The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Alireza Entezari - One of the best experts on this subject based on the ideXlab platform.
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Box Spline projection in non parallel geometry
International Symposium on Biomedical Imaging, 2019Co-Authors: Kai Zhang, Alireza EntezariAbstract:The pixel- and voxel-basis are common choices for image discretization in the context of computed tomography (CT). They can also be viewed as first-order Box Splines – a class of functions with closed-form X-ray and Radon transforms that can be computed efficiently. In this paper we derive a method for exact projection of Box Splines in a non-parallel geometry that can be used in fan-beam and cone-beam tomographic image reconstruction algorithms. We also provide efficient computational procedures for evaluation of the basis function in the projection domain.
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ISBI - Box Spline Projection in Non-Parallel Geometry
2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019), 2019Co-Authors: Kai Zhang, Alireza EntezariAbstract:The pixel- and voxel-basis are common choices for image discretization in the context of computed tomography (CT). They can also be viewed as first-order Box Splines – a class of functions with closed-form X-ray and Radon transforms that can be computed efficiently. In this paper we derive a method for exact projection of Box Splines in a non-parallel geometry that can be used in fan-beam and cone-beam tomographic image reconstruction algorithms. We also provide efficient computational procedures for evaluation of the basis function in the projection domain.
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A Box Spline Calculus for the Discretization of
2012Co-Authors: Alireza Entezari, Masih Nilchian, Michael UnserAbstract:B-Splines are attractive basis functions for the contin- uous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of Box Splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported Box Splines (tensor-product B-Splines, separable or not) in any number of dimensions. The proposed Box Spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an -direction Box Spline is generally a (nonuniform) polynomial Spline of degree . The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a Box Spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the Box Spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms.
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A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems
IEEE Transactions on Medical Imaging, 2012Co-Authors: Alireza Entezari, Masih Nilchian, Michael UnserAbstract:B-Splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of Box Splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported Box Splines (tensor-product B-Splines, separable or not) in any number of dimensions. The proposed Box Spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an -direction Box Spline is generally a (nonuniform) polynomial Spline of degree . The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a Box Spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the Box Spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms.
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Reconstruction of Irregularly-Sampled Volumetric Data in Efficient Box Spline Spaces
IEEE Transactions on Medical Imaging, 2012Co-Authors: Xie Xu, Alexander Singh Alvarado, Alireza EntezariAbstract:We present a variational framework for the reconstruction of irregularly-sampled volumetric data in, nontensor-product, Spline spaces. Motivated by the sampling-theoretic advantages of body centered cubic (BCC) lattice, this paper examines the BCC lattice and its associated Box Spline spaces in a variational setting. We introduce a regularization scheme for Box Splines that allows us to utilize the BCC lattice in a variational reconstruction framework. We demonstrate that by choosing the BCC lattice over the commonly-used Cartesian lattice, as the shift-invariant representation, one can increase the quality of signal reconstruction. Moreover, the computational cost of the reconstruction process is reduced in the BCC framework due to the smaller bandwidth of the system matrix in the Box Spline space compared to the corresponding tensor-product B-Spline space. The improvements in accuracy are quantified numerically and visualized in our experiments with synthetic as well as real biomedical datasets.
Ming-jun Lai - One of the best experts on this subject based on the ideXlab platform.
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Box Spline Wavelet Frames for Image Edge Analysis
SIAM Journal on Imaging Sciences, 2013Co-Authors: Weihong Guo, Ming-jun LaiAbstract:We present a new Box Spline wavelet frame and apply it for image edge analysis. The wavelet frame is constructed using a Box Spline of eight directions. It is tight and has seldom been used for applications. Due to the eight different directions, it can find edges of various types in detail quite well. In addition to step edges (local discontinuities in intensity), it is able to locate Dirac edges (momentary changes of intensity) and hidden edges (local discontinuity in intensity derivatives). The method is simple and robust to noise. Many numerical examples are presented to demonstrate the effectiveness of this method. Quantitative and qualitative comparisons with other edge detection techniques are provided to show the advantages of this wavelet frame. Our test images include synthetic images with known ground truth and natural, medical images with rich geometric information.
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Construction of trivariate compactly supported biorthogonal Box Spline wavelets
Journal of Approximation Theory, 2003Co-Authors: Ming-jun LaiAbstract:We give a formula for the duals of the masks associated with trivariate Box Spline functions. We show how to construct trivariate nonseparable compactly supported biorthogonal wavelets associated with Box Spline functions. The biorthogonal wavelets may have arbitrarily high regularities.
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Construction of Bivariate Compactly Supported Biorthogonal Box Spline Wavelets with Arbitrarily High Regularities
Applied and Computational Harmonic Analysis, 1999Co-Authors: Ming-jun LaiAbstract:Abstract We give a simple formula for the duals of the filters associated with bivariate Box Spline functions. We show how to construct bivariate non-separable compactly supported biorthogonal wavelets associated with Box Spline functions which have arbitrarily high regularities.
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bivariate Box Spline wavelets in sobolev spaces
SPIE's International Symposium on Optical Science Engineering and Instrumentation, 1998Co-Authors: Ming-jun LaiAbstract:We use bivariate BoxSpline functions to construct nonseparable wavelets in Sobolev spaces.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
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Bivariate Box Splines for image processing
Wavelet Applications in Signal and Image Processing IV, 1996Co-Authors: Ming-jun LaiAbstract:ABSTRACT Bivariate Box Splines for image interpolation, enhancement, digital filter design, sub- band coding bank, hexagonal filtering will be discussed. Some existing and new results will be presented. A computational method for Box Spline image interpolation and Box Splinedigital filters are included. Keywords: bivariate Box Splines, image interpolation, filter design, hexagonal Spline wavelets, Toeplitz matrices. 1. INTRODUCTION We are interested in applying bivariate Box Spline theory to image processing. Our studyis motivated by recent papers on the applications of B-Spline theory to signal and image processing (cf. [Chui'93}, [Aidroubi, Eden, Unser'94], [Aldroubi and Unser'93,'94], [Unser, Aidroubi, and Eden'93a, '93b]). It is well-known that the univariate B-Spline functions havefound successful applications in signal and image processing, especially, B-Spline wavelets for image compression (cf. [Chui and Wang'92].) Bivariate Box Spline functions are a
Torsten Moller - One of the best experts on this subject based on the ideXlab platform.
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Extensions of the Zwart-Powell Box Spline for Volumetric Data Reconstruction on the Cartesian Lattice
IEEE Transactions on Visualization and Computer Graphics, 2006Co-Authors: Alireza Entezari, Torsten MollerAbstract:In this article we propose a Box Spline and its variants for reconstructing volumetric data sampled on the Cartesian lattice. In particular we present a tri-variate Box Spline reconstruction kernel that is superior to tensor product reconstruction schemes in terms of recovering the proper Cartesian spectrum of the underlying function. This Box Spline produces a C2 reconstruction that can be considered as a three dimensional extension of the well known Zwart-Powell element in 2D. While its smoothness and approximation power are equivalent to those of the tri-cubic B-Spline, we illustrate the superiority of this reconstruction on functions sampled on the Cartesian lattice and contrast it to tensor product B-Splines. Our construction is validated through a Fourier domain analysis of the reconstruction behavior of this Box Spline. Moreover, we present a stable method for evaluation of this Box Spline by means of a decomposition. Through a convolution, this decomposition reduces the problem to evaluation of a four directional Box Spline that we previously published in its explicit closed form
Jörg Peters - One of the best experts on this subject based on the ideXlab platform.
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Symmetric Box-Splines on the A* n lattice
Journal of Approximation Theory, 2010Co-Authors: Jörg PetersAbstract:Sampling and reconstruction of generic multivariate functions is more efficient on non-Cartesian root lattices, such as the BCC (Body-Centered Cubic) lattice, than on the Cartesian lattice. We introduce a new nxn generator matrix A^* that enables, in n variables, efficient reconstruction on the non-Cartesian root lattice A"n^* by a symmetric Box-Spline family M"r^*. A"2^* is the hexagonal lattice and A"3^* is the BCC lattice. We point out the similarities and differences of M"r^* with respect to the popular Cartesian-shifted Box-Spline family M"r, document the main properties of M"r^* and the partition induced by its knot planes and construct, in n variables, the optimal quasi-interpolant of M"2^*.
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Fast and stable evaluation of Box-Splines via the BB-form
Numerical Algorithms, 2009Co-Authors: Jörg PetersAbstract:To repeatedly evaluate linear combinations of Box-Splines in a fast and stable way, in particular along knot planes, the Box-Spline is converted to and tabulated as piecewise polynomial in BB-form (Bernstein–Bézier-form). We show that the BB-coefficients can be derived and stored as integers plus a rational scale factor and derive a hash table for efficiently accessing the polynomial pieces. This pre-processing, the resulting evaluation algorithm and use in a widely-used ray-tracing package are illustrated for Splines based on two trivariate Box-Splines: the seven-directional Box-Spline on the Cartesian lattice and the six-directional Box-Spline on the face-centered cubic lattice.
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Fast and stable evaluation of Box-Splines via the BB-form
Numerical Algorithms, 2008Co-Authors: Minho Kim, Jörg PetersAbstract:To repeatedly evaluate linear combinations of Box-Splines in a fast and stable way, in particular along knot planes, the Box-Spline is converted to and tabulated as piecewise polynomial in BB-form (Bernstein–Bezier-form). We show that the BB-coefficients can be derived and stored as integers plus a rational scale factor and derive a hash table for efficiently accessing the polynomial pieces. This pre-processing, the resulting evaluation algorithm and use in a widely-used ray-tracing package are illustrated for Splines based on two trivariate Box-Splines: the seven-directional Box-Spline on the Cartesian lattice and the six-directional Box-Spline on the face-centered cubic lattice.
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Box Spline Reconstruction On The Face-Centered Cubic Lattice
IEEE Transactions on Visualization and Computer Graphics, 2008Co-Authors: Alireza Entezari, Jörg PetersAbstract:We introduce and analyze an efficient reconstruction algorithm for FCC-sampled data. The reconstruction is based on the 6-direction Box Spline that is naturally associated with the FCC lattice and shares the continuity and approximation order of the triquadratic B-Spline. We observe less aliasing for generic level sets and derive special techniques to attain the higher evaluation efficiency promised by the lower degree and smaller stencil-size of the C1 6-direction Box Spline over the triquadratic B-Spline.
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Blending Basic Implicit Shapes Using Trivariate Box Splines
1995Co-Authors: Jörg Peters, Michael WittmanAbstract:To blend be~ween simple implicit surfaces, such as the sphere, the cone, the cylinder and the torus, we propose La locally employ the zero set of a serendipitous trivariate Box Spline. This Box Spline is defined by seven directions that form a regular partition of space into tetrahedra. The resulting blend surface is curvature continuous. An approxlmateparametrization of the piecewise implicit surface of degree four is obtained by subdivision and sign comparison. Blending basic implicit shapes using trivariate Box Splines
Mollertorsten - One of the best experts on this subject based on the ideXlab platform.
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Extensions of the Zwart-Powell Box Spline for Volumetric Data Reconstruction on the Cartesian Lattice
IEEE Transactions on Visualization and Computer Graphics, 2006Co-Authors: Entezarialireza, MollertorstenAbstract:In this article we propose a Box Spline and its variants for reconstructing volumetric data sampled on the Cartesian lattice. In particular we present a tri-variate Box Spline reconstruction kernel...