Efficient EMD and Hilbert spectra computation for 3D geometry processing and analysis via space-filling curve

Abstract

Empirical Mode Decomposition (EMD) has proved to be an effective and powerful analytical tool for non-stationary time series and starts to exhibit its modeling potential for 3D geometry analysis. Yet, existing EMD-based geometry processing algorithms only concentrate on multiscale data decomposition by way of computing intrinsic mode functions. More in-depth analytical properties, such as Hilbert spectra, are hard to study for 3D surface signals due to the lack of theoretical and algorithmic tools. This has hindered much more broader penetration of EMD-centric algorithms into various new applications on 3D surface. To tackle this challenge, in this paper we propose a novel and efficient EMD and Hilbert spectra computational scheme for 3D geometry processing and analysis. At the core of our scheme is the strategy of dimensionality reduction via spacefilling curve. This strategy transforms the problem of 3D geometry analysis to 1D time series processing, leading to two major advantages. First, the envelope computation is carried out for 1D signal by cubic spline interpolation, which is much faster than existing envelope computation directly over B Jianping Hu 3D surface. Second, it enables us to calculate Hilbert spectra directly on 3D surface. We could take advantages of Hilbert spectra that contain a wealth of unexploited properties and utilize them as a viable indicator to guide our EMD-based 3D surface processing. Furthermore, to preserve sharp features, we develop a divide-and-conquer scheme of EMD by explicitly separating the feature signals from non-feature signals. Extensive experiments have been carried out to demonstrate that our new EMD and Hilbert spectra based method is both fast and powerful for 3D surface processing and analysis.

Novel and efficient computation of Hilbert–Huang transform on surfaces

“…Hu et al [21] developed a novel framework based on EMD using a measure of mean curvature as an input signal, which can help extract features of different scales of the original surface more precisely. Wang et al proposed an EMD and Hilbert spectra computational scheme to transform the problem of 3D surface analysis/processing to 1D time series processing by the strategy of dimensionality reduction via space-filling curve [24], enabling us to decompose the original signal into IMFs and use the 1D Hilbert transform of a function. However, this method tends to affect the overall accuracy of HHT computation because theoretically it shares common drawbacks of any sampling method in the aspect of information loss.…”

“…The success of HHT in processing nonlinear and non-stationary signal in 1D has motivated its generalization to 2D image [ [24], etc. The extension of EMD to high dimensional domains have been researched extensively where the critical step is the interpolation from extrema at each iteration where various interpolation methods can be used such as radial basis function, thin plate spline, cubic polynomial interpolation and so on.…”

“…The primary reason lies in the fact that 3D surfaces are irregular, curved, and possibly have arbitrarily complex structure which will bring forth much more challenges for the computation of the Hilbert transform on surfaces [19]. As a result, prior graphics-related efforts of HHT on surfaces mainly concentrate on identifying multiple intrinsic scales of the input signal via EMD in a global sense [21] [22] [23] or perform HHT computation via dimensionality reduction [24], and without an accurate and effective way to extract local information from each IMF, we are primarily confined to geometry signal processing such as denoising, smoothing, and enhancement.…”

Inverse Modelling of Incompressible Gas Flow in Subspace

“…Wang et al . [WHZQ15] applied EMD for 3D geometry processing. The difficulty of applying EMD on multi‐dimensional data lies in constructing the envelopes.…”

Interactive modeling of complex geometric details based on empirical mode decomposition for multi-scale 3D shapes

“…EMD was first proposed by Huang et al [9] for processing 1D nonlinear and non-stationary signals. It is a fully data-driven method and has been successfully used for 3D surface processing [10,11,12,13,14]. EMD can obtain different scale details of surface and avoid explicitly separating surface into base surface and detail part.…”