Direct Volume Rendering (DVR): Ray-casting Jian Huang This set of slides references slides used by...

Direct Volume Rendering (DVR): Ray-casting

Jian Huang

This set of slides references slides used by Prof. Torsten Moeller (Simon Fraser), Prof. Han-Wei

Shen (Ohio State).

Papers

• Tuy and Tuy, 1984, IEEE CG & A (one of the earliest volume rendering techniques)

• Levoy, 1988 IEEE CG&A, and later improvements

• Drebin, Carpenter, Hanrahan, 1988, SIGGRAPH

Direct: No conversion to surface geometry

Basic Idea

Based on the idea of ray tracing• Trace from eat each pixel as a ray into object space• Compute color value along the ray• Assign the value to the pixel

Data Representation•3D volume data are represented by a finite number of cross sectional slices (hence a 3D raster)•On each volume element (voxel), stores a data value (if it uses only a single bit, then it is a binary data set. Normally, we see a gray value of 8 to 16 bits on each voxel.)

N x 2D arraies = 3D array

Data Representation (2)

What is a Voxel? – Two definitions

A voxel is a cubic cell, whichhas a single value cover the entire cubic region

A voxel is a data pointat a corner of the cubic cellThe value of a point inside the cell is determined by interpolation

Viewing

Ray Casting

• Where to position the volume and image plane • What is a ‘ray’ • How to march a ray

Viewing (1)

1. Position the volumeAssuming the volume dimensions is w x w x w

We position the center of the volume at the world origin

y

(0,0,0) x

z

Volume center = [w/2,w/2,w/2](local space)

Translate T(-w/2,-w/2,-w/2)

(data to world matrix? world to data matrix )

Viewing (2)

2. Position the image planeAssuming the distance between the image plane and the volume center is D, and initially the center of the image plane is (0,0,-D)

y

(0,0,0) x

z

Image plane

Viewing (3)

3. Rotate the image planeA new position of the image plane can be defined in termsof three rotation angle with respect to x,y,z axes

Assuming the original view vector is [0,0,1], then the newview vector g becomes:

cossincossing = [0,0,1] 0 1 0 0 cos sinsincos sin 0 cossincos

Viewing (4)

y

(0,0,0) x

z

uv

E

S

E0 u0v0

+ S0

B

B = [0,0,0]S0 = [0,0,-D]u0 = [1,0,0]v0 = [0,1,0]

Now,R: the rotation matrix S = B – D x g U = [1,0,0] x R V = [0,1,0] x R

Viewing (5)

R: the rotation matrix S = B – D x g U = [1,0,0] x R V = [0,1,0] x R

+ S

Image Plane: L x L pixels

E

Then

E = S – L/2 x u – L/2 x v

So Each pixel (i,j) has coordinates

P = E + i x u + j x v

u

v

We enumerate the pixels by changingi and j (0..L-1)

Viewing (6)

4. Cast raysRemember for each pixel on the image planeP = E + i x u + j x vand the view vector g = [0,0,1] x RSo the ray has the equation:

Q = P + k (d x g) d: the sampling distance at each step

xx

dx

x p

Q

K = 0,1,2,…

Early Methods• Before 1988• Did not consider transparency• did not consider sophisticated light

transportation theory• were concerned with quick solutions• hence more or less applied to binary data

non-binary data - require sophisticated

classification/compositing methods!

Ray Tracing -> Ray Casting

• “another” typical method from traditional graphics

• Typically we only deal with primary rays -hence: ray-casting

• a natural image-order technique• as opposed to surface graphics - how do we

calculate the ray/surface intersection???• Since we have no surfaces - we need to

carefully step through the volume

Ray Casting• Stepping through the volume: a ray is cast

into the volume, sampling the volume at certain intervals

• The sampling intervals are usually equi-distant, but don’t have to be (e.g. importance sampling)

• At each sampling location, a sample is interpolated / reconstructed from the grid voxels

• popular filters are: nearest neighbor (box), trilinear (tent), Gaussian, cubic spline

• Along the ray - what are we looking for?

Example: Using the nearest neighbor kernel

In tuys’ paperQ = P + K x V (v=dxg)

At each step k, Q is roundedoff to the nearest voxel(like the DDA algorithm)

Check if the voxel is on the boundary or not (compareagainst a threshold)

If yes, perform shading

Basic Idea of Ray-casting Pipeline

- Data are defined at the corners of each cell (voxel)

- The data value inside the voxel is determined using interpolation (e.g. tri-linear)

- Composite colors and opacities along the ray path

- Can use other ray-traversal schemes as well

c1

c2

c3

Ray Traversal Schemes

Depth

IntensityMax

Average

Accumulate

First

Ray Traversal - First

Depth

Intensity

First

• First: extracts iso-surfaces (again!)done by Tuy&Tuy ’84

Ray Traversal - Average

Depth

Intensity

Average

• Average: produces basically an X-ray picture

Ray Traversal - MIP

Depth

IntensityMax

• Max: Maximum Intensity Projectionused for Magnetic Resonance Angiogram

Ray Traversal - Accumulate

Depth

Intensity

Accumulate

• Accumulate opacity while compositing colors: make transparent layers visible!Levoy ‘88

Raycasting

color

opacity

1.0

volumetric compositing

object (color, opacity)

Raycasting

color

opacity

Interpolationkernel

1.0



Raycasting

color c = c s s(1 - ) + c

opacity = s (1 - ) +

1.0


volumetric compositingInterpolationkernel

Raycasting

color

opacity

1.0



Raycasting

color

opacity



Volume Rendering PipelineAcquired values

Data preparation

Prepared values

classificationshading

Voxel colors

Ray-tracing / resampling Ray-tracing / resampling

Sample colors

compositing

Voxel opacities

Sample opacities

Image Pixels

DCH DVR Pipeline

DCH - PipelineOriginal data

Material percentage volumes

Color volumeOpacity volumeDensity volume

Gradient Shaded volume

Transformed volumeFinal image

Classification

Normals

Shading

shears

compositing

Common Components of General Pipeline

• Interpolation/reconstruction

• Classification or transfer function

• Gradient/normal estimation for shading– Question: are normals also interpolated?

Levoy - Interpolationeye

image pixel

viewing ray

voxel

sample point

trilinearinterpolation

Levoy - Interpolation (2)

binary smooth

Closest value Weighted average

Levoy – Gradient/Normals

• Central difference• per voxel

2

2

2

1,,1,,

,1,,1,

,,1,,1

kjikjiz

kjikjiy

kjikjix

vvG

vvG

vvG

X+1y-1,

Y+1

z-1

Levoy - Compositing

• Image order• back-to-front• using the over operator

)1(

)1(

inout

inout CCC

Levoy - Shading

• Phong Shading + Depth Cueing

))()((21

nsd

pap HxNkLxNk

xdkk

CkCxC

• Cp = color of parallel light source

• ka / kd / ks = ambient / diffuse / specular light coefficient

• k1, k2 = fall-off constants

• d(x) = distance to picture plane• L = normalized vector to light• H = normalized vector for maximum highlight

• N(xi) = surface normal at voxel xi

Classification

Classification: Mapping from data to opacities

Region of interest: high opaicity (more opaque) Rest: translucent or transparent

The opacity function is typically specified by the user

Levoy came up with two formula to compute opacity1. Isosurface

2. Region boundary (e.g. between bone and fresh)

Classification/Transfer Function

• Maps raw voxel value into presentable entities: color, intensity, opacity, etc.

Raw-data material (R, G, B, , Ka, Kd, Ks, ...)

• May require probabilistic methods (Drebin). Derive material volume from input. Estimate % of each material in all voxels. Pre-computed. AKA segmentation.

• Often use look-up tables (LUT) to store the transfer function that are discovered

Levoy - Classification

• Usually not only interested in a particular iso-surface but also in regions of “change”

• Feature extraction - High value of opacity exists in regions of change

• Transfer function (Levoy) - Saliency• Surface “strength”

Levoy - Classification

• Chemistry Data– only iso-value - loose information of layers– iso-range - could be too narrow or too wide– thickness of region should be constant– hence linear fall off of opacity– wider fall off for larger gradient

• Medical Data– assume at most two tissues meet– linear transition between opacities of “neighboring”

tissues– reflects linear combination of tissues within one

voxel

Opacity function (1)

Goal: visualize voxels that have a selected threshold value fv - No intermediate geometry is extracted - The idea is to assign voxels that have value fv the maximum opacity (say ) - And then create a smooth transition for the surrounding area from 1 to 0 -Levoy wants to maintain a constant thickness for the transition area.


Maintain a constant isosurface thickness

opacity = opacity = 0

Can we assign opacity based on function value instead of distance? (local operation: we don’t know wherethe isosurface is)

Yes – we can based on the value distance f – fv but we need to take into account the local gradient


Assign opacity based on value difference (f-fv) and local gradient

gradient: the value fall-off rate grad = fsAssuming a region has a constant gradient and the isosurfacetransition has a thickness R

opacity = F = fv

opacity = 0F = fv – grad * R

thickness = R

F = f(x)Then we interpolate the opacity

opacity = – * ( fv-f(x))/ (grad * R)

Levoy - Classification A

Opacity (xi)

Acquired value f (xi)

Gradient magnitude |f (xi)| Opacity (xi)


Gradient magnitude |f (xi)|

v

fv

i

ivvi xf

xff

rx

'

11

Levoy - Classification B

Opacity (xi)


Gradient magnitude |f (xi)|

vc

vbva

fvafvb

fvc

nn

n

n

nn

n

n

vv

ivv

vv

vivii ff

xff

ff

fxfxfx

1

1

1

1'

DCH - Material Percentage V.

• Probabilistic classifier• probability that a voxel has intensity I:

• pi - percentage of material

• Pi(I) - prob. that material i has value I

• Pi(I) given through statistics/physics

• pi then given by:

n

iii IPpIP

1

n

jj

ii

IP

IPIp

1

DCH - Classification• Like Levoy - assumes only two materials per voxel• that will lead to material percentage volumes

• from them we conclude color/opacity:

– where Ci=(iRi, iGi, iBi, i)

n

iiiCpC

1

DCH- Classification

Air Fat TissueBone

CT Number

Air Fat Tissue Bone

histogramConstituent’s Distributions

Material Assignment%

CT

Levoy - Improvements

• Levoy 1990• front-to-back with early ray termination

= 0.95• hierarchical oct-tree data structure

– skip empty cells efficiently

Volumetric Ray Integration

color

opacity


1.0

Direct Volume Rendering (DVR): Ray-casting Jian Huang This set of slides references slides used by...

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