373 research outputs found
Learning shape correspondence with anisotropic convolutional neural networks
Establishing correspondence between shapes is a fundamental problem in
geometry processing, arising in a wide variety of applications. The problem is
especially difficult in the setting of non-isometric deformations, as well as
in the presence of topological noise and missing parts, mainly due to the
limited capability to model such deformations axiomatically. Several recent
works showed that invariance to complex shape transformations can be learned
from examples. In this paper, we introduce an intrinsic convolutional neural
network architecture based on anisotropic diffusion kernels, which we term
Anisotropic Convolutional Neural Network (ACNN). In our construction, we
generalize convolutions to non-Euclidean domains by constructing a set of
oriented anisotropic diffusion kernels, creating in this way a local intrinsic
polar representation of the data (`patch'), which is then correlated with a
filter. Several cascades of such filters, linear, and non-linear operators are
stacked to form a deep neural network whose parameters are learned by
minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic
dense correspondences between deformable shapes in very challenging settings,
achieving state-of-the-art results on some of the most difficult recent
correspondence benchmarks
Geometric deep learning
The goal of these course notes is to describe the main mathematical ideas behind geometric deep learning and to provide implementation details for several applications in shape analysis and synthesis, computer vision and computer graphics. The text in the course materials is primarily based on previously published work. With these notes we gather and provide a clear picture of the key concepts and techniques that fall under the umbrella of geometric deep learning, and illustrate the applications they enable. We also aim to provide practical implementation details for the methods presented in these works, as well as suggest further readings and extensions of these ideas
Geometric deep learning: going beyond Euclidean data
Many scientific fields study data with an underlying structure that is a
non-Euclidean space. Some examples include social networks in computational
social sciences, sensor networks in communications, functional networks in
brain imaging, regulatory networks in genetics, and meshed surfaces in computer
graphics. In many applications, such geometric data are large and complex (in
the case of social networks, on the scale of billions), and are natural targets
for machine learning techniques. In particular, we would like to use deep
neural networks, which have recently proven to be powerful tools for a broad
range of problems from computer vision, natural language processing, and audio
analysis. However, these tools have been most successful on data with an
underlying Euclidean or grid-like structure, and in cases where the invariances
of these structures are built into networks used to model them. Geometric deep
learning is an umbrella term for emerging techniques attempting to generalize
(structured) deep neural models to non-Euclidean domains such as graphs and
manifolds. The purpose of this paper is to overview different examples of
geometric deep learning problems and present available solutions, key
difficulties, applications, and future research directions in this nascent
field
Deep Functional Maps: Structured Prediction for Dense Shape Correspondence
We introduce a new framework for learning dense correspondence between
deformable 3D shapes. Existing learning based approaches model shape
correspondence as a labelling problem, where each point of a query shape
receives a label identifying a point on some reference domain; the
correspondence is then constructed a posteriori by composing the label
predictions of two input shapes. We propose a paradigm shift and design a
structured prediction model in the space of functional maps, linear operators
that provide a compact representation of the correspondence. We model the
learning process via a deep residual network which takes dense descriptor
fields defined on two shapes as input, and outputs a soft map between the two
given objects. The resulting correspondence is shown to be accurate on several
challenging benchmarks comprising multiple categories, synthetic models, real
scans with acquisition artifacts, topological noise, and partiality.Comment: Accepted for publication at ICCV 201
Geometric deep learning for shape analysis: extending deep learning techniques to non-Euclidean manifolds
The past decade in computer vision research has witnessed the re-emergence of artificial neural networks (ANN), and in particular convolutional neural network (CNN) techniques, allowing to learn powerful feature representations from large collections of data. Nowadays these techniques are better known under the umbrella term deep learning and have achieved a breakthrough in performance in a wide range of image analysis applications such as image classification, segmentation, and annotation. Nevertheless, when attempting to apply deep learning paradigms to 3D shapes one has to face fundamental differences between images and geometric objects. The main difference between images and 3D shapes is the non-Euclidean nature of the latter. This implies that basic operations, such as linear combination or convolution, that are taken for granted in the Euclidean case, are not even well defined on non-Euclidean domains. This happens to be the major obstacle that so far has precluded the successful application of deep learning methods on non-Euclidean geometric data. The goal of this thesis is to overcome this obstacle by extending deep learning tecniques (including, but not limiting to CNNs) to non-Euclidean domains. We present different approaches providing such extension and test their effectiveness in the context of shape similarity and correspondence applications. The proposed approaches are evaluated on several challenging experiments, achieving state-of-the- art results significantly outperforming other methods. To the best of our knowledge, this thesis presents different original contributions. First, this work pioneers the generalization of CNNs to discrete manifolds. Second, it provides an alternative formulation of the spectral convolution operation in terms of the windowed Fourier transform to overcome the drawbacks of the Fourier one. Third, it introduces a spatial domain formulation of convolution operation using patch operators and several ways of their construction (geodesic, anisotropic diffusion, mixture of Gaussians). Fourth, at the moment of publication the proposed approaches achieved state-of-the-art results in different computer graphics and vision applications such as shape descriptors and correspondence
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