The reconstruction of images from measured data is an increasing field of research. For highly under-determined problems, template-based image reconstruction provides a way of compensating for the lack of sufficient data. A caveat of this approach is that dealing with different topologies of the template and the target image is challenging. In this paper, we propose a LDDMM-based image-reconstruction model that resolves this issue by adding a source term. On the theoretical side, we show that the model satisfies all criteria for being a well-posed regularization method. For the implementation, we pursue a discretize-then-optimize approach involving the proximal alternating linearized minimization algorithm, which is known to converge under mild assumptions. Our simulations with both artificial and real data confirm the robustness of the method, and its ability to successfully deal with topology changes even if the available amount of data is very limited.