Convolution | EPFL Graph Search

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus it is a cro

Spherical Convolutionnal Neural Networks: Empirical Analysis of SCNNs

Frédérick Matthieu Gusset

Convolutional neural networks (CNNs) are powerful tools in Deep Learning mainly due to their ability to exploit the translational symmetry present in images, as they are equivariant to translations. Other datasets present different types of symmetries (e.g. rotations), or lie on the sphere S2 (e.g. cosmological maps, omni-directional images, 3D models, ...). It is therefore of interest to design architectures that exploit the structure of the data and are equivariant to the 3D rotation group SO(3). Different architectures were designed to exploit these symmetries, such as 2D convolutions on planar projections, convolutions on the SO(3)group, or convolutions on graphs. The DeepSphere model approximates the sphere with a graph and performs graph convolutions. In this study, DeepSphere is evaluated against other spherical CNNs on different tasks.While the SO(3) convolution is equivariant to all rotations in SO(3), the graph convolutionis only equivariant to the rotations in S2 and invariant to the third rotation. Our experiments on SHREC-17 (a 3D shape retrieval task) show that DeepSphere achieves the same performance while being 40 times faster to train than Cohen et al. and 4 times faster than Esteves et al. Equivariance to the third rotation is an unnecessary price to pay. In order to prove these results, DeepSphere was tested on the similar dataset ModelNet40 (a shape classification task), and similar results as obtained by Esteves et al. were achieved. The odd behaviour with rotations (the performance worsens in presence of rotation perturbations) may be inherent to the task and the classes, instead of the models or the choice of the sampling scheme. Finally, regression tasks (both global and dense) were performed on GHCN-daily to prove the flexibility of DeepSphere with a non-hierarchical and irregular sampling of the sphere. The SCNN performed better than simply learning on the time series for each nodes.

2019

Resolution-robust Large Mask In painting with Fourier Convolutions

Anastasia Remizova

Modern image inpainting systems, despite the significant progress, often struggle with large missing areas, complex geometric structures, and high-resolution images. We find that one of the main reasons for that is the lack of an effective receptive field in both the inpainting network and the loss function. To alleviate this issue, we propose a new method called large mask inpainting (LaMa). LaMa is based on i) a new inpainting network architecture that uses fast Fourier convolutions (FFCs), which have the imagewide receptive field; ii) a high receptive field perceptual loss; iii) large training masks, which unlocks the potential of the first two components. Our inpainting network improves the state-of-the-art across a range of datasets and achieves excellent performance even in challenging scenarios, e.g. completion of periodic structures. Our model generalizes surprisingly well to resolutions that are higher than those seen at train time, and achieves this at lower parameter&time costs than the competitive baselines. The code is available at https://github.com/saic-mdal/lama.

IEEE COMPUTER SOC2022

Convolution Operators And Homomorphisms Of Locally Compact Groups

Let 1 < p < infinity, let G and H be locally compact groups and let c) be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of the Banach algebra CVp (G) of the p-convolution operators on G into CVp (H) which extends the usual definition of the image of a bounded measure by omega. We also discuss the uniqueness of this linear contraction onto important subalgebras of CVp(G). Even if G and H are abelian, we obtain new results. Let G(d) denote the group G provided with a discrete topology. As a corollary, we obtain, for every discrete measure, vertical bar parallel to mu vertical bar parallel to CVp(G)

Cambridge University Press2008

Spherical Convolutionnal Neural Networks: Empirical Analysis of SCNNs

Frédérick Matthieu Gusset

2019