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Multichannel reconstruction from nonuniform samples with application to image recovery

Dong Cheng, Kit Ian Kou

TL;DR

This work addresses reconstruction from nonuniform samples by extending multichannel interpolation to two nonuniform sampling models (recurrent and generic) and four closed-form formulas that incorporate both function and derivative samples. It provides FFT-based implementations, exact reconstruction under bandwidth constraints, and analytic error bounds for non-bandlimited signals, plus a practical image-recovery demonstration showing improvements over median filtering. The main contributions are the four nonuniform interpolation formulas, their closed-form interpolants, and the demonstrated benefits of including derivative data for improved accuracy in real-world signals and images. The results highlight the method's potential to robustly recover signals and degraded images from irregular sampling patterns in applications where sampling is nonuniform or interleaved, with practical FFT-based computation and error analyses supporting reliability.

Abstract

The multichannel trigonometric reconstruction from uniform samples was proposed recently. It not only makes use of multichannel information about the signal but is also capable to generate various kinds of interpolation formulas according to the types and amounts of the collected samples. The paper presents the theory of multichannel interpolation from nonuniform samples. Two distinct models of nonuniform sampling patterns are considered, namely recurrent and generic nonuniform sampling. Each model involves two types of samples: nonuniform samples of the observed signal and its derivatives. Numerical examples and quantitative error analysis are provided to demonstrate the effectiveness of the proposed algorithms. Additionally, the proposed algorithm for recovering highly corrupted images is also investigated. In comparison with the median filter and correction operation treatment, our approach produces superior results with lower errors.

Multichannel reconstruction from nonuniform samples with application to image recovery

TL;DR

This work addresses reconstruction from nonuniform samples by extending multichannel interpolation to two nonuniform sampling models (recurrent and generic) and four closed-form formulas that incorporate both function and derivative samples. It provides FFT-based implementations, exact reconstruction under bandwidth constraints, and analytic error bounds for non-bandlimited signals, plus a practical image-recovery demonstration showing improvements over median filtering. The main contributions are the four nonuniform interpolation formulas, their closed-form interpolants, and the demonstrated benefits of including derivative data for improved accuracy in real-world signals and images. The results highlight the method's potential to robustly recover signals and degraded images from irregular sampling patterns in applications where sampling is nonuniform or interleaved, with practical FFT-based computation and error analyses supporting reliability.

Abstract

The multichannel trigonometric reconstruction from uniform samples was proposed recently. It not only makes use of multichannel information about the signal but is also capable to generate various kinds of interpolation formulas according to the types and amounts of the collected samples. The paper presents the theory of multichannel interpolation from nonuniform samples. Two distinct models of nonuniform sampling patterns are considered, namely recurrent and generic nonuniform sampling. Each model involves two types of samples: nonuniform samples of the observed signal and its derivatives. Numerical examples and quantitative error analysis are provided to demonstrate the effectiveness of the proposed algorithms. Additionally, the proposed algorithm for recovering highly corrupted images is also investigated. In comparison with the median filter and correction operation treatment, our approach produces superior results with lower errors.

Paper Structure

This paper contains 15 sections, 2 theorems, 94 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier series
  4. Multichannel interpolation
  5. Multichannel interpolation of recurrent non-uniformly distributed data
  6. Recurrent nonuniform samples
  7. Recurrent nonuniform samples and derivatives
  8. Multichannel interpolation of generic non-uniformly distributed data
  9. Generic nonuniform samples
  10. Generic nonuniform samples and derivatives
  11. Numerical examples and error analysis
  12. Numerical examples
  13. Error analysis
  14. Application to image recovery
  15. Conclusion

Key Result

Lemma 4.1

Let $\widetilde{\mathbf{H}}$ be given above. Then its determinant satisfies the following recursive relation

Figures (9)

  • Figure 1: Illustration of interpolation for recurrent nonuniform samples and its consistency. The blue line is original function. The red dash-dot line is the interpolated result for the given data points.
  • Figure 2: Illustration of interpolation for recurrent nonuniform samples of a function and its derivative. The blue line is original function. The red dash-dot line is interpolated result for the given data points.
  • Figure 3: Illustration of interpolation for generic nonuniform samples. The blue line is original function. The red dash-dot line is interpolated result for the given data points.
  • Figure 4: Illustration of interpolation for generic nonuniform samples of a function and its derivative. The blue line is original function. The red circle is given data points. The red dash-dot line is interpolated result for the given data points.
  • Figure 5: Illustration of inclusion relations for the interpolation formulas.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma 4.1
  • Theorem 4.2