New Spline Quasi-Interpolant for Fitting 3-D Data on the Sphere: Applications to Medical Imaging

Ameur, El Bachir; Sbibih, D.; Almhdie, Ahmad; Léger, Christophe

doi:10.1109/lsp.2006.888261

Cited by 14 publications

(19 citation statements)

References 12 publications

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“…In comparison with the method introduced in [2], we remark that the surface provided by the proposed method better approximates the initial set of scattered points, and reduces the oscillations observed at the visible pole in Fig. 1 of [2].…”

Section: Examplementioning

confidence: 85%

Quadratic spherical spline quasi-interpolants on Powell–Sabin partitions

Lamnii¹,

Mraoui²,

Sbibih³

2009

Applied Numerical Mathematics

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Section: Examplementioning

confidence: 85%

Quadratic spherical spline quasi-interpolants on Powell–Sabin partitions

Lamnii¹,

Mraoui²,

Sbibih³

2009

Applied Numerical Mathematics

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“…where the matrices C k−1,l−1 , D (1) k−1,l−1 , D (2) k−1,l−1 and D (3) k−1,l−1 satisfy the following system of equations…”

Section: Algorithm Of Decomposition and Reconstruction: Applications mentioning

confidence: 99%

“…While reconstruction of the coefficient matrix C k,l from the coefficient matrices of its components is carried out by C k,l = P T k C k−1,l−1 P l + P T k D (1) k−1,l−1 Q l + Q T k D (2) k−1,l−1 P l + Q T k D (3) k−1,l−1 Q l . (7.2) On the other hand, at each step of the decomposition, many of the wavelet coefficients are small.…”

Section: Algorithm Of Decomposition and Reconstruction: Applications mentioning

confidence: 99%

“…Since the problem of the construction of F has been transformed to a rectangle H , it is natural to use tensor products for constructing the corresponding function f of the form (1.1) approach was proposed in [1] and [2] using 2π -periodic uniform algebraic trigonometric B-splines (UAT B-splines) of order four generated over the space spanned by {1, φ, cos(φ), sin(φ)} and the cubic polynomial B-splines. This approach provides bases for the present method.…”

Section: Introductionmentioning

confidence: 99%

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A multiresolution method for fitting scattered data on the sphere

et al. 2009

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In this work, we propose an efficient multiresolution method for fitting scattered data functions on a sphere S, using a tensor product method of periodic algebraic trigonometric splines of order 3 and quadratic polynomial splines defined on a rectangular map of S. We describe the decomposition and reconstruction algorithms corresponding to the polynomial and periodic algebraic trigonometric wavelets. As application of this method, we give an algorithm which allows to compress scattered data on spherelike surfaces. In order to illustrate our results, some numerical examples are presented.Keywords Multiresolution analysis · B-spline wavelets · UAT B-spline wavelets · Compression of a closed surface Mathematics Subject Classification (2000) 41A05 · 41A15 · 43A90 · 65D05 · 65D07 · 65D10 Communicated by Tom Lyche.This work is supported in part by PROTARS III, D 11/18 and AL N°MA/08/182. A. Lamnii · H. Mraoui · D. Sbibih ( ) Laboratoire MATSI, EST, Université Mohammed 1er,

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“…In volumetric imaging, it is used for rescaling 3-D volumes [10]. Fitting 3-D data on geometric shapes is also best done by taking the interpolation model into consideration [11]. Other applications where it plays a vital role include volume rendering for visualization of scalar fields [12]- [14], evaluation of image gradients [15], [16], and texture mapping where a 2-D image is painted on a 3-D surface [17], [18].…”

mentioning

confidence: 99%

Regularized Interpolation for Noisy Images

Ramani

Thévenaz

Unser

2010

IEEE Trans. Med. Imaging

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Abstract-Interpolation is the means by which a continuously defined model is fit to discrete data samples. When the data samples are exempt of noise, it seems desirable to build the model by fitting them exactly. In medical imaging, where quality is of paramount importance, this ideal situation unfortunately does not occur. In this paper, we propose a scheme that improves on the quality by specifying a tradeoff between fidelity to the data and robustness to the noise. We resort to variational principles, which allow us to impose smoothness constraints on the model for tackling noisy data. Based on shift-, rotation-, and scale-invariant requirements on the model, we show that the -norm of an appropriate vector derivative is the most suitable choice of regularization for this purpose. In addition to Tikhonov-like quadratic regularization, this includes edge-preserving total-variation-like (TV) regularization. We give algorithms to recover the continuously defined model from noisy samples and also provide a data-driven scheme to determine the optimal amount of regularization. We validate our method with numerical examples where we demonstrate its superiority over an exact fit as well as the benefit of TV-like nonquadratic regularization over Tikhonov-like quadratic regularization.

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New Spline Quasi-Interpolant for Fitting 3-D Data on the Sphere: Applications to Medical Imaging

Cited by 14 publications

References 12 publications

Quadratic spherical spline quasi-interpolants on Powell–Sabin partitions

Quadratic spherical spline quasi-interpolants on Powell–Sabin partitions

A multiresolution method for fitting scattered data on the sphere

Regularized Interpolation for Noisy Images

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