Efficient HPR-based Rendering of Point CloudsRenan Machado e Silva, Claudio Esperanca, Antonio Oliveira
Slides by: Roger Hernando
Introducction Fast HPR operator Results
Outline
1 IntroducctionPoint cloudsThe HPR operator
2 Fast HPR operatorFast HPRApproximate Convex HullPartial view dependant reconstrucction
3 ResultsFiguresConclusions
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Point clouds
Point Clouds
Describe sampled surfaces without storing mesh topology.Visibility not well defined.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Point clouds
Point Clouds
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
The HPR operator
HPR
Given P(point cloud) and C(viewpoint) associate with P acoordinate system in which C is placed at the origin, and P isspherically flipped O(n).
pi = pi + 2 (R − ||pi ||)pi||pi ||
Convex hull computation of the set of points P and theviewing point C . O(n log n)Each point it’s projection lays on the convex hull is labelled asvisible.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
The HPR operator
HPR
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
The HPR operator
HPRThe HPR is an approximate operator to determine the visiblepoints of a point cloud. the parameter R controls the falsenegatives and positives in the final solution.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Fast HPR
Fast HPR
The computation of the convex hull is O(n log n) but in theworst case is quadratic, making it unsuitable for large pointclouds.The main idea is to use an approximate convex hull algorithmthat can be executed in the GPU.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Approximate Convex Hull
Approximate Convex hull
Choose a point q inside the hull, for example the centroid.The plane is divided in k sectors covering an angle of 2π
k . Foreach sector establish a representative which maximizespi ∗ ds(extreme points).Refine the estimate for each sector, comparing therepresentative of the sector with the representatives of theneighbouring sectors.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Approximate Convex Hull
Defining sectorsDivide the frustum in a grid of k sectors.
Express the frustum in angular coordinates, and sample them√k times to create the grid:
γ ∈[−∆Φ
2 ,∆Φ2
]θ ∈
[π2 −
∆Φ2 ,
π2 + ∆Φ
2
]∆Φ = 2 ∗ sin−1 r
|C−Ce |
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Approximate Convex Hull
Compute sector candidate points
Transform all the points to be in a coordinate system whichhave as it’s origin C , compute the spherical flipping for eachpoint and assign each point to a sector.Compute the centroids of each sector to obtain sectordirections
−→ds
Obtain the points that maximize pi ∗−→ds being s the sector
which the point i belongs.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Approximate Convex Hull
Candidate point propagationIn this step a candidate point initially considered as a extreme for agiven sector may be replaced by a candidate assigned to one of theneighbouring sectors.
go through all sectors and see if a neighbouring representativecan replace or not the current representative of the sector.Repeat the process until no changes occur.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Partial view dependant reconstrucction
Reconstruction
If it’s need to reconstruct the model for rendering, we just simplehave to triangulate the points defining the approximate convexhull. Filtering the triangles to remove the invalid ones, and theones which it’s vertices are not likely to be contiguous(edge >threshold).
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Figures
Results
R parameter was determined by mere visual inspection, and k wasdetremined to yeld the number of visible points as the 95% of thatobtained by exact HPR algorithm.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Figures
Results
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Conclusions
Results
The use of approximate convex hull algorithm makes HPRoperator suitable for interactive rendering of point cloudsO(n + k).Possibility to calibrate visual detail simply using the parameterk.As k increases, the number of empty sectors increases too andleads to a waste of memory.
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds
Introducction Fast HPR operator Results
Conclusions
The end
Questions?
Slides by: Roger Hernando Efficient HPR-based Rendering of Point Clouds