Volume Rendering Reference Material The slides have used/adapted - - PowerPoint PPT Presentation

Volume Rendering

Reference Material

The slides have used/adapted material from the following references:

• 1. Advanced Animation and Rendering Techniques by Watt and Watt
• 2. Slides by Prof Parag Chaudhuri of Course CS 775 on Volume Graphics
• 3. Slides by Prof Bong Soo Sohn of Lecture on Direct Volume Rendering
• 4. Lecture Slides on Contouring by Prof Tao Ju (Course CSE554)

Volume Rendering

Bicubic Parametric Patches Polygons 3D Structured Objects 2D Image

I x y ( , )

Conventional Rendering

Volume Rendering

V x y z ( , , )

Scalar Field:

Volume Rendering

Volume Rendering

V x y z ( , , ) V x y z ( , , )

Structured 3D Model 3D Scalar Field 3D Scalar Field 2D Scalar Image

Volume Rendering

• Polylines (2D) or

meshes (3D) that tile the object boundary

• Smoother appearance
• Less storage (no

interior pieces)

Binary picture Boundary mesh

Volume Rendering

• Creates a binary picture from a grayscale image
• How to create a smooth boundary at the threshold?
• Such boundary is known as an iso-contour (or iso-curve,

iso-surface, etc.)

Digital image Binary picture Boundary curve

Volume Rendering

• Iso-contour: set of points at a particular function value
• Given a continuous function f defined on every point in

the space

• An iso-contour is all points where f evaluates to be some

value c (called the iso-value)

• Iso-curves: iso-contours of 2D

functions

2 2

) , ( y x y x f + = 1 ) , ( = y x f

} ) , ( | } , {{ c y x f y x =

Volume Rendering

• Iso-surfaces: iso-contours of 3D functions

} ) , , ( | } , , {{ c z y x f z y x =

2 2 2

) , , ( z y x z y x f + + = 1 ) , , ( = z y x f

Volume Rendering

Black curves: iso-contours Red arrows: gradient directions Green dots: critical points

• Closed
• With a well-defined inside and
• utside
• Inside: points above the iso-

value

• Outside: points below the iso-

value

• Orthogonal to gradient directions
• In general, a manifold

Volume Rendering

Iso-value = Grid point (pixel) Grid edge Grid cell

• Input
• A grid where each grid

point (pixel or voxel) has a value (color)

• An iso-value (threshold)
• Output
• A closed, manifold, non-

intersecting polyline (2D)

• r mesh (3D) that

separates grid points above or below the iso- value

Volume Rendering

• For each grid cell with a sign

change

• Create one vertex on each grid

edge with a sign change

• Connect vertices by lines

Volume Rendering

• Creating vertices
• Assuming the underlying, continuous function is linear
• n the grid edge
• Linearly interpolate the positions of the two grid points

{x0,y0} {x1,y1 } f f0 f1 {x,y}

) ( ) (

1 1 1

y y t y y x x t x x f f f t ! + = ! + = ! =

+

• t

1-t

Volume Rendering

• Connecting vertices by lines
• Lines shouldn’t intersect
• Each vertex is used once
• So that the polyline is a closed

manifold

• Two approaches
• Do a walk around the grid cell
• Connect consecutive pair of vertices
• Or, using a pre-computed look-up

table

• 2^4=16 entries
• For each sign combination at the 4

corners, it stores the indices of the grid edges whose vertices make up the lines.

1 2 3 4 1 3 4 2

“0 0 0 1” {{2,4}} “0 0 1 1” {{3,4}} “1 0 0 1” {{1,3},{2,4}}

Volume Rendering

Volume Rendering

Volume Rendering

C D D C E F E C E G D , , C H E 1 1 3 2 1 3 2 1

Marching Square

• Select a cell
• Calculate inside/outside

state of each vertex

• Determine which edge is

intersected

• Move to next cell till all

cells are

Volume Rendering

? @ @ ? A B A ? A C @ , , ? D A 1 1 3 2 1 3 2 1 No intersection Contour intersects 2 edges (no ambiguity) Contour intersects 2 edges (ambiguity) Contour intersects 2 edges (no ambiguity)

Volume Rendering

No intersection Contour intersects 2 edges (no ambiguity) Contour intersects 2 edges (ambiguity) Contour intersects 2 edges (no ambiguity) ? @ @ ? A B A ? A C @ , , ? D A 1 1 3 2 1 3 2 1

Volume Rendering

No intersection Contour intersects 2 edges (no ambiguity) Contour intersects 2 edges (ambiguity) Contour intersects 2 edges (no ambiguity) ? @ @ ? A B A ? A C @ , , ? D A 1 1 3 2 1 3 2 1

Volume Rendering

No intersection Contour intersects 2 edges (no ambiguity) Contour intersects 2 edges (ambiguity) Contour intersects 2 edges (no ambiguity) ? @ @ ? A B A ? A C @ , , ? D A 1 1 3 2 1 3 2 1

Volume Rendering

Volume Rendering

Direction of march Current cube Previously dealt with New vertex

Volume Rendering

• To calculate surface normal, we need to determine gradient

vector, g (derivative of the density function).

• To estimate the gradient vector at the surface of interest, we

first estimate the gradient vectors at the vertices and interpolate the gradient at the intersection.

• The gradient at cube vertex (i , j, k), is estimated using central

differences along the three coordinate axes by:

D (i, j, k) is the density at pixel (i, j) in slice k. !x, !y, !z are lengths of the cube edges

Volume Rendering

Volume Rendering

Volume Rendering

y j =

y j = +1 V x y z k ( , , ) =

Polygon structure generated by “skimming”

Volume Rendering

Volume Rendering

• scalar field

– like temperature in room at each (x,y,z)

• many applications

– medical (MRI, CT, PET) – geophysical – fluid dynamics

Volume Rendering

Direct Volume Rendering

• 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 arrayes = 3D array

Direct Volume Rendering

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

Direct Volume Rendering

• Maps voxel data values to optical properties
• Color/opacity map
• Emphasize or classify features of interest in the data
• Piecewise linear functions, Look-up tables, 1D, 2D
• GPU – simple shader functions, texture lookup tables

Voxel Data

• Density
• Temperature

Optical Properties

• Color
• Opacity

Direct Volume Rendering

• 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?

Volume Rendering

Light Source

D x y z ( , , )

t1 t2 t !

Eye Point

D x t y t z t D t I x t y t z t I t ( ( ), ( ), ( )) ( ) ( ( ), ( ), ( )) ( ) = =

Density: Illumination: Light scattered along R in direction of eye from point at t: R

I t D t P ( ) ( ) cos!

Volume Rendering

Attenuation of light scattered from point t:

exp ( ) ! " # $ $ % & ' '

(

) D s ds

t t

1

Summing the intensity of light arriving at the eye along R from all of the elements:

B D s ds I t D t P dt

t t t t

= ! " # $ $ % & ' ' ( ) * + * ,

• *

. *

/ / exp

( ) ( ) ( ) cos 1

1 1 2

Volume Rendering

R

V x y z ( , , )

C R k R k ( , ) ( , ) !

C R ( )

Image Plane

Volume Rendering

Determining the color of a voxel: C i j k

( , , )

Calculated using the Phong model, assuming that the normal is given by the gradient:

!V x y z

i j k

( , , ) f1 f2 f3 f4 ! f 1 !f 2 ! f 3 !f 4 !( ) X V X ( )

Determining the opacity: !( , , )

i j k

Volume Rendering

Cin Cout C R k R k ( , ) ( , ) ! C C R k C R k R k

• ut

in

= ! + ( ( , )) ( , ) ( , ) 1 " "

Transparency Term Opacity Term

C R C R k R k R i

i k K k K

( ) ( , ) ( , ) ( ( , )) = ! " # $ % & '

= + =

( )

* * 1

1

kth voxel along R

Volume Rendering

Volume Rendering

Depth Max Average Accumulate First Intensity

Volume Rendering

Depth Intensity First

Volume Rendering

Depth Intensity Average

• Average: produces basically an X-ray picture

Volume Rendering

Depth Intensity Max

• Max: Maximum Intensity Projection

used for Magnetic Resonance Angiogram

Volume Rendering

Depth Intensity Accumulate

• Accumulate opacity while compositing colors: make transparent layers

visible! Levoy ‘88

Volume Rendering

color

• pacity

1.0

• bject (color, opacity)

color

• pacity

1.0 volumetric compositing

• bject (color, opacity)

Volume Rendering

color

• pacity

Interpolation kernel 1.0

• bject (color, opacity)

volumetric compositing

Volume Rendering

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

• pacity ! = ! s (1 - !) + !

1.0

• bject (color, opacity)

volumetric compositing Interpolation kernel

Volume Rendering

color

• pacity

1.0

• bject (color, opacity)

volumetric compositing

Volume Rendering

color

• pacity

1.0

• bject (color, opacity)

volumetric compositing

Volume Rendering

color

• pacity

1.0

• bject (color, opacity)

volumetric compositing

Volume Rendering

color

• pacity

1.0

• bject (color, opacity)

volumetric compositing

Volume Rendering

color

• pacity

1.0

• bject (color, opacity)

volumetric compositing

Volume Rendering

color

• pacity
• bject (color, opacity)

volumetric compositing

Volume Rendering

Volume Rendering

• 1. Raycast – once per pixel
• 2. Sample – uniform intervals along ray
• 3. Interpolate – trilinear interpolate, apply transfer function
• 4. Accumulate – integrate optical properties

Volume Rendering

• 1. Raycast – once per pixel
• 2. Sample – uniform intervals along ray
• 3. Interpolate – trilinear interpolate, apply transfer function
• 4. Accumulate – integrate optical properties

• Phong Shading Model
• Requires surface normal vector
• What’s the normal vector of a voxel? Gradient
• Central differences between neighboring voxels

N s L d L a

n h k I n l k I k I ) ˆ ˆ ( ) ˆ ˆ ( ! + ! + =

Resulting = Ambient + Diffuse + Specular

x back front x bottom top x left right I I grad 2 ) ( , 2 ) ( , 2 ) ( ) ( ! ! ! = " =

Volume Rendering

Volume Rendering

Volume Rendering

Volume Rendering

Volume Rendering

Volume Rendering

Volume Rendering

Lacroute’94

3D Texture Mapping

Consider ray casting … x y z (top view)

3D Texture Mapping

x z y

• Render every xz slice in the volume as a texture-mapped polygon
• The proxy polygon will sample the volume data
• Per-fragment RGBA (color and opacity) as classification results
• The polygons are blended from back to front

Use pProxy geometry for sampling

3D Texture Mapping

3D Texture Mapping

Use Image-space axis-aligned slicing plane: the slicing planes are always parallel to the view plane

3D Texture Mapping

Volume Rendering

Volume Rendering

• forward method

– splat voxels onto screen – must interpolate to get pixel values

• reverse method

– is ray-casting through volume – resample along ray – collect intensity/gradient info for pixel

Volume Rendering

Compositing Order

• front-to-back

– keep running product for the term – keep running sum of terms

• back-to-front

– keep one variable – multiply by (1-a), add next I