Empirical mode decomposition on surfaces
Graphical abstract
Highlights
► This paper generalizes the classical concept of Empirical Mode Decomposition (EMD) from Euclidean space to the setting of surfaces. ► The generalized EMD on surfaces is used for filtering scalar functions defined over surfaces and surfaces themselves. ► A novel feature-preserving surface smoothing method is proposed based on extremal envelopes inspired by the EMD.
Introduction
Generalizing the classical signal processing approaches to surfaces is an important problem in digital geometry processing. Many applications, such as mesh smoothing, compression, and watermarking, have benefited immensely from such extensions [1], [2], [3]. One of the most significant developments involves extending the classical Fourier and harmonic analyses to the surface domain [4], [5], [6], [7]. The Fourier basis functions, which are smoothed harmonic functions, cannot adapt to the geometry so well. However, some applications, for example, compression and feature-preserving smoothing, benefit from adaptive basis functions.
Empirical Mode Decomposition (EMD) is a relatively new signal processing method [8]. In contrast to the Fourier and wavelet analysis, which project a signal onto a number of predefined basis functions, the EMD expresses a signal as an expansion of Intrinsic Mode Functions (IMFs) that are signal-dependent and estimated via an iterative procedure called a sifting process [9]. Thus, the EMD is fully adaptive and data-driven. Furthermore, the EMD overcomes the limitation of traditional Fourier and wavelet analyses that can only handle linear and stationary signals, and performs well for non-linear and non-stationary signals [8].
The elegant mathematical properties and popularity of EMD motivate its potential applications in surface filtering and analysis. In this paper, we extend the classical EMD from Euclidean space to the setting of surfaces, and make a preliminary attempt using an EMD-based method for feature-preserving surface smoothing. The generalization of EMD to surfaces is not a straightforward problem. Unlike the Euclidean space, surfaces are irregular, curved and do not admit a consistent parameterization. Thus, behaviors of IMFs on surfaces are different from that in Euclidean space, and generalizing the EMD to surfaces is non-trivial. Furthermore, the original 1D EMD is not edge-aware, thus the generalization of EMD cannot directly provide a feature-preserving surface smoothing method.
In this paper, we first extend EMD from Euclidean space to the setting of surfaces. The key of the extension is an envelope computation method that solves a bi-harmonic field with Dirichlet boundary conditions. Although the classical 1D EMD was not designed to be feature-preserving, we still develop a novel multiscale feature-preserving surface decomposition method based on extremal envelopes of EMD. This technique is inspired by a similar work on edge-aware image decomposition [10].
Section snippets
Related work
Here we give a brief background about signal processing on surfaces, empirical mode decomposition, and feature-preserving surface smoothing.
The original 1D EMD
The EMD algorithm extracts IMFs from signals by the sifting process, leaving the final residue as a constant or monotone trend. An IMF represents a generally simple oscillatory mode as a counterpart to the simple harmonic function. The first IMF is extracted from the given signal by the following algorithm.
- 1.
Find all local extrema of g.
- 2.
Interpolate all local minima (resp. maxima) by the cubic spline to obtain the lower envelope Envmin (resp. upper envelope Envmax).
- 3.
Compute the mean of
EMD on surfaces
In this paper, the surface S is discretized as a triangular mesh M = (V, E, F). denotes the set of vertices, E = {(i, j)∣ vi and vj are linked by an edge} denotes the set of edges, and F = {(i, j, k)∣vi, vj and vk are the three vertices of a triangle} denotes the set of faces. Each vertex vi ∈ V is represented using the Cartesian coordinates, denoted by vi = (vix, viy, viz)T. contains the x, y and z Cartesian coordinates of the n vertices. Let N(i) = {j∣(i, j) ∈
Results and applications of EMD on surfaces
In this section, several results and applications shows the efficiency and performance of the proposed EMD on surfaces. The proposed EMD is implemented in MATLAB, with some core codes wrapped in C++, and tested on the following hardware: Intel Pentium 1.60 GHz with 2.0 GB memory. The total computational times of the proposed EMD are reported in Table 1. The proposed method requires iterations for extracting each IMF, and in each iteration two linear systems are solved. Thus, it is a super-linear
Multiscale feature-preserving decomposition
Although the generalized EMD on surfaces can be used for surface smoothing as shown in Fig. 4e, it cannot preserve sharp geometry features. The reasons are twofold. First, the interpolation method of the generalized EMD, i.e., bi-harmonic fields with cotangent Laplace operator, is not feature-preserving. Second, the three coordinate functions are processed individually. However, geometry features are determined by the three coordinate functions together, not separately.
In this section, we
Conclusion and future work
An extension of EMD has been proposed for surface signals processing in this paper. The interpolation method for computing envelopes, which is the critical step of EMD, is performed by solving a bi-harmonic field with Dirichlet boundary conditions. The proposed interpolation method on surfaces is similar to the thin-plate spline in Euclidean space. Numerical experiments show that the proposed EMD on surfaces works well. Although the original 1D EMD and the proposed generalized EMD on surfaces
Acknowledgments
We thank the anonymous reviewers for their valuable comments and suggestions. We are also grateful to Prof. Hui Huang for discussions of the classical bilateral filtering. This research is supported in part by the National Natural Science Foundation of China-Guangdong Joint Fund (Grant No. U0935004), the National Natural Science Foundation of China (Grant Nos. 61173102 and 61173103), the Natural Sciences and Engineering Research Council of Canada (Grant No. 61137), and the China Scholarship
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2018, Computer Aided Geometric DesignCitation Excerpt :This problem is further addressed by restoring to extend extreme and structure measurement in Zang et al. (2015). In 3D geometry processing, inspired by the extreme-based image smoothing (Subr et al., 2009), H. Wang et al. (2012) resorted to an interpolation by minimizing a quadratic function with an edge-aware Laplace operator, which measures the similarity between the current vertex and its neighbors. However, the proposed feature-aware interpolation method is not sufficiently robust.