Conditionally complete sponges: new results on generalized lattices

Sponges were recently proposed as a generalization of lattices, focussing on joins/meets of sets, while letting go of associativity/transitivity. In this work we provide tools for characterizing and constructing sponges on metric spaces and groups. These are then used in a characterization of epigraph sponges: a new class of sponges on Hilbert spaces whose sets of left/right bounds are formed by the epigraph of a rotationally symmetric function. We also show that the so-called hyperbolic sponge generalizes to more than two dimensions.Comment: 24 page

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

Efficient 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 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 parallelizatio

Assessing Dynamic Balance Performance during Exergaming based on Speed and Curvature of Body Movements

Improving balance performance among the elderly is of utmost importance because of the increasing number of injuries and fatalities caused by fall incidences. Digital games controlled by body movements (exergames) have been proposed as a way to improve balance among older people. However, the assessment of balance performance in real-time during exergaming remains a challenging task. This assessment could be used to provide instantaneous feedback and automatically adjust the exergame difficulty. Such features could potentially increase the motivation of the player, thus augmenting the effectiveness of exergames. As clear differences in balance performance have been identified between older and younger people, distinguishing between older and younger adults can help identifying measures of balance performance. We used generalized linear models to investigate whether the assessment of balance performance based on movement speed can be improved by incorporating curvature of the movement trajectory into the analysis. Indeed, our results indicated that curvature improves the performance of the models. Five-fold cross validation indicated that our method is promising for the assessment of balance performance in real-time by showing more than 90% classification accuracy. Finally, this method could be valuable not only for exergaming but also for real-time assessment of body movements in sports, rehabilitation and medicine

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

Beyond scalar morphology: A theoretical framework for mathematical morphology on colour and tensor-valued images

Mathematische morfologie is een aanpak voor het verwerken van vormen, typisch toegepast op monochrome beelden. Helaas is het niet eenvoudig deze aanpak uit te breiden naar kleurenplaatjes, laat staan diffusie MRI scans van het brein. Het probleem is dat hoewel de theorie best uit te breiden is, dit vaak leidt tot tegenintuïtieve resultaten. In dit proefschrift wordt beargumenteerd dat dit komt omdat dergelijke uitbreidingen conflicteren met onze interpretatie van de gegevens (in tegenstelling tot computers ervaren mensen kleuren bijv. niet als combinaties van rood, groen en blauw). Door de gegevens anders te representeren, of een nieuw formalisme te hanteren, kunnen dit soort problemen vermeden worden. Hopelijk leidt dit in de toekomst tot eenvoudiger analyses van niet alleen kleurenbeelden, maar ook breinscans, vloeistofstromingen, en meer

Nonscalar Mathematical Morphology

Mathematical morphology is not always straightforward to generalize to situations where values in an image do not admit a natural total order. This mostly due to the limitations of the underlying formalism. Three state-of-the-art solutions for bypassing those limitations are discussed, with example applications ranging from color images interpreted as vector spaces to periodic and hyperbolic value spaces, and categorical data stemming from per-pixel classification of remote sensing images