use spherical harmonics basis, a solution to find the weights of the basis functions is obtained through solving a linear system of equations. To overcome computational challenges in using linear least squares to fit a large mesh using a large number of basis functions, regularisation is investigated for reducing the effects of numerical outliers and computational inefficiency. Once the mesh has been parametrised as a combination of spherical harmonic basis functions, the new mesh defined on a sphere can be remeshed to approximate the original surface. Further to this, the consequences of surface parametrisation as a solution to surface inpainting are addressed.

#### 2. Methods

##### 2.1. Spherical Harmonics

For a sufficiently smooth surface, represented by a function , an infinite series of spherical harmonic basis functions can be used to represent it in the following form [23]:where and are the polar and azimuth angles in a spherical coordinate system. As , this representation becomes an exact description of the surface . The spherical harmonic functions are defined by , with order and degree (), and is some weighting coefficient for . The degree describes the number of basis functions to be computed for each order.

Spherical harmonics are formed from the set of solutions to the 3D Laplace equation, given here as a combination of associated Legendre polynomials, :The associated Legendre polynomials, , are defined as follows:

The order specifies the number of polynomial terms that a harmonic function contains. Relatively smooth functions can be obtained using low order spherical harmonics, with increasing order corresponding to an increased number of frequencies captured by the spherical harmonics. The higher the order, the more precise the approximation. Figure 1 shows the spherical harmonics for even orders up to —note that is the reflection of .