generalization to 3d · 320322: graphics and visualization 726 visualization and computer graphics...

320322: Graphics and Visualization 725

Visualization and Computer Graphics LabJacobs University

Generalization to 3D

• Coming back to our original problem of extractingisosurfaces, we can apply the idea of the marchingsquares algorithm to volumetric cells.

• We still compute the intersection of edges with theisocontour (now isosurface).

• The intersection points on the edges are obtained as a linear combination as before.

• The intersection points are connected by triangles.• Connectivity becomes more complicated.



Generalization to 3D

• We can still establish a case table with all thepossible configuration for inside/outside-properties.

• As each cell has 8 vertices, we obtain 28 = 256 cases.• The cells are cubes (or cuboids).• Hence, the 3D algorithm is called Marching Cubes.



Example cases

• Due to (rotational and inside-outside) symmetries, wecan reduce the case table from 256 cases to 15 cases.



Reduced marching cubes case table



Ambiguous faces

• Do ambiguous faces occur?

• Yes, in 6 of the 15 reduced cases.• Solution: Use asymptotic decider on each ambiguous

face.



Ambiguous faces

• If we want to have a fast solution, can we just pick anyof the two choices?

• One has to make sure that the picking is doneconsistently such that a manifold surface is generated.

• This can be achieved using a case table with 256 cases.



Ambiguous cells

• Is there also something like an ambiguous cell?

No ambiguous face Still, there is a second possibility

This is called a tunnel case



Tunnel case



3D asymptotic decider

• The same observation as in 2D can be done in 3D using a trilinear interpolation within the cube.

• One computes the intersection of the asymptotes of the hyperboloid and evaluates the function valuethere.

• Then, the analogous decision can be made.



Marching Cubes



9.3 Direct Volume Rendering



Direct volume rendering

• Direct volume rendering is an alternative to surfaceextraction for visualizing volumetric shapes.

• Direct volume rendering does not extract anygeometry but directly displays the visible volumefrom a given viewpoint.

• This saves the surface extraction step.• On the other hand, it is view-dependent, i.e., the

entire computation needs to be executed again aftereach transformation step.



Generalization of color mapping

• Direct volume rendering can be regarded as a generalization of color mapping.

• Again, the range of the scalar field f is mapped to color values.

• However, now it is done for all the samples within thevolume.

• In order to decide, what parts of the volume arevisible, we also need to assign opacities.

• Hence, the color mapping is extended from the rangeof f to RGBA values.



Approach

• Feature extraction:Define opaque (and semi-transparent) regions.

• Displaying the feature:Render the opaque regions visible from the viewpoint.

• Interaction:Allow for viewing transformations and for changingthe color mapping (including opacity values).



Ray casting

• The most famous direct volume rendering approach isobtained by casting rays to the viewpoint through anypixel of the viewing plane:

viewerviewing plane

bounding box of volume

ray

pixel



Ray casting

• Light with intensity I0 enters the volume at some entry point.• The light traverses the volume and loses intensities depending

on the property of the traversed medium.• The denser the medium, the more light is absorbed.• Density is regulated using the assigned opacities.• The light that exits the volume is what is observed by the

viewer.

viewerviewing plane

bounding box of volume

ray

pixel

light I0

entry into volumeexit from volume



Ray casting

• In order to determine, how much light arrives on a point on the ray, one needs to integrate the light attenuation from the entry into the volume (position0 on the ray) to the current position L.

light I0



Ray casting• The attenuation is described by an exponential

decrease.• Hence, the intensity I(L) at distance L from the ray‘s

entry into the volume is described by

where μ(t) describes the density at position t alongthe ray.

I0I(L)



Ray casting

• The attenuation term comprises light absorption and light scattering.

• For light scattering, light is not absorbed but deflected.• On the other hand, light that arrives the ray at position

L from a different direction, may be reflected into thedirection of the viewer.



Ray casting

• Taking this incoming scattering effect into account, wemodify the light intensity computation at point L to

• We integrate the incoming light C(s) along the ray fromits entry s=0 to the position s=L.

• How much of the incoming light is scattered depends on the density μ(s) at position s.

• From position s to position L, the incoming scattered light C(s) μ(s) is again attenuated (exponential decrease).

generalization to 3d · 320322: graphics and visualization 726 visualization and computer graphics...

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