Unstructured Volume Rendering

Jian Huang, CS 594, Spring 2002

This set of slides reference slides developed by Prof. Torsten Moeller, SFU, Canada

Grid Types

uniform rectilinearregular curvilinear

Structured Grids:

regular irregular hybrid curved

Unstructured Grids:

Unstructured Grids• still assume scalar data• translation - as before

translation matrix still the same• classification - as before

because it is still a scalar data type• shading - same shading model(s)• interpolation - problematic• compositing - still BTF or FTB

problem - how to sort?

Ray-Casting• Garrity 1990• needs to figure out, where the ray is going• then needs to figure out how to integrate

along that ray• assumes arbitrary polyhedra (as opposed to

just tetrahedra as many other algorithm)• image order• FTB

Ray-Casting - basic algorithm

• For each ray– find intersection with first cell

– interpolate a color, opacity value

– while not outside the volume

• find exit point of ray in cell

• interpolate a color, opacity value

• integrate opacity/color along ray within cell

Ray-Casting - intersection point

• just look at all faces• compute the intersection with all faces of the

current cell and take the nearest• requires cell organized data structure• first cell - only consider faces that belong to only

one cell

A

D

C

B

Ray-Casting - intersection point

• Intersection of ray defined by point d and direction vector e:

en

dnant

acabn

d

a

c

b e

Ray-Casting - rendering integral

• classify scalar values at nodes into color/opacity• it then linearly interpolates these values within a

plane for the intersection points• integrate along the cell by multiplying the average

with the distance (another linear interpolation at work)

• needs the gradient for shading

A

D

C

B

Ray-Casting - gradients

• Vertices are not nicely aligned in x/y/z direction• hence, we need to solve an equation system using

any three pairs of neighboring points

2

1

0

222

111

000

f

f

f

f

f

f

zyx

zyx

zyx

z

y

x

Ray-Casting - conclusion

• Bottleneck - size of image (# of pixels)• results for 5122 image:

960 cells - 81.6 seconds1490 cells - 147.6 seconds16,896 cells - 149.2 seconds381,548 cells - 591.6 seconds

• may miss small polyhedra!!

Cell Projection• Max, Hanrahan, Crawfis 1990• Shirley, Tuchman 1990• object order (cannot miss small structures)• makes sense if one cell projects to many pixels• based on tetrahedral decomposition

Cell Projection - basic algorithm

• Decompose volume into tetrahedral cells

• classify each tetrahedron according to its projected profile relative to a viewpoint

• find the positions of the tetrahedra vertices after the perspective transformation

• decompose projections of the tetrahedra into triangles

• find color and opacity values for the triangle vertices using ray integration in the original world coordinates

• scan convert the triangles on a graphics workstation

Cell Projection - tetrahedra

• Decompose volume into tetrahedral cells

• 5 or 6 tetrahedra are possible

• want least amount of computation

• for 5 - need to make sure that there will be no “cracks”

Cell Projection - tetrahedra

• for 5 - need to make sure that there will be no “cracks”

• I.e. edges need to line up properly

Cell Projection - projection

• classify each tetrahedron according to its projected profile relative to a viewpoint

• find the positions of the tetrahedra vertices after the perspective transformation

Projected Tetrahedra - decompose

• decompose projections of the tetrahedra into triangles

Cell Integration• find color and opacity values for the triangle vertices using

ray integration in the original world coordinates

• Sabella assumes, that volume density (x,y,z) glows with an energy of C per unit length and absorbs with an optical density of (C and are constants for a fixed material

dtduuPtPCIt

a

b

a

exp

Cell Integration• How can we evaluate the rendering integral:

DC

duuPC

dtduuPdt

dC

dtduuPtPC

I

b

a

t

a

b

a

t

a

b

a

exp1

exp1

exp

exp

Cell Integration• How can we evaluate D?• linear interpolation (A=P(a) - P is the ray equation):

• and now:

AfSBfSg

hgaShgbSg

dvvduhgu

duuPfduuPD

hgb

hga

b

a

b

a

b

a

hgu

abauAfBfAfuPf

duutSt

0

Cell Integration• How can we find S?

• (u) is basically the transfer function

• it’s integral S(t) over a linear function could be pre-computed

• there is no restriction to the transfer function really

• scan convert the triangles on a graphics workstation

duutSt

0

Projected Tetrahedra• Shirley et al:

• find color and opacity values for the triangle vertices using ray integration in the original world coordinates at the thickest point

• scan convert the triangles on a graphics workstation

• then linearly interpolate this integral on the scanline

• can be implemented in hardware

0

I

Cell Integration• Max et al:

• the integral is computed for each pixel/object

• scanline along the front and back face to determine front and back values of S(u).

• then we evaluate I with our formula

Depth Sorting• But we need to repeat all these steps for all cells• how to sort those cells?• Max et al -> topological sort• first: create a directed graph

– nodes - cells– edges - faces– direction of edge means “in front of”

Depth Sorting• do a topological sort of this graph (in linear time - linear in

the order of the number of edges)

• only works if there are no cycles!

Rendering by Slicing• Yagel et al 1996

• uses the texture mapping hardware

• only implemented for tetrahedral grids

• cannot use the same algorithm as for regular grids where the slices are aligned along major axis

• we have to intersect the underlying geometry

Slicing - basic algorithm

• For any change in the viewing parameters– transform grid points into image space– determine zmin and zmax

– dz = (zmin-zmax)/NumSlices– for i:=1 to NumSlices do

• slice the transformed cells with plane zslice

• render 2D polymesh and compose (hardware)• zslice := zslice +dz

• for any change in the viewing parameters re-render polylist

Slicing - adaptive slicing

• Problem - might miss small detail

• idea: just keep (adaptively) subdividing until we sliced every polyhedra in that particular interval

• the compositing step must take care of different distance of slices

Slicing - progressive slicing

• Problem: How many slices do we need for a good image?

• Idea: time-quality tradeoff

• start with very few slices -> interactive display

• as long as the user isn’t changing the viewing parameters, keep refining -> hence progressively better images!

Splatting• Mao, Hong, Kaufman 1995/96• extends the regular splatting ideas to irregular

domain• based on a resampling step• based on Poisson distribution of new sample

points

Splatting - basic algorithm

• For each cell– compute a random point

– compute its extent (sphere/ellipse) by interpolating the gradients from the vertices

– see whether it fits in the cell

• sort each ellpsoid/sphere• splat each ellipsoid/sphere

Splatting - Poisson distribution

• Poisson distribution

Splatting - basic algorithm

• Pre-processing takes very long (days!!!)• number of points increases dramatically• alternative - just compute approximation of “bean-

shapes” at the vertices (ellipsoids)

Splatting - sorting

• Mao doesn’t report anything specific• can be done by quicksort• better - topological sort