Mathematical Morphology on Tensor Data Using the Loewner Ordering

The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensor-valued data. Based upon the Loewner ordering for symmetric matrices this chapter extends the maximum and minimum operation to the tensor-valued setting. This provides the ground to establish matrix-valued analogues of the basic morphological operations ranging from erosion/dilation to top hats. In contrast to former attempts to develop a morphological machinery for matrices, the novel definitions of maximal/minimal matrices depend continuously on the input data, a property crucial for the construction of morphological derivatives such as the Beucher gradient or a morphological Laplacian. These definitions are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DTMRI data. The morphological operations resulting from a component-wise maximum/minimum of the matrix channels disregarding their strong correlation fail to be rotational invariant. Experiments on DT-MRI images as well as on indefinite matrix data illustrate the properties and performance of our morphological operators.

[1]  Alfred M. Bruckstein,et al.  Regularized Laplacian Zero Crossings as Optimal Edge Integrators , 2003, International Journal of Computer Vision.

[2]  Joachim Weickert,et al.  Curvature-Driven PDE Methods for Matrix-Valued Images , 2006, International Journal of Computer Vision.

[3]  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.

[4]  Maher Moakher,et al.  Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization , 2006, Visualization and Processing of Tensor Fields.

[5]  Joachim Weickert,et al.  Median Filtering of Tensor-Valued Images , 2003, DAGM-Symposium.

[6]  Thomas Brox,et al.  PDEs for Tensor Image Processing , 2006, Visualization and Processing of Tensor Fields.

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

[8]  H. Heijmans.  Morphological image operators , 1994 .

[9]  Lucas J. van Vliet,et al.  A nonlinear laplace operator as edge detector in noisy images , 1989, Comput. Vis. Graph. Image Process..

[10]  Joachim Weickert,et al.  Tensor Field Interpolation with PDEs , 2006, Visualization and Processing of Tensor Fields.

[11]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[12]  G. Matheron.  Éléments pour une théorie des milieux poreux , 1967 .

[13]  Adrian S. Lewis,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[14]  Henry P. Kramer,et al.  Iterations of a non-linear transformation for enhancement of digital images , 1975, Pattern Recognit..

[15]  Kristel Michielsen,et al.  Morphological image analysis , 2000 .

[16]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[17]  Juan Ruiz-Alzola,et al.  Tensor Field Regularization using Normalized Convolution and Markov Random Fields in a Bayesian Framework , 2006, Visualization and Processing of Tensor Fields.

[18]  Robert M. Haralick,et al.  Digital Step Edges from Zero Crossing of Second Directional Derivatives , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  David H. Laidlaw,et al.  An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications , 2006, Visualization and Processing of Tensor Fields.

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

[21]  Joachim Weickert,et al.  Morphological Operations on Matrix-Valued Images , 2004, ECCV.