Abstract
This paper addresses the problem of simultaneous fusion and denoising of an ensemble of multifocused noisy source images using statistical approach. The central theme of the paper is to develop a novel generalized Bayesian framework based on maximum a posteriori (MAP) estimation technique to obtain the fused image from the noisy observations using a multiscale wavelet transform. A mathematically tractable multivariate a priori function is used in the MAP estimator to derive the closed-form expression of the fusion rule for the wavelet coefficients of noisy images. Experiments are carried out on a number of test-sets having an ensemble of multifocused source images with varying noise strengths to evaluate the performance of the proposed MAP-based fusion method as compared to the existing methods. Results show that the performance of the proposed method is better than that of the other wavelet or principal component analysis-based methods in terms of various metrics such as the structural similarity, peak signal-to-noise ratio and cross-entropy, uses of which are common both in the areas of fusion and denoising. In addition, the proposed method yields excellent results in terms of visual quality even in the case of non-Gaussian noise as well as computational load.
Similar content being viewed by others
References
S. Arivazhagan, L. Ganesan, T.G.S. Kumar, A modified statistical approach for image fusion using wavelet transform. Signal Image Video Process. 3, 137–144 (2009)
L. Cao, L. Jin, H. Tao, G. Li, Z. Zhuang, Y. Zhang, Multi-focus image fusion based on spatial frequency in discrete cosine transform domain. IEEE Signal Process. Lett. 22(2), 220–224 (2015)
Z. Chen, Y. Zheng, B.R. Abidi, D.L. Page, M.A. Abidi, A combinational approach to the fusion, de-noising and enhancement of dual-energy X-ray luggage images, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition: Workshops, vol. 3., San Diego, 2005, p. 2
H. Choi, J.K. Romberg, R.G. Baraniuk, N.G. Kingsbury, Hidden Markov tree modeling of complex wavelet transforms, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 1, Istambul, 2000, pp. 133–136
I. De, B. Chanda, A simple and efficient algorithm for multifocus image fusion using morphological wavelets. Signal Process. 86, 924–936 (2006)
I. De, B. Chanda, Multi-focus image fusion using a morphology-based focus measure in a quad-tree structure. Inf. Fusion 14, 136–146 (2013)
D.L. Donoho, De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)
D.L. Donoho, I.M. Johnstone, Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432), 1200–1224 (1995)
B. Escalante-Ramírez, The Hermite transform as an efficient model for local image analysis: an application to medical image fusion. Comput. Electr. Eng. 34, 99–110 (2008)
D. Fasbender, J. Radoux, P. Bogaert, Bayesian data fusion for adaptable image pansharpening. IEEE Trans. Geosci Remote Sens. 46(6), 1847–1857 (2008)
N.C. Giri, Introduction to Probability and Statistics, 2nd edn. (CRC Press, Boca Raton, 1993)
A.B. Hamza, Y. He, H. Krim, A. Willsky, A multiscale approach to pixel-level image fusion. Integr. Comput. Aided Eng. 12, 135–146 (2005)
K. Kotwal, S. Chaudhuri, A Bayesian approach to visualization-oriented hyperspectral image fusion. Inf. Fusion 14, 349–360 (2013)
H. Li, Y. Chai, H. Yin, G. Liu, Multifocus image fusion and denoising scheme based on homogeneity similarity. Opt. Commun. 285, 91–100 (2012)
H. Li, B.S. Manjunath, S.K. Mitra, Multisensor image fusion using the wavelet transform. Gr. Models Image Process. 57(3), 235–245 (1995)
S. Li, J.S. Taylor, Comparison and fusion of multiresolution features for texture classification. Pattern Recognit. Lett. 26, 633–638 (2005)
S. Li, B. Yang, Multifocus image fusion by combining curvelet and wavelet transform. Pattern Recognit. Lett. 29, 1295–1301 (2008)
J. Liang, Y. He, D. Liu, X. Zeng, Image fusion using higher order singular value decomposition. IEEE Trans. Image Process. 21(5), 2898–2909 (2012)
J. Liu, P. Moulin, Information-theoretic analysis of interscale and intrascale dependencies between image wavelet coefficients. IEEE Trans. Image Process. 10(11), 1647–1658 (2001)
Z. Liu, P. Song, J. Zhang, J. Wang, Bidimensional empirical mode decomposition for the fusion of multispectral and panchromatic images. Int. J. Remote Sens. 28(18), 4081–4093 (2007)
A. Loza, D. Bull, N. Canagarajah, A. Achim, Non-Gaussian model-based fusion of noisy images in the wavelet domain. Comput. Vis. Image Underst. 114, 54–65 (2010)
C. Ludusan, O. Lavialle, Multifocus image fusion and denoising: a variational approach. Pattern Recognit. Lett. 33, 1388–1396 (2012)
K. Mardia, Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519–530 (1970)
M.K. Mihçak, I. Kozintsev, K. Ramchandran, P. Moulin, Low-complexity image denoising based on statistical modeling of wavelet coefficients. IEEE Signal Process. Lett. 6(12), 300–303 (1999)
G. Pajares, J.M. Cruz, A wavelet-based image fusion tutorial. Pattern Recognit. 37, 1855–1872 (2004)
S. Pertuz, D. Puig, M.A. Garcia, A. Fusiello, Generation of all-in-focus images by noise-robust selective fusion of limited depth-of-field images. IEEE Trans. Image Process. 22(3), 1242–1251 (2013)
V.S. Petrović, C.S. Xydeas, Gradient-based multiresolution image fusion. IEEE Trans. Image Process. 13(2), 228–237 (2004)
G. Piella, A general framework for multiresolution image fusion: from pixels to regions. Inf. Fusion 4, 259–280 (2003)
S. Prasad, W. Li, J.E. Fowler, L.M. Bruce, Information fusion in the redundant-wavelet-transform domain for noise-robust hyperspectral classification. IEEE Trans. Geosci. Remote Sens. 50(9), 3474–3486 (2012)
T. Pu, G.Q. Ni, Contrast-based image fusion using the discrete wavelet transform. Opt. Eng. 39(8), 2075–2082 (2000)
S.M.M. Rahman, M.O. Ahmad, M.N.S. Swamy, Contrast-based fusion of noisy images using discrete wavelet transform. IET Image Process. 4(5), 374–384 (2010)
R. Redondo, F. Šroubek, S. Fischer, G. Cristóbal, Multifocus image fusion using the log-Gabor transform and a multisize windows technique. Inf. Fusion 10, 163–171 (2009)
S. Roy, T. Howlader, S.M.M. Rahman, Image fusion technique using multivariate statistical model for wavelet coefficients. Signal Image Video Process. 7(2), 355–365 (2013)
M. Sasikala, M. Kumaravel, A comparative analysis of feature based image fusion method. Inf. Technol. J. 6(8), 1224–1230 (2007)
A. Stein, Use of single- and multi-source image fusion for statistical decision-making. Int. J. Appl. Earth Obs. Geoinf. 6, 229–239 (2005)
A. Toet, Hierarchical image fusion. Mach. Vis. Appl. 3, 1–11 (1990)
T. Wan, C. Zhu, Z. Qin, Multifocus image fusion based on robust principal component analysis. Pattern Recognit. Lett. 34(9), 1001–1008 (2013)
Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 3(4), 600–612 (2004)
J. Wu, J. Liu, J. Tian, H. Huang, Multi-scale image data fusion based on local deviation of wavelet transform, in Proceedings of the IEEE International Conference on Intelligent Mechatronics and Automation. Chengdu, 2004, pp. 677–680
W. Xueyun, X. Huaping, L. Jingwen, W. Pengbo, Comparison of diverse approaches for synthetic aperture radar images pixel fusion under different precision registration. IET Image Process. 5(8), 661–670 (2011)
L. Yang, B.L. Guo, W. Ni, Multimodality medical image fusion based on multiscale geometric analysis of contourlet transform. Neurocomputing 72, 203–211 (2008)
H. Yesou, Y. Besnus, Y. Rolet, Extraction of spectral information from Landsat TM data and merger with SPOT panchromatic imagery—a contribution to the study of geological structures. ISPRS J. Photogramm. Remote Sens. 48(5), 23–36 (1993)
S. Yin, L. Cao, Y. Ling, G. Jin, Fusion of noisy infrared and visible images based on anisotropic bivariate shrinkage. Infrared Phys. Technol. 54, 13–20 (2011)
C. Yunhao, D. Lei, L. Jing, L. Xiaobing, S. Peijun, A new wavelet-based image fusion method for remotely sensed data. Int. J. Remote Sens. 27(7), 1465–1476 (2006)
Q. Zhang, B. Guo, Multifocus image fusion using the nonsubsampled contourlet transform. Signal Process. 89, 1334–1346 (2009)
Y. Zhang, Methods for image fusion quality assesment: a review, comparison and analysis. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. XXXVII(B7), 1101–1109 (2008)
Z. Zhang, R.S. Blum, A categorization of multiscale-decomposition-based image fusion schemes with a performance study for a digital camera application. Proc. IEEE 87, 1315–1326 (1999)
Y. Zheng, E.A. Essock, B.C. Hansen, Advanced discrete wavelet transform fusion algorithm and its optimization by using the metric of image quality index. Opt. Eng. 44(3), 037003-1–037003-12 (2005)
Y. Zheng, Z. Qin, L. Shao, X. Hou, A novel objective image quality metric for image fusion based on Renyi entropy. Inf. Technol. J. 7(6), 930–935 (2008)
X. Zhou, W. Wang, R. Liu, Compressive sensing image fusion algorithm based on directionlets. EURASIP J. Wirel. Commun. Netw. 2014, 1–6 (2014)
Acknowledgments
The authors would like to give thanks to the anonymous reviewers for their valuable comments that were useful to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
The joint PDF of \(X_1\), \(X_2\), and \(X_{\text {f}}\) is given by [33]
Since the noise is uncorrelated with the reference images as well as the fused image
Consider the transformation of variables \(Y_1=X_1+N\), \(Y_2=X_2+N\) and \(Y_3=N\). The transformation is one-to-one, so there exists a unique inverse \(X_1=Y_1-Y_3\), \(X_2=Y_2-Y_3\) and \(N=Y_3\). The Jacobian of the transformation is
Then the quadvariate joint PDF can be obtained as
Expanding the squared terms in the first exponent and collecting the terms that contain \(y_3^2\) and \(y_3\) gives
The required PDF \(p_{Y_1,Y_2,X_{\text {f}}}\big (y_1,y_2,x_{\text {f}}\big )\) is obtained after integrating out \(y_3\) and noting that \(\lambda _1=\sqrt{\frac{1}{\lambda _3\sigma _{nn}}}\)
Rights and permissions
About this article
Cite this article
Jhohura, F.T., Howlader, T. & Rahman, S.M.M. Bayesian Fusion of Ensemble of Multifocused Noisy Images. Circuits Syst Signal Process 34, 2287–2308 (2015). https://doi.org/10.1007/s00034-014-9956-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-014-9956-5