Efficient HPR-based Rendering of Point Clouds

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


Point clouds

Point Clouds

Describe sampled surfaces without storing mesh topology.Visibility not well defined.



Point clouds

Point Clouds



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.



The HPR operator

HPR



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.



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.



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.




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 |




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.




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.



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).



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.



Figures

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.



Conclusions

The end

Questions?


Software

Efficient HPR-based Rendering of Point Clouds