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A fast algorithm for bidimensional EMD

In this letter, we describe a new method for bidimensional empirical mode decomposition (EMD). This decomposition is based on Delaunay triangulation and on piecewise cubic polynomial interpolation. Particular attention is devoted to boundary conditions that are crucial for the feasibility of the bidimensional EMD. The study of the behavior of the decomposition on a different kind of image shows its efficiency in terms of computational cost, and the decomposition of Gaussian white noises leads to bidimensional selective filter banks.

A New Formulation for Empirical Mode Decomposition Based on Constrained Optimization

The empirical mode decomposition (EMD) is an algorithmic construction that aims at decomposing a signal into several modes called intrinsic mode functions. In this letter, we present a new approach for the EMD based on the direct construction of the mean envelope of the signal. The definition of the mean envelope is achieved through the resolution of a quadratic programming problem with equality and inequality constraints. Some numerical experiments conclude this letter, and comparisons are carried out with the classical EMD.

Second-Order Synchrosqueezing Transform or Invertible Reassignment? Towards Ideal Time-Frequency Representations

This paper considers the analysis of multicomponent signals, defined as superpositions of real or complex modulated waves. It introduces two new post-transformations for the short-time Fourier transform that achieve a compact time-frequency representation while allowing for the separation and the reconstruction of the modes. These two new transformations thus benefit from both the synchrosqueezing transform (which allows for reconstruction) and the reassignment method (which achieves a compact time-frequency representation). Numerical experiments on real and synthetic signals demonstrate the efficiency of these new transformations, and illustrate their differences.

An Alternative Formulation for the Empirical Mode Decomposition

The Empirical Mode Decomposition (EMD) is a relatively new adaptive method for multicomponent signal representation which allows for analyzing nonlinear and nonstationary signals. In spite of its lack of mathematical foundations, very few papers are dedicated to defining new decompositions that would preserve the interesting properties of the EMD while improving the mathematical setting. The new decomposition based on direct constrained optimization we introduce in this article is an attempt in that direction.

Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows

We investigate the use of compactly supported divergence-free wavelets for the representation of solutions of the Navier–Stokes equations. After reviewing the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for two and three-dimensional incompressible flows. We also propose a new method for practically computing the wavelet Helmholtz decomposition of any (even compressible) flow; this decomposition, which allows the incompressible part of the flow to be separated from its orthogonal complement (the gradient component of the flow) is the key point for developing divergence-free wavelet schemes for Navier–Stokes equations. Finally, numerical tests validating our approach are presented.In this paper, we investigate the use of compactly supported divergencefree waveletsfor the representation of the Navier-Stokes solution. Afte r r minding the theoretical construction of divergence-free wavelet vect ors, we present in detail the bases and corresponding fast algorithms for 2D and 3D incompressi ble flows. In order to compute the nonlinear term, we propose a new method which provides in practice with the Hodge decomposition of any flow: this decomposition enables us to s eparate the incompressible part of the flow from its orthogonal complement, which correspond s to the gradient component of the flow. Finally we show numerical tests to validate our appr oach. Submitted to:Journal of Turbulence

The fourier-based synchrosqueezing transform

The short-time Fourier transform (STFT) and the continuous wavelet transform (CWT) are extensively used to analyze and process multicomponent signals, i.e. superpositions of modulated waves. The synchrosqueezing is a post-processing method which circumvents the uncertainty relation inherent to these linear transforms, by reassigning the coefficients in scale or frequency. Originally introduced in the setting of the CWT, it provides a sharp, concentrated representation, while remaining invertible. This technique received a renewed interest with the recent publication of an approximation result related to the application of the synchrosqueezing to multi-component signals. In the current paper, we adapt the formulation of the synchrosqueezing to the STFT and state a similar theoretical result to that obtained in the CWT framework. The emphasis is put on the differences with the CWT-based synchrosqueezing with numerical experiments illustrating our statements.

Direct Numerical Simulation of Turbulence Using Divergence-Free Wavelets

We present a numerical method based on divergence-free wavelets to solve the incompressible Navier–Stokes equations. We introduce a new scheme which uses anisotropic (or generalized) divergence-free wavelets and which needs only fast wavelet transform algorithms. We prove its stability and show convincing numerical experiments.

Effective construction of divergence-free wavelets on the square

We present an effective construction of divergence-free wavelets on the square, with suitable boundary conditions. Since 2D divergence-free vector functions are the curl of scalar stream-functions, we simply derive divergence-free multiresolution spaces and wavelets by considering the curl of standard biorthogonal multiresolution analyses (BMRAs) on the square. The key point of the theory is that the derivative of a 1D BMRA is also a BMRA, as established by Jouini and Lemarie-Rieusset (1993) [16]. We propose such construction in the context of generic compactly supported wavelets, which allows fast algorithms. Examples illustrate the practicality of the method.

Towards better digital pathology workflows: programming libraries for high-speed sharpness assessment of Whole Slide Images

BackgroundSince microscopic slides can now be automatically digitized and integrated in the clinical workflow, quality assessment of Whole Slide Images (WSI) has become a crucial issue. We present a no-reference quality assessment method that has been thoroughly tested since 2010 and is under implementation in multiple sites, both public university-hospitals and private entities. It is part of the FlexMIm R&D project which aims to improve the global workflow of digital pathology. For these uses, we have developed two programming libraries, in Java and Python, which can be integrated in various types of WSI acquisition systems, viewers and image analysis tools.MethodsDevelopment and testing have been carried out on a MacBook Pro i7 and on a bi-Xeon 2.7GHz server. Libraries implementing the blur assessment method have been developed in Java, Python, PHP5 and MySQL5. For web applications, JavaScript, Ajax, JSON and Sockets were also used, as well as the Google Maps API. Aperio SVS files were converted into the Google Maps format using VIPS and Openslide libraries.ResultsWe designed the Java library as a Service Provider Interface (SPI), extendable by third parties. Analysis is computed in real-time (3 billion pixels per minute). Tests were made on 5000 single images, 200 NDPI WSI, 100 Aperio SVS WSI converted to the Google Maps format.ConclusionsApplications based on our method and libraries can be used upstream, as calibration and quality control tool for the WSI acquisition systems, or as tools to reacquire tiles while the WSI is being scanned. They can also be used downstream to reacquire the complete slides that are below the quality threshold for surgical pathology analysis. WSI may also be displayed in a smarter way by sending and displaying the regions of highest quality before other regions. Such quality assessment scores could be integrated as WSIs metadata shared in clinical, research or teaching contexts, for a more efficient medical informatics workflow.

Helmholtz-Hodge decomposition on [0,1] d by divergence-free and curl-free wavelets

This paper deals with the Helmholtz-Hodge decomposition of a vector field in bounded domain. We present a practical algorithm to compute this decomposition in the context of divergence-free and curl-free wavelets satisfying suitable boundary conditions. The method requires the inversion of divergence-free and curl-free wavelet Gram matrices. We propose an optimal preconditioning which allows to solve the systems with a small number of iterations. Finally, numerical examples prove the accuracy and the efficiency of the method.