1 Introduction
In most areas that involve representation of discrete virtual surfaces as 3D meshes, there has been an increasing trend in working with higher precision. This has led to the generation of meshes comprised of a large number of elements, for which the processing, visualization and storage of has become a challenge. The task of transmission of these geometric models over communication networks can lead to a large amount of storage space and put a considerable strain on network resources.
The research on mesh compression started in the direction of singlerate compression methods that constructed a compact representation of an input mesh as a whole. However, the need for transmission of large geometric models led to the development of algorithms based on progressive mesh compression where a mesh can be transmitted and reconstructed progressively, that is, with increasing levels of detail. This enables the rendering of the 3D model in progression where the finer details of the model can be processed in a cumulative fashion.
The information contained in a mesh is generally divided into two categories: the geometry information, which is the position of each vertex of the mesh in the 3D Euclidean space and the connectivity or topological information, which describes the incidence relations between the mesh vertices. Since the geometric information comprises a dominant part of the mesh, most recent algorithms focused on its efficient compression. Complete reviews, summarizing work on mesh compression can be found in alliez2005recent ; peng2005technologies ; maglo20153d .
1.1 Related Results
Fourier analysis has been used extensively in many signal processing areas. By projecting the data into the frequency domain, one can retrieve a good approximation of the original signal with only a few Fourier coefficients. A very prominent application is the JPEG wallace1992jpeg image compression method based on the closely related 2D Discrete Cosine Transform.
Karni and Gotsman karni2000spectral
proposed a generalization of the Fourier basis on discrete graphs in order to compress mesh vertex positions. This is achieved by projecting the coordinate vectors onto the orthonormal basis obtained from the spectral decomposition of the combinatorial Laplacian of the shape. For smooth meshes, most of the signal energy is contained in the low frequencies of the mesh spectrum and hence, the coefficients corresponding to the lower frequencies are sufficient to build a good approximation of the original mesh. BenChen and Gotsman
ben2005optimalityproved that for certain geometric models, spectral compression is optimal and is equivalent to Principal Component Analysis.
The idea of using a representative basis for approximating a mesh has been explored in detail in multiple prior works. karni20013d proposed to use a fixed basis independent of the mesh connectivity. sorkine2005geometry proposed a geometryaware basis that considers both the connectivity and the geometry of the mesh for mesh compression. Mahadevan mahadevan2007adaptive replaced the manifold harmonics by a diffusion wavelet bases and showed an improvement in compression performance. This was attributed to the rich multiscale nature of the wavelet basis.
Methods dealing with basis design for meshes have also been explored in other applications.In order to overcome the nonlocal support of the Fourier eigenfunctions, compressed eigenfunctions have been adapted from mathematical physics to shape analysis neumann2014compressed ; bronstein2016consistent .
Anisotropic phenomena, such as diffusion, has been explored in different image processing applications weickert1998anisotropic and has recently demonstrated impressive results in image compression peter2015optimised . Anisotropic Laplacians have been introduced in order to improve the performance compared to the classical isotropic Laplace operator. Kovnatsky et al. kovnatsky
modified the metric tensor of the manifold to take into account photometric information. Andreux et al.
andreux2014anisotropic used principal curvatures in order to construct anisotropic tensor for the task of shape segmentation and shape matching.Keeping the anisotropic information out of the metric leads to lower order of derivatives, a property that yields numerically more robust operators. Hildebrandt et al. hildebrandt2012modal designed a new kind of eigenvibrations constructed as the Hessians of surface energies using extrinsic curvatures and deformation energies. Similarly, Iglesias and Kimmel iglesias2012schrodinger used artificial surface textures on shapes to define elliptic operators that give birth to a new family of diffusion distances. This operator, also called Hamiltonian, was recently explored in ChoukrounSK16 , which provides a broad theoretical foundation of the operator and its use in different shape analysis tasks, including data driven potential optimization, compressed manifold modes and shape matching. In this paper, we extend this work by analyzing the operator for the task of mesh compression and provide a powerful sparse representation framework based on the Hamiltonian operator, that outperforms previous spectral compression methods.
Given a basis, a plain spectral truncation of the signal in that basis is a fairly simple method for representation. Instead, sparse approximation techniques have been proposed and have become very popular. The basic idea is to estimate a given signal as a
sparse linear combination of a large pool of constituent vectors  called a dictionary. These vectors, or atoms, are selected such that the coefficients of representation are sparse. The rationale is that high dimensional signals generally possess intrinsic structures that are better represented in a lowerdimensional linear subspace. Although these dictionaries have traditionally been populated with complete orthogonal bases, redundant or overcomplete dictionaries allow greater flexibility in design by better capturing the intrinsic characteristics of a signal.The main difficulty with sparse algorithms is the availability of a rich representative dictionary. This seems trivial for signals defined over regularly and consistently sampled domains, like images and speech, but it is not straightforward to extend the idea to nonflat domains like meshes of surfaces in 3D or general graphs. The use of redundant representations for mesh representation and compression have started to emerge in peyre2005surface ; tosic2006progressive ; krivokuca2013sparse ; zhong2014sparse . We extend the ideas discussed in these papers by choosing the eigenfunctions of a dataaware operator as the constituents of a large overcomplete dictionary to represent the mesh.
1.2 Contribution
In this paper, we use the spectral geometry of the Hamiltonian operator for approximation of meshes representing surfaces in 3D. The operator is obtained by modifying the Laplacian with a potential function that defines the rate of oscillation of the harmonics on different regions of the surface. We use a simple and efficient construction of the potential function using a vertex ordering scheme. This modulates the Fourier basis of the 3D mesh to focus on crucial regions of the shape having fine geometric structures equivalent to highfrequencies in the Fourier domain. We optimize our approximation using a sparse representation framework by building optimized dictionaries in order to sparsely encode mesh geometry.
In contrast to ChoukrounSK16 , which suggests a simplistic truncation scheme for the representation of discrete surfaces, this paper provides an elaborate treatise on the the design of the Hamiltonian operator to tackle the specific task of spectral mesh compression. We develop a novel multiresolution dictionary learning mechanism which takes advantage of the adaptable nature of the Hamiltonian to design a context basis. The combination of the spectral geometry of the Hamiltonian in conjunction with a sparse approximation approach outperforms existing spectral compression schemes as demonstrated in section 5 and illustrated in Figure 1.
2 Background
2.1 Laplace Beltrami Operator
Consider a parametrized manifold with boundary . The space of squareintegrable functions on is denoted by with the standard inner product , where is an area element induced by the Riemannian metric .
The Laplace Beltrami Operator (LBO) acting on a scalar function , is defined as
(1) 
where represents the intrinsic gradient. The LBO is a selfadjoint operator and hence admits a spectral decomposition: , such that,
(2)  
where , and the Kronecker delta function. In case has a boundary, we add Neumann boundary condition to . The LBO eigendecomposition can be obtained by minimizing the Dirichlet energy
(3)  
s.t. 
Therefore, the LBO eigenfunctions can be seen as an extension of the smooth Fourier harmonics in Euclidean space to manifolds and are also referred to as Manifold HarmonicsLevy06 ; ValletL08 .
2.2 Graph Laplacian
Let be an undirected graph of vertices representing a sampled version of a manifold . Let the graph edges be equipped with nonnegative weights , defining the weighted adjacency matrix of the graph as
(4) 
The diagonal vertex degree matrix is defined as
(5) 
The unnormalized graph Laplacian is defined as the matrix . For properties of the Laplacian, the reader is referred to mohar1991laplacian ; chung1997spectral .
The combinatorial graph Laplacian is constructed by solely considering the connectivity of the mesh, where the edge weights are defined as . Therefore, the degree matrix is populated with the valence of each vertex. While other geometric discretizations of the Laplacian on graphs exist pinkall1993computing , the combinatorial Laplacian is used in this paper, since the mesh topology can be efficiently encoded .
The discrete manifold harmonics are then obtained by solving the spectral decomposition of the combinatorial Laplacian
(6) 
Here,
is the matrix of eigenvectors where each column
is one eigenvector andis the diagonal matrix of eigenvalues where each element
is one eigenvalue.2.3 Spectral Mesh Compression
Given a complete basis on , any function can be expressed as
(7) 
where is the ith (manifold) Fourier coefficient of . Since the manifold harmonics form a complete basis, Karni and Gotsman karni2000spectral proposed to use them for spectral mesh compression. Considering the mesh coordinate functions
as functions defined on the vertices of the mesh, the basic idea of spectral mesh compression is to compute the Fourier transform of the coordinate functions and truncate the highfrequency coefficients. Thus, if
is a coordinate function, it can be approximated using the graph Laplacian decomposition as(8) 
with and . By using the graph connectivity information and constructing the manifold harmonics as a representative basis, the geometric information of the mesh can be encoded efficiently through the obtained coefficients. Figure 2 illustrates spectral reconstructions as applied to a toy example of a 2D curve.
Simple truncation as showed in eq. (8) uses a restrictive assumption of only focusing on the lower frequencies and is analogous to lowpass filtering of the signal. In general, the lower frequencies encapsulate most of signal energy and the reconstructed mesh tends to preserve the global features of the original. However, local geometry and fine details of the mesh corresponding to high frequencies are generally missing and require a high number of eigenvectors for their preservation if a truncation approach is followed.
In the next section we propose a more pragmatic approach of sparsity where the data itself searches for its best constituents (frequencies) with the objective of attaining sparsity in a certain basis of representation. This leads to an improved compression strategy.
2.4 Sparse Approximation of Mesh Coordinates
As motivated in the last section, classical methods are based on the premise that family of signals are best represented using only the first few components of an optimal orthonormal basis. The concept of sparsity provides an alternative perspective for representation.
Consider a graph and a normalized overcomplete () dictionary containing atoms . We aim to approximate any given signal on by
(9) 
with the coefficient representation of the input signal with respect to the dictionary . In order to achieve significant reduction in storage we should assume that the number of nonzero elements of the coefficient vector should satisfy . Thus, the sparse approximation of the signal can be obtained by solving the sparse coding problem
(10)  
Let us be given a multiple channels signals , the new coefficient matrix is designated by with . The new sparse approximation is obtained by solving the simultaneous sparse approximation
(11)  
where denotes the Frobenius norm. In our mesh compression configuration, the input signal is defined by the mesh coordinates such that .
Since sparse coding solution is a combinatorial NPhard problem, well known relaxations are generally used mallat1993matching ; pati1993orthogonal . In this paper Simultaneous Orthogonal Matching Pursuit algorithm tropp2006algorithms is used for sparse approximation of the mesh geometry. Pursuit algorithms are based on a greedy approach of solving for the optimization in (11) by incrementally selecting atoms from the dictionary that minimize the residual error. The orthogonal matching pursuit is one such way of picking the atoms where at each stage of the selection, the residual error vector is orthogonal to the atoms already selected in the support and hence will not be chosen again. Simultaneous OMP is a scheme of atom selection such that multiple signals have the same support in the dictionary , i.e. simultaneously solving three sparse problems by imposing same support constraint.
The success of a sparsity based algorithm depends on the availability of a representative dictionary. However, developing dictionaries for the representation of functions defined over graphs is a nontrivial task, since they do not possess a fixed structure as in Euclidean domains (grid). As a result, elements composed from fixed spectral bases have been used for populating the atoms of the dictionary krivokuca2013sparse .
The manifold harmonics can be used because of their stability and compactness, that is, low frequencies contain most of the signal energy. However, fine local structures are not well represented using this basis due to the global nature of their support. In zhong2014sparse , it was shown that a dictionary constructed from spectral graph wavelets hammond2011wavelets show better compression performance as compared to the standard Laplacian basis, since it encapsulates fine details and some dominant highfrequency structures of the mesh. However, since the locality of the basis cannot be controlled, the wavelet dictionaries are highly overcomplete, yielding large dictionaries which enforce higher compression rates. These shortcomings motivate the need for finding a basis which can be modulated in a datadependent manner. This will lead to an optimal localglobal structure tradeoff, thereby requiring a smaller cardinality of representation. In the next section we borrow an operator from quantum mechanics which has precisely the aforementioned properties.
3 Hamiltonian Operator
3.1 Definition
A Hamiltonian operator , also called Schrödinger operator, is an operator acting on a scalar function on a manifold that has the form
(12) 
where is a potential function and . The Hamiltonian plays a fundamental role in the field of quantum mechanics via the famous Schrödinger equation that describes the wave motion of a particle. The spectral decomposition of the Hamiltonian is defined by
(13) 
where denotes the energy of a particle at the stationary eigenstate . Here, represents the wave function of the particle such that
is interpreted as the probability distribution of finding the particle at a given position
. Since the Hamiltonian is a symmetric operator, its eigenfunctions form a complete orthonormal basis on the manifold . An illustration of the Hamiltonian basis is given in figure 3.The Hamiltonian eigendecomposition from (13) is obtained from the EulerLagrange equation of
(14)  
s.t. 
The parameter controls the potential energy and defines the tradeoff between local and global support of the basis. Larger values of will give solutions that concentrate on the low potential regions, while a smaller will give solutions that better minimize the total energy at the expense of more extended wave functions.
In its discrete setting, the Hamiltonian basis is obtained by solving
(15) 
with . Here denotes the previously described graph Laplacian, and a diagonal matrix that is defined by the potential scalar function values at vertices . Since only modifies the diagonal of the sparse Laplacian, the Hamiltonian is also a sparse matrix with the same nonzero entries. Thus, there is no increase in the computational cost of the decomposition compared to that of the Laplacian.
3.2 Analysis and Visualization of the Operator
Among the numerous reasons that motivated the selection of the Laplacian basis for shape analysis, one ubiquitous is its efficiency in representing smooth functions. This property is defined by the following observation for functions with bounded gradient magnitudes:
(16) 
This result can be established as follows,
(17)  
The representation error for a function using the Laplacian basis is bounded by a factor of the squared magnitude of its gradient. This result was subsequently proved to be optimal for representing smooth functions over surfaces in aflalo2015optimality , which says that there exists no other basis with such a bound for all possible bounded gradient functions.
To gain further insight, we simulate this result in a discrete setting by analyzing the gradient operators for a simple one dimension case. In matrix form, since is a positive semidefinite matrix, there exists a matrix such that and the discrete equivalent of result 17 is:
(18) 
It is interesting to perform a similar analysis on the eigenfunctions of the Hamiltonian and Figure 4 shows a visualization of the gradient operator of the Laplacian and the induced gradient of the Hamiltonian. The result in equation (18) gets modified to account for the action of the potential.
(19) 
with such that . Thus the Hamiltonian basis can be considered as optimal for the representation of functions which have a bounded weighted gradient. The weights depend on the potential function and its action in different regions.
The gradient matrix for the 1D case is the result of a simple finite difference scheme of one dimensional forward differentiation yielding a circulant populated with the kernel circulated along the diagonal. The Hamiltonian has the effect of escalating the cost of a strong gradient exclusively in regions of high potential. This shows that coefficients of the induced gradient matrix are larger in the high potential regions as compared to anywhere else in the domain. The nonzero values of the offdiagonal elements suggest that the linear operation loses its shift invariance property and the effect of a derivative is no more an exclusive property of its local neighborhood but also depends on its global positioning imposed by the potential function. Thus the Hamiltonian operator advocates measuring smoothness differently for different regions of the domain and this is a useful property to exploit in a compression setup.
The Hamiltonian operator can improve the compression performance by modifying the harmonics in order to emphasize designated regions of interest. We propose a method for choosing a potential function which does not require any additional encoding. We construct a dictionary based on the eigendecomposition of both the Laplacian and the Hamiltonian that can focus on difficult reconstruction areas of the harmonics. Thus, by designing high vibrations in selected regions of the shape, our dictionary is much less redundant than the wavelets proposed in mahadevan2007adaptive , has a better ability to encode high frequency details and achieves better compression performance.
4 Hamiltonian Operator for Spectral Mesh Compression
4.1 Potential Design
Consider a mesh (graph) with and the coordinates matrix . Given the Laplacian eigenvectors matrix , one can design a fixed dictionary composed of the full harmonics basis matrix and solve the simultaneous sparse coding problem of (11) where . Then, a norm reconstruction error can be obtained at each vertex of the mesh. This vertex error profile over the mesh provides an indication of difficult regions that can be enhanced through the potential.
Since a fixed potential must be used to avoid additional encoding, we permute the arbitrary indexing of the mesh according to this error. The vertices are sorted in ascending order such that vertices having small indexes correspond to points with a small error while vertices with the larger index correspond to regions with higher distortion in approximation. It can be done trivially in complexity and has no influence on the mesh itself since the ordering has no physical meaning.
Given the Hamiltonian , we propose to use a fixed potential matrix defined as
(20) 
normalized such that . We illustrate the influence of a linear potential in one dimension in Figure 5. Since we sorted the indexes according to the error, a designed hierarchical energy of the potential is defined over the shape while the potential encoding itself is avoided. At the decoder side, the Hamiltonian can be built using the mesh connectivity (Laplacian) and the encoded scalar coupled with the fixed linear potential described above.
4.2 Dictionary Construction
We construct a dictionary built from the eigendecomposition of both the Laplacian and the Hamiltonian to benefit from their global and local properties. The potential is defined according to the representation error of the Laplacian basis as described in the previous section. We encode multiple constants in order to obtain a multiresolution of the basis. The regularization constants are found via a direct search on a given domain.
Thus, our dictionary can be obtained as:
(21) 
where the subdictionary is the matrix of eigenvectors of the Hamiltonian and . The constant is obtained by solving
(22)  
The algorithm is summarized in Algorithm 1. Given a compression ratio requirement, the number of atoms is calculated according to the formulas elucidated in Section 5.2.
4.3 Discussion
Algorithm 1 is designed to take advantage of the two principle components of our contribution: sparsity and datadependent basis. The potential design described in section 4.1 alters the Laplacian basis to focus on errorprone regions of the shape. The extent to which such an alteration must be enforced is encoded in the coefficients: which are sequentially optimized in Algorithm 1. The final dictionary is comprised of the regular Laplacian eigenvectors appended with these Hamiltonian eigenvectors. The results in section 5 demonstrate that this provides a distinct improvement in shapereconstruction errors.
Another advantage of our approach is that the potential design is based on mesh connectivity information alone. The vertex reordering enables us to encode information about the errorprone regions of the shape, which are encoded in the eigenvectors of the Hamiltonian. This reordering scheme avoids any secondary encoding of information and hence provides a distinct gain in saving of information units for compression.
5 Experimental Results
5.1 Evaluation Metric
In order to measure the loss resulting from an estimated approximation, Karni and Gotsman karni2000spectral used a visual metric that captures the visual difference between the original mesh and its reconstruction . The measure is a linear combination of the RMS geometric distance between corresponding vertices in both models and a visual metric capturing the smoothness of the reconstruction. The error at each vertex is defined as
(23) 
The visual metric is defined as
(24) 
with is the set of indexes of the neighbors of vertex , and is the geometric distance between vertices and . A global representation error is obtained by summing over the mesh vertices. We normalize the error by the area of the shape for uniform comparison between different shapes.
5.2 Compression Ratio
The compression ratio calculation is obtained as a ratio of the net information before and after the dictionary encoding zhong2014sparse . If each coordinate takes up to bits, then for a mesh of vertices, the total uncompressed information is . Now, assuming that the dictionary is known in advance, or can be constructed from the connectivity information of the mesh, on both the encoder and decoder sides, let be the total number of selected atoms each taking bits to encode each coordinate function, making the total information equal to . The information required to store the indexes of the coefficients which is given by , The compression ratio is given as
(25) 
The function is used since it can be more efficient to represent the selected atoms with a bit vector of size instead of the indexes themselves. The compression ratio in the coefficient truncation scheme is given by
(26) 
since only the ordered coefficients must be encoded. The compression ratio of the Hamiltonian dictionary can be easily extended by adding to the numerator of (25), with the number of regularization coefficients of the Hamiltonian and the number of bits required to encode each one of them.
5.3 Shape partition
Spectral methods, including the one proposed in this paper are hinged on the eigendecomposition of a large matrix. This time complexity operation can be computationally demanding especially when the number of points is large. Therefore, as motivated in karni2000spectral , we resort to mesh partitioning, where we segment a mesh into smaller constituents and compress each segment independently. We use the METIS karypis1998fast package for fast graph partitioning and segmentation. Therefore, the time complexity of the proposed framework can be seen as linear in the number of partitions of fixed size.
The Neumann boundary conditions induced by the graph Laplacian may cause a discrepancy between the reconstructed segments along shared boundaries. Mesh partitioning may result in a socalled edgeeffects karni2000spectral , a degradation of the reconstruction along the submesh boundaries. However, compressing smaller meshes separately has the advantage of better capturing local properties of the mesh even with low number of coefficients. Also, from our experiments, this degradation is not visually sensitive even for low compression rate and especially not with the suggested method. We partition the mesh into submeshes of few hundred vertices. This ensures both faster computation of the coefficients and fewer difficult areas for reconstruction the Hamiltonian operator can focus on.
5.4 Results
Figures 6 and 7 show a visual comparison of the shape approximation results. We propose two visualizations. The first row presents a regular reconstruction with the three different methods for a fixed compression ratio. The second row shows a profile of the error plot over the original mesh to highlight regions of erroneous representation. Clearly the MHB has a fairly large error profile over the surface. The results with the spectral graph wavelets show an improvement as compared to the manifold harmonics. However, the sparse approximation with the Hamiltonian basis shows a distinct positive difference as compared to the other two methods both in the reconstruction and error profiles.
Figures 8, 9 and 10 show compression results of our method compared with other spectral techniques on different shapes. We present results according to four prominent spectral methods. First with the plain truncation scheme of the Manifold Harmonic Basis (MHB) as suggested in karni2000spectral . Second, the truncation scheme of the Hamiltonian with optimal potential as suggested in ChoukrounSK16 . Third, MHB with sparse coding i.e. using the SOMP algorithm (MHBSOMP) on the manifold harmonics dictionary. Fourth, we present Spectral graph wavelets with sparse coding (MHB+SGWSOMP) using dictionaries defined with the harmonics and the graph wavelets as presented in zhong2014sparse .We used 32 bits as for single bitprecision in the computation of compression ratio. An illustration of the mesh partition result and reconstruction performance between the methods is present in Figure 11.
We can see that our algorithm shows a considerable improvement in compression performance by generating smaller errors over the entire span of compression requirements for different shapes, even under nonuniform triangulations. We highlight the fact we leverage benefits in optimizing on both fronts: a superior basis and the improved sparse coding mechanism for representation. Using only the Hamiltonian basis in a simple truncation scheme as suggested in ChoukrounSK16 shows improvement over standard spectral techniques making it comparable to the pursuit based solutions, but the major gain comes only after its use in conjunction with sparse representation.
Finally, in order to gain a global assessment of the performance of our method in comparison to generic nonspectral mesh compression algorithms (motivated from maglo20153d ), we compiled an average bitspervertex (bpv) analysis in table 1, where the proposed method bpv is obtained on the same shapes tested in karni2000spectral and with the same reconstruction error threshold. However, since other prominent nonspectral methods like khodakovsky2000progressive ; guskov2000normal involve some form of remeshing, the comparison may not be entirely appropriate, but provides a high level comparison between the compression types. The general comparison shows that spectral algorithms show lower compression rates as compared to methods like khodakovsky2000progressive and guskov2000normal . However, our approach of using the spectral route along with sparse representations shows considerable gains without any any assumption on the mesh connectivity.
Algorithm  Total Compression rates (bpv)  Compression Type 

Progressive Meshes hoppe1996progressive  37  Vertex split 
Wavemesh valette2004wavelet  19  Irregular Wavelets 
Spectral Compression karni2000spectral  19  Spectral 
Incremental parametric refinement valette2009progressive  15  Parametric refinement 
Spectral Hamiltonian (proposed method)  12  Spectral 
Wavelet Compression khodakovsky2000progressive  8  Remeshing based 
Normal Meshes guskov2000normal  6  MeshSequence compression 
6 Conclusion and Future Work
We introduced the Hamiltonian operator for spectral mesh compression. This operator uses the concept of a potential function whose action leads to the modulation of the spectral geometry of that domain. We use this operation to modify the standard manifold harmonics of the 3D surface by emphasizing the role of regions with errors that emerge as a result of compression with classical harmonics. We maximize compression performance by constructing meaningful dictionaries from this basis in a sparse approximation framework and show a distinct improvement from previous spectral techniques.
In future work, we intend to explore the design and construction of newer meaningful potential functions for 3D surfaces. The use of the Schrödinger operator for generic graph structures or discrete manifolds is an unchartered territory that can lead to interesting spectral embeddings.
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