Source-linked AI summary
Empirical wavelet transform
Jerome Gilles
TL;DR
EMD adapts to signal content but lacks a clear theoretical basis. This paper constructs adaptive wavelets by detecting Fourier supports and designing a corresponding filter bank, yielding the empirical wavelet transform. Experiments report mode separation and time-frequency representations that compare favorably with EMD on the presented signals.
Problem
EMD is adaptable for decomposing signals into modes, but its ad-hoc process is mathematically difficult to model and interpret.
Method
The paper detects Fourier supports associated with signal modes, builds an adaptive wavelet filter bank, and defines the empirical wavelet transform and its inverse.
Results
Experiments show that EWT separates simulated components more closely than EMD and avoids artifacts observed in EMD-based low-frequency time-frequency representations.
Takeaways & Limitations
EWT provides signal-adapted wavelets whose dilation factors are detected empirically rather than imposed by a prescribed scheme.
Abstract
from arXiv · showhide
Some recent methods, like the Empirical Mode Decomposition (EMD), propose to decompose a signal accordingly to its contained information. Even though its adaptability seems useful for many applications, the main issue with this approach is its lack of theory. This paper presents a new approach to build adaptive wavelets. The main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank. This construction leads us to a new wavelet transform, called the empirical wavelet transform. Many experiments are presented showing the usefulness of this method compared to the classic EMD.
I. INTRODUCTION
The paper motivates adaptive signal representations by contrasting signal-independent transforms with methods that adapt to a signal’s contained information. It introduces empirical wavelets as a theoretically grounded alternative to EMD, using Fourier-spectrum supports to extract modes.
- Adaptive representations construct a basis from information contained in the analyzed signal, unlike Fourier and classic wavelet transforms.
- The proposed empirical wavelet transform segments the Fourier spectrum into mode supports and applies corresponding filters.
- Empirical wavelets are constructed from distinct Fourier supports to form a tight frame or orthonormal basis and provide a transform with an inverse.
- EMD adaptively separates signal modes and non-stationary components, but its ad-hoc process is mathematically difficult to model and interpret.
- Wavelet packets provide adaptive time-frequency tilings but retain a prescribed subdivision scheme and higher-order variational EMD approaches can be sensitive to noise.
B. Wavelets approaches
Wavelet methods represent signals through localized time-frequency filters, but conventional adaptive approaches impose fixed subdivisions or require difficult segmentation. The empirical construction adapts wavelet supports to spectral information in the analyzed signal.
- Wavelet foundations: Wavelet transforms compute inner products with translated and dilated mother wavelets, equivalently applying one filter per scale.The dyadic case tiles the time-frequency plane with scales s = 2^j.
- Existing adaptive approaches: Wavelet packets adapt time-frequency tiling through successive refinements, but their constant subdivision ratio limits adaptability.This approach uses a basis-pursuit framework.
- Existing adaptive approaches: Malvar-Wilson wavelets segment the temporal signal, whereas brushlets segment its Fourier spectrum using a more complicated construction with prescribed subdivisions.Both methods seek adaptive representations by separating regions containing different spectral information.
- Empirical wavelets: The proposed method builds empirical wavelets as bandpass filters whose supports follow spectral information and partition the Fourier axis into contiguous segments.The segmentation uses boundaries ω_n, transition regions of width 2τ_n, and empirical scaling and wavelet functions.
- Empirical wavelets: The transition width can be chosen proportionally to each boundary frequency as τ_n = γω_n, yielding supports defined by (1 − γ)ω_n ≤ |ω| ≤ (1 + γ)ω_n.The parameter satisfies 0 < γ < 1.
- Empirical wavelets: Figure 4 illustrates Fourier-domain scaling and wavelet functions for ν_n = 1, with γ = 0.5 for the scaling function and γ = 0.2 for the wavelet.The wavelet example uses ν_n+1 = 2.5.
B. Segmentation of the Fourier spectrum
Fourier segmentation supplies the empirical wavelets’ adaptability by locating spectral maxima and using them to determine band boundaries. The procedure retains the requested number of modes when enough maxima exist and adjusts it otherwise.
- Boundary detection: The method detects local spectral maxima, sorts them by decreasing magnitude, and excludes the fixed endpoints 0 and π.The number of segments N is initially assumed given, requiring N − 1 additional boundaries.
- Boundary detection: When M ≥ N, the algorithm keeps the first N − 1 detected maxima to define the wanted number of segments.Here M denotes the number of detected maxima.
- Boundary detection: When M < N, it keeps all detected maxima, resets N, and places each boundary midway between consecutive maxima together with 0 and π.The resulting boundaries are denoted ω_n.
C. Frame
The empirical filter construction can form a tight frame when the transition parameter is chosen appropriately. The frame analysis accounts for Fourier periodicity and non-overlapping transition regions.
- Tight-frame condition: Properly choosing γ yields a tight frame consisting of the empirical scaling function and empirical wavelets.The frame is stated to be a tight frame of L2(R).
- Periodicity: Because the Fourier representation is 2π-periodic, the proof can focus on the interval [0, 2π] and include reflected copies of the spectral segments.The reflected segment Λ^σ(n) is centered at 2π − ν_n.
- Tight-frame condition: The transition-region argument requires consecutive transition areas not to overlap, with the resulting condition determined by the smallest transition region.The sufficient condition is expressed as an upper bound on γ.
- Example: For boundaries {0, 1.5, 2, 2.8, π}, the example uses γ = 0.05, below the theoretical bound γ < 0.057.This example is presented as an empirical filter bank partitioning.
D. Empirical wavelet transform
The Empirical Wavelet Transform applies the constructed empirical filters through inner products, producing detail and approximation coefficients and supporting reconstruction. It also provides a representation of the empirical modes.
- Transform coefficients: The EWT detail coefficients are obtained from inner products of the signal with the empirical wavelets.These coefficients are denoted W_E f(n,t).
- Transform coefficients: The approximation coefficients are obtained from the inner product with the empirical scaling function.The paper denotes them by W_E f(0,t).
- Reconstruction: The empirical wavelet and scaling functions are defined in the Fourier domain, and the reconstruction is then obtained from the transform coefficients.The coefficient formulas correspond to convolution in the temporal domain and multiplication in the Fourier domain.
- Empirical modes: The empirical mode f_k is expressed within this transform formalism using the empirical wavelet representation.This connects the transform coefficients to the modes defined earlier in the paper.
A. Test signals
The experiments use three artificial signals and two real signals to test EWT across distinct component structures and real-world waveforms.
- The test set includes three artificial signals and real ECG and seismic waveform signals.The artificial signals are fSig1, fSig2, and fSig3; the real signals are fSig4 and fSig5.
- fSig1 is formed by summing three distinct components over t ∈ [0, 1].
- fSig2 is formed by summing three distinct components over t ∈ [0, 1].
- fSig3 contains three distinct components but only two additive components: fc1 and the product fc2fc3.
B. Comparison EMD vs. EWT
EWT isolates spectral modes using a prescribed number of wavelets, whereas EMD can overestimate modes and separate information that belongs to the same component. EWT outputs remain closer to the known components, with some signal-specific separations and decomposition effects.
- EWT uses a fixed number of modes N, while EMD automatically estimates the number of modes.The EWT output consists of filtering with one scaling function and N wavelets.
- EMD overestimates modes for the simulated signals and separates information that originally belongs to the same component.Its outputs are difficult to interpret against the known true components except at high frequencies.
- EWT detects spectral modes and produces components close to the original ones, but it separates initially shared modes in fSig2.The separated fSig2 modes have significant individual energy and can therefore be considered independent modes.
- For fSig3, EWT decomposes fc1 into two modes whose sum is shown separately.
- The oscillating patterns in fSig4 require cardiologist input for medical interpretation.
- EWT-based time-frequency representations avoid low-frequency artifacts seen in the Hilbert-Huang transform and appear more consistent for the seismic signal.The seismic EMD representation is described as sparse and difficult to interpret during the earthquake.
VI. AUTOMATIC DETECTION OF THE NUMBER OF MODES
The paper presents a simple Fourier-spectrum threshold for estimating the number of modes when no prior mode count is available. Threshold values around 0.3–0.4 provide a reported trade-off between excessive detection and spectral separation.
- Estimating the appropriate number of modes is difficult when no a priori information is available.The paper presents a simple estimate but states that deeper analysis is needed for a more robust method.
- The method retains Fourier-spectrum maxima above MM + α(M1 − MM) to select significant spectral bands.The maxima are sorted by decreasing magnitude and normalized to [0, 1].
- α values around 0.3 and 0.4 give consistent detected-band results across the tested signals.These values represent a trade-off between too much detection and good separation of information in the Fourier spectrum.
VII. EXTENSION TO IMAGES
The paper extends EWT to images through a tensor-product construction that processes rows and columns using adaptively detected Fourier supports. On a synthetic image, the resulting subbands isolate AM-FM components in separate Fourier domains.
- The 2D-EWT extends the one-dimensional transform to images using a tensor product.
- Directly processing rows and columns creates issues because their Fourier supports may differ in number and location.Supports with the same band number can be far apart across rows or columns.
- The 2D procedure averages row and column spectrum magnitudes, detects boundaries, builds filter banks, and filters first along rows and then columns.With NR and NC detected bands, the procedure produces (NR + 1)(NC + 1) subband images.
- With a maximum of two bands per direction, the synthetic-image experiment produces subbands that isolate each AM-FM component in separate Fourier domains.The resulting subband images and empirical Fourier-domain tiling are shown in the experiment.
VIII. CONCLUSION
The paper introduces empirical wavelets whose filter bank is adapted to Fourier supports detected in the processed signal. Experiments show separation of AM-FM modes, with EWT producing a more consistent decomposition than EMD, while future work remains open for broader filters and applications.
- VIII. CONCLUSION: EWT builds a wavelet filter bank from Fourier supports detected in the processed signal spectrum.The construction yields a tight frame of wavelets and defines the EWT and its inverse.
- VIII. CONCLUSION: The empirical wavelets use dilation factors detected from the signal rather than a prescribed scheme.In the temporal domain, they remain dilated versions of a single mother wavelet.
- VIII. CONCLUSION: Experiments on simulated and real signals show that EWT separates different AM-FM modes composing the input signal.
- VIII. CONCLUSION: Compared with EMD, EWT gives a more consistent decomposition, whereas EMD generally exhibits too many difficult-to-interpret modes.
- VIII. CONCLUSION: Future work includes independent-support filters, higher-dimensional extensions, Fourier-spectrum segmentation, and applications such as denoising and deconvolution.The paper describes direct Fourier-spectrum segmentation as difficult and still open, with denoising and deconvolution deferred to future work.