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Modelling and Data Analysis
2023. Vol. 13, no. 4, 7–22
https://doi.org/10.17759/mda.2023130401
ISSN: 2219-3758 / 2311-9454 (online)

Construction of a Parametric Family of Wavelets and Its Use in Image Processing

 
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Abstract

This article is devoted to the construction of a parametric family of biorthogonal wavelets according to the lifting scheme and subdivision schemes, and the use of such a family in the problem of image rendering when part of the pixel data in the image is missing or overwritten in some way. The parametric family of wavelets provides a parametric family of filters for restoring damaged images. With such a recovery, the desired wavelet is selected not from any general considerations, but from a parametric family in the process of solving an optimization problem.

General Information

Keywords: wavelet, lifting scheme, subdivision scheme, image processing

Journal rubric: Data Analysis

OpenAlex citations: 1

OpenAlex trends: Image and Signal Denoising Methods, Advanced Image Fusion Techniques, Advanced Computational Techniques in Science and Engineering

Information about the work in OpenAlex

Number of citations: 1

Topics

Image and Signal Denoising Methods

This cluster of papers encompasses a wide range of techniques and algorithms for image denoising, including sparse representations, wavelet transform, deep learning with convolutional neural networks, non-local means, and methods specific to handling different types of noise such as Gaussian, Poisson, and salt-and-pepper noise. The applications also extend to hyperspectral imaging and the use of anisotropic diffusion for speckle reduction.

Number of works: 91964  |  Total number of citations: 1528626

Topic detailsв OpenAlex

Advanced Image Fusion Techniques

This cluster of papers focuses on the fusion of multispectral and hyperspectral images using techniques such as wavelet transform, sparse representation, convolutional neural networks, and pansharpening. The research covers methods for remote sensing, image quality assessment, and applications in fields such as medical imaging.

Number of works: 33005  |  Total number of citations: 456746

Topic detailsв OpenAlex

Advanced Computational Techniques in Science and Engineering

This cluster of papers focuses on the intersection of Internet of Things (IoT) and healthcare systems, including topics such as remote monitoring, algorithmic modeling, wireless sensor networks, telemedicine, biomedical signal processing, and digital information management. The papers also touch upon the application of IoT in smart city healthcare systems and the use of corpus analysis in technical writing classrooms.

Number of works: 19217  |  Total number of citations: 82460

Topic detailsв OpenAlex

Work details in OpenAlex

Article type: scientific article

DOI: https://doi.org/10.17759/mda.2023130401

Received 28.09.2023

Accepted

Published

For citation: Bityukov, Y.I., Bityukov, P.Y. (2023). Construction of a Parametric Family of Wavelets and Its Use in Image Processing. Modelling and Data Analysis, 13(4), 7–22. (In Russ.). https://doi.org/10.17759/mda.2023130401

© Bityukov Y.I., Bityukov P.Y., 2023

License: CC BY-NC 4.0

References

  1. Frazier Michael W. An introduction to wavelets through linear algebra. 1999. Springer. 503 p.
  2. Bertalmio M., Bertozzi A., and Sapiro G., Navier Stokes, fluid-dynamics and image and video inpainting, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1 (2001), pp. 355-362.
  3. Chan T.F., Shen J., and Zhou H.M., Total variation wavelet inpainting, J. Math. Imaging Vision, 25 (2006), pp. 107-125.
  4. Cai J.F., Chan R.H., and Shen Z., A framelet-based image inpainting algorithm, Applied and Computational Harmonic Analysis 24 (2008), no. 2, 131–149.
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  6. Nira Dyn, Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials. Tutorials on Multiresolution in Geometric Modelling, 2002, 51–68
  7. Blatter K. Vejvlet-analiz. Osnovy teorii. Moskva, 2004. – 280 p. (In Russ)
  8. Novikov I.YA., Protasov V.YU., Skopina M.A.. Teoriya vspleskov. M.: FIZMATLIT. 2005. 612 p. (In Russ)
  9. Sweldens Wim, The lifting scheme: A custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis, volume3, issue 2, 1996, pp. 186-200.
  10. Sweldens Wim, The lifting scheme: A new philosophy in biorthogonal wavelets construction. In Wavelets Application in Signal and Image Processing III, volume 2569 of Processing of the SPIE, pp. 68-79. SPIE, Bellingham, WA, 1995.

Information About the Authors

Yuri I. Bityukov, Doctor of Engineering, associate professor, Professor of the Department of Probability Theory and Computer Modeling, Moscow Aviation Institute (National Research University), Moscow, Russian Federation, ORCID: https://orcid.org/0009-0008-6384-0564, e-mail: yib72@mail.ru

Pavel Y. Bityukov, Master's Student, Moscow Power Engineering Institute (National Research University), Moscow, Russian Federation, ORCID: https://orcid.org/0009-0000-8697-7047, e-mail: p.bityukoff@yandex.ru

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