# Geometric Generative Models based on Morphological Equivariant PDEs and GANs

arXiv: https://arxiv.org/abs/2403.14897 · HTML: https://mlsanity.com/paper/2403.14897/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 80 references that are in the map.
- First posted on arXiv 22 Mar 2024.

## Builds on (most cited first, 30 of 37)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Generative Adversarial Text to Image Synthesis](https://mlsanity.com/paper/1605.05396/) (2016)
- [Unsupervised Image-to-Image Translation Networks](https://mlsanity.com/paper/1703.00848/) (2017)
- [Generative Image Inpainting with Contextual Attention](https://mlsanity.com/paper/1801.07892/) (2018)
- [StackGAN: Text to Photo-realistic Image Synthesis with Stacked Generative Adversarial Networks](https://mlsanity.com/paper/1612.03242/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [Adversarial Feature Learning](https://mlsanity.com/paper/1605.09782/) (2016)
- [Toward Multimodal Image-to-Image Translation](https://mlsanity.com/paper/1711.11586/) (2017)
- [Data Augmentation Generative Adversarial Networks](https://mlsanity.com/paper/1711.04340/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [EnhanceNet: Single Image Super-Resolution Through Automated Texture Synthesis](https://mlsanity.com/paper/1612.07919/) (2016)
- [Semantic Image Inpainting with Deep Generative Models](https://mlsanity.com/paper/1607.07539/) (2016)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Multi-Modal Imitation Learning from Unstructured Demonstrations using Generative Adversarial Nets](https://mlsanity.com/paper/1705.10479/) (2017)
- [Attentive Group Equivariant Convolutional Networks](https://mlsanity.com/paper/2002.03830/) (2020)
- [Generative Multi-Adversarial Networks](https://mlsanity.com/paper/1611.01673/) (2016)
- [3D G-CNNs for Pulmonary Nodule Detection](https://mlsanity.com/paper/1804.04656/) (2018)
- [PDE-based Group Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2001.09046/) (2020)
- [RAN4IQA: Restorative Adversarial Nets for No-Reference Image Quality Assessment](https://mlsanity.com/paper/1712.05444/) (2017)
- [Scale Equivariant U-Net](https://mlsanity.com/paper/2210.04508/) (2022)
- [A heterogeneous group CNN for image super-resolution](https://mlsanity.com/paper/2209.12406/) (2022)
- [Generating images with recurrent adversarial networks](https://mlsanity.com/paper/1602.05110/) (2016)

## Built on by (most cited first, 1 of 1)

- [Symmetry Matters: Auditing and Symmetrizing 3D Generative Models](https://mlsanity.com/paper/2512.18953/) (2025)

## Abstract (arXiv)

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.