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.

翻译：内容与图像生成旨在通过提取纹理、边缘及其他细微图像结构等特定特征，从含噪信息中创建或生成数据。本文聚焦于生成模型，主要解决两大问题：一是改进多尺度层面下固有几何特征的提取能力；二是通过网络的等变性降低模型复杂度并提供几何可解释性。为此，我们提出一种基于等变偏微分方程（PDE）的几何生成模型，该模型结合了形态学算子与生成对抗网络（GAN），用于群卷积神经网络（G-CNNs），即PDE-G-CNN架构。等变形态学PDE层由定义在黎曼流形上的多尺度膨胀与腐蚀运算构成，而群对称性则定义在李群上。我们利用李群结构在各层中合理整合等变性，并借助黎曼度规实现多尺度形态学运算。李群中的每个点与流形中的唯一一点相关联，这有助于我们从李群不变的张量场导出黎曼流形上的度规，使得诱导度规具有相同的对称性。通过将所提出的形态学等变卷积应用于PDE-G-CNNs中，向经典CNN引入非线性，从而构建了所提出的几何形态学生成对抗网络（GM-GAN）。GM-GAN在MNIST数据集上进行了评估，并与GAN进行了对比。初步结果表明，GM-GAN模型优于经典GAN。