Goossens, Bart

Luong, Hiep

Pizurica, Aleksandra

Ruzic, Tijana

English

Ghent University Academic Bibliography

Adaptive Non-Local Means Filtering of Imagescorrupted by Colored NoiseBart Goossens, Tijana Ruzˇic´, Hieˆp Luong, Aleksandra PizˇuricaSupervisor(s): Wilfried PhilipsAbstract— Recently, the NLMeans Filter hasbeen proposed by Buades et al. for the suppres-sion of white Gaussian noise. This filter exploitsthe repetitive character of structures in an image,unlike conventional denoising algorithms, whichtypically operate in a local neighborhood. In thispaper, we present an extension of this technique tothe noise reduction of colored (correlated) noise,which is applicable in more realistic scenarios. Vi-sual and PSNR results show a significant improve-ment in denoising performance compared to re-cent wavelet-based techniques.I. INTRODUCTIONDigital imaging devices inevitably producenoise, originating from the analog circuitry inthese devices. Noise reduction by means ofdigital post-processing is often desirable, butalso very challenging. In this paper, we focuson the design of a noise reduction method forstationary Gaussian noise, that preserves imagedetails and that has a high visual quality.In the last decades, numerous and diversedenoising methods have been proposed, e.g.[1–6]. These methods only exploit the spa-tial redundancy in a local neighborhood and aretherefore referred to as local methods.More recently, a number of non-local meth-ods have been developed. In contrast to lo-cal techniques these methods rely on the pres-ence of similar patterns and features in the im-age. This relatively new class of denoisingmethods originates from the Non-Local Means(NLMeans) filter, introduced by Buades et al.[7] and is designed for white noise. Unfortu-nately, in practice the image noise is mostly—————————————————–The authors are with the department of Telecom-munications and Information Processing of GhentUniversity (UGent-TELIN-IPI-IBBT).colored and the NLMeans filter yields poor re-sults in this case. For this reason, we extendthis filter to deal with colored noise.The remainder of this paper is as follows:in Section II we briefly present the NLMeansfilter. In Section III we describe how theNLMeans filter is adapted to colored noise. Vi-sual results are given in Section IV. Finally,Section V concludes this paper.II. NON-LOCAL MEANS FILTERINGWe assume that the image is corrupted byzero-mean white stationary Gaussian noise.The denoised value of the local neighborhoodat position i in the image is robustly estimatedas the weighted average of all neighborhoods inthe image1, which can be seen as a linear spa-tially adaptive filter:xˆi = arg minxNXj=1ρ(x − yj) (1)=PNj=1 w(i, j)yjPNj=1 w(i, j)(2)where yj denotes a local neighborhood vectorat position j in the observed noisy image andN is the number of pixels in the image. Theweights w(i, j) depend on the similarity be-tween the neighborhoods. For the Leclerc ro-bust function ρ(r) (see [8]) the weights are:w(i, j) = exp−||yi − yj ||22h2(3)where h is a constant, proportional to the noisestandard deviation σn. The weights decay at an—————————————————–1A one dimensional index is used here, such as inraster scanningexponential rate, which results in large weightsfor similar neighborhoods and small weightsfor non-similar neighborhoods.III. PROPOSED EXTENSION TO COLOREDNOISEIn this Section, we assume that the imagenoise is colored. Applying the NLMeans filterwithout modifications to images corrupted bycorrelated noise yields poor denoising perfor-mance, because of the model mismatch. For-tunately, a robust estimator for correlated noisecan be obtained by replacing the Euclidean dis-tance ||yi − yj || by the Mahalanobis distanceq(yi − yj)T C−1n (yi − yj), where Cn is thenoise covariance matrix. This gives the follow-ing estimator:xˆi = arg minxNXj=1ρC−1/2n (x − yj)(4)where (·)1/2 is the square root of a positive def-inite matrix. The weights now become:w(i, j) = exp −||C−1/2n (yi − yj)||22h2!(5)The correlatedness of the noise only affects theweights w(i, j) and not the averaging. Hence,Eq. 2 can be used to estimate the noise-free im-age, but with weights defined in Eq. 5.IV. RESULTSIn Figure 1, the visual performance of theproposed method is compared to the wavelet-based state-of-the-art (local) method for col-ored noise from [4]. To be able to computethe objective quality in terms of PSNR, artifi-cial noise is generated and added to a noise-freeimage. The proposed technique brings an im-provement both visually as in PSNR: there arealmost no ringing artifacts and reconstructededges are much sharper.V. CONCLUSIONBy exploiting the self-similarity of images, asignificant gain in denoising performance is ob-tained compared to purely local filtering meth-(PSNR=16.1 dB) (PSNR=24.8 dB) (PSNR=25.4 dB)(PSNR=18.6 dB) (PSNR=29.3 dB) (PSNR=30.7 dB)Figure 1. Denoising results for two images. Fromleft to right: the noisy image, the restored imageusing [4], the restored image using the proposedmethod.ods. In this work, we have presented an exten-sion of the NLMeans filter that offers a superiorvisual and objective quality compared to state-of-the-art local methods for the restoration ofimages corrupted by colored noise.REFERENCES[1] L. Rudin and S. Osher, “Total variation basedimage restoration with free local constraints,” inProc. of IEEE International Conference on ImageProcessing (ICIP), Nov. 1994, vol. 1, pp. 31–35.[2] C. Tomasi and R. Manduchi, “Bilateral filteringfor gray and color images,” in Proceedings In-ternational conference on computer vision, 1998,pp. 839–846.[3] D. L. Donoho, “De-Noising by Soft-Thresholding,” IEEE Trans. Inform. Theory, vol.41, pp. 613–627, May 1995.[4] J. Portilla, V. Strela, M. Wainwright, and E.P. Si-moncelli, “Image denoising using scale mixturesof gaussians in the wavelet domain,” IEEE Trans-actions on image processing, vol. 12, no. 11, pp.1338–1351, 2003.[5] A. Pizˇurica and W. Philips, “Estimating the prob-ability of the presence of a signal of interest inmultiresolution single- and multiband image de-noising,” IEEE Transactions on image process-ing, vol. 15, no. 3, pp. 654–665, 2006.[6] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazar-ian, “Image denoising by sparse 3d transform-domain collaborative filtering,” IEEE Transac-tions on image processing, vol. 16, no. 8, pp.2080–2095, 2007.[7] A. Buades, B. Coll., and J.M Morel, “A non lo-cal algorithm for image denoising,” in Proc. Int.Conf. Computer Vision and Pattern Recognition(CVPR), 2005, vol. 2, pp. 60–65.[8] M. Elad, “On the Origin of the Bilateral Filter andWays to Improve it,” IEEE Transactions on Im-age Processing, vol. 11, no. 10, pp. 1141–1151,2002.

Adaptive non-local means filtering of images corrupted by colored noise

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Adaptive non-local means filtering of images corrupted by colored noise

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