دانلود رایگان مقاله محاسبات کارآمد و جدید تبدیل هیلبرت هوانگ بر روی سطوح

Abstract

Hilbert–Huang Transform (HHT) has proven to be extremely powerful for signal processing and analysis in 1D time series, and its generalization to regular tensor-product domains (e.g., 2D and 3D Euclidean space) has also demonstrated its widespread utilities in image processing and analysis. Compared with popular Fourier transform and wavelet transform, the most prominent advantage of Hilbert–Huang Transform (HHT) is that, it is a fully data-driven, adaptive method, especially valuable for handling non-stationary and nonlinear signals. Two key technical elements of Hilbert–Huang transform are: (1) Empirical Mode Decomposition (EMD) and (2) Hilbert spectra computation. HHT's uniqueness results from its capability to reveal both global information (i.e., Intrinsic Mode Functions (IMFs) enabled by EMD) and local information (i.e., the computation of local frequency, amplitude (energy) and phase information enabled by Hilbert spectra computation) from input signals. Despite HHT's rapid advancement in the past decade, its theory and applications on surfaces remain severely under-explored due to the current technical challenge in conducting Hilbert spectra computation on surfaces. To ameliorate, this paper takes a new initiative to compute the Riesz transform on 3D surfaces, a natural generalization of Hilbert transform in higher-dimensional cases, with a goal to make the theoretic breakthrough. The core of our theoretic and computational framework is to fully exploit the relationship between Riesz transform and fractional Laplacian operator, which can enable the computation of Riesz transform on surfaces via eigenvalue decomposition of Laplacian matrix. Moreover, we integrate the techniques of EMD and our newly-proposed Riesz transform on 3D surfaces by monogenic signals to compute Hilbert spectra, which include the space-frequency-energy distribution of signals defined over 3D surfaces and characterize key local feature information (e.g., instantaneous frequency, local amplitude, and local phase). Experiments and applications in spectral geometry processing and prominent feature detection illustrate the effectiveness of the current computational framework of HHT on 3D surfaces, which could serve as a solid foundation for upcoming, more serious applications in graphics and geometry computing fields.

6. Conclusion and future work

In this paper, we have proposed a novel data-driven analysis theory on surfaces and have made our initial efforts to apply such theory to a few graphics applications towards more quantitative and accurate analysis. In particular, we have extended the popular Hilbert–Huang transform, a fully data-driven adaptive method, to handle signals defined over 3D surfaces. The technical core of our computational framework is to compute the Riesz transform on 3D surfaces by eigenvalue decomposition of Laplacian matrix, which utilizes the relationship between Riesz transform and fractional Laplacian operator. Furthermore, we integrate our Riesz transform with the numerical techniques of EMD on 3D surfaces by way of monogenic signals, in order to compute Hilbert spectra of any input signal defined over 3D surfaces. Hilbert spectra include the space-frequency-energy distribution of signals defined over 3D surfaces and can characterize key local feature information in a more quantitative and accurate manner. As a result, critical local information such as instantaneous frequency, local amplitude, and local phase, could aid many graphics applications, opening up a new way to process geometric and/or non-geometric information on surfaces. Within our novel data-driven computational framework enabled by HHT on surfaces, potential applications are abundant. Through comprehensive experiments, we are able to accommodate many modeling, analysis, processing functionalities, such as 3D surface spectral processing and prominent feature detection. Extensive comparisons on popularly-used geometric models have effectively demonstrated that our surface processing method based on novel Hilbert–Huang transform on 3D surfaces can produce excellent results. Continuing to broaden the HHT’s application scope to contribute to more application sub-fields would require much more research endeavors in the near future. We expect more research results on this subject to be documented in the near future by us and colleagues. Ultimately, many effective, robust analysis on a large variety of scientific datasets generated from numerous digital content paradigms would enhance the overall data processing capabilities.