Bandlimited Reconstruction of Multidimensional Images From Irregular Samples

We examine different sampling lattices and their respective bandlimited spaces for reconstruction of irregularly sampled multidimensional images. Considering an irregularly sampled dataset, we demonstrate that the non-tensor-product bandlimited approximations corresponding to the body-centered cubic and face-centered cubic lattices provide a more accurate reconstruction than the tensor-product bandlimited approximation associated with the commonly-used Cartesian lattice. Our practical algorithm uses multidimensional sinc functions that are tailored to these lattices and a regularization scheme that provides a variational framework for efficient implementation. Using a number of synthetic and real data sets we record improvements in the accuracy of reconstruction in a practical setting.

[1]  Gerald E. Farin,et al.  Natural neighbor extrapolation using ghost points , 2009, Comput. Aided Des..

[2]  F. Marvasti.  Nonuniform sampling : theory and practice , 2001 .

[3]  Gregory M. Nielson,et al.  Scattered data modeling , 1993, IEEE Computer Graphics and Applications.

[4]  Bernd Hamann,et al.  Discrete Sibson interpolation , 2006, IEEE Transactions on Visualization and Computer Graphics.

[5]  Markus H. Gross,et al.  Shape modeling with point-sampled geometry , 2003, ACM Trans. Graph..

[6]  Eduard Gröller,et al.  Efficient reconstruction from non-uniform point sets , 2008, The Visual Computer.

[7]  Jared Tanner,et al.  Fast Reconstruction Methods for Bandlimited Functions from Periodic Nonuniform Sampling , 2006, SIAM J. Numer. Anal..

[8]  Hans R. Künsch,et al.  Optimal lattices for sampling , 2005, IEEE Transactions on Information Theory.

[9]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[10]  A.K. Krishnamurthy,et al.  Multidimensional digital signal processing , 1985, Proceedings of the IEEE.

[11]  Lars Linsen,et al.  A Narrow Band Level Set Method for Surface Extraction from Unstructured Point-based Volume Data , 2011 .

[12]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[13]  David Middleton,et al.  Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces , 1962, Inf. Control..

[14]  Wenxing Ye,et al.  A Geometric Construction of Multivariate Sinc Functions , 2012, IEEE Transactions on Image Processing.

[15]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[16]  R. Vershynin,et al.  A Randomized Kaczmarz Algorithm with Exponential Convergence , 2007, math/0702226.

[17]  Sung Yong Shin,et al.  Scattered Data Interpolation with Multilevel B-Splines , 1997, IEEE Trans. Vis. Comput. Graph..

[18]  Alexander Singh-Alvarado,et al.  Reconstruction of Irregularly-Sampled Volumetric Data in Efficient Box Spline Spaces , 2012, IEEE Transactions on Medical Imaging.

[19]  Wenxing Ye,et al.  Tomographic reconstruction of diffusion propagators from DW-MRI using optimal sampling lattices , 2010, 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro.

[20]  Bernd Hamann,et al.  Reconstruction of B-spline surfaces from scattered data points , 2000, Proceedings Computer Graphics International 2000.

[21]  Rik Van de Walle,et al.  Accepted for Publication in Ieee Transactions on Image Processing Hex-splines: a Novel Spline Family for Hexagonal Lattices , 2022 .

[22]  Alireza Entezari,et al.  Optimal sampling lattices and trivariate box splines , 2007 .

[23]  M. Unser.  Sampling-50 years after Shannon , 2000, Proceedings of the IEEE.

[24]  Valerio Lucarini,et al.  Three-Dimensional Random Voronoi Tessellations: From Cubic Crystal Lattices to Poisson Point Processes , 2008, ArXiv.

[25]  Matthias Zwicker,et al.  Pointshop 3D: an interactive system for point-based surface editing , 2002, SIGGRAPH.

[26]  David S. Ebert,et al.  Enhancing the Interactive Visualization of Procedurally Encoded Multifield Data with Ellipsoidal Basis Functions , 2006, Comput. Graph. Forum.

[27]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

[28]  Akram Aldroubi,et al.  Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces , 2001, SIAM Rev..

[29]  Michael Unser,et al.  Reconstruction of Large, Irregularly Sampled Multidimensional Images. A Tensor-Based Approach , 2011, IEEE Transactions on Medical Imaging.

[30]  Michael Unser,et al.  Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution , 2005, IEEE Transactions on Image Processing.

[31]  Dimitri Van De Ville,et al.  Practical Box Splines for Reconstruction on the Body Centered Cubic Lattice , 2008, IEEE Transactions on Visualization and Computer Graphics.

[32]  Steve Marschner,et al.  An evaluation of reconstruction filters for volume rendering , 1994, Proceedings Visualization '94.

[33]  Zuowei Shen,et al.  Scattered data reconstruction by regularization in B-spline and associated wavelet spaces , 2009, J. Approx. Theory.

[34]  Matthew Dylan Tisdall.  Development and validation of algorithms for MRI signal component estimation , 2007 .

[35]  Minh N. Do,et al.  A Computable Fourier Condition Generating Alias-Free Sampling Lattices , 2009, IEEE Transactions on Signal Processing.

[36]  Hanspeter Pfister,et al.  Volume MLS Ray Casting , 2008, IEEE Transactions on Visualization and Computer Graphics.

[37]  A. W. M. van den Enden,et al.  Discrete Time Signal Processing , 1989 .

[38]  Yao Wang,et al.  Second-order derivative-based smoothness measure for error concealment in DCT-based codecs , 1998, IEEE Trans. Circuits Syst. Video Technol..

[39]  Manojkumar Saranathan,et al.  ANTHEM: anatomically tailored hexagonal MRI. , 2007, Magnetic resonance imaging.

[40]  James F. Blinn.  Jim Blinn's corner: dirty pixels , 1998 .