Elementarym-harmonic cardinal B-splines

Rabut, Christophe

doi:10.1007/bf02142205

Cited by 58 publications

(49 citation statements)

References 11 publications

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“…Fortunately, when the set of spline knots forms a uniform grid, the resulting cardinal polyharmonic splines admit a simple representation in terms of B-spline-like shift-invariant basis functions [6,7]. In particular, for γ > d/2, a cardinal spline function s(x) with knot spacing T admits the following Shannon-like representation…”

Section: Multiresolution Polyharmonic Splinesmentioning

confidence: 99%

Fractional Laplacian pyramids

Delgado-Gonzalo

Tafti

Unser

2009

2009 16th IEEE International Conference on Image Processing (ICIP)

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We provide an extension of the L2-spline pyramid (Unser et al., 1993) using polyharmonic splines. We analytically prove that the corresponding error pyramid behaves exactly as a multi-scale Laplace operator. We use the multiresolution properties of polyharmonic splines to derive an efficient, non-separable filterbank implementation. Finally, we illustrate the potentials of our pyramid by performing an estimation of the parameters of multivariate fractal processes.

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Section: Multiresolution Polyharmonic Splinesmentioning

confidence: 99%

Fractional Laplacian pyramids

Delgado-Gonzalo

Tafti

Unser

2009

2009 16th IEEE International Conference on Image Processing (ICIP)

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“…which generalizes Rabut's γ-harmonic B-splines [15,19]. It is possible to show that for γ > d 4 the lattice shifts of φ2γ are squareintegrable and satisfy the necessary conditions for constituting a Mallat-type multi-resolution analysis (MRA) [20][21][22], meaning that they a) form a partition of unity; b) ful l a two-scale relation of the form…”

Section: Polyharmonic Multi-resolution Analysismentioning

confidence: 99%

Innovation modelling and wavelet analysis of fractal processes in bio-imaging

Tafti

Ville

Unser

2008

2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro

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Growth and form in biology are often associated with some level of fractality. Fractal characteristics have also been noted in a number of imaging modalities. These observations make fractal modelling relevant in the context of bio-imaging.In this paper, we introduce a simple and yet rigorous innovation model for multi-dimensional fractional Brownian motion (fBm) and provide the computational tools for the analysis of such processes in a multi-resolution framework. The key point is that these processes can be whitened by application of the appropriate fractional Laplacian operator which has a corresponding polyharmonic wavelet. We examine the case of MRI and mammography images through comparison with theoretical results, which underline the suitability of fractal models in the study of bio-textures.

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“…First, as a scaling function, the polyharmonic B-spline (Rabut, 1992) is deployed. These basis functions can be regarded as the true multi-dimensional extension of 1D B-splines, spanning the same space as radial basis functions; i.e., functions of the form q(x 1 , x 2 ) = (x 1 2 + x 2 2 ) (cÀ2)/2 , where c is the order of the polyharmonic B-spline.…”

Section: Appendix a Essential Characteristics Of The 2d Polyharmonicmentioning

confidence: 99%

Determining significant connectivity by 4D spatiotemporal wavelet packet resampling of functional neuroimaging data

2006

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An active area of neuroimaging research involves examining functional relationships between spatially remote brain regions. When determining whether two brain regions exhibit significant correlation due to true functional connectivity, one must account for the background spatial correlation inherent in neuroimaging data. We define background correlation as spatiotemporal correlation in the data caused by factors other than neurophysiologically based functional associations such as scanner induced correlations and image preprocessing. We develop a 4D spatiotemporal wavelet packet resampling method which generates surrogate data that preserves only the average background spatial correlation within an axial slice, across axial slices, and through each voxel time series, while excluding the specific correlations due to true functional relationships. We also extend an amplitude adjustment algorithm which adjusts our surrogate data to closely match the amplitude distribution of the original data. Our method improves upon existing wavelet-based methods and extends them to 4D. We apply our resampling technique to determine significant functional connectivity from resting state and motor task fMRI datasets. D

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Elementarym-harmonic cardinal B-splines

Cited by 58 publications

References 11 publications

Fractional Laplacian pyramids

Fractional Laplacian pyramids

Innovation modelling and wavelet analysis of fractal processes in bio-imaging

Determining significant connectivity by 4D spatiotemporal wavelet packet resampling of functional neuroimaging data

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