Time Discrete Geodesics in Deep Feature Spaces for Image Morphing

Image morphing in computer vision amounts to computing a visually appealing transition of two images. A prominent model for image morphing originally proposed by Trouvé, Younes and coworkers is image metamorphosis. Here, the space of images is endowed with a Riemannian metric that separately quantifies the contributions due to transport and image intensity variations along a transport path. Geodesic curves in this Riemannian space of images give rise to morphing transitions. The classical metamorphosis model considers images as square-integrable functions on some image domain and thus is non-sensitive to image features such as sharp interfaces or fine texture patterns. To resolve this drawback, we treat images not as intensity maps, but rather as maps into some feature space that encode local structure information. In the simplest case, color intensities are such feature vectors. To appropriately treat local structures and semantic information, deep convolutional neural network features are investigated. The resulting model is formulated directly in terms of a variational time discretization developed for the classical metamorphosis model by Berkels, Effland and Rumpf. The key ingredient is a mismatch energy that locally approximates the squared Riemannian distance and consists of a regularization energy of the time discrete flow and a dissimilarity energy that measures the feature vector modulation along discrete transport paths. The spatial discretization is based on a finite difference and a stable spline interpolation. A variety of numerical examples demonstrates the robustness and versatility of the proposed method for real images using a variant of the iPALM algorithm for the minimization of the fully discrete energy functional.