During the late 1990s, Huang introduced the algorithm called Empirical Mode Decomposition, which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands. EMD is known for limitations like sensitivity to noise and sampling. These limitations could only partially be addressed by more mathematical attempts to this decomposition problem, like synchrosqueezing, empirical wavelets or recursive variational decomposition. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for an ensemble of modes and their respective center frequencies, such that the modes collectively reproduce the input signal, while each being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into multiple, adaptive bands. Our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand. The variational model is efficiently optimized using an alternating direction method of multipliers approach. Preliminary results show attractive performance with respect to existing mode decomposition models. In particular, our proposed model is much more robust to sampling and noise. Finally, we show promising practical decomposition results on a series of artificial and real data.

Variational Mode Decomposition

IEEE Transactions on Pattern Analysis and Machine Intelligence

Proceedings of the American Mathematical Society

Research on variational mode decomposition and its application in detecting rub-impact fault of the rotor system

Wavelet transform based on inner product in fault diagnosis of rotating machinery: A review

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.

/pdf/empirical-wavelet-transform-490hebsyq6.pdf

Empirical Wavelet Transform

A recently developed approach, called “empirical wavelet transform,” aims to build one-dimensional (1D) adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to two-dimensional (2D) signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood--Paley wavelets, ridgelets, and curvelets) and show that it is possible to build their empirical counterparts. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.

2D Empirical Transforms. Wavelets, Ridgelets, and Curvelets Revisited

In this paper, we present an algorithm to automatically detect meaningful modes in a histogram. The proposed method is based on the behavior of local minima in a scale-space representation. We show that the detection of such meaningful modes is equivalent in a two classes clustering problem on the length of minima scale-space curves. The algorithm is easy to implement, fast and does not require any parameter. We present several results on histogram and spectrum segmentation, grayscale image segmentation and color image reduction.

A parameterless scale-space approach to find meaningful modes in histograms — Application to image and spectrum segmentation

The non-uniformity is a time-dependent noise caused by the lack of sensor equalization. We present here the detailed algorithm and online demo of the non-uniformity correction method by midway infrared equalization. This method was designed to suit infrared images. Nevertheless, it can be applied to images produced for example by scanners, or by push-broom satellites. This single image method works on static images, is fully automatic, has no user parameter, and requires no registration. It needs no camera motion compensation, no closed aperture sensor equalization and is able to correct for a fully non-linear non-uniformity.

/pdf/non-uniformity-correction-of-infrared-images-by-midway-10682izrhe.pdf

Non-uniformity Correction of Infrared Images by Midway Equalization

A novel approach is presented to recover an image degraded by atmospheric turbulence. Given a sequence of frames affected by turbulence, we construct a
variational model to characterize the static image. The optimization problem is solved by Bregman Iteration and the operator splitting method.
Our algorithm is simple, efficient, and can be easily generalized for different scenarios.

Jérôme Gilles

Papers

Empirical Wavelet Transform

2D Empirical Transforms. Wavelets, Ridgelets, and Curvelets Revisited

A parameterless scale-space approach to find meaningful modes in histograms — Application to image and spectrum segmentation

Non-uniformity Correction of Infrared Images by Midway Equalization

Non rigid geometric distortions correction - Application to atmospheric turbulence stabilization