Purpose: Semantic segmentation and landmark detection are fundamental tasks of medical image processing, facilitating further analysis of anatomical objects. Although deep learning-based pixel-wise classification has set a new-state-of-the-art for segmentation, it falls short in landmark detection, a strength of shape-based approaches. Methods: In this work, we propose a dense image-to-shape representation that enables the joint learning of landmarks and semantic segmentation by employing a fully convolutional architecture. Our method intuitively allows the extraction of arbitrary landmarks due to its representation of anatomical correspondences. We benchmark our method against the state-of-the-art for semantic segmentation (nnUNet), a shape-based approach employing geometric deep learning and a CNN-based method for landmark detection. Results: We evaluate our method on two medical dataset: one common benchmark featuring the lungs, heart, and clavicle from thorax X-rays, and another with 17 different bones in the paediatric wrist. While our method is on pair with the landmark detection baseline in the thorax setting (error in mm of $2.6\pm0.9$ vs $2.7\pm0.9$), it substantially surpassed it in the more complex wrist setting ($1.1\pm0.6$ vs $1.9\pm0.5$). Conclusion: We demonstrate that dense geometric shape representation is beneficial for challenging landmark detection tasks and outperforms previous state-of-the-art using heatmap regression. While it does not require explicit training on the landmarks themselves, allowing for the addition of new landmarks without necessitating retraining.}
