83 research outputs found
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
We propose an efficient approach for the grouping of local orientations
(points on vessels) via nilpotent approximations of sub-Riemannian distances in
the 2D and 3D roto-translation groups and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . In a qualitative validation we show that the norms provide
accurate approximations of the true sub-Riemannian distances, and we discuss
their relations to the fundamental solution of the sub-Laplacian on .
The quantitative experiments further confirm the accuracy of the
approximations. Quantitative results are obtained by evaluating perceptual
grouping performance of retinal blood vessels in 2D images and curves in
challenging 3D synthetic volumes. The results show that 1) sub-Riemannian
geometry is essential in achieving top performance and 2) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)
We present a new flexible wavefront propagation algorithm for the boundary
value problem for sub-Riemannian (SR) geodesics in the roto-translation group
with a metric tensor depending on a smooth
external cost , , computed from
image data. The method consists of a first step where a SR-distance map is
computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system
derived via Pontryagin's Maximum Principle (PMP). Subsequent backward
integration, again relying on PMP, gives the SR-geodesics. For
we show that our method produces the global minimizers. Comparison with exact
solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics.
We present numerical computations of Maxwell points and cusp points, which we
again verify for the uniform cost case . Regarding image
analysis applications, tracking of elongated structures in retinal and
synthetic images show that our line tracking generically deals with crossings.
We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven
Sub-Riemannian Geodesics in SE(2)
Regular SE(3) Group Convolutions for Volumetric Medical Image Analysis
Regular group convolutional neural networks (G-CNNs) have been shown to
increase model performance and improve equivariance to different geometrical
symmetries. This work addresses the problem of SE(3), i.e., roto-translation
equivariance, on volumetric data. Volumetric image data is prevalent in many
medical settings. Motivated by the recent work on separable group convolutions,
we devise a SE(3) group convolution kernel separated into a continuous SO(3)
(rotation) kernel and a spatial kernel. We approximate equivariance to the
continuous setting by sampling uniform SO(3) grids. Our continuous SO(3) kernel
is parameterized via RBF interpolation on similarly uniform grids. We
demonstrate the advantages of our approach in volumetric medical image
analysis. Our SE(3) equivariant models consistently outperform CNNs and regular
discrete G-CNNs on challenging medical classification tasks and show
significantly improved generalization capabilities. Our approach achieves up to
a 16.5% gain in accuracy over regular CNNs.Comment: 10 pages, 1 figure, 2 tables, accepted at MICCAI 2023. Updated
version to camera ready version
PDE-based Group Equivariant Convolutional Neural Networks
We present a PDE-based framework that generalizes Group equivariant
Convolutional Neural Networks (G-CNNs). In this framework, a network layer is
seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients
become the layer's trainable weights. Formulating our PDEs on homogeneous
spaces allows these networks to be designed with built-in symmetries such as
rotation in addition to the standard translation equivariance of CNNs.
Having all the desired symmetries included in the design obviates the need to
include them by means of costly techniques such as data augmentation. We will
discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space
setting while also going into the specifics of our primary case of interest:
roto-translation equivariance.
We solve the PDE of interest by a combination of linear group convolutions
and non-linear morphological group convolutions with analytic kernel
approximations that we underpin with formal theorems. Our kernel approximations
allow for fast GPU-implementation of the PDE-solvers, we release our
implementation with this article in the form of the LieTorch extension to
PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for
linear convolution a morphological convolution is specified by a kernel that we
train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as
max/min-pooling and ReLUs as they are already subsumed by morphological
convolutions.
We present a set of experiments to demonstrate the strength of the proposed
PDE-G-CNNs in increasing the performance of deep learning based imaging
applications with far fewer parameters than traditional CNNs.Comment: 27 pages, 18 figures. v2 changes: - mentioned KerCNNs - added section
Generalization of G-CNNs - clarification that the experiments utilized
automatic differentiation and SGD. v3 changes: - streamlined theoretical
framework - formulation and proof Thm.1 & 2 - expanded experiments. v4
changes: typos in Prop.5 and (20) v5/6 changes: minor revisio
On genuine invariance learning without weight-tying
In this paper, we investigate properties and limitations of invariance
learned by neural networks from the data compared to the genuine invariance
achieved through invariant weight-tying. To do so, we adopt a group theoretical
perspective and analyze invariance learning in neural networks without
weight-tying constraints. We demonstrate that even when a network learns to
correctly classify samples on a group orbit, the underlying decision-making in
such a model does not attain genuine invariance. Instead, learned invariance is
strongly conditioned on the input data, rendering it unreliable if the input
distribution shifts. We next demonstrate how to guide invariance learning
toward genuine invariance by regularizing the invariance of a model at the
training. To this end, we propose several metrics to quantify learned
invariance: (i) predictive distribution invariance, (ii) logit invariance, and
(iii) saliency invariance similarity. We show that the invariance learned with
the invariance error regularization closely reassembles the genuine invariance
of weight-tying models and reliably holds even under a severe input
distribution shift. Closer analysis of the learned invariance also reveals the
spectral decay phenomenon, when a network chooses to achieve the invariance to
a specific transformation group by reducing the sensitivity to any input
perturbation
Can strong structural encoding reduce the importance of Message Passing?
The most prevalent class of neural networks operating on graphs are message
passing neural networks (MPNNs), in which the representation of a node is
updated iteratively by aggregating information in the 1-hop neighborhood. Since
this paradigm for computing node embeddings may prevent the model from learning
coarse topological structures, the initial features are often augmented with
structural information of the graph, typically in the form of Laplacian
eigenvectors or Random Walk transition probabilities. In this work, we explore
the contribution of message passing when strong structural encodings are
provided. We introduce a novel way of modeling the interaction between feature
and structural information based on their tensor product rather than the
standard concatenation. The choice of interaction is compared in common
scenarios and in settings where the capacity of the message-passing layer is
severely reduced and ultimately the message-passing phase is removed
altogether. Our results indicate that using tensor-based encodings is always at
least on par with the concatenation-based encoding and that it makes the model
much more robust when the message passing layers are removed, on some tasks
incurring almost no drop in performance. This suggests that the importance of
message passing is limited when the model can construct strong structural
encodings
Latent Field Discovery In Interacting Dynamical Systems With Neural Fields
Systems of interacting objects often evolve under the influence of field
effects that govern their dynamics, yet previous works have abstracted away
from such effects, and assume that systems evolve in a vacuum. In this work, we
focus on discovering these fields, and infer them from the observed dynamics
alone, without directly observing them. We theorize the presence of latent
force fields, and propose neural fields to learn them. Since the observed
dynamics constitute the net effect of local object interactions and global
field effects, recently popularized equivariant networks are inapplicable, as
they fail to capture global information. To address this, we propose to
disentangle local object interactions -- which are equivariant
and depend on relative states -- from external global field effects -- which
depend on absolute states. We model interactions with equivariant graph
networks, and combine them with neural fields in a novel graph network that
integrates field forces. Our experiments show that we can accurately discover
the underlying fields in charged particles settings, traffic scenes, and
gravitational n-body problems, and effectively use them to learn the system and
forecast future trajectories.Comment: NeurIPS 2023. https://github.com/mkofinas/aethe
Attentive Group Equivariant Convolutional Networks
Although group convolutional networks are able to learn powerful
representations based on symmetry patterns, they lack explicit means to learn
meaningful relationships among them (e.g., relative positions and poses). In
this paper, we present attentive group equivariant convolutions, a
generalization of the group convolution, in which attention is applied during
the course of convolution to accentuate meaningful symmetry combinations and
suppress non-plausible, misleading ones. We indicate that prior work on visual
attention can be described as special cases of our proposed framework and show
empirically that our attentive group equivariant convolutional networks
consistently outperform conventional group convolutional networks on benchmark
image datasets. Simultaneously, we provide interpretability to the learned
concepts through the visualization of equivariant attention maps.Comment: Proceedings of the 37th International Conference on Machine Learning
(ICML), 202
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