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by Jos Roerdink

The processing, analysis, and visualization of tensor images has become very important in many application domains, such as brain imaging and seismology. In a tensor image the value at each pixel is not just a scalar (as in a grey scale image), but a matrix or tensor, hence the name. In the project COMOTI – Connected Morphological Operators for Tensor Images, funded by the Dutch National Science Foundation (NWO), we address the development of techniques for morphological filtering and visualization of tensor fields. Potentially, this could lead to new tools for the analysis of brain connectivity and diagnosis of connectivity-related disorders.

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A prime motivation for our work is brain imaging, where diffusion tensor magnetic resonance imaging (DTI) enables the in-vivo exploration of nerve fibre bundles. This allows the determination and visualization of anatomical connections between brain regions. Whereas the theoretical and algorithmic foundations for morphological operators on scalar images are well developed, this is not the case for tensor images.

Figure 1: Illustrative visualization of DTI fiber tracts.  Source: Scientific Visualization and Computer Graphics group, University of Groningen (Ref. [3]).

Our goals are the development of new mathematical methods and efficient algorithms for analysing and understanding tensor data, and evaluating these methods in selected application domains.

Our work is founded on the complete lattice framework of mathematical morphology. In this methodology, shapes of objects in binary (black and white) images are detected by moving small test shapes – structuring elements – of various forms and sizes over the image and recording the pixel locations where certain logical relations between the image and the structuring element are satisfied. In this way, it is possible to define a wealth of morphological operations that can extract various shapes and patterns from the image. Morphological image filters are nonlinear, which makes their mathematical analysis more complex.

This approach can be extended to grey scale images. A unified framework is provided by complete lattice theory, where the existence of a partial order on the space of images is essential. For the binary case this partial order is given by set inclusion, while in the grey scale case the (complete) order on the image values can be extended to a partial order on the image functions themselves. Also, it is very important to pay attention to invariance properties of the resulting operators, such as translation, rotation, or scale invariance.

Extending morphological filters to tensor data is difficult, since there is no obvious way to define a partial order on tensors. We started with the simpler case of colour images, and are now extending the results to tensor images. By using the frame (overcomplete basis) concept it is possible to construct operators that are invariant to a given group of transformations. For example, we obtained saturation- and rotation-invariant frames for colour images, leading to more intuitive and better quality results.

Of particular interest is the study of connectivity, which allows us to group pixels of the same grey value into connected components. Connectivity-preserving (“connected”, for short) filters either remove or keep each connected component, but do not change its shape. However, the human observer sometimes interprets a single connected component as multiple visual entities, or vice versa. To deal with such cases hyperconnectivity and associated hyperconnected filters have been proposed. The axiomatization of hyperconnected filters is a very recent development, and extending this to tensor data will allow us to explore the full potential of such filters.

The processing of tensor images is computationally very demanding. Hence we use approximation techniques that achieve a compromise between accuracy and speed, and also foresee the use of parallel processing and special hardware such as graphical processing units (GPUs).

Although this project has a strong theoretical orientation, we also aim to test and validate our methods in various application domains, notably neuroscience, material science, and seismology.

Future work concerns the development of morphological operators for higher order tensors, which are relevant for High Angular Resolution Diffusion Imaging (HARDI), where data for a large number of gradient directions are measured in an MRI scanner.

The project started in 2010 and will run until 2014. Its research team consists of PhD student Jasper van de Gronde, Dr. Michael Wilkinson, and Prof. Dr. Jos Roerdink (PI) of the Johann Bernoulli Institute for Mathematics and Computer Science of the University of Groningen. In this project, we are collaborating with Dr. J. Angulo, Center for Mathematical Morphology (CMM), Fontainebleau, France; and with Dr. R. Duits, Department of Mathematics and Computer Science, Eindhoven University of Technology, the Netherlands.


[1] J.J. van de Gronde, J.B.T.M. Roerdink: “Group-invariant frames for colour morphology”, in proc. ISMM 2013, LNCS 7883, 267-278
[2] J.B.T.M. Roerdink: “Group morphology”, Pattern Recog. 33(6), 877-895, 2000
[3] M.H. Everts et al.: “Depth-Dependent Halos: Illustrative Rendering of Dense Line Data”, IEEE TVCG 15(6), 1299–1306, 2009.

Please contact:
Jos Roerdink
Scientific Visualization and Computer Graphics, Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen
Tel. +31 50 363 3931
E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

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