Content and image generation consist in creating or generating data from
noisy information by extracting specific features such as texture, edges, and
other thin image structures. We are interested here in generative models, and
two main problems are addressed. Firstly, the improvements of specific feature
extraction while accounting at multiscale levels intrinsic geometric features;
and secondly, the equivariance of the network to reduce its complexity and
provide a geometric interpretability. To proceed, we propose a geometric
generative model based on an equivariant partial differential equation (PDE)
for group convolution neural networks (G-CNNs), so called PDE-G-CNNs, built on
morphology operators and generative adversarial networks (GANs). Equivariant
morphological PDE layers are composed of multiscale dilations and erosions
formulated in Riemannian manifolds, while group symmetries are defined on a Lie
group. We take advantage of the Lie group structure to properly integrate the
equivariance in layers, and are able to use the Riemannian metric to solve the
multiscale morphological operations. Each point of the Lie group is associated
with a unique point in the manifold, which helps us derive a metric on the
Riemannian manifold from a tensor field invariant under the Lie group so that
the induced metric has the same symmetries. The proposed geometric
morphological GAN (GM-GAN) is obtained by using the proposed morphological
equivariant convolutions in PDE-G-CNNs to bring nonlinearity in classical CNNs.
GM-GAN is evaluated on MNIST data and compared with GANs. Preliminary results
show that GM-GAN model outperforms classical GAN.

基于偏微分方程和形态学操作的几何生成对抗网络模型（GM-GAN）在 MNIST 数据集上展现了比传统生成对抗网络更好的性能。

基于形态等变偏微分方程和生成对抗网络的几何生成模型

Geometric Generative Models based on Morphological Equivariant PDEs and  GANs

PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) utilize solvers of
geometrically meaningful evolution PDEs as substitutes for the conventional
components in G-CNNs. PDE-G-CNNs offer several key benefits all at once: fewer
parameters, inherent equivariance, better performance, data efficiency, and
geometric interpretability. In this article we focus on Euclidean equivariant
PDE-G-CNNs where the feature maps are two dimensional throughout. We call this
variant of the framework a PDE-CNN. We list several practically desirable
axioms and derive from these which PDEs should be used in a PDE-CNN. Here our
approach to geometric learning via PDEs is inspired by the axioms of classical
linear and morphological scale-space theory, which we generalize by introducing
semifield-valued signals. Furthermore, we experimentally confirm for small
networks that PDE-CNNs offer fewer parameters, better performance, and data
efficiency in comparison to CNNs. We also investigate what effect the use of
different semifields has on the performance of the models.

PDE-CNNs, a variant of PDE-based Group Convolutional Neural Networks, offer fewer parameters, better performance, and data efficiency compared to CNNs, while utilizing semifield-valued signals for geometric interpretability.

PDE-CNNs：公理推导及应用

PDE-CNNs: Axiomatic Derivations and Applications

Research in 3D semantic segmentation has been increasing performance metrics,
like the IoU, by scaling model complexity and computational resources, leaving
behind researchers and practitioners that (1) cannot access the necessary
resources and (2) do need transparency on the model decision mechanisms. In
this paper, we propose SCENE-Net, a low-resource white-box model for 3D point
cloud semantic segmentation. SCENE-Net identifies signature shapes on the point
cloud via group equivariant non-expansive operators (GENEOs), providing
intrinsic geometric interpretability. Our training time on a laptop is 85~min,
and our inference time is 20~ms. SCENE-Net has 11 trainable geometrical
parameters and requires fewer data than black-box models. SCENE--Net offers
robustness to noisy labeling and data imbalance and has comparable IoU to
state-of-the-art methods. With this paper, we release a 40~000 Km labeled
dataset of rural terrain point clouds and our code implementation.

本文提出了一种低资源的白盒子模型 SCENE-Net，利用类群不变非扩张算子（GENEOs）在点云中识别签名形状，提供固有几何可解释性，在保证模型决策机制透明性的同时，使训练时间在 85 分钟、推理时间在 20ms，需要的可训练几何参数仅 11 个，数据要求较黑盒子模型更少。SCENE-Net 对噪声标签和数据不平衡具有鲁棒性，与最先进的方法具有可比的 IoU。同时，本文公布了 40000 公里的农村地形点云标记数据集和代码实现。