This paper proposes an approach for the semantic seg- mentation and structural parsing of modular furniture items, such as cabinets, wardrobes, and bookshelves, into so called interaction elements. Such a segmentation into functional units is challenging not only due to the visual similarity of the different elements but also because of their often uniformly colored and low-texture appearance. Our method addresses these challenges by merging structural and appearance likelihoods of each element and jointly op- timizing over shape, relative location, and class labels us- ing Markov Chain Monte Carlo (MCMC) sampling. We propose a novel concept called rectangle coverings which provides a tight bound on the number of structural elements and hence narrows down the search space. We evaluate our approach’s performance on a novel dataset of furniture items and demonstrate its applicability in practice.