5,950 research outputs found
H\"older Regularity of Geometric Subdivision Schemes
We present a framework for analyzing non-linear -valued
subdivision schemes which are geometric in the sense that they commute with
similarities in . It admits to establish
-regularity for arbitrary schemes of this type, and
-regularity for an important subset thereof, which includes all
real-valued schemes. Our results are constructive in the sense that they can be
verified explicitly for any scheme and any given set of initial data by a
universal procedure. This procedure can be executed automatically and
rigorously by a computer when using interval arithmetics.Comment: 31 pages, 1 figur
A global approach to the refinement of manifold data
A refinement of manifold data is a computational process, which produces a
denser set of discrete data from a given one. Such refinements are closely
related to multiresolution representations of manifold data by pyramid
transforms, and approximation of manifold-valued functions by repeated
refinements schemes. Most refinement methods compute each refined element
separately, independently of the computations of the other elements. Here we
propose a global method which computes all the refined elements simultaneously,
using geodesic averages. We analyse repeated refinements schemes based on this
global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
Algorithms and error bounds for multivariate piecewise constant approximation
We review the surprisingly rich theory of approximation of functions of many vari- ables by piecewise constants. This covers for example the Sobolev-PoincarĀ“e inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions
Learning single-image 3D reconstruction by generative modelling of shape, pose and shading
We present a unified framework tackling two problems: class-specific 3D
reconstruction from a single image, and generation of new 3D shape samples.
These tasks have received considerable attention recently; however, most
existing approaches rely on 3D supervision, annotation of 2D images with
keypoints or poses, and/or training with multiple views of each object
instance. Our framework is very general: it can be trained in similar settings
to existing approaches, while also supporting weaker supervision. Importantly,
it can be trained purely from 2D images, without pose annotations, and with
only a single view per instance. We employ meshes as an output representation,
instead of voxels used in most prior work. This allows us to reason over
lighting parameters and exploit shading information during training, which
previous 2D-supervised methods cannot. Thus, our method can learn to generate
and reconstruct concave object classes. We evaluate our approach in various
settings, showing that: (i) it learns to disentangle shape from pose and
lighting; (ii) using shading in the loss improves performance compared to just
silhouettes; (iii) when using a standard single white light, our model
outperforms state-of-the-art 2D-supervised methods, both with and without pose
supervision, thanks to exploiting shading cues; (iv) performance improves
further when using multiple coloured lights, even approaching that of
state-of-the-art 3D-supervised methods; (v) shapes produced by our model
capture smooth surfaces and fine details better than voxel-based approaches;
and (vi) our approach supports concave classes such as bathtubs and sofas,
which methods based on silhouettes cannot learn.Comment: Extension of arXiv:1807.09259, accepted to IJCV. Differentiable
renderer available at https://github.com/pmh47/dir
- ā¦