Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution

We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified poin...

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Bibliographic Details
Published in:IEEE transactions on image processing Vol. 14; no. 4; pp. 450 - 460
Main Authors: Arigovindan, M., Suhling, M., Hunziker, P., Unser, M.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.04.2005
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Applied sciences
Artificial Intelligence
B-splines
Computer Simulation
Cost function
Differential algebraic equations
Exact sciences and technology
Image Enhancement - methods
Image Interpretation, Computer-Assisted - methods
Image processing
Image reconstruction
Image resolution
Information Storage and Retrieval - methods
Information, signal and communications theory
Interpolation
Linear systems
Models, Statistical
multigrid algorithm
multiresolution reconstruction
nonuniform interpolation
Nonuniform sampling
Numerical Analysis, Computer-Assisted
Pattern Recognition, Automated - methods
radial basis functions (RBFs)
Reproducibility of Results
Sample Size
Sensitivity and Specificity
Signal processing
Signal Processing, Computer-Assisted
Signal resolution
Spline
Studies
Subtraction Technique
Telecommunications and information theory
thin-plate splines
variational reconstruction
B spline
Performance evaluation
Image processing
Thin plate
Numerical method
Multiresolution analysis
Image reconstruction
Spline
multigrid algorithm
Linear equation
Multigrid
radial basis functions (RBFs)
variational reconstruction
Fast algorithm
Localization
multiresolution reconstruction
Large scale system
B-splines
nonuniform interpolation
Image method
Radial basis function
Interpolation
Linear system
Analytical method
thin-plate splines
ISSN:1057-7149, 1941-0042
Online Access:Get full text
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Summary:We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified points and 2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the two-scale relation for B-splines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples.
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ISSN:1057-7149
1941-0042
DOI:10.1109/TIP.2004.841203