372 research outputs found
Morphological Scale-Space Operators for Images Supported on Point Clouds
International audienceThe aim of this paper is to develop the theory, and to propose an algorithm, for morphological processing of images painted on point clouds, viewed as a length metric measure space . In order to extend morphological operators to process point cloud supported images, one needs to define dilation and erosion as semigroup operators on . That corresponds to a supremal convolution (and infimal convolution) using admissible structuring function on . From a more theoretical perspective, we introduce the notion of abstract structuring functions formulated on length metric Maslov idempotent measurable spaces, which is the appropriate setting for . In practice, computation of Maslov structuring function is approached by a random walks framework to estimate heat kernel on , followed by the logarithmic trick
Non-Redundant Spectral Dimensionality Reduction
Spectral dimensionality reduction algorithms are widely used in numerous
domains, including for recognition, segmentation, tracking and visualization.
However, despite their popularity, these algorithms suffer from a major
limitation known as the "repeated Eigen-directions" phenomenon. That is, many
of the embedding coordinates they produce typically capture the same direction
along the data manifold. This leads to redundant and inefficient
representations that do not reveal the true intrinsic dimensionality of the
data. In this paper, we propose a general method for avoiding redundancy in
spectral algorithms. Our approach relies on replacing the orthogonality
constraints underlying those methods by unpredictability constraints.
Specifically, we require that each embedding coordinate be unpredictable (in
the statistical sense) from all previous ones. We prove that these constraints
necessarily prevent redundancy, and provide a simple technique to incorporate
them into existing methods. As we illustrate on challenging high-dimensional
scenarios, our approach produces significantly more informative and compact
representations, which improve visualization and classification tasks
Perturbation of the Eigenvectors of the Graph Laplacian: Application to Image Denoising
The original contributions of this paper are twofold: a new understanding of
the influence of noise on the eigenvectors of the graph Laplacian of a set of
image patches, and an algorithm to estimate a denoised set of patches from a
noisy image. The algorithm relies on the following two observations: (1) the
low-index eigenvectors of the diffusion, or graph Laplacian, operators are very
robust to random perturbations of the weights and random changes in the
connections of the patch-graph; and (2) patches extracted from smooth regions
of the image are organized along smooth low-dimensional structures in the
patch-set, and therefore can be reconstructed with few eigenvectors.
Experiments demonstrate that our denoising algorithm outperforms the denoising
gold-standards
Constructing Desirable Scalar Fields for Morse Analysis on Meshes
Morse theory is a powerful mathematical tool that uses the local differential properties of a manifold to make conclusions about global topological aspects of the manifold. Morse theory has been proven to be a very useful tool in computer graphics, geometric data processing and understanding. This work is divided into two parts. The first part is concerned with constructing geometry and symmetry aware scalar functions on a triangulated -manifold. To effectively apply Morse theory to discrete manifolds, one needs to design scalar functions on them with certain properties such as respecting the symmetry and the geometry of the surface and having the critical points of the scalar function coincide with feature or symmetry points on the surface. In this work, we study multiple methods that were suggested in the literature to construct such functions such as isometry invariant scalar functions, Poisson fields and discrete conformal factors. We suggest multiple refinements to each family of these functions and we propose new methods to construct geometry and symmetry-aware scalar functions using mainly the theory of the Laplace-Beltrami operator. Our proposed methods are general and can be applied in many areas such as parametrization, shape analysis, symmetry detection and segmentation. In the second part of the thesis we utilize Morse theory to give topologically consistent segmentation algorithms
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