Jasper J. van de Gronde

Group-Invariant Frames for Colour Morphology

In theory, there is no problem generalizing morphological operators to colour images. In practice, it has proved quite tricky to define a generalization that “makes sense”. This could be because many generalizations violate our implicit assumptions about what kind of transformations should not matter. Or in other words, to what transformations operators should be invariant. As a possible solution, we propose using frames to explicitly construct operators invariant to a given group of transformations. We show how to create saturation- and rotation-invariant frames, and demonstrate how group-invariant frames can improve results.

Group-Invariant Colour Morphology Based on Frames

Mathematical morphology is a very popular framework for processing binary or grayscale images. One of the key problems in applying this framework to color images is the notorious false color problem. We discuss the nature of this problem and its origins. In doing so, it becomes apparent that the lack of invariance of operators to certain transformations (forming a group) plays an important role. The main culprits are the basic join and meet operations, and the associated lattice structure that forms the theoretical basis for mathematical morphology. We show how a lattice that is not group invariant can be related to another lattice that is. When all transformations in a group are linear, these lattices can be related to one another via the theory of frames. This provides all the machinery to let us transform any (grayscale or color) morphological filter into a group-invariant filter on grayscale or color images. We then demonstrate the potential for both subjective and objective improvement in selected tasks.

Path-Based Mathematical Morphology on Tensor Fields

Traditional path-based morphology allows finding long, approximately straight, paths in images. Although originally applied only to scalar images, we show how this can be a very good fit for tensor fields. We do this by constructing directed graphs representing such data, and then modifying the traditional path opening algorithm to work on these graphs. Cycles are dealt with by finding strongly connected components in the graph. Some examples of potential applications are given, including path openings that are not limited to a specific set of orientations.

Sponges for Generalized Morphology

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.

Frames, the Loewner order and eigendecomposition for morphological operators on tensor fields

Rotation invariance is an important property for operators on tensor fields, but up to now, most methods for morphology on tensor fields had to either sacrifice rotation invariance, or do without the foundation of mathematical morphology: a lattice structure. Recently, we proposed a framework for rotation-invariant mathematical morphology on tensor fields that does use a lattice structure. In addition, this framework can be derived systematically from very basic principles. Here we show how older methods for morphology on tensor fields can be interpreted within our framework. On the one hand this improves the theoretical underpinnings of these older methods, and on the other this opens up possibilities for improving the performance of our method. We discuss commonalities and differences of our method and two methods developed by Burgeth et al.

Frames for Tensor Field Morphology

We propose to apply our recently developed frame-based framework for group-invariant morphology to the problem of tensor field morphology. Group invariance (and particularly rotation invariance) have been, and are, motivated to be relevant for filtering tensor fields. This leads to the development of a rotation-invariant frame for tensors, which can be used to easily define rotation-invariant morphological operators on tensor fields. We also show how our method can be used to filter structure tensor fields.

Visual Data Exploration for Balance Quantification in Real-Time During Exergaming

Unintentional injuries are among the ten leading causes of death in older adults; falls cause 60% of these deaths. Despite their effectiveness to improve balance and reduce the risk of falls, balance training programs have several drawbacks in practice, such as lack of engaging elements, boring exercises, and the effort and cost of travelling, ultimately resulting in low adherence. Exergames, that is, digital games controlled by body movements, have been proposed as an alternative to improve balance. One of the main challenges for exergames is to automatically quantify balance during game-play in order to adapt the game difficulty according to the skills of the player. Here we perform a multidimensional exploratory data analysis, using visualization techniques, to find useful measures for quantifying balance in real-time. First, we visualize exergaming data, derived from 400 force plate recordings of 40 participants from 20 to 79 years and 10 trials per participant, as heat maps and violin plots to get quick insight into the nature of the data. Second, we extract known and new features from the data, such as instantaneous speed, measures of dispersion, turbulence measures derived from speed, and curvature values. Finally, we analyze and visualize these features using several visualizations such as a heat map, overlapping violin plots, a parallel coordinate plot, a projection of the two first principal components, and a scatter plot matrix. Our visualizations and findings suggest that heat maps and violin plots can provide quick insight and directions for further data exploration. The most promising measures to quantify balance in real-time are speed, curvature and a turbulence measure, because these measures show age-related changes in balance performance. The next step is to apply the present techniques to data of whole body movements as recorded by devices such as Kinect.

Generalized Morphology using Sponges

Abstract 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.

Fast Computation of Greyscale Path Openings

Path openings are morphological operators that are used to preserve long, thin, and curved structures in images. They have the ability to adapt to local image structures, which allows them to detect lines that are not perfectly straight. They are applicable in extracting cracks, roads, and more. Although path openings are very efficient to implement for binary images, the greyscale case is more problematic. This study provides an analysis of the main existing greyscale algorithm, and shows that although its time complexity can be quadratic in the number of pixels, this is optimal in terms of the output (if the full opening transform is created). Also, it is shown that under many circumstances the worst-case running time is much less than quadratic. Finally, a new algorithm is provided, which has the same time complexity, but is simpler, faster in practice and more amenable to parallelization.

Efficient Computation of Greyscale Path Openings

Abstract Path openings are morphological operators that are used to preserve long, thin, and curved structures in images. They have the ability to adapt to local image structures,which allows them to detect lines that are not perfectly straight. They are applicable in extracting cracks, roads, and similar structures. Although path openings are very efficient to implement for binary images, the greyscale case is more problematic. This study provides an analysis of the main existing greyscale algorithm, and shows that although its time complexity can be quadratic in the number of pixels, this is optimal in terms of the output (if the full opening transform is created). Also, it is shown that under many circumstances the worst-case running time is much less than quadratic. Finally, a new algorithm is provided,which has the same time complexity, but is simpler, faster in practice and more amenable to parallelization