![Figure 1. Stages of the basic HPR operator [26]. Figures have been suitably scaled for illustration.](/figures/figure-1-stages-of-the-basic-hpr-operator-26-figures-have-za8tvmmw.webp)
![Figure 10. Reconstruction outputs on raw scanned data for the coati model using a combination of AMLS [18] and cocone [12] (left), and robust HPR based smoothing (Section 4.4) with reconstruction (Algorithm 2). While AMLS-cocone generate large triangles in poorly sampled regions our approach provides a more natural result leaving such regions empty. Similarly, by construction, our method avoids self intersections present in AMLS-cocone based reconstruction. The raw scan is thick due to noise and a direct reconstruction is unnatural. Hence instead of an interpolatory solution, we choose approximation as with AMLS method. For smoothing the point cloud for robust HPR, we use connectivity based on robust HPR (see also Figure 14). On such noisy scans original HPR based quick-and-dirty reconstruction breaks down, since visibility and convex-hull based connectivity estimates from single views have significant errors (see Figure 9).](/figures/figure-10-reconstruction-outputs-on-raw-scanned-data-for-the-1nai9nmw.webp)
![Figure 9. Typical behavior of original HPR and robust HPR on raw scanned data from a given viewpoint (not shown), with the hidden points in pink and visible points in yellow. Notice that the boundary between visible and hidden regions is sharp with robust HPR, indicating few misclassifications. In absence of ground truth, we show the AMLS [18] + cocone [12] based reconstruction and use the surface for visibility computation (left column) for comparison. Notice the result also has misclassifications. We highlight some regions where original HPR clearly produces a large number of misclassifications. The scanner noise parameter σ is estimated using a calibration plane.](/figures/figure-9-typical-behavior-of-original-hpr-and-robust-hpr-on-1m8jbea9.webp)
Q2. What is the underlying idea of the algorithm?
The underlying idea is to greedily extend the solution using weighted atomic elements while maintaining the desired properties using a graph theoretic formulation.
Q3. How many vertices can be added to the heap?
Since the weight of any vertex is updated at most twice and the total number of vertices is bounded by |V ||C|, there are at most 2|V ||C| operations on the heap.
Q4. What is the way to determine visibility for a point set?
since a point set typically corresponds to an underlying surface, one can first reconstruct this surface, identify the visible part from the specified viewpoint, and then mark points as visible if they lie on the visible surface parts.
Q5. How do the authors obtain a robust visibility operator?
By suitably relaxing the condition of points lying on the convex hull to include points near the convex hull, the authors arrive at a robust visibility operator.
Q6. How many times do connectivity edges appear in the convex hull?
Similar to weighted visibility of points, the authors assign weights to such connectivity edges/triangles proportional to the number of times they appears in the convex hull over different values of R.
Q7. What is the problem of determining the visible parts of a polygonal model?
Given a polygonal model, the problem of correctly and efficiently identifying the hidden faces or determining the visible parts of a model from a specified viewpoint has received significant attention since the early days of computer graphics [5], [7].
Q8. What is the weight of a vertex?
Thus the weight of a vertex not only depends on the visibility information but also on its connectivity to other vertices that have already been chosen, and may change during the course of the algorithm.
Q9. What are the problems that arise when a point set is inverted?
The authors observe that such problems arise as slight input perturbations can result in significant changes in the structure of the corresponding convex hull.