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Generative models have recently gained increasing attention in image generation and editing tasks. However, they often lack a direct connection to object geometry, which is crucial in sensitive domains such as computational anatomy, biology, and robotics. This paper presents a novel framework for Image Generation informed by Geodesic dynamics (IGG) in deformation spaces. Our IGG model comprises two key components: (i) an efficient autoencoder that explicitly learns the geodesic path of image transformations in the latent space; and (ii) a latent geodesic diffusion model that captures the distribution of latent representations of geodesic deformations conditioned on text instructions. By leveraging geodesic paths, our method ensures smooth, topology-preserving, and interpretable deformations, capturing complex variations in image structures while maintaining geometric consistency. We validate the proposed IGG on plant growth data and brain magnetic resonance imaging (MRI). Experimental results show that IGG outperforms the state-of-the-art image generation/editing models with superior performance in generating realistic, high-quality images with preserved object topology and reduced artifacts. Our code is publicly available at https://github.com/nellie689/IGG. 

Abstract The decline in global plant diversity has raised concerns about its implications for carbon fixation and global greenhouse gas emissions (GGE), including carbon dioxide (CO₂), nitrous oxide (N₂O) and methane (CH₄). Therefore, we conducted a comprehensive meta‐analysis of 2103 paired observations, examining GGE, soil organic carbon (SOC) and plant carbon in plant mixtures and monocultures. Our findings indicate that plant mixtures decrease soil N₂O emissions by 21.4% compared to monocultures. No significant differences occurred between mixtures and monocultures for soil CO₂emissions, CH₄emissions or CH₄uptake. Plant mixtures exhibit higher SOC and plant carbon storage than monocultures. After 10 years of vegetation development, a 40% reduction in species richness decreases SOC content and plant carbon storage by 12.3% and 58.7% respectively. These findings offer insights into the intricate connections between plant diversity, soil and plant carbon storage and GGE—a critical but previously unexamined aspect of biodiversity–ecosystem functioning.