point cloud rendering - jacobs university bremen · point cloud rendering • the size of the...

320322: Graphics and Visualization 207

Visualization and Computer Graphics LabJacobs University

Point cloud rendering• The size of the spheres must be determined locally.• It can be estimated by computing distances to

neighboring points.– natural neighbors– k-nearest neighbors with fixed k– k-nearest neighbors with variable k– k-nearest neighbors with angle criterion



Splatting

• Rather than drawing isotropic extensions, one wouldlike to draw extensions only in directions tangential to the surface.

• Spheres are replaced by spherical disks.• The disks are called splats.• This approach is called splatting.



Splatting



Splat generation

• Splats can be computed as the intersection betweenthe sphere and the tangential plane in the respectivepoint.

• Question: How can we obtain the tangential plane?



Estimating tangential plane

Fit a plane through and its neighborssuch that the distances from the points to the planeis minimal in the least squares sense.



Estimating tangential plane

Theorem:The normal of the fitting plane P is the third principaldirection/component that is computed by theeigenvector to the smallest eigenvalue of matrix

with .



Estimating tangential planeProof: Q is symmetric and positive semi-definit

-> Q is diagonizable->

with eigenvalues-> eigenvectors and span P-> for any rotation

-> Every rotation S increases the squared distance to P.



Splatting



Anisotropic sampling

• In case of anisotropic sampling, circular splats can bereplaced by elliptical splats.



Splat transitions

• To produce nicer transitions between splats, thesplats are drawn with increasing transparencytowards their border.

• Transparency transition can be determined usingGaussian filters.



Splat-based rendering



2.4 Implicit Surfaces



Implicit surfaces

Definition:• Implicit surfaces are surfaces that are given as a

solution to a mathematical equation.

Example:• Circle

in parametric form can be described as the kernelF(x,y) = 0 of the implicit function



Implicit surfaces

• Simple implicit functions that are easy and fast to evaluate can be used to replace piecewise linear object representations by meshes.

• The most simple non-planar surfaces are quadrics.



Ellipsoid

with being the radii.



Ellipsoid

Ellipsoid with z=constant: cylinder



Elliptical Paraboloid



Hyperbolic Paraboloid



Hyperboloid

α > 0: one sheet α = 0: conesα < 0: two sheets



Rendering

Implicit surfaces can be rendered by:• sampling the implicit equation at discrete positions,

i.e., converting it into an explicit mesh representation,• ray tracing (see later).



Sweep geometry

• Sweep geometry is a (semi-)implicit method, where a 2D structure is implicitly extruded to a 3D objectusing rotation.

• Construction:– Given a 2D contour in form of a closed curve or a curve with

endpoints on the rotation axis.– Sweep the contour around the rotation axis by 360° to form

a rotationally symmetric 3D object.



Sweep geometry

• Curve with endpoints on the rotation axis:

• Result: Boundary representation of genus 0 object.

explicit curve representation:sequence of line segments



Sweep geometry

• Closed curve:

• Result: Boundary representation of genus 1 object.

implicit curve representation:quadric



Constructive solid geometry

Idea:• Use a set of objects with simple representation and a

set of efficient operations to combine the simple objects to more complex ones.

• Only store pointers to simple objects and operations.• Hence, this can also be considered as a (semi-)implicit

representation.



Constructive solid geometryBasis objects represented by their boundaries:• Polyhedra: cuboids, tetrahedra, …• Implicitly defined objects: quadrics, sweep geometry, …

Operations operate on solids (volumetric objects):• Boolean operations: intersection, union, difference,

xor,…

Combining constructions:• Many basis objects can be successively combined using

Boolean operations.• The combinations are stored in a binary tree.




Transformations:• In addition to the operations, we need to store

transformations for the objects that are appliedbefore the objects get combined.



Constructive solid geometry (CSG)



CSG

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