Modelling the generalised median correspondence through an edit distance.

On the one hand, classification applications modelled by structural pattern recognition, in which elements are represented as strings, trees or graphs, have been used for the last thirty years. In these models, structural distances are modelled as the correspondence (also called matching or labelling) between all the local elements (for instance nodes or edges) that generates the minimum sum of local distances. On the other hand, the generalised median is a well-known concept used to obtain a reliable prototype of data such as strings, graphs and data clusters. Recently, the structural distance and the generalised median has been put together to define a generalise median of matchings to solve some classification and learning applications. In this paper, we present an improvement in which the Correspondence edit distance is used instead of the classical Hamming distance. Experimental validation shows that the new approach obtains better results in reasonable runtime compared to other median calculation strategies

Generalised median of graph correspondences.

A graph correspondence is defined as a function that maps the elements of two attributed graphs. Due to the increasing availability of methods to perform graph matching, numerous graph correspondences can be deducted for a pair of attributed graphs. To obtain a representative prototype for a set of data structures, the concept of the median has been largely employed, as it has proven to deliver a robust sample. Nonetheless, the calculation of the exact (or generalised) median is known to be an NP-complete problem for most domains. In this paper, we present a method based on an optimisation function to calculate the generalised median graph correspondence. This method makes use of the Correspondence Edit Distance, which is a metric that considers the attributes and the local structures of the graphs to obtain more interesting and meaningful results. Experimental validation shows that this approach is capable of obtaining the generalised median in a comparable runtime with respect to state-of-the-art methods on artificial data, while maintaining the success rate for a real-application case

Obtaining the consensus of multiple correspondences between graphs through online learning.

In structural pattern recognition, it is usual to compare a pair of objects through the generation of a correspondence between the elements of each of their local parts. To do so, one of the most natural ways to represent these objects is through attributed graphs. Several existing graph extraction methods could be implemented and thus, numerous graphs, which may not only differ in their nodes and edge structure but also in their attribute domains, could be created from the same object. Afterwards, a matching process is implemented to generate the correspondence between two attributed graphs, and depending on the selected graph matching method, a unique correspondence is generated from a given pair of attributed graphs. The combination of these factors leads to the possibility of a large quantity of correspondences between the two original objects. This paper presents a method that tackles this problem by considering multiple correspondences to conform a single one called a consensus correspondence, eliminating both the incongruences introduced by the graph extraction and the graph matching processes. Additionally, through the application of an online learning algorithm, it is possible to deduce some weights that influence on the generation of the consensus correspondence. This means that the algorithm automatically learns the quality of both the attribute domain and the correspondence for every initial correspondence proposal to be considered in the consensus, and defines a set of weights based on this quality. It is shown that the method automatically tends to assign larger values to high quality initial proposals, and therefore is capable to deduce better consensus correspondences

Correspondence edit distance to obtain a set of weighted means of graph correspondences.

Given a pair of data structures, such as strings, trees, graphs or sets of points, several correspondences (also referred in literature as labellings, matchings or assignments) can be defined between their local parts. The Hamming distance has been largely used to define the dissimilarity of a pair of correspondences between two data structures. Although it has the advantage of being simple in computation, it does not consider the data structures themselves, which the correspondences relate to. In this paper, we extend the definitions of a recently presented distance between correspondences based on the concept of the edit distance, which we called Correspondence edit distance. Moreover, we present an algorithm to compute the set of weighted means between a pair of graph correspondences. Both the Correspondence edit distance and the computation of the set of weighted means are necessary for the calculation of a more representative prototype between a set of correspondences. In the validation section, we show how the use of the Correspondence edit distance increases the quality of the set of weighted means compared to using the Hamming distance

Pushing the boundaries for automated data reconciliation in official statistics

Jacco Daalmans presenteert een nieuwe wiskundige methode voor het ‘consistent maken’ van gegevens (data reconciliation) in officiële statistieken door statistische bureaus zoals het CBS. Wanneer één statistiek op meerdere plekken gepubliceerd wordt, dan moeten de uitkomsten hetzelfde zijn. Een aantal mag niet in de ene publicatie anders zijn dan in een andere. Een verschil in uitkomsten levert verwarring op. Eén van de oorzaken van verschillende uitkomsten is dat cijfers die een statistisch bureau verzamelt en verwerkt, uit tal van bronnen komen die meestal niet consistent zijn. Statistieken moeten ook aan onderlinge relaties voldoen. Een voorbeeld is dat twaalf maandcijfers op moeten tellen tot één jaarcijfer. Wanneer hier niet aan is voldaan, kan men naast het gegeven jaarcijfer, een alternatief jaarcijfer afleiden door de onderliggende maandcijfers op te tellen. Ook dan geen eenduidigheid over het ‘ware’ cijfer. Dit strookt niet met het doel van statistische bureaus om onbetwistbare statistieken te leveren. Om consistente statistische uitkomsten te maken is het nodig om gegevens aan te passen. De uitkomsten van verschillende statistieken worden een klein beetje gewijzigd, om ze daarmee beter op elkaar af te stemmen, bijvoorbeeld om te zorgen dat twaalf maandcijfers optellen tot een jaarcijfer. In het proefschrift wordt een nieuwe wiskundige methode voor formele data integratie ontwikkeld en toegepast. De nieuwe methode wordt nu op het CBS toegepast voor het samenstellen van Nationale Rekeningen. Dit gaat om zeer grote en gedetailleerde tabellen, die kunnen worden gezien als boekhouding van een land. Het bruto binnenlands product (BBP) is het bekendste cijfer uit de Nationale Rekeningen. Het aanpassen van data uit de Nationale Rekening is lastig omdat het gaat om zeer veel gegevens die onderling een sterke samenhang hebben. Stel bijvoorbeeld dat uit een confrontatie van bronnen blijkt dat de productie van een bepaald product naar boven moet worden bijgesteld. Dit betekent dat ook meer grondstof moet zijn verbruikt en dit betekent dan weer dat er ook meer van die grondstof moet zijn geproduceerd, of geïmporteerd, etc. Vóór de ingebruikneming van de wiskundige methode hingen correcties meer af van informele methoden zoals inschattingen van experts. De introductie van de formele, wiskundige methode vergroot de reproduceerbaarheid en transparantie van de statistiek. De methode die in het proefschrift is voorgesteld is vergeleken met een andere methode, die door een ander statistisch bureau is toegepast. De nieuwe methode heeft een belangrijke eigenschap waaraan de andere methode niet voldoet. Kortgezegd houdt die eigenschap in dat de resultaten hetzelfde moeten blijven als de richting van de tijd zou worden omgekeerd. Het proefschrift onderzoekt verder toepassingen van data integratiemethoden buiten het traditionele toepassingsgebied van de Nationale Rekeningen, zoals bij de volkstelling en bij bedrijfseconomische statistieken. De voorgestelde methoden blijken sommige problemen van de huidig toegepaste methoden te vermijden

Inferring Geodesic Cerebrovascular Graphs: Image Processing, Topological Alignment and Biomarkers Extraction

A vectorial representation of the vascular network that embodies quantitative features - location, direction, scale, and bifurcations - has many potential neuro-vascular applications. Patient-specific models support computer-assisted surgical procedures in neurovascular interventions, while analyses on multiple subjects are essential for group-level studies on which clinical prediction and therapeutic inference ultimately depend. This first motivated the development of a variety of methods to segment the cerebrovascular system. Nonetheless, a number of limitations, ranging from data-driven inhomogeneities, the anatomical intra- and inter-subject variability, the lack of exhaustive ground-truth, the need for operator-dependent processing pipelines, and the highly non-linear vascular domain, still make the automatic inference of the cerebrovascular topology an open problem. In this thesis, brain vessels’ topology is inferred by focusing on their connectedness. With a novel framework, the brain vasculature is recovered from 3D angiographies by solving a connectivity-optimised anisotropic level-set over a voxel-wise tensor field representing the orientation of the underlying vasculature. Assuming vessels joining by minimal paths, a connectivity paradigm is formulated to automatically determine the vascular topology as an over-connected geodesic graph. Ultimately, deep-brain vascular structures are extracted with geodesic minimum spanning trees. The inferred topologies are then aligned with similar ones for labelling and propagating information over a non-linear vectorial domain, where the branching pattern of a set of vessels transcends a subject-specific quantized grid. Using a multi-source embedding of a vascular graph, the pairwise registration of topologies is performed with the state-of-the-art graph matching techniques employed in computer vision. Functional biomarkers are determined over the neurovascular graphs with two complementary approaches. Efficient approximations of blood flow and pressure drop account for autoregulation and compensation mechanisms in the whole network in presence of perturbations, using lumped-parameters analog-equivalents from clinical angiographies. Also, a localised NURBS-based parametrisation of bifurcations is introduced to model fluid-solid interactions by means of hemodynamic simulations using an isogeometric analysis framework, where both geometry and solution profile at the interface share the same homogeneous domain. Experimental results on synthetic and clinical angiographies validated the proposed formulations. Perspectives and future works are discussed for the group-wise alignment of cerebrovascular topologies over a population, towards defining cerebrovascular atlases, and for further topological optimisation strategies and risk prediction models for therapeutic inference. Most of the algorithms presented in this work are available as part of the open-source package VTrails

Dynamical models and machine learning for supervised segmentation

This thesis is concerned with the problem of how to outline regions of interest in medical images, when the boundaries are weak or ambiguous and the region shapes are irregular. The focus on machine learning and interactivity leads to a common theme of the need to balance conflicting requirements. First, any machine learning method must strike a balance between how much it can learn and how well it generalises. Second, interactive methods must balance minimal user demand with maximal user control. To address the problem of weak boundaries,methods of supervised texture classification are investigated that do not use explicit texture features. These methods enable prior knowledge about the image to benefit any segmentation framework. A chosen dynamic contour model, based on probabilistic boundary tracking, combines these image priors with efficient modes of interaction. We show the benefits of the texture classifiers over intensity and gradient-based image models, in both classification and boundary extraction. To address the problem of irregular region shape, we devise a new type of statistical shape model (SSM) that does not use explicit boundary features or assume high-level similarity between region shapes. First, the models are used for shape discrimination, to constrain any segmentation framework by way of regularisation. Second, the SSMs are used for shape generation, allowing probabilistic segmentation frameworks to draw shapes from a prior distribution. The generative models also include novel methods to constrain shape generation according to information from both the image and user interactions. The shape models are first evaluated in terms of discrimination capability, and shown to out-perform other shape descriptors. Experiments also show that the shape models can benefit a standard type of segmentation algorithm by providing shape regularisers. We finally show how to exploit the shape models in supervised segmentation frameworks, and evaluate their benefits in user trials