A Nonlinear Structure Tensor with the Diffusivity Matrix Composed of the Image Gradient

We propose a nonlinear partial differential equation (PDE) for regularizing a tensor which contains the first derivative information of an image such as strength of edges and a direction of the gradient of the image. Unlike a typical diffusivity matrix which consists of derivatives of a tensor data, we propose a diffusivity matrix which consists of the tensor data itself, i.e., derivatives of an image. This allows directional smoothing for the tensor along edges which are not in the tensor but in the image. That is, a tensor in the proposed PDE is diffused fast along edges of an image but slowly across them. Since we have a regularized tensor which properly represents the first derivative information of an image, the tensor is useful to improve the quality of image denoising, image enhancement, corner detection, and ramp preserving denoising. We also prove the uniqueness and existence of solution to the proposed PDE.

[1]  Rachid Deriche,et al.  Orthonormal Vector Sets Regularization with PDE's and Applications , 2002, International Journal of Computer Vision.

[2]  Joost van de Weijer,et al.  Adaptive Structure Tensors and their Applications , 2006, Visualization and Processing of Tensor Fields.

[3]  J. Bigun,et al.  Optimal Orientation Detection of Linear Symmetry , 1987, ICCV 1987.

[4]  Johan Wiklund,et al.  Multidimensional Orientation Estimation with Applications to Texture Analysis and Optical Flow , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  Xue-Cheng Tai,et al.  Image Inpainting Using a TV-Stokes Equation , 2007 .

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

[7]  Otmar Scherzer,et al.  Inverse Problems, Image Analysis, and Medical Imaging , 2002 .

[8]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[9]  T. Brox,et al.  Diffusion and regularization of vector- and matrix-valued images , 2002 .

[10]  Ullrich Köthe,et al.  Edge and Junction Detection with an Improved Structure Tensor , 2003, DAGM-Symposium.

[11]  Thomas Brox,et al.  Nonlinear structure tensors , 2006, Image Vis. Comput..

[12]  Stanley Osher,et al.  Numerical Methods for p-Harmonic Flows and Applications to Image Processing , 2002, SIAM J. Numer. Anal..

[13]  J. Weickert.  Scale-Space Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor , 1994 .

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

[15]  Rachid Deriche,et al.  Image coupling, restoration and enhancement via PDE's , 1997, Proceedings of International Conference on Image Processing.

[16]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[17]  Knut-Andreas Lie,et al.  Image Processing Based on Partial Differential Equations , 2007 .

[18]  Ron Kimmel,et al.  StereographicCombing a Porcupine or Studies on Direction Diffusion in Image Processing , 2004, SIAM J. Appl. Math..

[19]  Joachim Weickert,et al.  Coherence-Enhancing Diffusion Filtering , 1999, International Journal of Computer Vision.

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

[21]  J. Schnabel,et al.  Nonlinear smoothing for reduction of systematic and random errors in diffusion tensor imaging , 2000, Journal of magnetic resonance imaging : JMRI.

[22]  L. Álvarez,et al.  Signal and image restoration using shock filters and anisotropic diffusion , 1994 .

[23]  Joachim Weickert,et al.  Coherence-Enhancing Shock Filters , 2003, DAGM-Symposium.

[24]  Joachim Weickert,et al.  A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion , 2001, International Journal of Computer Vision.

[25]  Andrew P. Witkin,et al.  Analyzing Oriented Patterns , 1985, IJCAI.

[26]  Jean-Michel Morel,et al.  The staircasing effect in neighborhood filters and its solution , 2006, IEEE Transactions on Image Processing.

[27]  Byung Il Lee,et al.  Conductivity image reconstruction from defective data in MREIT: numerical Simulation and animal experiment , 2006, IEEE Transactions on Medical Imaging.

[28]  Xue-Cheng Tai,et al.  Noise removal using smoothed normals and surface fitting , 2004, IEEE Transactions on Image Processing.

[29]  Rachid Deriche,et al.  Diffusion tensor regularization with constraints preservation , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

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