Fast fourier transforms: a tutorial review and a state of the art
Signal Processing
Approximate GCD and its application to ill-conditioned algebraic equations
ISCM '90 Proceedings of the International Symposium on Computation mathematics
Displacement structure: theory and applications
SIAM Review
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Approximate polynomial greatest common divisors and nearest singular polynomials
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
GCD of polynomials and Bezout matrices
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Detection and validation of clusters of polynomial zeros
Journal of Symbolic Computation - Special issue: validated numerical methods and computer algebra
On approximate GCDs of univariate polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
When are two numerical polynomials relatively prime?
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Computation of approximate polynomial GCDs and an extension
Information and Computation
Journal of Symbolic Computation
Approximate factorization of multivariate polynomials via differential equations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
The approximate GCD of inexact polynomials
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Symbolic-numeric sparse interpolation of multivariate polynomials
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Structured matrix-based methods for polynomial ∈-gcd: analysis and comparisons
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On exact and approximate interpolation of sparse rational functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
A structured rank-revealing method for Sylvester matrix
Journal of Computational and Applied Mathematics
An iterative method for calculating approximate GCD of univariate polynomials
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
IEEE Transactions on Signal Processing
QR factoring to compute the GCD of univariate approximate polynomials
IEEE Transactions on Signal Processing
Blind image deconvolution using a robust GCD approach
IEEE Transactions on Image Processing
Blind identification of multichannel FIR blurs and perfect image restoration
IEEE Transactions on Image Processing
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Calculating approximate GCD of multiple univariate polynomials using approximate syzygies
ACM Communications in Computer Algebra
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The problem of blind image deconvolution can be solved by computing approximate greatest common divisors (GCD) of polynomials. The bivariate polynomials corresponding to the z-transforms of several blurred images have an approximate GCD corresponding to the z-transform of the original image. Since blurring functions as cofactors have very low degree in general, this GCD will be of high degree. On the other hand, if we only have one blurred image and want to identify the original scene, the blurred image can be partitioned such that each part completely contains the blurring function, hence the blurring function becomes the GCD which is of low degree. Therefore, we design a specialized algorithm for computing GCDs of polynomials to recover true images in two different cases. The new algorithm is based on the fast GCD algorithm for univariate polynomials and the Fast Fourier Transform (FFT) algorithm. The complexity of our specialized algorithm for identifying both the true image and the blurring functions from blurred images of size n x n is O(n2 log(n)) in the case of blurring functions of very low degree. The algorithm has been implemented in Maple and can extract true images of hundreds by hundreds pixel images from blurred images in a few seconds.