Robust regression analysis using the weighted median model for improved denoising of MR data in image processing
Öz
This paper presents a comprehensive analysis of a multidimensional regression model using the weighted median as the regression function. The model is formulated as an optimization problem within the framework of the $L_1$-norm error fitting approach, exhibiting robustness to outliers, a critical advantage in various applications where data might be contaminated by extreme values. The core of the investigation focuses on the regression and objective functions of the proposed model. A detailed mathematical study reveals that the optimization problem inherent in the model can be effectively discretized, leading to computationally tractable solutions. The study's findings are further validated through a rigorous exploration of the model's application in the context of image denoising, a significant problem in image processing. Specifically, the model addresses the challenging task of impulse noise removal in Magnetic Resonance images. By integrating the proposed model into well-established adaptive denoising methods, this work demonstrates that significant improvements in image quality reconstruction and noise suppression are easily achievable. The results highlight the model's efficacy in balancing the competing demands of preserving essential image features while effectively reducing noise artifacts. This research offers a novel approach for robust regression analysis and provides a robust tool for image denoising, particularly in scenarios involving impulse noise. The mathematical underpinnings, along with the demonstrated practical application, contribute significantly to the field of robust statistical modeling and image processing.
Anahtar Kelimeler
functional analysis image processing optimization regression analysis robust regression
Kaynakça
- [1] P. Bloomfield and W. L. Steiger, Least Absolute Deviations: Theory, Applications, and Algorithms, Birkhäuser, Boston, 1983.
- [2] Y. Dodge, An introduction to $L_1$-norm based statistical data analysis, Comput. Stat. Data Anal.5(4), 239253, 1987.
- [3] R. C. Gonzalez and R. E. Woods, Digital Image Processing, Third Edition, Pearson Prentice Hall, New Jersey, 2008.
- [4] C. Gurwitz, Weighted median algorithms for $L_1$ approximation, BIT 30(2), 301-310, 1990.
- [5] H. Ibrahim, N. S. Pik Kong and T. F. Ng, Simple adaptive median filter for the removal of impulse noise from highly corrupted images, IEEE Trans. Consum. Electron. 54(4), 1920-1927, 2008.
- [6] C. J. J. Sheela and G. Suganthi, An efficient denoising of impulse noise from MRI using adaptive switching modified decision based unsymmetric trimmed median filter, Biomed. Signal Process. Control 55, 101657, 2020.
- [7] S. J. Ko and Y. H. Lee, Center Weighted Median filters and their applications and image enhancements, IEEE Trans. Circuits syst. 38(9), 984-993, 1991.
- [8] G. Y. Lee, G. L. Ra, G. L. Ra, G. S. Kim, H. H. Moon and J. S. Jeong, Probability Mass Function-Based Adaptive Median Filtering for Acoustic Radiation Force Impulse Imaging: A Feasibility Study, IEEE Access 11(11), 2169-3536, 2023.
- [9] M. M. Ali, R. Imon, I. Ali and H. M. Yousof, Statistical Outliers and Related Topics, CRR Press, Taylor & Francis Group, 2025.
- [10] V. Novoselac, The Component Weighted Median Absolute Deviations Problem, Eur. J. Pure Appl. Math. 13(4), 964-976, 2020.
- [11] V. Novoselac and Z. Pavic, Cluster detection in noisy environment by using the modified EM algorithm, Croat. Oper. Res. Rev. 9(2), 223-234, 2018.
- [12] V. Novoselac and Z. Pavic, Optimal Solution Properties of an Overdetermined System of Linear Equations, Eur. J. Pure Appl. Math. 12(4), 1360-1370, 2019.
- [13] M. R. Osborne, Finite Algorithms in Optimization and Data Analysis, Department of Statistics, Australian National University, Camberra, John Wiley, 1985.
- [14] Y. W. Park, Optimization for $L_1$-Norm Error Fitting via Data Aggregation, INFORMS J. Comput. 33(1), 120-140, 2021.
- [15] Z. Pavic and V. Novoselac, Investigating an overdetermined system of linear equations by using convex functions, Hacet. J. Math. Stat. 46(5), 865-874, 2017.
- [16] P. Qiu, Image Processing and Jump Regression Analysis, John Wiley & Sons, 2005.
- [17] P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection, Wiley, New York, 2003.
- [18] K. Sabo and R. Scitovski, The best least absolute deviations line-properties and two efficient methods for its derivation, Anziam J. 50(2), 185-198, 2008.
- [19] T. Sun and Y. Neuvo, Detail-preserving median based filters in image processing, Pattern Recognit. Lett. 15(4), 341-347, 1994.
- [20] K. K. V. Toh and N. A. Mat Isa, Noise Adaptive Fuzzy Switching Median Filter for Salt-and-Pepper Noise Reduction, IEEE Signal Process. Lett. 17(3), 281-284, 2010.
- [21] A. Toprak and . Güler, Suppression of Impulse Noise in Medical Images with the Use of Fuzzy Adaptive Median Filter. J. Med. Syst., 30(6), 465471, 2006.
- [22] I. Vazler, K. Sabo and R. Scitovski, Weighted median of the data in solving least absolute deviations problems, Comm. Statist. Theory Methods 41(8), 1455-1465, 2012.
- [23] S. Zhang and M. A. Karim, A new impulse detector for switching median filters, IEEE Signal Process. Lett. 9(11), 360-363, 2002.
Öz
Kaynakça
- [1] P. Bloomfield and W. L. Steiger, Least Absolute Deviations: Theory, Applications, and Algorithms, Birkhäuser, Boston, 1983.
- [2] Y. Dodge, An introduction to $L_1$-norm based statistical data analysis, Comput. Stat. Data Anal.5(4), 239253, 1987.
- [3] R. C. Gonzalez and R. E. Woods, Digital Image Processing, Third Edition, Pearson Prentice Hall, New Jersey, 2008.
- [4] C. Gurwitz, Weighted median algorithms for $L_1$ approximation, BIT 30(2), 301-310, 1990.
- [5] H. Ibrahim, N. S. Pik Kong and T. F. Ng, Simple adaptive median filter for the removal of impulse noise from highly corrupted images, IEEE Trans. Consum. Electron. 54(4), 1920-1927, 2008.
- [6] C. J. J. Sheela and G. Suganthi, An efficient denoising of impulse noise from MRI using adaptive switching modified decision based unsymmetric trimmed median filter, Biomed. Signal Process. Control 55, 101657, 2020.
- [7] S. J. Ko and Y. H. Lee, Center Weighted Median filters and their applications and image enhancements, IEEE Trans. Circuits syst. 38(9), 984-993, 1991.
- [8] G. Y. Lee, G. L. Ra, G. L. Ra, G. S. Kim, H. H. Moon and J. S. Jeong, Probability Mass Function-Based Adaptive Median Filtering for Acoustic Radiation Force Impulse Imaging: A Feasibility Study, IEEE Access 11(11), 2169-3536, 2023.
- [9] M. M. Ali, R. Imon, I. Ali and H. M. Yousof, Statistical Outliers and Related Topics, CRR Press, Taylor & Francis Group, 2025.
- [10] V. Novoselac, The Component Weighted Median Absolute Deviations Problem, Eur. J. Pure Appl. Math. 13(4), 964-976, 2020.
- [11] V. Novoselac and Z. Pavic, Cluster detection in noisy environment by using the modified EM algorithm, Croat. Oper. Res. Rev. 9(2), 223-234, 2018.
- [12] V. Novoselac and Z. Pavic, Optimal Solution Properties of an Overdetermined System of Linear Equations, Eur. J. Pure Appl. Math. 12(4), 1360-1370, 2019.
- [13] M. R. Osborne, Finite Algorithms in Optimization and Data Analysis, Department of Statistics, Australian National University, Camberra, John Wiley, 1985.
- [14] Y. W. Park, Optimization for $L_1$-Norm Error Fitting via Data Aggregation, INFORMS J. Comput. 33(1), 120-140, 2021.
- [15] Z. Pavic and V. Novoselac, Investigating an overdetermined system of linear equations by using convex functions, Hacet. J. Math. Stat. 46(5), 865-874, 2017.
- [16] P. Qiu, Image Processing and Jump Regression Analysis, John Wiley & Sons, 2005.
- [17] P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection, Wiley, New York, 2003.
- [18] K. Sabo and R. Scitovski, The best least absolute deviations line-properties and two efficient methods for its derivation, Anziam J. 50(2), 185-198, 2008.
- [19] T. Sun and Y. Neuvo, Detail-preserving median based filters in image processing, Pattern Recognit. Lett. 15(4), 341-347, 1994.
- [20] K. K. V. Toh and N. A. Mat Isa, Noise Adaptive Fuzzy Switching Median Filter for Salt-and-Pepper Noise Reduction, IEEE Signal Process. Lett. 17(3), 281-284, 2010.
- [21] A. Toprak and . Güler, Suppression of Impulse Noise in Medical Images with the Use of Fuzzy Adaptive Median Filter. J. Med. Syst., 30(6), 465471, 2006.
- [22] I. Vazler, K. Sabo and R. Scitovski, Weighted median of the data in solving least absolute deviations problems, Comm. Statist. Theory Methods 41(8), 1455-1465, 2012.
- [23] S. Zhang and M. A. Karim, A new impulse detector for switching median filters, IEEE Signal Process. Lett. 9(11), 360-363, 2002.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | İstatistiksel Veri Bilimi, Matematikte Optimizasyon, Kombinatorik ve Ayrık Matematik (Fiziksel Kombinatorik Hariç), Operatör Cebirleri ve Fonksiyonel Analiz |
| Bölüm | İstatistik |
| Yazarlar | |
| Erken Görünüm Tarihi | 17 Mart 2025 |
| Yayımlanma Tarihi | 28 Nisan 2025 |
| Gönderilme Tarihi | 29 Mart 2024 |
| Kabul Tarihi | 12 Mart 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 54 Sayı: 2 |