# American Institute of Mathematical Sciences

May  2013, 7(2): 499-521. doi: 10.3934/ipi.2013.7.499

## A geometry guided image denoising scheme

 1 10900 Euclid Avenue, Cleveland, OH 44106-7058, United States, United States

Received  August 2011 Revised  January 2013 Published  May 2013

During image denoising, it is often difficult to balance between the removal of noise and the preservation of contrast and fine features, especially when the noise is excessive. We propose to efficiently balance the two using segmentation and more general geometry extraction transforms. Explained in the nonlocal-means (NL-means) framework, we introduce a mutual position function to ensure the averaging is only taken over pixels in the same segmentation phase, and provide selection schemes for convolution kernel and weight function to further improve the performance. To address unreliable segmentation due to more excessive noise, we use a feature extraction transform that is more general than segmentation and less sensitive to noise. Unlike most denoising approaches that only work for one type of noise and/or involve heuristic parameter tuning, the proposed method comes with an automatic parameter selection scheme, and can be easily adapted for various types of noise, ranging from Gaussian, Poisson, Rician to ultrasound noise. Comparison with the original NL-means as well as ROF, BM3D, and K-SVD on various simulated data, MRI and SEM images, indicates potentials of the proposed method.
Citation: Weihong Guo, Jing Qin. A geometry guided image denoising scheme. Inverse Problems & Imaging, 2013, 7 (2) : 499-521. doi: 10.3934/ipi.2013.7.499
##### References:
 [1] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577. doi: 10.1002/cpa.3160420503. [2] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629. doi: 10.1109/34.56205. [3] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [4] M. Bertalmio, V. Caselles, B. Rougé and A. Solé, TV based image restoration with local constraints,, Journal of Scientific Computing, 19 (2003), 95. doi: 10.1023/A:1025391506181. [5] S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721. [6] P. Saint-Marc, J. S. Chen and G. Medioni, Adaptive smoothing: A general tool for early vision,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13 (1991), 514. [7] S. M. Smith and J. M. Brady, SUSAN - A new approach to low level image processing,, International Journal of Computer Vision, 23 (1995), 45. [8] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images,, in, (1998), 839. doi: 10.1109/ICCV.1998.710815. [9] M. Elad, On the origin of the bilateral filter and ways to improve it,, IEEE Transactions on Image Processing, 11 (2002), 1141. doi: 10.1109/TIP.2002.801126. [10] S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization,, SIAM J. Sci. Comput., 24 (2003), 1754. doi: 10.1137/S1064827501397792. [11] A. Buades, B. Coll and J. M. Morel, A non-local algorithm for image denoising,, IEEE Computer Society, 2 (2005), 60. doi: 10.1109/CVPR.2005.38. [12] S. Kindermann, S. Osher and P. W. Jones, Deblurring and Denoising of Images by Nonlocal Functionals,, Multiscale Modelling and Simulation, 4 (2005), 1091. doi: 10.1137/050622249. [13] T. Brox and D. Cremers, Iterated nonlocal means for texture restoration,, Scale Space and Variational Methods in Computer Vision, (2008), 13. doi: 10.1007/978-3-540-72823-8_2. [14] C. Kervrann, J. Boulanger and P. Coupé, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,, Scale Space and Variational Methods in Computer Vision, (2007), 520. doi: 10.1007/978-3-540-72823-8_45. [15] B. Goossens, Q. Luong, A. Pizurica and W. Philips, An improved non-local denoising algorithm,, in, (2008). [16] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Modeling and Simulation, 7 (2008), 1005. doi: 10.1137/070698592. [17] C-A. Deledalle, L. Denis and F. Tupin, Iterative wieghted maximum likelihood denoising with probabilistic patch-based weights,, Transactions on Image Processing, 18 (2009), 2661. doi: 10.1109/TIP.2009.2029593. [18] D. Peter, V. Govindan and A. Mathew, Nonlocal-means image denoising techonology using robust M-estimator,, Journal of Computer Science and Technology, 25 (2010), 623. [19] N. Wiest-Daesslé, S. Prima, P. Coupé, S. P. Morrissey and C. Barillot, Rician noise removal by non-local means filtering for low signal-to-noise ratio MRI: Applications to DT-MRI,, in, (2008), 171. doi: 10.1007/978-3-540-85990-1_21. [20] P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images,, Medical Imaging, 27 (2008), 425. doi: 10.1109/TMI.2007.906087. [21] C-A. Deledalle, F. Tupin and L. Denis, Poisson NL means: Unsupervised non local means for Poisson noise,, in, (2010). doi: 10.1109/ICIP.2010.5653394. [22] S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,, Journal of Computational Physics, 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. [23] T. F. Chan and L. A. Vese, Active contours without edges,, Image Processing, 10 (2001), 266. doi: 10.1109/83.902291. [24] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the mumford and shah model,, International Journal of Computer Vision, 50 (2002), 271. [25] K. Krishnamoorthy, "Handbook of Statistical Distributions with Applications,", Chapman and Hall/CRC Press, (2006). doi: 10.1201/9781420011371. [26] T. Gasser, L. Sroka and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression,, Biometrika, 73 (1986), 625. doi: 10.1093/biomet/73.3.625. [27] M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation,, IEEE Transactions on Image Processing, 14 (2005), 2091. doi: 10.1109/TIP.2005.859376. [28] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity,, Image Processing, 13 (2004), 600. doi: 10.1109/TIP.2003.819861. [29] E. Candés and D. Donoho, "Curvelets: A Surprisingly Effective Nonadaptive Representation of Objects with Edges,", in, (). [30] K. Guo, G. Kutyniok and D. Labate, "Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators,", Wavelets and Splines (Athens, (2005). [31] E. A. Nadaraya, On estimating regression,, Theory Probab. Appl., 9 (1964), 141. doi: 10.1137/1109020. [32] F. J. Anscombe, The transformation of Poisson, binomial and negative-binomial data,, Biometrika, 35 (1948), 246. [33] M. D. DeVore, A. D. Lanterman and J. A. O'Sullivan, ATR performance of a rician model for SAR images,, in, 4050 (2000), 34. doi: 10.1117/12.395589. [34] J. Sijbers, A. J. Den Dekker, P. Scheunders and D. Van Dyck, Maximum-likelihood estimation of Rician distribution parameters,, Medical Imaging, 17 (1998), 357. doi: 10.1109/42.712125. [35] T. Loupas, W. N. McDicken and P. L. Allan, An adaptive weighted median filter for speckle suppression in medical ultrasonic images,, Circuits and Systems, 36 (1989), 129. doi: 10.1109/31.16577. [36] K. Krissian, Speckle-constrained anisotropic diffusion for ultrasound images,, in, (2005), 547. [37] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising with block-matching and 3D filtering,, in, 6064 (2006). doi: 10.1117/12.643267. [38] M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries,, Image Processing, 15 (2006), 3736. doi: 10.1109/TIP.2006.881969. [39] P. Coupé, P. Hellier, C. Kervrann and C. Barillot, Bayesian non local means-based speckle filtering,, in, (2008), 1291. doi: 10.1109/ISBI.2008.4541240.

show all references

##### References:
 [1] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math., 42 (1989), 577. doi: 10.1002/cpa.3160420503. [2] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629. doi: 10.1109/34.56205. [3] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. [4] M. Bertalmio, V. Caselles, B. Rougé and A. Solé, TV based image restoration with local constraints,, Journal of Scientific Computing, 19 (2003), 95. doi: 10.1023/A:1025391506181. [5] S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721. [6] P. Saint-Marc, J. S. Chen and G. Medioni, Adaptive smoothing: A general tool for early vision,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13 (1991), 514. [7] S. M. Smith and J. M. Brady, SUSAN - A new approach to low level image processing,, International Journal of Computer Vision, 23 (1995), 45. [8] C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images,, in, (1998), 839. doi: 10.1109/ICCV.1998.710815. [9] M. Elad, On the origin of the bilateral filter and ways to improve it,, IEEE Transactions on Image Processing, 11 (2002), 1141. doi: 10.1109/TIP.2002.801126. [10] S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization,, SIAM J. Sci. Comput., 24 (2003), 1754. doi: 10.1137/S1064827501397792. [11] A. Buades, B. Coll and J. M. Morel, A non-local algorithm for image denoising,, IEEE Computer Society, 2 (2005), 60. doi: 10.1109/CVPR.2005.38. [12] S. Kindermann, S. Osher and P. W. Jones, Deblurring and Denoising of Images by Nonlocal Functionals,, Multiscale Modelling and Simulation, 4 (2005), 1091. doi: 10.1137/050622249. [13] T. Brox and D. Cremers, Iterated nonlocal means for texture restoration,, Scale Space and Variational Methods in Computer Vision, (2008), 13. doi: 10.1007/978-3-540-72823-8_2. [14] C. Kervrann, J. Boulanger and P. Coupé, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal,, Scale Space and Variational Methods in Computer Vision, (2007), 520. doi: 10.1007/978-3-540-72823-8_45. [15] B. Goossens, Q. Luong, A. Pizurica and W. Philips, An improved non-local denoising algorithm,, in, (2008). [16] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Modeling and Simulation, 7 (2008), 1005. doi: 10.1137/070698592. [17] C-A. Deledalle, L. Denis and F. Tupin, Iterative wieghted maximum likelihood denoising with probabilistic patch-based weights,, Transactions on Image Processing, 18 (2009), 2661. doi: 10.1109/TIP.2009.2029593. [18] D. Peter, V. Govindan and A. Mathew, Nonlocal-means image denoising techonology using robust M-estimator,, Journal of Computer Science and Technology, 25 (2010), 623. [19] N. Wiest-Daesslé, S. Prima, P. Coupé, S. P. Morrissey and C. Barillot, Rician noise removal by non-local means filtering for low signal-to-noise ratio MRI: Applications to DT-MRI,, in, (2008), 171. doi: 10.1007/978-3-540-85990-1_21. [20] P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images,, Medical Imaging, 27 (2008), 425. doi: 10.1109/TMI.2007.906087. [21] C-A. Deledalle, F. Tupin and L. Denis, Poisson NL means: Unsupervised non local means for Poisson noise,, in, (2010). doi: 10.1109/ICIP.2010.5653394. [22] S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,, Journal of Computational Physics, 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. [23] T. F. Chan and L. A. Vese, Active contours without edges,, Image Processing, 10 (2001), 266. doi: 10.1109/83.902291. [24] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the mumford and shah model,, International Journal of Computer Vision, 50 (2002), 271. [25] K. Krishnamoorthy, "Handbook of Statistical Distributions with Applications,", Chapman and Hall/CRC Press, (2006). doi: 10.1201/9781420011371. [26] T. Gasser, L. Sroka and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression,, Biometrika, 73 (1986), 625. doi: 10.1093/biomet/73.3.625. [27] M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation,, IEEE Transactions on Image Processing, 14 (2005), 2091. doi: 10.1109/TIP.2005.859376. [28] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity,, Image Processing, 13 (2004), 600. doi: 10.1109/TIP.2003.819861. [29] E. Candés and D. Donoho, "Curvelets: A Surprisingly Effective Nonadaptive Representation of Objects with Edges,", in, (). [30] K. Guo, G. Kutyniok and D. Labate, "Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators,", Wavelets and Splines (Athens, (2005). [31] E. A. Nadaraya, On estimating regression,, Theory Probab. Appl., 9 (1964), 141. doi: 10.1137/1109020. [32] F. J. Anscombe, The transformation of Poisson, binomial and negative-binomial data,, Biometrika, 35 (1948), 246. [33] M. D. DeVore, A. D. Lanterman and J. A. O'Sullivan, ATR performance of a rician model for SAR images,, in, 4050 (2000), 34. doi: 10.1117/12.395589. [34] J. Sijbers, A. J. Den Dekker, P. Scheunders and D. Van Dyck, Maximum-likelihood estimation of Rician distribution parameters,, Medical Imaging, 17 (1998), 357. doi: 10.1109/42.712125. [35] T. Loupas, W. N. McDicken and P. L. Allan, An adaptive weighted median filter for speckle suppression in medical ultrasonic images,, Circuits and Systems, 36 (1989), 129. doi: 10.1109/31.16577. [36] K. Krissian, Speckle-constrained anisotropic diffusion for ultrasound images,, in, (2005), 547. [37] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising with block-matching and 3D filtering,, in, 6064 (2006). doi: 10.1117/12.643267. [38] M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries,, Image Processing, 15 (2006), 3736. doi: 10.1109/TIP.2006.881969. [39] P. Coupé, P. Hellier, C. Kervrann and C. Barillot, Bayesian non local means-based speckle filtering,, in, (2008), 1291. doi: 10.1109/ISBI.2008.4541240.
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