\section{Related work}
\label{sec:related}
%Background on medical imaging??

% Related work sull'image generation in generale
In recent years, image generation has been advancing rapidly, evolving from GANs~\cite{heusel_gans_2017}, which produce high quality images but lack diversity and can suffer from mode collapse~\cite{che2017moderegularizedgenerativeadversarial}, to DDPMs~\cite{ho_denoising_2020}, which generate more diverse images but at the cost of lower resolution. Latent Diffusion Models address these limitations by enabling high-resolution image generation exploiting latent spaces, therefore improving efficiency and scalability~\cite{rombach_high-resolution_2022}. 
Recently, Flow Matching emerged as a novel generative modelling paradigm that subsumes DDPMs and enables more robust and stable training compared to diffusion models~\cite{lipman2023flowmatchinggenerativemodeling}. FM models learn to map noise to data by matching probability flow trajectories between two distributions. A particular instance of FM, Optimal Transport Flow Matching, implements this mapping as a straight line between samples drawn from the two distributions. In the domain of 2D natural image generation, this approach outperforms diffusion models in terms of both likelihood and sample quality~\cite{lipman2023flowmatchinggenerativemodeling, pmlr-v235-esser24a}.
However, OTFM has not yet been extensively studied in other domains, such as 3D medical image generation.
% Beyond 2D natural image generation, these techniques have been adapted for a variety of tasks involving three dimensions such as video generation~\cite{blattmann_align_2023, ho2022imagenvideohighdefinition}, 3D object synthesis~\cite{zhang20233dshape2vecset3dshaperepresentation, hui2022neuralwaveletdomaindiffusion3d}, and \textcolor{red}{3D medical} data generation. 

The medical domain poses its own challenges related both to the 3D nature of most medical imaging techniques, such as CTs and MRIs, and to the sensitivity of clinical applications, which require anatomically plausible synthetic data to avoid introducing harmful biases.
%GANs
The generation of 3D medical data has closely followed the advances in 2D natural image generation. GANs have been widely used for 3D image generation across different modalities and anatomical regions: to generate 3D Time-of-Flight Magnetic Resonance Angiography patches, brain MRI, thorax CTs, liver and spine CTs~\cite{SUBRAMANIAM2022102396, sun_9770375, kim_10452780}.
%DDPMs
However, due to their unstable training that often results in mode collapse, GANs have increasingly been replaced by diffusion-based models.
% proved to be a valid alternative and have been exploited for different medical data generation tasks. 
\citet{pinaya_brain_2022} leveraged LDMs to generate synthetic data from high-resolution 3D brain MRIs. \citet{khader_denoising_2023} proposed a similar approach using four different datasets with about $1000$ elements each, showing that it is possible to train generators with datasets of limited size.
% obtaining good results in generating novel data.
\citet{friedrich_wdm_2024} applied diffusion on wavelet decomposed 3D images for improving efficiency. \citet{wang20243dmeddiffusion3dmedical} presented a patch-wise autoencoder and a novel denoiser to generate both MRIs and CT scans.
% What are the limitations of DDPM models? Slow inference (we are not exploiting that)? Margins for improvement for images to be clinically valid? (we could use this but is a bit strong)
Recently, \citet{yazdani_flow_2025} tested OTFM with 2D echocardiographic images and 3D MRI data, but the study related to 3D data addresses only low resolution volumes ($138$x$169$x$138$) and quantitative evaluation is limited to generation quality metrics, which may not necessarily transfer to utility in clinical downstream tasks. 
Conversely, we rely on OTFM to train a generative model with high resolution 3D craniofacial skeletal data ($456$x$352$x$512$).
We compare its performance with that of a diffusion-based approach to assess whether the improvements observed in general 2D image generation extend to 3D medical image generation. Furthermore, we validate the generated synthetic data by evaluating their impact on two clinical downstream tasks.


\iffalse
Clustering related works:
\begin{enumerate}
    \item Cluster on Image generation (general)
    \item Cluster on Flow Matching (Lipmann, etc.) what did we change? We applied it to 3D medical images
    \item Cluster on medical image generation (2d and 3d). what did we change? We used FM instead of diffusion and shown improvements. We changed the architecture a little bit??
    \item (Cluster on high resolution 3d image generation??). what did we do? We ablated different resolutions and upscaling techniques to monitor improvements.
\end{enumerate}
\fi