We propose a new method for reconstructing an implicit surface from an un-oriented point set. While existing methods often involve non-trivial heuristics and require additional constraints, such as normals or labelled points, we introduce a direct definition of the function from the points as the solution to a constrained quadratic optimization problem. The definition has a number of appealing features: it uses a single parameter (parameter-free for exact interpolation), applies to any dimensions, commutes with similarity transformations, and can be easily implemented without discretizing the space. More importantly, the use of a global smoothness energy allows our definition to be much more resilient to sampling imperfections than existing methods, making it particularly suited for sparse and non-uniform inputs. CCS Concepts: • Computing methodologies → Point-based models; Volumetric models.

in St. Louis for sampling distinct topologies in this space. Besides specifying topological constraints, the user can steer the algorithm interactively, such as by scribbling. We demonstrate, on synthetic and biological shapes, how our algorithm opens up new opportunities for topology-aware modeling in the multi-labeled context.

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Surface reconstruction is one of the central problems in computer graphics. Existing research on this problem has primarily focused on improving the geometric aspects of the reconstruction (e.g., smoothness, features, element quality, etc.), and little attention has been paid to ensure it also has desired topological properties (e.g., connectedness and genus). In this paper, we propose a novel and general optimization method for surface reconstruction under topological constraints. The input to our method is a prescribed genus for the reconstructed surface, a partition of the ambient volume into cells, and a set of possible surface candidates and their associated energy within each cell. Our method computes one candidate per cell so that their union is a connected surface with the prescribed genus that minimizes the total energy. We formulate the task as an integer program, and propose a novel solution that combines convex relaxations within a branch and bound framework. As our method is oblivious of the type of input cells, surface candidates, and energy, it can be applied to a variety of reconstruction scenarios, and we explore two of them in the paper: reconstruction from cross-section slices and iso-surfacing an intensity volume. In the first scenario, our method outperforms an existing topology-aware method particularly for complex inputs and higher genus constraints. In the second scenario, we demonstrate the benefit of topology control over classical topology-oblivious methods such as Marching Cubes.

Introduction-Craniosynostosis is typically corrected surgically within the first year of life through cranial vault reconstruction. These procedures often leaves open calvarial defects at the time of surgery, which are anticipated to close over time in a large proportion of cases. However, residual calvarial defects may result as long-term sequelae from cranial vault remodeling. When larger defects are present, they may necessitate further reconstruction for closure.Better understanding of the calvarial osseous healing process may help to identify which defects will resolve or shrink to acceptable size and which will require further surgery. Our study aims to assess the long-term changes in defect size after cranial vault reconstruction for craniosynostosis.Methods-One year post-operative and long-term computed tomography scans were retrieved from the craniofacial anomalies archive. Analysis used custom software. All defects above the size of 1 cm 2 were analyzed and tracked for calvarial location, surface area, and circularity. Monte Carlo simulation was performed to model the effect of initial defect size on the rate of defect closure.Results-We analyzed a total of 74 defects. The average initial defect surface area was 3.27 ± 3.40 cm 2 . The average final defect surface area was 1.71 ± 2.54 cm 2 . The average percent decrease was 55.06 ± 28.99 %. There was a significant difference in the percentage decrease of defects in the parietal and fronto-parietal locations: 68.4% and 43.7%, respectively (p = 0.001). Monte Carlo simulation results suggest that less than 10% of defects above the size of 9 cm 2 will close to the size of 2.5 cm 2 or less.Conclusions-We describe and make available a novel, validated method of measuring cranial defects. We find that the large majority of initial defects greater than 9cm 2 remain at least one square inch in size (2.5cm 2 ) one year post-operatively. Additionally, there appear to be regional differences in closure rates across the cranium, with front-parietal defects closing more slowly than those in the parietal region. This information will aid surgeons in the decision-making process regarding cranioplasty after craniosynostosis correction.