\begin{figure}[!t]
    \centering
    \includegraphics[width=0.95\linewidth]{Figure/title_fig.png}
    \caption{Generated images for the 17 planar groups using the prompt \emph{stained-glass mosaic fragments...}. Annotated symmetry transformations demonstrate that our patterns exhibit perceptually perfect preservation of the target symmetries.}
    \label{fig:title_fig}
\end{figure}

\section{Introduction}
\label{sec:introduction}

In visual arts and manufacturing engineering, symmetry plays an important role. It shapes geometric aesthetics and serves as a geometric prior that facilitates manufacturing processes, for example by enabling mold reuse, while also helping satisfy physical requirements. Classic examples range from wallpaper art to reflection-symmetric patterns in paper-cutting crafts formed by folding and unfolding, as well as periodic designs in architectural ornaments and lattice structures. All planar symmetry groups are illustrated in \cref{fig:title_fig}. In these tasks, designers often seek to impose constraints associated with planar symmetry groups while preserving spatial continuity and avoiding abrupt transitions.

However, existing multimodal large language models (MLLMs) fail to produce strictly symmetric images even with rich textual and visual guidance. Previous work such as \citet{bergmannLearningTextureManifolds2017a} studies periodic texture generation, but does not provide a approach with exact symmetry across planar groups. We therefore seek symmetry-embedded representations and begin with the symmetrization of general 2D representations. A naive approach is to define the representation only on the asymmetric unit and extend it to the full plane via group transformations.

Unfortunately, for continuous representation, extending a function from the asymmetric unit by symmetry can introduce boundary discontinuities for non-reflective transformations: points just inside the boundary remain fixed, while points just outside are abruptly mapped to the opposite side. To resolve this, we propose a symmetric continuous representation framework that embeds any planar group into an affine reflection group to preserve boundary continuity, and constructs a continuous $G$-invariant field via a combination of high-symmetry coefficients and low-symmetry bases.

Based on this representation, we develop a unified pipeline for symmetry-constrained controllable generation with diffusion priors. Given a target planar group, we optimize the parameters of the proposed symmetric representation with respect to the loss functions. This separates symmetry constraints from other task-specific objectives: symmetry is handled by the representation, while other task-specific objectives can be imposed through loss functions. As a result, the same framework can be applied across diverse design tasks without collecting symmetric data or training symmetry-specific generative models.

We validate the versatility of our framework through three visual design tasks: 
(i) Pattern Design: Generating symmetric RGB images that align with given text descriptions (\textit{Visual Semantic Constraints}).
(ii) Paper-Cutting Design: Generating globally connected, symmetric binary masks subject to volume constraints and text semantics (\textit{Visual + Connectivity Constraints}).
(iii) Topology Design: Generating connected binary masks optimized for mechanical properties under volume constraints, alongside text-aligned stylized images (\textit{Visual + Connectivity + Mechanical Constraints}). In addition to visual design, we further extend the same idea of symmetry control to metamaterial design based on diffusion prior. Experimental results demonstrate that our framework achieves stable symmetry control under physical constraints.

\textbf{Organization.}
\cref{sec:formulation} details the theoretical construction of the symmetric representation based on affine reflection group embeddings. \cref{sec:computation} presents the computation of the required bases. \cref{sec:gen_obj_and_loss} introduces the unified generative framework and loss functions. \cref{sec:experiment} elaborates on the implementation and results of the three applications.


