TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Freq.: MONTHLY - Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

Solving linear inverse problem via mixture regularization

, , *

* Corresponding author · click or hover a name for details

pp. 2045–2057Vol. 28Issue 5August 2025DOI: 10.47974/JIM-2270XML
Received:
05 Nov 2024
Published Online:
27 Aug 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-2270
Pages:
2045–2057

Abstract

The iterative shrinkage/thresholding algorithm (ISTA) with the regularizer gives good efficiency to solve the linear inverse problem. In fact, the maximum a posteriori (MAP) estimator with the probability density function (PDF) and ISTA with the regularizer are similar, and we imply that PDF is similar to the regularizer. In general cases, the mixture PDF created from many ordinary PDFs gives better efficiency than the ordinary PDF. Consequently, we present the simple method based on the linear combination of various ordinary regularizers to create the mixture regularizer, and also present the simple-mixture regularizer for ISTA to solve the linear inverse problem. Experimental results of the generated and real signals show that our method outperforms the state-of-the-art methods.

Keywords

Subject Classifications

47A5260H5065F22

References

[1] A. Kanungo, M. Mittal, and L. Dewan, “Wavelet based PID controller using GA optimization and scheduling for feedback systems,”  J. Interdiscip. Math., vol. 23, no. 1, pp. 145-152 (2020).
[2] M. A. Ansari, R. Mehrotra, and R. Agrawal, “Detection and classification of brain tumor in MRI images using wavelet transform and support vector machine,”  J. Interdiscip. Math., vol. 23, no. 5, pp. 955-966 (2020).
[3] Y. Singh and C. S. Rai, “An independent component analysis technique for blind source separation,”  J. Interdiscip. Math., vol. 5, no. 3, pp. 231-241 (2002).
[4] P. Shah, P. Tandel, and J. Prajapati, “A study on unsteady flow through porous media: A cubic spline collocation approach,”  J. Interdiscip. Math., vol. 24, no. 5, pp. 1375-1386 (2021).
[5] P. Kittisuwan and P. Akkaraekthalin, “Improved low-rank matrix approximation in multivariate case,” Int. J. Comput. Methods, vol. 22, no. 1, p. 2450044 (2025).
[6] J. He, B. Peng, Z. Feng, S. Zhong, B. He, and G. Wang, “A Gaussian mixture unscented Rauch-Tung-Striebel smoothing framework for trajectory reconstruction,”  IEEE Trans. Ind. Inform., vol. 20, no. 5, pp. 7481-7491 (2024).
[7] H. Liu, L. Li, J. Lu, and S. Tan, “Group sparsity mixture model and its application on image denoising,”  IEEE Trans. Signal Process., vol. 31, pp. 5677-5690 (2022).
[8] F. Shi and I. W. Selesnick, “An elliptically contoured exponential mixture model for wavelet based image denoising,”  Appl. Comput. Harmon. Anal., vol. 23, no. 1, pp. 131-151 (2007).
[9] Y. Lai,  et. al., “Positive data modelling using mixture of mixtures of inverted beta distributions,”  IEEE Access, vol. 7, pp. 38146-38155 (2019).
[10] C. F. J. Wu, “On the convergence properties of the EM algorithm,”  Ann. Statist., vol. 11, no. 1, pp. 95-103 (1983).
[11] A. Blake and A. Zisserman,  Visual Reconstruction. Cambridge, MA, USA: MIT Press (1987).
[12] M. Nikolova, M. K. Ng, and C.-P. Tam, “Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction,”  IEEE Trans. Image Process., vol. 19, no. 12, pp. 3073-3088 (2010).
[13] A. Lanza, S. Morigi, and F. Sgallari, “Convex image denoising via nonconvex regularization with parameter selection,”  J. Math. Imag. Vis., vol. 56, no. 2, pp. 195-220 (2016).
[14] I. W. Selesnick and I. Bayram, “Sparse signal estimation by maximally sparse convex optimization,”  IEEE Trans. Signal Process., vol. 62, no. 5, pp. 1078-1092 (2014).
[15] M. M.-Mohammadi, C. R. Rojas, and B. Wahlberg, “A class of nonconvex penalties preserving overall convexity in optimization-based mean filtering,”  IEEE Trans. Signal Process., vol. 64, no. 24, pp. 6650-6664 (2016).
[16] I. W. Selesnick and I. Bayram, “Enhanced sparsity by non-separable regularization,” IEEE Trans. Signal Process., vol. 64, no. 9, pp. 2298-2313 (2016).
[17] I. W. Selesnick and M. Farshchian, “Sparse signal approximation via non-separable regularization,” IEEE Trans. Signal Process., vol. 65, no. 10, pp. 2561-2575 (2017).
[18] I. W. Selesnick and A. Parekh, “Convex denoising using non-convex tight frame regularization,”  IEEE Signal Process. Lett., vol. 22, no. 10, pp. 1786-1790 (2015).
[19] R. Gribonval, R. Jenatton and F. Bach, “Sparse and spurious: Dictionary learning with noise and outliers,”  IEEE Trans. Inf. Theory, vol. 61, no. 11, pp. 6298-6319 (2015).
[20] I. W. Selesnick, Sparse signal restoration, available online: https://eeweb.engineering. nyu.edu/iselesni/lecturenotes/index.html.
[21] L. Sendur and I. W. Selesnick, “Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency,” IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2744-2756 (2002).
[22] P. Y. Chen and I. W. Selesnick, “Group-sparse signal denoising: Non-convex regularization, convex optimization,”  IEEE Trans. Signal Process., vol. 62, no. 13, pp. 3464-3478 (2014).
[23] D. L. Donoho, A. Maleki and M. Shahram, Wavelab 850, available online: http://www-stat.stanford.edu/7Ewavelab/.
[24] X. H. Wang, R. S. H. Istepanian, and Y. H. Song, “Microarray image enhancement by denoising using stationary wavelet transform,”  IEEE Trans. Nanobioscience, vol. 2, no. 4, pp. 184-189 (2003).
[25] P. Kittisuwan and P. Akkaraekthalin, “Fused lasso algorithm based on novel non-convex regularization in sparse domain for audio signal enhancement,”  Fluct. Noise Lett., vol. 22, no. 2, p. 2350010 (2023).
[26] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,”  Biometrika, vol. 81, no. 3, pp. 425-455 (1994).
[27] H. Liu, W. Wang, C. Xiang, L. Han and H. Nie, “A de-noising method using the improved wavelet threshold function based on noise variance estimation,”  Mech. Syst. Signal Process., vol. 99, pp. 30-46 (2018).
[28] Z. Zhang, Y. Xu, J. Yang, X. Li and D. Zhang, “A survey of sparse representation: Algorithms and applications,” IEEE Access, vol. 3, pp. 490-530 (2015).
[29] P. Kittisuwan, “Relation between penalized least squares regression and Bayesian estimation in AWGN based on novel penalty function of Pareto density,” ICT Express, vol. 9, pp. 326-332 (2023).
[30] P. Kittisuwan and W. Thaiwirot, “Novel non-convex regularization for generating double threshold value in penalized least squares regression,” Fluct. Noise Lett., vol. 21, no. 6, p. 2250056

Views: 126Downloads: 5Citations: 0