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Sampling and reconstruction of shapes with algebraic boundaries

Fatemi, M ; Sharif University of Technology | 2016

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  1. Type of Document: Article
  2. DOI: 10.1109/TSP.2016.2591505
  3. Publisher: Institute of Electrical and Electronics Engineers Inc , 2016
  4. Abstract:
  5. We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of 1{p le; 0} to the image plane, where 1 denotes the indicator function and p is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial p. We show that the image parameters - i.e., the polynomial coefficients'satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel. The benefits are threefold: 1) it relaxes the requirements on the sampling kernels; 2) produces annihilation equations that are robust at numerical precision; and 3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels
  6. Keywords:
  7. Signals with finite rate of innovation (FRI) ; Algebra ; Binary images ; Bins ; Polynomial approximation ; Polynomials ; Sampling ; Signal reconstruction ; Algebraic curves ; Finite rate of innovations (FRI) ; Generalized moments ; Numerical experiments ; Polynomial coefficients ; Reconstruction process ; Sampling and reconstruction ; Signals with finite rate of innovations ; Image sampling
  8. Source: IEEE Transactions on Signal Processing ; Volume 64, Issue 22 , 2016 , Pages 5807-5818 ; 1053587X (ISSN)
  9. URL: http://ieeexplore.ieee.org/document/7523408/?reload=true