Anisotropic Diffusion is very efficient non-linear image processing PDE based technique which simultaneously restore images and enhance image features for 2-D or, 3-D images. This technique is described by local eigenvalues and local eigenvectors of the anisotropic diffusion tensor field where anisotropic diffusion coefficients are depending on direction and position. Here, mathematical analysis of robust anisotropic diffusion (RAD) filter for ultrasound (US) image has been discussed in this paper. It includes probabilistic memory mechanism and speckle statistics models of tissues characterization and adapts the anisotropic diffusion tensor to the ultrasound image iteratively. Higher frequency absorbed by tissue and skin but cannot penetrate deeply in comparison to lower frequency which give poorer image quality by echo signals, so we get an inferior quality image with some clinical information loss. This clinical information loss is restored by iterative process of various state-of-the-art filters, but discussed RAD filter shows better performance in terms of measured MSE and SSIM index, with including memory mechanism and speckle statistics, and preserves the relevant tissue details for clinical purposes.

Ultrasound imaging, speckle filter, anisotropic diffusion, tensor, Volterra equations.

[1] J.-S. Lee, �Digital image enhancement and noise filtering by use of local statistics,� IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-2, no. 2, pp. 165�168, Mar. 1980.
[2] D. T. Kuan, A. A. Sawchuk, T. C. Strand, and P. Chavel, �Adaptive noise smoothing filter for images with signal-dependent noise,� IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-7, no. 2, pp. 165�177, Mar. 1985.
[3] K. Krissian, C.-F. Westin, R. Kikinis, and K. G. Vosburgh, �Oriented speckle reducing anisotropic diffusion,� IEEE Trans. Image Process., vol. 16, no. 5, pp. 1412�1424, May 2007.
[4] V. Frost, J. Stiles, K. Shanmugan, and J. Holzman, �A model for radar images and its application to adaptive digital filtering of multiplicative noise,� IEEE Trans. PAMI, vol. 4, no. 2, pp. 157�166, 1982.
[5] S. Gupta, R. Chauhan, and S. Sexana, �Wavelet-based statistical approach for speckle reduction in medical ultrasound images.� Med Biol Eng Comput, vol. 42, no. 2, pp. 189�92, Mar 2004.
[6] L. Rudin, P.-L. Lions, and S. Osher, �Multiplicative denoising and deblurring: Theory and algorithms,� in Geometric Level Set Methods in Imaging, Vision and Graphics. Springer-Verlag, 2003, ch. 6, pp. 103�120.
[7] Y. Yu and S. Acton, �Edge detection in ultrasound imagery using the instantaneous coefficient of variation,� IEEE Transactions on Image Processing, vol. 13, no. 12, pp. 1640�1655, 2004.
[8] S. Aja-Fern�andez and C. Alberola-L�opez, �On the estimation of the coefficient of variation for anisotropic diffusion speckle filtering,� IEEE Transactions on Image Processing, in press, vol. 15, no. 9, pp. 2694� 2701, sep 2005.
[9] K. Z. Abd-Elmoniem, A.-B. M. Youssef, and Y. M. Kadah, �Real-time speckle reduction and coherence enhancement in ultrasound imaging via nonlinear anisotropic diffusion,� IEEE Trans Biomed Eng, vol. 49, no. 9, pp. 997�1014, Sep 2002.
[10] J. Weickert, Anisotropic Diffusion in Image Processing. Stuttgart, Germany: Teubner, 1998.
[11] P. Perona and J. Malik, �Scale-space and edge detection using anisotropic diffusion,� IEEE Trans. Pattern Anal. Mach. Intell., vol. 12, no. 7, pp. 629�639, Jul. 1990.
[12] Jean-Marie Mirebeau, Jer^ome Fehrenbach, Laurent Risser, and Shaza Tobji,�Anisotropic Diffusion in ITK�, arXiv: 1503.00992v1 [cs.CV], March, 2015.
[13] Y. Yu and S. T. Acton, �Speckle reducing anisotropic diffusion,� IEEE Trans. Image Process., vol. 11, no. 11, pp. 1260�1270, Nov. 2002.
[14] K. Krissian, �Flux-based anisotropic diffusion applied to enhancement of 3d angiogram,� IEEE Trans. Medical Imaging, vol. 21, no. 11, pp. 1440�1442, Nov. 2002.
[15] K. Krissian, C.-F. Westin, R. Kikinis, and K. G. Vosburgh, �Oriented speckle reducing anisotropic diffusion,� IEEE Trans. Image Process., vol. 16, no. 5, pp. 1412�1424, May 2007.
[16] G. H. Cottet and M. E. Ayyadi, �A Volterra type model for image processing,� IEEE Trans. Image Process., vol. 7, no. 3, pp. 292�303, Mar. 1998.
[17] F. Destrempes, J. Meunier, M. F. Giroux, G. Soulez, and G. Cloutier, �Segmentation in ultrasonic B-mode images of healthy carotid arteries using mixtures of Nakagami distributions and stochastic optimization,� IEEE Trans. Med. Imag., vol. 28, no. 2, pp. 215�229, Feb. 2009.
[18] Jorgen Arendt Jensen�s website, http://field-ii.dk/ for Field � II Simulation Program.
[19] M. Nachtegeal, �Fuzzy techniques in image processing�, volume 52, SPRINGER-Verlag, New York, 2000.
[20] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, �Image Quality Assessment: From error visibility to structural similarity�, IEEE Transaction on Image Processing, 13(3), pp. 1 � 14, March 2000.
[21] Sumit Kushwaha and Rabindra Kumar Singh, �Study of Various Image Noises and their Behaviors�, International Journal of Computer Sciences and Engineering (IJCSE), vol. 3, issue 3, March 2015, E-ISSN: 2347-2693.
[22] G. Ramos-Llord�n, G. Vegas-S�nchez-Ferrero, M. Martin-Fernandez, C. Alberola-L�pez and S. Aja-Fern�ndez, "Anisotropic Diffusion Filter With Memory Based on Speckle Statistics for Ultrasound Images," in IEEE Transactions on Image Processing, vol. 24, no. 1, pp. 345-358, Jan. 2015.