Harmonic active contours.

We propose a segmentation method based on the geometric representation of images as 2-D manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and results in a segmentation criterion that exploits the directional information of image gradients to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: 1) the natural coupling of image channels proposed for anisotropic diffusion and 2) the ability of subjective surfaces to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours, the active contours without edges, and the robust edge integrator) to higher dimensional spaces, non-flat images, and feature spaces. Our experiments show that the proposed technique improves the segmentation of multi-channel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.

GeoKernels: Modeling of Spatial Data on GeoManifolds

This paper presents a review of methodology for semi-supervised modeling with kernel methods, when the manifold assumption is guaranteed to be satisfied. It concerns environmental data modeling on natural manifolds, such as complex topographies of the mountainous regions, where environmental processes are highly influenced by the relief. These relations, possibly regionalized and nonlinear, can be modeled from data with machine learning using the digital elevation models in semi-supervised kernel methods. The range of the tools and methodological issues discussed in the study includes feature selection and semisupervised Support Vector algorithms. The real case study devoted to data-driven modeling of meteorological fields illustrates the discussed approach.

Machine learning algorithms for spatial data. Case studies: Environmental pollution, natural hazards, renewable resources

This paper presents general problems and approaches for the spatial data analysis using machine learning algorithms. Machine learning is a very powerful approach to adaptive data analysis, modelling and visualisation. The key feature of the machine learning algorithms is that they learn from empirical data and can be used in cases when the modelled environmental phenomena are hidden, nonlinear, noisy and highly variable in space and in time. Most of the machines learning algorithms are universal and adaptive modelling tools developed to solve basic problems of learning from data: classification/pattern recognition, regression/mapping and probability density modelling. In the present report some of the widely used machine learning algorithms, namely artificial neural networks (ANN) of different architectures and Support Vector Machines (SVM), are adapted to the problems of the analysis and modelling of geo-spatial data. Machine learning algorithms have an important advantage over traditional models of spatial statistics when problems are considered in a high dimensional geo-feature spaces, when the dimension of space exceeds 5. Such features are usually generated, for example, from digital elevation models, remote sensing images, etc. An important extension of models concerns considering of real space constrains like geomorphology, networks, and other natural structures. Recent developments in semi-supervised learning can improve modelling of environmental phenomena taking into account on geo-manifolds. An important part of the study deals with the analysis of relevant variables and models' inputs. This problem is approached by using different feature selection/feature extraction nonlinear tools. To demonstrate the application of machine learning algorithms several interesting case studies are considered: digital soil mapping using SVM, automatic mapping of soil and water system pollution using ANN; natural hazards risk analysis (avalanches, landslides), assessments of renewable resources (wind fields) with SVM and ANN models, etc. The dimensionality of spaces considered varies from 2 to more than 30. Figures 1, 2, 3 demonstrate some results of the studies and their outputs. Finally, the results of environmental mapping are discussed and compared with traditional models of geostatistics.