Mathematical Morphology

Mathematical morphology is concerned with the analysis of shapes of objects in images. This means e.g. that by morphological filtering it is possible to simplify structures in an image depending on their scale, or just to keep structures up to a selected scale, or to enhance flow-like structures. The field of mathematical morphology arised in the mid of the sixties, and today it is one of the most successful areas in image processing.

There exist two basic processes in mathematical morphology: dilation and erosion. In a dilation process, at each pixel the given grey value is changed making use of the maximum of grey values within a certain pixel mask. This means, light structures are enhanced, and dark structures vanish. In an erosion process this role of light and dark is switched. All other filters rely on a suitable combination of dilation/erosion.

On the modelling side, there are two ways to realise the basic processes dilation/erosion. The classic way is to define a mask, or a set of masks, over which the maximum/minimum is computed. In our group we mainly follow an alternative proceeding using partial differential equations (PDEs). The arising PDEs are hyperbolic, and they are variations of so-called Eikonal equations arising e.g. in physics in the field of geometric optics.

The research efforts in our group proceed in three main directions.

• Establishing a concept of morphology of tensor-valued data, as arising e.g. in DT-MRI (diffusion tensor magnetic resonance imaging).
  
  
• Construction of high-quality numerical schemes for PDE-based morphology.
  
  
• Theoretical investigation of the relation between the scale space concepts of morphological and linear scale spaces.
  
  

Contributions in Morphology for Tensor-valued Data

The processing of matrix-valued data sets (so-called tensor fields) is becoming an increasingly important task in modern digital imaging since the tensor concept provides an adequate description of anisotropic behaviour: Prominent examples are the diffusive behaviour of water molecules in tissue, visualised by diffusion tensor magnetic resonance imaging (DT-MRI), or the so-called structure tensor used in fields ranging from motion analysis to texture segmentation. The tensors that arise in these ways have real-valued entries and are symmetric. For general information on filtering such data, see also this webpage.

• Tensor-Valued Morphology, Algebraic-Geometric Approaches
  Dilation/erosion rely on maximum and minimum operations. Their generalisation to the tensor-valued setting is not straightforward due to the lack of suitable ordering relations for symmetric matrices. In [4] we have proposed such extensions exploiting analytic-algebraic and geometric properties of symmetric matrices while preserving positive definiteness of the initial data. Unfortunately the proposed approaches have their weaknesses when it comes to computational issues. This led to the development of more appropriate extensions of dilation and erosion operations for matrix-valued data.
• Tensor-Valued Morphology, Ordering-Based Approach
  In the matrix valued setting there is a partial ordering, the so-called Loewner ordering, that is suitable for the definition of appropriate maximal and minimal elements of a set of matrices. The task to determine these extremal elements is already non-trivial in the case of 2 by 2 matrices [4] and becomes more intricate in higher dimensions [5],[6]; it requires tools from convex analysis and computational geometry. Nevertheless, the dilation and erosion operations inferred from the ordering approach exhibit the desired rotational invariance, preservation of positive definiteness, and continuous dependence on the initial matrix data. This makes it possible to define morphological derivatives and shock filters for matrix-fields.
• The PDE-based Approach
  Dilation/erosion can be described by nonlinear hyperbolic first order PDEs as introduced by R. van den Boomgaard in 1992. Based on novel mathematical notions for symmetric matrices, in [7] we were able to find the truly matrix-valued counterpart to these nonlinear morphological PDEs. For the appropriate numerical treatment of those novel matrix-valued PDEs we devised matrix-valued generalisations of known real-valued schemes, see especially [9].

Numerical Contributions

• Shock-Capturing Schemes for PDEs in Greyscale Morphology
  In 1992 R. van den Boomgaard proposed nonlinear hyperbolic partial differential equations that mimic the dilation and erosion of images with structuring elements of increasing size. It is quite challenging to devise numerical schemes that are capable of following a corresponding evolution of edges adequately, that means, with as little numerical blurring as possible. In [8] we develop a novel flux corrected transport (FCT) scheme that can accurately cope with the peculiarities of the morphological processes of dilation and erosion. Later on, this method has been extended to deal with general structuring elements [10]. In the mentioned works, it is shown that the method performs better than other PDE-based schemes in the field. Also, the FCT scheme is competitive to set-based, algebraic solutions.
• Investigating Relations between Fully Discrete and Continuous-Scale Morphology
  In 2005, Lerallut et al. have proposed a fully discrete means to incorporate tonal differences into adaptive morphological filters. In [13] we validate that such morphological amoebas are a discretisation of so-called self-snakes, a PDE-based process somewhat similar to mean curvature motion.
• PDE-based Methods for Tensor-Valued Morphology
  Relying on the developed PDEs for tensor-valued morphology, we devised the first high-resolution-type schemes for morphology of tensor fields [9]. Also, the FCT-scheme introduced in [8] for processing grey value images has been successfully extended to this setting showing superior resolution of edge-like structures compared with other methods.
• PDE-based Anisotropic Morphology
  Relying on the developed PDEs for grey-valued morphology as well as for tensor-valued morphology, we developed the concept of anisotropic morphology for enhancing flow-like, coherent image structures. Beginning with the investigation in [12], we refined the modeling as well as the numerics in subsequent steps [11], [10].

Contributions in Scale Space Theory

• Explaining the Logarithmic Relation between Linear and Morphological Systems
  In 1994 Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological equivalent of the Fourier transform. As a result an almost logarithmic connection between linear and morphological systems became apparent. In [1] we give an explanation for this relation by revealing that morphology, in essence, is linear systems theory based on the (max,+)-algebra resp. the (min,+)-algebra. The fundamental operations of dilation and erosion turn out to be convolutions with respect to these algebras. The so-called Cramer transform as the conjugate of the logarithmic Laplace transform establishes a homeomorphism between morphological and Gaussian scale space. These articles provide a step towards the unification of linear and morphological scale-space theory.

• Families of Morphological Scale Spaces
  In [2] morphological dilation and erosion scale spaces with parabolic structuring functions are embedded into two-parameter families of scale spaces which include the Gaussian scale space as a limit case. The scale space families are obtained by deforming the algebraic operations underlying the morphological operations. Aspects of these scale space families such as continuity, invariance and separability are discussed. The article enriches the picture of structural analogies between the classes of scale spaces mentioned above.

1. B. Burgeth, J. Weickert:
   An Explanation for the Logarithmic Connection between Linear and Morphological Systems.
   In L.D. Griffin, M Lillholm (Eds.): Scale Space Methods in Computer Vision. Lecture Notes in Computer Science, Vol. 2695, Springer, Berlin, 325-339, 2003.

2. M. Welk:
   Families of Generalised Morphological Scale Spaces.
   In L.D. Griffin, M Lillholm (Eds.): Scale Space Methods in Computer Vision. Lecture Notes in Computer Science, Vol. 2695, Springer, Berlin, 770-784, 2003.
   

3. B. Burgeth, M. Welk, C. Feddern, J. Weickert:
   Morphological operations on matrix-valued images.
   In T. Pajdla, J. Matas (Eds.): Computer Vision - ECCV 2004. Lecture Notes in Computer Science, Vol. 3024, 155-167, Springer, Berlin, 2004.

4. B. Burgeth, M. Welk, Ch. Feddern, J. Weickert:
   Mathematical Morphology on Tensor Data Using the Loewner Ordering.
   In J. Weickert, H. Hagen (Eds.): Visualization and Processing of Tensor Fields. Mathematics and Visualization, 357-367, Springer, Berlin, 2006.
   Revised version of Technical Report No. 160, Department of Mathematics, Saarland University, Saarbrücken, Germany, December 2005.
   

5. B. Burgeth, N. Papenberg, A. Bruhn, M. Welk, C. Feddern, J. Weickert:
   Morphology for higher-dimensional tensor data via Loewner ordering.
   In C. Ronse, L. Najman, E. Decencière (Eds.): Mathematical Morphology: 40 Years On. Computational Imaging and Vision, Vol. 30, Springer, Dordrecht, 407–416, 2005.
   

6. B. Burgeth, N. Papenberg, A. Bruhn, M. Welk, J. Weickert:
   Mathematical Morphology for Tensor Data Induced by the Loewner Ordering in Higher Dimensions.
   Signal Processing, Vol. 87, No. 2, 291-308, February 2007.
   Revised version of Technical Report No. 161, Department of Mathematics, Saarland University, Saarbrücken, Germany, December 2005.
   
7. B. Burgeth, A. Bruhn, S. Didas, J. Weickert, M. Welk:
   Morphology for Tensor Data: Ordering versus PDE-Based Approach.
   Image and Vision Computing, Vol. 25, No. 4, 496-511, 2007.
   Revised version of Technical Report No. 162, Department of Mathematics, Saarland University, Saarbrücken, Germany, December 2005.
   
8. M. Breuß, J. Weickert:
   A shock-capturing algorithm for the differential equations of dilation and erosion.
   Journal of Mathematical Imaging and Vision, Vol. 25, 187-201, (2006).
   Revised version of Technical Report No. 153, Department of Mathematics, Saarland University, Saarbrücken, Germany, September 2005.
   

9. B. Burgeth, M. Breuß, S. Didas, J. Weickert:
   PDE-based morphology for matrix fields: Numerical solution schemes.
   Tensors in Image Processing and Computer Vision, Advances in Pattern recognition, pp. 125-150. Springer, London, 2009.
   Revised version of Technical Report No. 220, Department of Mathematics, Saarland University, Saarbrücken, Germany, September 2008.

10. M. Breuß, J. Weickert:
    Highly accurate PDE-based morphology for general structuring elements.
    In X.-C. Tai et al. (Eds.): Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, Vol. 5567, 758 - 769,
    © Springer-Verlag Berlin Heidelberg 2009.
11. L. Pizarro, B. Burgeth, M. Breuß, J. Weickert
    A directional Rouy-Tourin scheme for adaptive matrix-valued morphology.
    In M.H.F. Wilkinson and J.B.T.M. Roerdink (Eds.): Proc. Ninth International Symposium on Mathematical Morphology (ISMM 2009), Lecture Notes in Computer Science, vol. 5720, pp. 250 -260, Springer, Berlin, 2009.
    © Springer-Verlag Berlin Heidelberg 2009.
    
12. M. Breuß, B. Burgeth, J. Weickert:
    Anisotropic continuous-scale morphology.
    In Proceedings of the 3rd Iberian Conference on Pattern Recognition and Image Analysis, IbPRIA, June 6–8, 2007, Girona, Spain, Lecture Notes in Computer Science, Springer, Berlin, 2007.
    
13. M. Welk, M. Breuß, O. Vogel:
    Morphological amoebas are self-snakes.
    To appear in Journal of Mathematical Imaging and Vision.
    Revised version of Technical Report No. 259, Department of Mathematics, Saarland University, Saarbrücken, Germany, February 2010.
14. M. Breuß, J. Weickert:
    Highly Accurate Schemes for PDE-Based Morphology with General Convex Structuring Elements.
    To appear in International Journal of Computer Vision.
    Revised version of Technical Report No. 236, Department of Mathematics, Saarland University, Saarbrücken, Germany, May 2009.
15. B. Burgeth, L. Pizarro, M. Breuß, J. Weickert:
    Adaptive continuous-scale morphology for matrix fields.
    To appear in International Journal of Computer Vision.
    Revised version of Technical Report No. 237, Department of Mathematics, Saarland University, Saarbrücken, Germany, May 2009.

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