An iterative algorithm for homology computation on simplicial shapes

We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes. We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators.To the best of our knowledge, this is the first algorithm based on the constructive Mayer–Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.     

Calculation of simplicial depth estimators for polynomial regression with applications

A fast algorithm for calculating the simplicial depth of a single parameter vector of a polynomial regression model is derived. Additionally, an algorithm for calculating the parameter vectors with maximum simplicial depth within an affine subspace of the parameter space or a polyhedron is presented. Since the maximum simplicial depth estimator is not unique, l₁ and l₂ methods are used to make the estimator unique. This estimator is compared with other estimators in examples of linear and quadratic regression. Furthermore, it is shown how the maximum simplicial depth can be used to derive distribution-free asymptotic α-level tests for testing hypotheses in polynomial regression models. The tests are applied on a problem of shape analysis where it is tested how the relative head length of the fish species Lepomis gibbosus depends on the size of these fishes. It is also tested whether the dependency can be described by the same polynomial regression function within different populations.     

Computing Bessel functions of the second kind in extreme parameter regimes

A useful method of computing the integral order Bessel functions of the second kind Y_n(x+iy) when either, the absolute value of the real part, or the imaginary part of the argument z=x+iy is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when x∼0 (or y∼0) and is useful because a number existing algorithms and methods fail to give correct results for small x or small y. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.     

Mapping two Eucalyptus subgenera using multiple endmember spectral mixture analysis and continuum-removed imaging spectrometry data

Successful discrimination of a variety of natural and urban landscape components has been achieved with remote sensing data using multiple endmember spectral mixture analysis (MESMA). MESMA is a spectral matching algorithm that addresses spectral variability by allowing multiple reference spectra (i.e., endmembers) to represent each material class. However, materials that have a high-degree of spectral similarity between classes, such as similar plant-types or closely related plant species, and large variations in albedo present an ongoing challenge for accurate class discrimination with imaging spectrometry. Continuum removal (CR) analysis may improve class separability by emphasizing individual absorption features across a normalized spectrum. The spectral and structural characteristics common to most Eucalyptus trees make them notoriously difficult to discriminate in closed-canopy forests with imaging spectrometry. We evaluated whether CR applied to hyperspectral remote sensing data improved the performance of MESMA in classifying and mapping nine eucalypt tree species according to the two major Eucalyptus subgenera, Eucalyptus (common name “monocalypt”) and Symphyomyrtus (common name “symphyomyrtle”). Mixed-canopies comprised of monocalypts and symphyomyrtles are common in Australia, although their spatial distribution is not random. The ability to map these functional types on a landscape-scale could provide important information about ecosystem processes, landscape disturbance history and wildlife habitat. We created a spectral library of 229 pixels from 37 symphyomyrtle tree canopies and 406 pixels from 62 monocalypt tree canopies selected from HyMap imagery and verified with field data. Based on these reference data, we achieved overall classification accuracies at the subgenera-level of 75% (Kappa 0.48) for non-CR spectra and 83% (Kappa 0.63) for the CR spectra. We found that continuum-removal improved the classification performance of most endmember-models, although a larger portion of pixels remained unmodeled with the CR spectra (2%) compared to the non-CR spectra (0%). We utilized a new method for model optimization and created maps of monocalypt and symphyomyrtle distribution in our study area based on our best performing endmember-models. Our vegetation maps were largely consistent with our expectations of subgenera distribution based on our knowledge of the region.     

New bounds on D-ary optimal codes

We propose a simple method that, given a symbol distribution, yields upper and lower bounds on the average code length of a D-ary optimal code over that distribution. Thanks to its simplicity, the method permits deriving analytical bounds for families of parametric distributions. We demonstrate this by obtaining new bounds, much better than the existing ones, for Zipf and exponential distributions when D>2.     

A parametrization of piecewise linear Lyapunov functions via linear programming

In this paper, we present a parametrization of piecewise linear (PWL) Lyapunov functions. To this end, we consider the class of all continuous PWL functions defined over a simplicial partition. We take advantage of a recently developed high level canonical PWL (HL CPWL) representation, which expresses the PWL function in a compact and closed form. Once the parametrization problem is properly stated, we focus on its application to the stabiilty analysis of dynamic systems. We consider uncertain non-linear systems and extend the sector condition obtained by Ohta et al. In addition, we propose a method of selecting an optimal candidate. One of the main advantages of this approach is that the parametrization and choice of the Lyapunov candidate, as well as the stability analysis, result in linear programming problems.     

On the direct decomposability of strong negations and S-implication operators on product lattices

In this paper, we discuss the direct product of strong negations. We characterize strong negations on product lattices that are direct products of strong negations. Furthermore, we present a method for constructing t-norms on product lattices which are not direct products. This method is quite general and useful to generate a wide class of such t-norms and solves an open problem posed in 2003 by Jenei and De Baets. Finally, we study direct decompositions of S-implication operators on product lattices.     

A multifractal simulation model for the distribution of VMS deposits in the Spanish segment of the Iberian Pyrite Belt

The Iberian Pyrite Belt (IPB) contains one of the greatest known concentrations of volcanic massive sulfides (VMS) on Earth. Investigation of the spatial attributes of the mineralization using multifractal methods, integrated within a GIS, provides new information, of potential interest to understand the distribution of VMS deposits. A new probability map was generated from three binary maps by weighting the geological constraints of the VMS deposits using the weights-of-evidence method (WofE). The resulting map can be considered a prospectivity map of the Spanish segment of the IPB. The aim of this study was to simulate the distribution of VMS deposits using a simple multifractal model, a multiplicative scaling cascade, based on the De Wijs scheme of the ore concentration. The application was developed and implemented in a GIS. Multifractal functions such as the generalized dimension (D_q) and the f(α) spectrum are different depending on whether or not the geological constraints are considered. The use of the probability map together with the results from the multifractal analysis has generated realistic simulations of the spatial distribution of VMS mineralization in the Spanish segment of the IPB.     

Object-oriented change detection for the city of Harare,Zimbabwe

Object building and the extraction of homogeneous landscape units on which spatial statistics can be applied is useful in assessing land use and land cover change. Object-oriented processing techniques are becoming more popular compared to traditional pixel-based image analysis. A hierarchical image segmentation approach was adopted to extract the objects from multi-temporal Landsat images over Zimbabwe. The spatial arrangement of t₀ and t₁ objects was independent as the segmentation process was independently applied, although object change of t₁ was based on t₀ boundaries. We applied a Standardized, Object Oriented, Automatic Classification (SOOAC) method based on fuzzy logic. The error matrix for the TM image had an overall accuracy of 95.6% and a KIA value of 94.7%, the ETM showed slightly lower overall accuracy. Various LULC changes were identified over the 13 year period per object and also per class, mainly vegetation decrease. Object-oriented change information is necessary in decision support systems and uncertainty management strategies. This approach addresses some of the major issues in object-based GIS change analysis as it is based on stable object geometry.     

The data type of spatial objects

A spatial object consists of data assigned to points in a space. Spatial objects, such as memory states and three dimensional graphical scenes, are diverse and ubiquitous in computing. We develop a general theory of spatial objects by modelling abstract data types of spatial objects as topological algebras of functions. One useful algebra is that of continuous functions, with operations derived from operations on space and data, and equipped with the compact-open topology. Terms are used as abstract syntax for defining spatial objects and conditional equational specifications are used for reasoning. We pose a completeness problem: Given a selection of operations on spatial objects, do the terms approximate all the spatial objects to arbitrary accuracy? We give some general methods for solving the problem and consider their application to spatial objects with real number attributes.     

Measuring incompatibility between Atanassov’s intuitionistic fuzzy sets

The aim of this paper is to establish an axiomatic definition of incompatibility measure in the framework of Atanassov’s intuitionistic fuzzy sets and use geometrical methods to build some families of such incompatibility measures. First, we construct several functions to measure incompatibility for an intuitionistic t-norm that can be represented by an adequate t-norm and t-conorm. Additionally, we establish some relations between some particular cases of these functions. Similarly, we then obtain incompatibility measures for a family of non-representable intuitionistic t-norms.     

A multi-resolution topological representation for non-manifold meshes

We address the problem of representing and processing 3D objects, described through simplicial meshes, which consist of parts of mixed dimensions, and with a non-manifold topology, at different levels of detail. First, we describe a multi-resolution model, that we call a non-manifold multi-tessellation (NMT), and we consider the selective refinement query, which is at the heart of several analysis operations on multi-resolution meshes. Next, we focus on a specific instance of a NMT, generated by simplifying simplicial meshes based on vertex-pair contraction, and we describe a compact data structure for encoding such a model. We also propose a new data structure for two-dimensional simplicial meshes, capable of representing both connectivity and adjacency information with a small memory overhead, which is used to describe the mesh extracted from an NMT through selective refinement. Finally, we present algorithms to efficiently perform updates on such a data structure.     

A unified subdivision approach for multi-dimensional non-manifold modeling

This paper presents a new unified subdivision scheme that is defined over a k-simplicial complex in n-D space with k≤3. We first present a series of definitions to facilitate topological inquiries during the subdivision process. The scheme is derived from the double (k+1)-directional box splines over k-simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a regular mesh. The subdivision rules are modified by spatial averaging to guarantee C¹ smoothness near extraordinary cases. Within a single framework, we combine the subdivision rules that can produce 1-, 2-, and 3-manifolds in arbitrary n-D space. Possible solutions for non-manifold regions between the manifolds with different dimensions are suggested as a form of selective subdivision rules according to user preference. We briefly describe the subdivision matrix analysis to ensure a reasonable smoothness across extraordinary topologies, and empirical results support our assumption. In addition, through modifications, we show that the scheme can easily represent objects with singularities, such as cusps, creases, or corners. We further develop local adaptive refinement rules that can achieve level-of-detail control for hierarchical modeling. Our implementation is based on the topological properties of a simplicial domain. Therefore, it is flexible and extendable. We also develop a solid modeling system founded on our subdivision schemes to show potential benefits of our work in industrial design, geometric processing, and other applications.     

A note on decisive symmetric games

Binary voting systems, usually represented by simple games, constitute a main DSS topic. A crucial feature of such a system is the easiness with which a proposal can be collectively accepted, which is measured by the “decisiveness index” of the corresponding game. We study here several functions related to the decisiveness of any simple game. The analysis, including the asymptotic behavior as the number n of players increases, is restricted to decisive symmetric games and their compositions, and it is assumed that all players have a common probability p to vote for the proposal. We show that, for n large enough, a small variation, either positive or negative, in p when p = 1/2 takes the decisiveness to quickly approach, respectively, 1 or 0. Moreover, we analyze the speed of the decisiveness convergence.     

Mapping understory vegetation using phenological characteristics derived from remotely sensed data

Understory vegetation is an important component in forest ecosystems not only because of its contributions to forest structure, function and species composition, but also due to its essential role in supporting wildlife species and ecosystem services. Therefore, understanding the spatio-temporal dynamics of understory vegetation is essential for management and conservation. Nevertheless, detailed information on the distribution of understory vegetation across large spatial extents is usually unavailable, due to the interference of overstory canopy on the remote detection of understory vegetation. While many efforts have been made to overcome this challenge, mapping understory vegetation across large spatial extents is still limited due to a lack of generality of the developed methods and limited availability of required remotely sensed data. In this study, we used understory bamboo in Wolong Nature Reserve, China as a case study to develop and test an effective and practical remote sensing approach for mapping understory vegetation. Using phenology metrics generated from a time series of Moderate Resolution Imaging Spectroradiometer data, we characterized the phenological features of forests with understory bamboo. Using maximum entropy modeling together with these phenology metrics, we successfully mapped the spatial distribution of understory bamboo (kappa: 0.59; AUC: 0.85). In addition, by incorporating elevation information we further mapped the distribution of two individual bamboo species, Bashania faberi and Fargesia robusta (kappa: 0.68 and 0.70; AUC: 0.91 and 0.92, respectively). Due to its generality, flexibility and extensibility, this approach constitutes an improvement to the remote detection of understory vegetation, making it suitable for mapping different understory species in different geographic settings. Both biodiversity conservation and wildlife habitat management may benefit from the detailed information on understory vegetation across large areas through the applications of this approach.     

Classification of functional data: A segmentation approach

We suggest a classification and feature extraction method on functional data where the predictor variables are curves. The method, called functional segment discriminant analysis (FSDA), combines the classical linear discriminant analysis and support vector machine. FSDA is particularly useful for irregular functional data, characterized by spatial heterogeneity and local patterns like spikes. FSDA not only reduces the computation and storage burden by using a fraction of the spectrum, but also identifies important predictors and extracts features. FSDA is highly flexible, easy to incorporate information from other data sources and/or prior knowledge from the investigators. We apply FSDA to two public domain data sets and discuss the understanding developed from the study.     

Minimizing the sum of the k largest functions in linear time

Given a collection of n functions defined on , and a polyhedral set , we consider the problem of minimizing the sum of the k largest functions of the collection over Q. Specifically we focus on collections of linear functions and several classes of convex, piecewise linear functions which are defined by location models. We present simple linear programming formulations for these optimization models which give rise to linear time algorithms when the dimension d is fixed. Our results improve complexity bounds of several problems reported recently by Tamir [Discrete Appl. Math. 109 (2001) 293-307], Tokuyama [Proc. 33rd Annual ACM Symp. on Theory of Computing, 2001, pp. 75-84] and Kalcsics, Nickel, Puerto and Tamir [Oper. Res. Lett. 31 (1984) 114-127].     

Use of SPOT 5 for mapping seagrasses: An application to Posidonia oceanica

The SPOT 5 satellite was launched in May 2002; it provides multispectral imagery with a spatial resolution of 10 m and fused imagery with a spatial resolution of 2.5 m. These types of satellite imagery were used for mapping beds of Posidonia oceanica in the Mediterranean Sea, where it is a dominant species forming monospecific beds in a structurally simple environment (four classes: sand, photophilous algae on rock, patchy seagrass beds and continuous seagrass beds). Supervised classifications by depth range were made of both types of image. A direct comparison of overall accuracy between SPOT 2.5 m and SPOT 10 m revealed that this tool provided accurate mapping in both cases (between 73 and 96% accuracy). Although SPOT 2.5 m provides lower overall accuracy than SPOT 10 m, it is a very useful tool for the mapping of P. oceanica, as it allows the patchiness of the formations to be better taken into account. The opportunity to use a reliability scale, which takes into account the effects of extrinsic factors on the processing of the images, confirmed the usefulness of the option of using a reduced pixel size in order to obtain an improved match between the results from mapping and field observations.     

A spatially constrained fuzzy hyper-prototype clustering algorithm

We present in this paper a fuzzy clustering algorithm which can handle spatially constraint problems often encountered in pattern recognition. The proposed method is based on the notions of hyperplanes, the fuzzy c-means, and spatial constraints. By adding a spatial regularizer into the fuzzy hyperplane-based objective function, the proposed method can take into account additionally important information of inherently spatial data. Experimental results have demonstrated that the proposed algorithm achieves superior results to some other popular fuzzy clustering models, and has potential for cluster analysis in spatial domain.     

Use of high-resolution satellite imagery in an integrated model to predict the distribution of shade coffee tree hybrid zones

In New Caledonia (21°S, 165°E), shade-grown coffee plantations were abandoned for economic reasons in the middle of the 20th century. Coffee species (Coffea arabica, C. canephora and C. liberica) were introduced from Africa in the late 19th century, they survived in the wild and spontaneously cross-hybridized. Coffee species were originally planted in native forest in association with leguminous trees (mostly introduced species) to improve their growth. Thus the canopy cover over rustic shade coffee plantations is heterogeneous with a majority of large crowns, attributed to leguminous trees. The aim of this study was to identify suitable areas for coffee inter-specific hybridization in New Caledonia using field based environmental parameters and remotely sensed predictors. Due to the complex structure of tropical vegetation, remote sensing imagery needs to be spatially accurate and to have the appropriate bands for monitoring vegetation cover. Quickbird panchromatic (black and white) imagery at 0.6 to 0.7 m spatial resolutions and multispectral imagery at 2.4 m spatial resolution were pansharpened and used for this study. The two most suitable remotely sensed indicators, canopy heterogeneity and tree crown size, were acquired by the sequential use of tree crown detection (neural network), image processing (such as textural analysis) and classification. All models were supervised and trained on learning data determined by human expertise. The final model has two remotely sensed indicators and three physical parameters based on the Digital Elevation Model: elevation, slope and water flow accumulation. Using these five predictive variables as inputs, two modelling methods, a decision tree and a neural network, were implemented. The decision tree, which showed 96.9% accuracy on the test set, revealed the involvement of ecological parameters in the hybridization of Coffea species. We showed that hybrid zones could be characterized by combinations of modalities, underlining the complexity of the environment concerned. For instance, forest heterogeneity and large crown size, steep slopes (> 53.5%) and elevation between 194 and 429 m asl, are favourable factors for Coffea inter-specific hybridization. The application of the neural network on the whole area gave a predictive map that distinguished the most suitable areas by means of a nonlinear continuous indicator. The map provides a confidence level for each area. The most favourable areas were geographically localized, providing a clue for the detection and conservation of favourable areas for Coffea species neo-diversity.