Optimal Filters from Calibration Data for Image Deconvolution with Data Acquisition Error

Abstract

Data acquisition errors due to dead pixels or other hardware defects can cause undesirable artifacts in imaging applications. Compensating for these defects typically requires knowledge such as a defective pixel map, which can be difficult or costly to obtain and which is not necessarily static. However, recent calibration data is readily available in many applications. In this paper, we compute optimal filters for image deconvolution with denoising using only this calibration data, by minimizing the empirical Bayes risk. We derive a bound on how the reconstruction changes as the number of dead pixels grows. We show that our approach is able to reconstruct missing information better than standard filtering approaches and is robust even in the presence of a large number of defects and to defects that arise after calibration.

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Notes

  1. 1.

    The extension to a matrix A that has more rows than columns, or that has rank less than n, is straightforward. In that case, replace every Σ −1 by the pseudo-inverse Σ and every summation from i=1 to n by a summation i=1 to \(\operatorname {rank} ( \mathbf{A} )\), so that the number of filter factors is reduced to \(\operatorname {rank} (\mathbf{A} )\).

  2. 2.

    Any type of blurring operator A (spatially variant or invariant) may be considered, with the only restriction that the singular value decomposition should be available.

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Acknowledgements

We are grateful to the referees for helpful suggestions. The work of J.C. was partially supported by NSF grant DMS 0902322. The work of D.P.O. was partially supported by NSF grant DMS 1016266. The satellite images were obtained from NASA’s website: www.nasa.gov.

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Correspondence to Julianne Chung.

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Chung, J., Chung, M. & O’Leary, D.P. Optimal Filters from Calibration Data for Image Deconvolution with Data Acquisition Error. J Math Imaging Vis 44, 366–374 (2012). https://doi.org/10.1007/s10851-012-0332-4

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Keywords

  • Inverse problems
  • Deconvolution
  • Dead pixels
  • Spectral filtering
  • Missing data artifacts
  • Optimal filter
  • Defective pixel
  • Hot pixel
  • Camera artifact
  • CCD defect
  • Data acquisition error
  • MRI artifact
  • CT scan artifact
  • Empirical risk
  • Bayes risk
  • Bayesian risk
  • Machine learning