Continuous extension in topological digital spaces

[EN] We give necessary and sufficient conditions for the existence of a continuous extension from a smallest-neighborhood space (Alexandrov space) X to the Khalimsky line. Using this result, we classify the subsets A X such that every continuous function A ! Zbcan be extended to all of X. We also consider the more general case ofbmappings X ! Y between smallest-neighborhood spaces, and prove abdigital no-retraction theorem for the Khalimsky planeMelin, E. (2008). Continuous extension in topological digital spaces. Applied General Topology. 9(1):51-66. doi:10.4995/agt.2008.1869.SWORD51669

THE LOGIC OF TIME AND THE CONTINUUM IN KANT’S CRITICAL PHILOSOPHY

We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers

The logic and topology of Kant's temporal continuum

In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises

Automated Morphometric Characterization of the Cerebral Cortex for the Developing and Ageing Brain

Morphometric characterisation of the cerebral cortex can provide information about patterns of brain development and ageing and may be relevant for diagnosis and estimation of the progression of diseases such as Alzheimer's, Huntington's, and schizophrenia. Therefore, understanding and describing the differences between populations in terms of structural volume, shape and thickness is of critical importance. Methodologically, due to data quality, presence of noise, PV effects, limited resolution and pathological variability, the automated, robust and time-consistent estimation of morphometric features is still an unsolved problem. This thesis focuses on the development of tools for robust cross-sectional and longitudinal morphometric characterisation of the human cerebral cortex. It describes techniques for tissue segmentation, structural and morphometric characterisation, cross-sectional and longitudinally cortical thickness estimation from serial MRI images in both adults and neonates. Two new probabilistic brain tissue segmentation techniques are introduced in order to accurately and robustly segment the brain of elderly and neonatal subjects, even in the presence of marked pathology. Two other algorithms based on the concept of multi-atlas segmentation propagation and fusion are also introduced in order to parcelate the brain into its multiple composing structures with the highest possible segmentation accuracy. Finally, we explore the use of the Khalimsky cubic complex framework for the extraction of topologically correct thickness measurements from probabilistic segmentations without explicit parametrisation of the edge. A longitudinal extension of this method is also proposed. The work presented in this thesis has been extensively validated on elderly and neonatal data from several scanners, sequences and protocols. The proposed algorithms have also been successfully applied to breast and heart MRI, neck and colon CT and also to small animal imaging. All the algorithms presented in this thesis are available as part of the open-source package NiftySeg

On Improving Generalization of CNN-Based Image Classification with Delineation Maps Using the CORF Push-Pull Inhibition Operator

Deployed image classification pipelines are typically dependent on the images captured in real-world environments. This means that images might be affected by different sources of perturbations (e.g. sensor noise in low-light environments). The main challenge arises by the fact that image quality directly impacts the reliability and consistency of classification tasks. This challenge has, hence, attracted wide interest within the computer vision communities. We propose a transformation step that attempts to enhance the generalization ability of CNN models in the presence of unseen noise in the test set. Concretely, the delineation maps of given images are determined using the CORF push-pull inhibition operator. Such an operation transforms an input image into a space that is more robust to noise before being processed by a CNN. We evaluated our approach on the Fashion MNIST data set with an AlexNet model. It turned out that the proposed CORF-augmented pipeline achieved comparable results on noise-free images to those of a conventional AlexNet classification model without CORF delineation maps, but it consistently achieved significantly superior performance on test images perturbed with different levels of Gaussian and uniform noise

Boundaries in digital spaces

[EN] Intuitively, a boundary in an N-dimensional digital space is a connected component of the (N − 1)-dimensional surface of a connected object. In this paper we make these concepts precise, and show that the boundaries so specified have properties that are intuitively desirable. We provide some efficient algorithms for tracking such boundaries. We illustrate that the algorithms can be used, in particular, for computer graphic display of internal structures (such as the skull and the spine) in the human body based on the output of medical imaging devices (such as CT scanners). In the process some interesting mathematical results are proven regarding “digital Jordan boundaries,” such as a specification of a local condition that guarantees the global condition of “Jordanness.”The research of the author is currently supported by NIH grant HL070472 and NSF grant DMS0306215.Herman, GT. (2007). Boundaries in digital spaces. Applied General Topology. 8(1):93-149. doi:10.4995/agt.2007.1918.SWORD931498

Preface Digitization in Khalimsky spaces

This Licentiate Thesis consists of an introduction below and the following two papers. (i) Digital straight lines in the Khalimsky plane (to appear in Mathematica Scandinavica) (ii) Digital curves and surfaces in Khalimsky spaces The second paper is an extension of a conference paper (How to find a Khalimsky-continuous approximation of a real-valued function, Combinatoria