A comparison of state-of-the-art diffusion imaging techniques for smoothing medical/non-medical image data

Partial differential equations (PDEs) have dominated image processing research. The three main reasons for their success are: (1) their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; (2) their ability to solve PDEs in the level set framework using finite difference methods; and (3) their easy extension to a higher dimensional space. The paper is an attempt to summarize PDEs and their solutions applied to image diffusion. The paper first presents the fundamental diffusion equation. Next, the multi-channel anisotropic diffusion imaging is presented, followed by tensor non-linear anisotropic diffusion. We also present the anisotropic diffusion based on PDE and the Tukey/Huber weight function for image noise removal. The paper also covers the recent growth of image denoising using the curve evolution approach and image denoising using histogram modification based on PDE. Finally, the paper presents non-linear image denoising. Examples covering both synthetic and real world images are presented.

[1]  Steven Haker,et al.  Geometric pde's in computer vision , 1999 .

[2]  Rachid Deriche,et al.  Coupled Geodesic Active Regions for Image Segmentation: A Level Set Approach , 2000, ECCV.

[3]  Joachim Weickert,et al.  Relations Between Regularization and Diffusion Filtering , 2000, Journal of Mathematical Imaging and Vision.

[4]  Antonin Chambolle,et al.  Partial differential equations and image processing , 1994, Proceedings of 1st International Conference on Image Processing.

[5]  John F. Canny,et al.  A Computational Approach to Edge Detection , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  P. Lions,et al.  Axioms and fundamental equations of image processing , 1993 .

[7]  Andrew P. Witkin,et al.  Scale-Space Filtering , 1983, IJCAI.

[8]  Christoph Schnörr,et al.  A Study of a Convex Variational Diffusion Approach for Image Segmentation and Feature Extraction , 1998, Journal of Mathematical Imaging and Vision.

[9]  G. Sapiro,et al.  Histogram Modification via Differential Equations , 1997 .

[10]  Alan S. Willsky,et al.  Image segmentation and edge enhancement with stabilized inverse diffusion equations , 2000, IEEE Trans. Image Process..

[11]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[12]  Benjamin B. Kimia,et al.  Shocks from images: propagation of orientation elements , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[13]  Kecheng Liu,et al.  Shape recovery algorithms using level sets in 2-D/3-D medical imagery: a state-of-the-art review , 2002, IEEE Transactions on Information Technology in Biomedicine.

[14]  P. Olver,et al.  Conformal curvature flows: From phase transitions to active vision , 1996, ICCV 1995.

[15]  J. Sethian,et al.  An O(N log N) algorithm for shape modeling. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Joachim Weickert,et al.  PDE - Based Preprocessing of Medical Images , 2000, Künstliche Intell..

[17]  Ivan Bajla,et al.  Nonlinear filtering of magnetic resonance tomograms by geometry-driven diffusion , 1998, Machine Vision and Applications.

[18]  Bart M. ter Haar Romeny,et al.  Geometry-Driven Diffusion in Computer Vision , 1994, Computational Imaging and Vision.

[19]  Guillermo Sapiro,et al.  Experiments on geometric image enhancement , 1994, Proceedings of 1st International Conference on Image Processing.

[20]  J. S. Suri,et al.  Leaking prevention in fast level sets using fuzzy models: an application in MR brain , 2000, Proceedings 2000 IEEE EMBS International Conference on Information Technology Applications in Biomedicine. ITAB-ITIS 2000. Joint Meeting Third IEEE EMBS International Conference on Information Technol.

[21]  Guillermo Sapiro,et al.  Contrast Enhancement via Image Evolution Flow , 1997, CVGIP Graph. Model. Image Process..

[22]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[23]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[24]  Max A. Viergever,et al.  Scale-Space Theory in Computer Vision , 1997 .

[25]  Kaleem Siddiqi,et al.  Geometric heat equation and nonlinear diffusion of shapes and images , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[26]  L. Evans,et al.  Motion of level sets by mean curvature IV , 1995 .

[27]  Jayant Shah,et al.  A common framework for curve evolution, segmentation and anisotropic diffusion , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[28]  Daniel Rueckert,et al.  Knowledge-based tensor anisotropic diffusion of cardiac magnetic resonance images , 1999, Medical Image Anal..

[29]  Kaleem Siddiqi,et al.  Area and length minimizing flows for shape segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[30]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[31]  Guillermo Sapiro,et al.  From active contours to anisotropic diffusion: connections between basic PDE's in image processing , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[32]  Azriel Rosenfeld,et al.  Robust regression methods for computer vision: A review , 1991, International Journal of Computer Vision.

[33]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[34]  Robert M. Haralick,et al.  Left ventricle longitudinal axis fitting and its apex estimation using a robust algorithm and its performance: a parametric apex model , 1997, Proceedings of International Conference on Image Processing.

[35]  J. Suri,et al.  Advanced algorithmic approaches to medical image segmentation: state-of-the-art application in cardiology, neurology, mammography and pathology , 2001 .

[36]  Sameer Singh,et al.  Computer Vision and Pattern Recognition Techniques for 2-D and 3-D MR Cerebral Cortical Segmentation (Part I): A State-of-the-Art Review , 2002, Pattern Analysis & Applications.

[37]  Jasjit S. Suri,et al.  White Matter/Gray Matter Boundary Segmentation Using Geometric Snakes: A Fuzzy Deformable Model , 2001, ICAPR.

[38]  Guido Gerig,et al.  Nonlinear anisotropic filtering of MRI data , 1992, IEEE Trans. Medical Imaging.

[39]  Alessandro Sarti,et al.  A geometric model for 3-D confocal image analysis , 1998, IEEE Transactions on Biomedical Engineering.

[40]  K Siddiqi,et al.  Parts of Visual Form: Psychophysical Aspects , 1996, Perception.

[41]  Guillermo Sapiro,et al.  Robust anisotropic diffusion , 1998, IEEE Trans. Image Process..

[42]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[43]  Joachim Weickert.  Fast Segmentation Methods Based on Partial Differential Equations and the Watershed Transformation , 1998, DAGM-Symposium.

[44]  Joachim Weickert,et al.  A Review of Nonlinear Diffusion Filtering , 1997, Scale-Space.

[45]  Decision Systems.,et al.  Classification and uniqueness of invariant geometric flows , 1993 .

[46]  Wenyuan Xu,et al.  Behavioral analysis of anisotropic diffusion in image processing , 1996, IEEE Trans. Image Process..

[47]  Anthony J. Yezzi,et al.  A geometric snake model for segmentation of medical imagery , 1997, IEEE Transactions on Medical Imaging.

[48]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[49]  Sameer Singh,et al.  Advanced Algorithmic Approaches to Medical Image Segmentation , 2002, Advances in Computer Vision and Pattern Recognition.

[50]  Simon R. Arridge,et al.  Multi-Spectral Probabilistic Diffusion Using Bayesian Classification , 1997, Scale-Space.

[51]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[52]  Joachim Weickert,et al.  Scale-Space Theories in Computer Vision , 1999, Lecture Notes in Computer Science.

[53]  Jean-Michel Morel,et al.  Variational methods in image segmentation , 1995 .

[54]  Joachim Weickert,et al.  Fast Parallel Algorithms for a Broad Class of Nonlinear Variational Diffusion Approaches , 2001, Real Time Imaging.

[55]  Guillermo Sapiro,et al.  Geodesic Active Contours , 1995, International Journal of Computer Vision.

[56]  Pietro Perona,et al.  Orientation diffusions , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.