Direct Visibility of Point Sets

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Direct Visibility of Point Sets

• Sagi Katz, Ayellet Tal, Ronen Basri

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Why use points?

• Points offer a uniform way to represent object
  data.
• Topology free, non-manifold
• May be oriented or not
• Easy to split!
• Sampling tends to generate them
• Range scanner data, weather data,

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Why not to

• Problems with points
• Topology free
• Non-manifold
• Difficult to interpret
• A de facto standard has yet to emerge.

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Rendering for Point Clouds

• Point clouds have seen an increase in popularity
  in recent years.
• So has data!
• Rendering point clouds requires the determination
  of the visible set of points.
• Traditionally this is done via surface
  reconstruction.

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Surface reconstruction

• Many viable methods
• Moving Least Squares
• Probably the best when the user can provide
  topology.
• Implicit functions
• Point splatting
• ...
• All are expensive and have significant trouble
  cases!

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Direct Rendering

• We know the data is there, we just have to find
  it.
• Topology should be free.
• Points dont occlude so visibility might not be
  cheap.

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Direct Visibility of Point Sets

• A simple hidden point removal operator is
  proposed.
• Just two steps
• Flipping
• Convex hull extraction
• Proofs of behavior are given.

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Direct Visibility of Point Sets

• Claims
• Correct in the limit.
• Not dependent on sample resolution.
• Works for sparse data.
• Fast O(n log n)
• Has only one user parameter.

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The HPR Operator

• Step one Inversion
• Create a sphere of radius R centered on the
  viewpoint.
• Reflect each point within the view frustum
  monotonically across the sphere.

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The HPR Operator

• Step one Inversion
• Step two Convex Hull
• Extract the convex hull of the new points the
  view frustum.
• Visible points will be on the convex hull.

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The HPR Operator

• Step one Inversion
• Step two Convex Hull
• Surely its not that easy!
• Takes O(n log n) per viewpoint
• Compare to p log n and n

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The HPR Operator

• Step one Inversion
• Step two Convex Hull
• Surely its not that easy!
• R must be chosen carefully.

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Spherical Flipping

• Spherical flipping is used for inversion.
• PiPi2(R-Pi)(Pi/Pi)
• Bounded by two spheres at R,2R
• Other functions are possible
• The proofs are for spherical flipping.

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Convex hull recovery

• A point is considered visible if there is enough
  empty space around the point.
• To board
• r(a) (2R((ri-2R)sin(b)) / (sin(ab))
• For the point to be visible bibjltconst
• This is related to the union-of-balls
  reconstruction method.

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Convex hull recovery

• Correctness in the limit of infinite samples.
• Using p skips maximum empty space discovery.
• This is what makes the algorithm fast.
• They have a heuristic for choosing R.

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Artifacts and errors

• e-visibility
• A point is marked visible if some point within e
  would be visible.
• Low curvature
• Only convex patches with sufficiently low
  curvature are marked visible.

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Artifacts and error
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Artifacts and errors
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HPR is topology independent
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Conclusions

• HPR is okay for some things.
• Nicely topology independent
• Has errors that are characterized
• Very easy to implement
• Unfortunately it is slow.

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Comments and questions

• The visibility operator here wasnt a ray or
  beam!
• Can it be generalized to other visibility
  queries?
• Can it be accelerated?
• Is this general?