El-Hadji Samba Diop

Analysis of Intrinsic Mode Functions: A PDE Approach

The empirical mode decomposition is a powerful tool for signal processing. Because of its original algorithmic, recent works have contributed to its theoretical framework. Following these works, some mathematical contributions on its comprehension and formalism are provided. In this paper, the so called local mean is computed in such a way that it allows the use of differential calculus on envelopes. This new formulation makes us prove that iterations of the sifting process are well approximated by the resolution of partial differential equations (PDE). Intrinsic mode functions are originally defined in a intuitive way. Herein, a mathematical characterization of modes is given with the proposed PDE-based approach.

Image contrast enhancement based on 2D Teager-Kaiser operator

This paper describes a new method based on the 2D Teager- Kaiser Energy Operator (2DTKEO) for image contrast enhancement. The 2DTKEO reflects better the local activity than the amplitude of a classical edges detection operator. This quadratic filter is used to enhance high frequency information which is then combined with image gray values to estimate the edge strength value used in the enhancement process. This value is the average of the gray values by the energy activity at each pixel. Different examples of images are provided to demonstrate the Performance of the proposed method is demonstrated on synthetic and real images and the results compared to histogram equalization and to an edge- based contrast method.

Speech signal noise reduction by EMD

In this paper, a speech signal noise reduction based on a multiresolution approach referred to as Empirical Mode Decomposition (EMD) [1] is introduced. The proposed speech denoising method is a fully data-driven approach. Noisy signal is decomposed adaptively into oscillatory components called Intrinsic Mode Functions (IMFs), using a temporal decomposition called sifting process. The basic principle of the method is to reconstruct the signal with IMFs previously thresholded using a shrinkage function. The denoising method is applied to speech with different noise levels and the results are compared to wavelet shrinkage. The study is limited to signals corrupted by additive white Gaussian noise.

A PDE characterization of the intrinsic mode functions

For the first time, a proof of the sifting process (SP) and so the empirical mode decomposition (EMD), is given. For doing this, lower and upper envelopes are modeled in a more convenient way that helps us prove the convergence of the SP towards a solution of a partial differential equation (PDE). We also prove that such a PDE has a unique solution, which ensures the uniqueness of the EMD decomposition. The new formulation of envelopes has another benefit. In fact, it removes interpolation problems and related issues. Not only helps the modelization of envelopes to give a mathematical framework on the EMD, but also, as confirmed by the numerical simulations, the PDE-based EMD improves a lot the classical EMD.

A joint 2D AM–FM estimation based on higher order Teager–Kaiser energy operators

In this work, an image demodulation algorithm based on two-dimensional higher order Teager–Kaiser (TK) operators is presented. We show quantitatively and qualitatively that the introduction of higher orders in TK operator improves amplitude modulation (AM) and frequency modulation (FM) estimation results, compared to classical approaches such as the Discrete Energy Separation Algorithm (DESA) or the Analytic Signal (AS) method. Indeed, for a wide class of images, obtained demodulation errors for both the amplitude and frequency are numerically lower than the obtained ones with the DESA and AS method. The proposed method is illustrated on both synthetic and real images. Moreover, it turns out for some real images that the algorithm is very efficient in the sense that it tracks the most significant part in images and segments regions of interests, particularly, the AM counterpart. Finally, an application of our approach to the segmentation of mines’ shadows in Sonar images is presented. This is very important for both civil and military applications.

Empirical Mode Decomposition and some operators to estimate Instantaneous Frequency: A comparative study

The aim of the present work is to illustrate how the empirical mode decomposition (EMD) can be associated with demodulation methods to estimate the instantaneous frequency (IF) of multicomponent non stationary signals. The IF estimation of three methods, namely, Hilbert-Huang transform (HHT) [10], EMD-DESA (Discrete Energy Separation Algorithm)also known as Teager-Huang transform (THT) [2] [5], and B-spline version of the EMD-DESA called EMD-ESA-BS [1] are compared on AM-FM signal corrupted with an additive white Gaussian noise of varying signal to noise ratio (SNR). The obtained results show that for very low SNR the EMD-ESA-BS with regularization performs better than the HHT, the THT and the EMD-ESA- BS without regularization. For high SNR values all the methods globally have the same behavior, but the EMD-ESA-BS without regularization gives the best estimation.

Teager-Kaiser Energy bi-level thresholding

In this paper an automatic bi-level thresholding method of grey scale images using the Teager-Kaiser Energy Operator (TKEO) is proposed. The TKEO, a quadratic filter, is used to enhance high frequency information which is then combined with image grey values to estimate the threshold value. More precisely, this threshold is the average of the grey values by the energy at each pixel. The TKEO reflects better the local activity than the amplitude of the gradient. The performance of the thresholding is illustrated on different images and results compared to Otsu [1] and Kitler et al. [2] methods respectively.

Higher order Teager-Kaiser operators for image analysis: Part I - A monocomponent image demodulation

We present in this paper a new narrowband image demodulation method. Our approach is based on the 2D higher order Teager-Kaiser operators (HOTKO). We show that the introduction of higher orders in the Teager-Kaiser operator, improves a lot the demodulation results, in comparison to the Discrete Energy Separation Algorithm (DESA) and the Analytic Image (AI) method. More precisely, for synthetic images, we show that the approximation errors on both the amplitude and the frequency components are much more lower with our proposed demodulation method than the DESA and the AI method. Moreover, it turns out that for the presented real images, the algorithm is so efficient, especially the amplitude counterpart, that it tracks the most important parts in images, and segments the regions of interest. We show how the algorithm could be used in Sonar images for extracting minesshadows, which is very important for both military and civil applications.

Higher order Teager-Kaiser operators for image analysis: PART II - a multicomponent image demodulation

We present in this paper a new real image demodulation algorithm. Because of the multicomponent aspects of real images, we first split the image into narrow band components by the means of a Gabor filterbank. Each narrowband component is then demodulated by using the 2D higher order Teager-Kaiser operators (HOTKO). The demodulation process ends by a pixel-by-pixel selection achieved through a dominant component selection basis, in order to get a better image analysis. The whole procedure is presented here, and the algorithms effectiveness is demonstrated on a large class of various images.

Two-Dimensional Curvature-Based Analysis of Intrinsic Mode Functions

A novel approach in modeling the empirical mode decomposition (EMD) is proposed here, allowing a perfect image recovery and, for instance, a straightforward extension for multidimensional