Geodesic Distance Estimation Research Articles

Practical Computation of the Cut Locus on Discrete Surfaces

AbstractWe present a novel method to compute the cut locus of a distance function encoded on a polygonal mesh. Our method exploits theoretical findings about the cut locus and – with a combination of analytic, geometric and topological tools – it is able to compute a topologically correct and geometrically accurate approximation of it. Our result can be either restricted to the mesh edges, or aligned with the real cut locus. Both outputs may be useful for practical applications. We also provide a convenient tool to optionally prune the weak branches of the cut locus, simplifying its structure. Our approach supersedes prior art, in that it is easier to use and also orders of magnitude faster. In fact, it depends on just one parameter, and it flawlessly operates on meshes with high genus and very high element count at interactive rates. We experiment with different datasets and methods for geodesic distance estimation. We also present applications to local and global surface parameterization.

Connecting dots: from local covariance to empirical intrinsic geometry and locally linear embedding

Local covariance structure under the manifold setup has been widely applied in the machine learning society. Based on the established theoretical results, we provide an extensive study of two relevant manifold learning algorithms, empirical intrinsic geometry (EIG) and the locally linear embedding (LLE) under the manifold setup. Particularly, we show that without an accurate dimension estimation, the geodesic distance estimation by EIG might be corrupted. Furthermore, we show that by taking the local covariance matrix into account, we can more accurately estimate the local geodesic distance. When understanding LLE based on the local covariance structure, its intimate relationship with the curvature suggests a variation of LLE depending on the "truncation scheme". We provide a theoretical analysis of the variation.

Validating forest types based on geological and land‐form features in central Amazonia

AbstractQuestionsAre topography‐based forest types floristically consistent between sites in central Amazonia? Do broad landform and geological features control site‐specific edaphic and floristic variation and therefore obfuscate the floristic classification based on local topographical classes? Is model‐based clustering a useful tool for floristic classification?LocationNon‐inundated forest of central Amazonia, north of the Amazon River.MethodsWe analysed species presence–absence of a group of terrestrial monocot herbs (Zingiberales) in 123 plots (250 × 2 m) concentrated in three sites of non‐inundated forests. Distances between plots were 1–140 km. Floristic patterns were extracted by dimensionality reduction using geodesic floristic distance. We applied a model‐based cluster analysis (MC) coupled with the Bayesian information criterion to determine the best floristic classification. We used geometric and non‐geometric internal evaluators to compare the performance of MC to the agglomerative hierarchical clustering method UPGMA. The floristic clusters were tested for differences in edaphic and topographic features. Landform‐geological classes were defined based on geological maps and a digital elevation model. We used the Kappa index and ANOVA to evaluate the agreement between landform–geological classes, floristic clusters and environmental features.ResultsThe best MC solution found four floristic clusters. Differences in soil chemical properties, which were linked with lithological classes and broad land‐form features, explained abrupt floristic changes and floristic differences between the same topographical habitats of different sites. Within poor soils, floristic classes defined by elevation along the soil catena (upland and valley forests) were fuzzy. Valley sandy forest was not floristically consistent across sites due to subtle edaphic variation. Using a non‐geometric internal evaluator, MC coupled with geodesic floristic distance estimation performed better overall than UPGMA.Main conclusionsGeological classes defined by lithology and broad landform features control the major variation of edaphic and floristic patterns in central Amazonia. MC proved to be a useful method to classify and interpret floristic patterns. Revised vegetation maps that account for lithology, broad land‐form features and edaphic conditions would therefore be a better proxy for regional floristic variation than the presently used simple classes based on position along the catena.

Visualising the structure of document search results: a comparison of graph theoretic approaches

Previous work has shown that distance-similarity visualisation or 'spatialisation' can provide a potentially useful context in which to browse the results of a query search, enabling the user to adopt a simple local foraging or 'cluster growing' strategy to navigate through the retrieved document set. However, faithfully mapping feature-space models to visual space can be problematic owing to their inherent high dimensionality and non-linearity. Conventional linear approaches to dimension reduction tend to fail at this kind of task, sacrificing local structural in order to preserve a globally optimal mapping. In this paper the clustering performance of a recently proposed algorithm called isometric feature mapping (Isomap), which deals with non-linearity by transforming dissimilarities into geodesic distances, is compared to that of nonmetric multidimensional scaling (MDS). Various graph pruning methods, for geodesic distance estimation, are also compared. Results show that Isomap is significantly better at preserving local structural detail than MDS, suggesting it is better suited to cluster growing and other semantic navigation tasks. Moreover, it is shown that applying a minimum-cost graph pruning criterion can provide a parameter-free alternative to the traditional K-neighbour method, resulting in spatial clustering that is equivalent to or better than that achieved using an optimal-K criterion.

Visualising the Structure of Document Search Results: A Comparison of Graph Theoretic Approaches

Previous work has shown that distance-similarity visualisation or ‘spatialisation’ can provide a potentially useful context in which to browse the results of a query search, enabling the user to adopt a simple local foraging or ‘cluster growing’ strategy to navigate through the retrieved document set. However, faithfully mapping feature-space models to visual space can be problematic owing to their inherent high dimensionality and non-linearity. Conventional linear approaches to dimension reduction tend to fail at this kind of task, sacrificing local structural in order to preserve a globally optimal mapping. In this paper the clustering performance of a recently proposed algorithm called isometric feature mapping (Isomap), which deals with non-linearity by transforming dissimilarities into geodesic distances, is compared to that of non-metric multidimensional scaling (MDS). Various graph pruning methods, for geodesic distance estimation, are also compared. Results show that Isomap is significantly better at preserving local structural detail than MDS, suggesting it is better suited to cluster growing and other semantic navigation tasks. Moreover, it is shown that applying a minimum-cost graph pruning criterion can provide a parameter-free alternative to the traditional K-neighbour method, resulting in spatial clustering that is equivalent to or better than that achieved using an optimal- K criterion.

A robust multimodal approach for emotion recognition

Emotion recognition is one of the latest challenges in intelligent human/computer communication. Most of previous work on emotion recognition focused on extracting emotions from visual or audio information separately. A novel approach is presented in this paper, including both visual and audio from video clips, to recognize the human emotion. The Facial Animation Parameters (FAPs) compliant facial feature tracking based on GASM (GPU based Active Shape Model) is performed on the video to generate two vector streams which represent the expression feature and the visual speech one. To extract effective speech features, based on geodesic distance estimation, we develop an enhanced Lipschitz embedding to embed high dimensional acoustic features into low dimensional space. Combined with the visual vectors, the audio vector is extracted in terms of low dimensional features. Then, a tripled Hidden Markov Model is introduced to perform the recognition which allows the state asynchrony of the audio and visual observation sequences while preserving their natural correlation over time. The experimental results show that this approach outperforms the conventional approaches for emotion recognition.

Improving geodesic distance estimation based on locally linear assumption

Geodesic distance estimation for data lying on a manifold is an important issue in many applications of nonlinear dimensionality reduction. In this paper, a method aiming at improving the precision of geodesic distance estimation is proposed. The method is constructed on the basic principle, locally linear assumption, underlying the manifold data. It presumes that the locally linear patch, expressed as a convex combination of neighbors of a vertex, approximately resides on the manifold, as well as the local neighborhood edge does. The proposed method essentially extends the search area from local edges, employed by existing methods, to local patches. This naturally leads to a more accurate geodesic distance estimation. An efficient algorithm for the method is constructed, and its computational complexity is also analyzed. Experiment results also show that the proposed method outperforms the existing methods in geodesic distance estimation.

Building k edge-disjoint spanning trees of minimum total length for isometric data embedding

Isometric data embedding requires construction of a neighborhood graph that spans all data points so that geodesic distance between any pair of data points could be estimated by distance along the shortest path between the pair on the graph. This paper presents an approach for constructing k-edge-connected neighborhood graphs. It works by finding k edge-disjoint spanning trees the sum of whose total lengths is a minimum. Experiments show that it outperforms the nearest neighbor approach for geodesic distance estimation.

Gait pattern analysis using selforganizing neural networks

Emotion recognition is one of the latest challenges in intelligent human/computer communication. Most of previous work on emotion recognition focused on extracting emotions from visual or audio information separately. A novel approach is presented in this paper, including both visual and audio from video clips, to recognize the human emotion. The Facial Animation Parameters (FAPs) compliant facial feature tracking based on GASM (GPU based Active Shape Model) is performed on the video to generate two vector streams which represent the expression feature and the visual speech one. To extract effective speech features, based on geodesic distance estimation, we develop an enhanced Lipschitz embedding to embed high dimensional acoustic features into low dimensional space. Combined with the visual vectors, the audio vector is extracted in terms of low dimensional features. Then, a tripled Hidden Markov Model is introduced to perform the recognition which allows the state asynchrony of the audio and visual observation sequences while preserving their natural correlation over time. The experimental results show that this approach outperforms the conventional approaches for emotion recognition.