Mathematical morphology for tensor data induced by the Loewner orderingin higher dimensions

Positive semidefinite matrix fields are becoming increasingly important in digital imaging. One reason for this tendency consists of the introduction of diffusion tensor magnetic resonance imaging (DTMRI). In order to perform shape analysis, enhancement or segmentation of such tensor fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the matrix-valued setting. We start by presenting novel definitions for the maximum and minimum of a set of matrices since these notions lie at the heart of the morphological operations. In contrast to naive approaches like the component-wise maximum or minimum of the matrix channels, our approach is based on the Loewner ordering for symmetric matrices. The notions of maximum and minimum deduced from this partial ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. We introduce erosion, dilation, opening, closing, top hats, morphological derivatives, shock filters, and mid-range filters for positive semidefinite matrix-valued images. These morphological operations incorporate information simultaneously from all matrix channels rather than treating them independently. Experiments on DT-MRI images with ball- and rod-shaped structuring elements illustrate the properties and performance of our morphological operators for matrix-valued data

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 paper 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 DT-MRI 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

Morphology for matrix data : ordering versus PDE-based approach

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: One is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data

A Unified Algebraic Framework for Fuzzy Image Compression and Mathematical Morphology

In this paper we show how certain techniques of image processing, having different scopes, can be joined together under a common "algebraic roof"

Morphological shape generation through user-controlled group metamorphosis

Morphological shape design is interpreted in this paper as a search for new shapes from a particular application domain represented by a set of selected shape instances. This paper proposes a new foundation for morphological shape design and generation. In contrast to existing generative procedures, an approach based on a user-controlled metamorphosis between functionally based shape models is presented. A formulation of the pairwise metamorphosis is proposed with a variety of functions described for the stages of deformation, morphing and offsetting. This formulation is then extended to the metamorphosis between groups of shapes with user-defined, dynamically correlated and weighted feature elements. A practical system was implemented in the form of plugin to Maya and tested by an industrial designer on a group of representative shapes from a particular domain. © 2013 Elsevier Ltd

PDE-based morphology for matrix fields : numerical solution schemes

Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrix-valued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets

Model-Based Edge Detector for Spectral Imagery Using Sparse Spatiospectral Masks

Two model-based algorithms for edge detection in spectral imagery are developed that specifically target capturing intrinsic features such as isoluminant edges that are characterized by a jump in color but not in intensity. Given prior knowledge of the classes of reflectance or emittance spectra associated with candidate objects in a scene, a small set of spectral-band ratios, which most profoundly identify the edge between each pair of materials, are selected to define a edge signature. The bands that form the edge signature are fed into a spatial mask, producing a sparse joint spatiospectral nonlinear operator. The first algorithm achieves edge detection for every material pair by matching the response of the operator at every pixel with the edge signature for the pair of materials. The second algorithm is a classifier-enhanced extension of the first algorithm that adaptively accentuates distinctive features before applying the spatiospectral operator. Both algorithms are extensively verified using spectral imagery from the airborne hyperspectral imager and from a dots-in-a-well midinfrared imager. In both cases, the multicolor gradient (MCG) and the hyperspectral/spatial detection of edges (HySPADE) edge detectors are used as a benchmark for comparison. The results demonstrate that the proposed algorithms outperform the MCG and HySPADE edge detectors in accuracy, especially when isoluminant edges are present. By requiring only a few bands as input to the spatiospectral operator, the algorithms enable significant levels of data compression in band selection. In the presented examples, the required operations per pixel are reduced by a factor of 71 with respect to those required by the MCG edge detector

A Learning Framework for Morphological Operators using Counter-Harmonic Mean

We present a novel framework for learning morphological operators using counter-harmonic mean. It combines concepts from morphology and convolutional neural networks. A thorough experimental validation analyzes basic morphological operators dilation and erosion, opening and closing, as well as the much more complex top-hat transform, for which we report a real-world application from the steel industry. Using online learning and stochastic gradient descent, our system learns both the structuring element and the composition of operators. It scales well to large datasets and online settings.Comment: Submitted to ISMM'1