Novel and efficient computation of Hilbert-Huang transform on surfaces

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.