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

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

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

Adaptive continuous-scale morphology for matrix fields

In this article we consider adaptive, PDE-driven morphological operations for 3D matrix fields arising e.g. in diffusion tensor magnetic resonance imaging (DT-MRI). The anisotropic evolution is steered by a matrix constructed from a structure tensor for matrix valued data. An important novelty is an intrinsically one-dimensional directional variant of the matrix-valued upwind schemes such as the Rouy-Tourin scheme. It enables our method to complete or enhance anisotropic structures effectively. A special advantage of our approach is that upwind schemes are utilised only in their basic one-dimensional version. No higher dimensional variants of the schemes themselves are required. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method

Median and related local filters for tensor-valued images

We develop a concept for the median filtering of tensor data. The main part of this concept is the definition of median for symmetric matrices. This definition is based on the minimisation of a geometrically motivated objective function which measures the sum of distances of a variable matrix to the given data matrices. This theoretically wellfounded concept fits into a context of similarly defined median filters for other multivariate data. Unlike some other approaches, we do not require by definition that the median has to be one of the given data values. Nevertheless, it happens so in many cases, equipping the matrix-valued median even with root signals similar to the scalar-valued situation. Like their scalar-valued counterparts, matrix-valued median filters show excellent capabilities for structure-preserving denoising. Experiments on diffusion tensor imaging, fluid dynamics and orientation estimation data are shown to demonstrate this. The orientation estimation examples give rise to a new variant of a robust adaptive structure tensor which can be compared to existing concepts. For the efficient computation of matrix medians, we present a convex programming framework. By generalising the idea of the matrix median filters, we design a variety of other local matrix filters. These include matrix-valued mid-range filters and, more generally, M-smoothers but also weighted medians and \alpha-quantiles. Mid-range filters and quantiles allow also interesting cross-links to fundamental concepts of matrix morphology

Generalized Morphology using Sponges

Mathematical morphology has traditionally been grounded in lattice theory. For non-scalar data lattices often prove too restrictive, however. In this paper we present a more general alternative, sponges, that still allows useful definitions of various properties and concepts from morphological theory. It turns out that some of the existing work on “pseudo-morphology” for non-scalar data can in fact be considered “proper” mathematical morphology in this new framework, while other work cannot, and that this correlates with how useful/intuitive some of the resulting operators are

Modeling And Detection Of Uterine Contractions Using Magnetomyography

In this dissertation, we develop a novel mathematical framework for modeling and analyzing uterine contractions using biomagnetic measurements. The study of myometrium contractility during pregnancy is relevant to the field of reproductive assessment. Its clinical importance is grounded in the need for a better understanding of the bioreproduction mechanisms. For example, in the last decade the number of preterm labors has increased significantly. Preterm birth can cause health problems or even be fatal for the fetus if it happens too early, and, at the same time, it imposes significant financial burdens on health care systems. Therefore, it is critical to develop models and statistical tools that help to monitor non-invasively the uterine activities during pregnancy. We derive a forward electromagnetic model of uterine contractions during pregnancy. Existing models of myometrial contractions approach the problem either at an organ level or lately at a cellular level. At the organ level, the models focus on generating contractile forces that closely resemble clinical measurements of normal intrauterine pressure during contractions in labor. At the cellular level, the models focus on predicting the changes of ionic concentrations in a uterine myocyte during a contraction, and, as a consequence, on modeling the transmembrane potential evolution as a function of time. In this work, we propose an electromagnetic modeling approach taking into account electrophysiological and anatomical knowledge jointly at the cellular, tissue, and organ levels. Our model aims to characterize myometrial contractions using magnetomyography: MMG) and electromyography: EMG) at different stages of pregnancy. In particular, we introduce a four-compartment volume conductor geometry, and we use a bidomain approach to model the propagation of the myometrium transmembrane potential on the human uterus. The bidomain approach is given by a set of reaction-diffusion equations. The diffusion part of the equations governs the spatial evolution of the transmembrane potential, and the reaction part is given by the local ionic current cell dynamics. Here we introduce a modified version of the Fitzhugh-Nagumo: FHN) equation for modeling ionic currents in each myocyte, assuming a plateau-type transmembrane potential. We incorporate the anisotropic nature of the uterus by considering conductivity tensors in our model. In particular, we propose a general approach to design the conductivity-tensor orientation and to estimate the conductivity-tensor values in the extracellular and intracellular domains for any uterine shape. We use finite element methods: FEM) to solve our model, and we illustrate our approach by presenting a numerical example to model a uterine contraction at term. Our results are in good agreement with the values reported in the experimental technical literature, and these are potentially important as a tool for helping in the characterization of contractions and for predicting labor. We propose an automatic, robust, single-channel statistical detector of uterine MMG contractions. One common restriction of previous techniques is that algorithm parameters, such as the detection threshold and the window length of analysis need to be calibrated experimentally, based on a particular data set. Therefore, the detection performance might change from patient to patient, for example, because of differences in the pregnancy stage and tissue conductivities. In contrast, the proposed algorithm does not require the use of a sliding window of analysis, and the detection threshold is determined analytically; thus, it does not need to be calibrated. Our detection algorithm consists of two stages: In the first stage, we segment the measurements using a multiple change-point estimation algorithm and assuming a piecewise constant time-varying autoregressive model of the measurements; In the second stage, we apply the non-supervised K-means cluster algorithm to classify each time segment, using the RMS and FOZC as candidate features. As a result a discrete-time binary decision signal is generated indicating the presence of a contraction. Moreover, since each single channel detector provides local information regarding the presence of a contraction, we propose a spatio-temporal estimator of the magnetic activity generated by uterine contractions. The algorithm, when evaluated with real MMG measurements, detects uterine activity much earlier than the patient begins to sense it. It also enables visualizing the relative location of the origin of uterine contraction and quantifying the amount of energy delivered during a contraction. These results are important in obstetrics, e.g., as a tool for helping to characterize contractions and to predict labor. For the aforementioned problem of multiple change-point estimation, a class of one-dimensional segmentation, we also compute fundamental mathematical results for minimal bounds on mean-square error estimation. Indeed, if an estimator is available, the evaluation of its performance depends on knowing whether it is optimal or if further improvement is still possible. In our segmentation problem the parameters are discrete therefore the conventional Cramer-Rao bound does not apply. Hence, we derive Barankin-type lower bounds, the greatest lower bound on the covariance of any unbiased estimator, which are applicable to discrete parameters. The computation of the bound is challenging, as it requires finding the supremum on a finite set of symmetric matrices with respect to the Loewner ordering, which is not a lattice order. Therefore, we discuss the existence of the supremum, propose a minimal upper-bound by using tools from convex geometry, and compute closed-form solutions for the Barankin information matrix for several distributions. The results have broad biomedical applications, such as DNA sequence segmentation, MEG and EEG segmentation, and uterine contraction MMG detection, and they also have applications for signal segmentation in general, such as speech segmentation and astronomical data analysis

A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields

Coherence-enhancing diffusion filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flow-like features in images. The completion of line-like structures is also a major concern in diffusion tensor magnetic resonance imaging (DT-MRI). This medical image acquisition technique outputs a 3D matrix field of symmetric 3×3-matrices, and it helps to visualise, for example, the nerve fibers in brain tissue. As any physical measurement DT-MRI is subjected to errors causing faulty representations of the tissue corrupted by noise and with visually interrupted lines or fibers. In this paper we address that problem by proposing a coherence-enhancing diffusion filtering methodology for matrix fields. The approach is based on a generic structure tensor concept for matrix fields that relies on the operator-algebraic properties of symmetric matrices, rather than their channel-wise treatment of earlier proposals. Numerical experiments with artificial and real DT-MRI data confirm the gap-closing and flow-enhancing qualities of the technique presented

Graph-based morphological processing of multivariate microscopy images and data bases

International audienceThe extension of lattice based operators to manifolds is still a challenging theme in mathematical morphology. In this paper, we propose to explicitly construct complete lattices and replace each element of a manifold by its rank suitable for classical morphological processing. Manifold learning is considered as the basis for the construction of a complete lattice. The whole processing of multivariate functions is expressed on graphs to have a formalism that can be applied on images, region adjacency graphs, and image databases. Several examples in microscopy do illustrate the benefits of the proposed approach