State-of-the-art Style Transfer methods often leverage pre-trained encoders optimized for discriminative tasks, which may not be ideal for image synthesis. This can result in significant artifacts and loss of photorealism. Motivated by the ability of multiscale geometric image representations to capture fine-grained details and global structure, we propose GIST: Geometric-based Image Style Transfer, a novel Style Transfer technique that exploits the geometric properties of content and style images. GIST replaces the standard Neural Style Transfer autoencoding framework with a multiscale image expansion, preserving scene details without the need for post-processing or training. Our method matches multiresolution and multidirectional representations such as Wavelets and Contourlets by solving an optimal transport problem, leading to an efficient texture transferring. Experiments show that GIST is on-par or outperforms recent photorealistic Style Transfer approaches while significantly reducing the processing time with no model training.