An orthogonal wavelet representation of multivalued images

2003 ◽  
Vol 12 (6) ◽  
pp. 718-725 ◽  
Author(s):  
P. Scheunders
Keyword(s):  
Orthogonal Wavelet ◽  
Wavelet Representation ◽  
Multivalued Images

scholarly journals Wavelet tools to study intermittency: application to vortex bursting

2009 ◽  
Vol 636 ◽  
pp. 427-453 ◽  
Author(s):  
JORI RUPPERT-FELSOT ◽  
MARIE FARGE ◽  
PHILIPPE PETITJEANS
Keyword(s):  
Wavelet Transform ◽  
Particle Image ◽  
Three Dimensional ◽  
Local Spectrum ◽  
Orthogonal Wavelet ◽  
Wavelet Representation ◽  
Image Velocimetry ◽  
Laminar Channel Flow ◽  
Steady Laminar ◽  
Orthogonal Wavelet Transform

This paper proposes statistical tools adapted to study highly unsteady and inhomogeneous flows, such as vortex bursting. For this, we use the wavelet representation in which each coefficient keeps track of both location and scale, in contrast to Fourier representation which requires keeping the phase of all coefficients to preserve the spatial structure of the flow. Based on the continuous wavelet transform, we propose several diagnostics, such as the local spectrum and the local intermittency measure. We also use the orthogonal wavelet transform to split each flow realization into coherent and incoherent contributions, which are then analysed independently and from which we define the coherency measure. We apply these wavelet tools to analyse the bursting of a three-dimensional stretched vortex immersed in a steady laminar channel flow. The time evolution of the velocity field is measured by particle image velocimetry during several successive bursts.

scholarly journals Non-equispaced B-spline wavelets

2016 ◽  
Vol 14 (06) ◽  
pp. 1650056 ◽  
Author(s):  
Maarten Jansen
Keyword(s):  
Wavelet Transforms ◽  
Orthogonal Wavelet ◽  
B Spline ◽  
Wavelet Representation ◽  
Spline Wavelets ◽  
New Construction ◽  
Irregular Point ◽  
Interpolating Wavelets ◽  
New Framework ◽  
Orthogonal Wavelet Transform

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen–Daubechies–Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.

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