Computer Graphics 2 Lecture 13: Ray-Tracing Techniques

Computer Graphics 2Lecture 13:

Ray-Tracing Techniques

Benjamin Mora 1University of Wales

Swansea

Dr. Benjamin Mora

Content

2Benjamin MoraUniversity of Wales

Swansea

• What is Ray-Tracing?

• Mathematics for ray-tracing intersections.

• Common Ray-Tracing effects.

• Other Ray-Tracing Algorithms.

• Current Ray-Tracing Technology.

What Is Ray-Tracing ?


Swansea

What is Ray Tracing ?


Swansea

• The alternative to rasterization-based methods (like OpenGL or DirectX).– Image-Order algorithm.

– Every pixel is processed separately.

• Slower than rasterization methods in the case of few primitives to render.

• Rarely used in games, widely used for realistic computer graphics.

• www.povray.org.

http://www.povray.org/



Swansea

Jaime Vives Piqueres (2004), http://www.povray.org/i/hof/office-13.jpg



Swansea

Jaime Vives Piqueres (2001), http://www.povray.org/i/hof/office-13.jpg



Swansea

viewlocation

image plane

primaryray

secondaryrays



Swansea

• Basic (or Naïve )Algorithm (Whitted 1980):FOR every pixel DO

FOR every primitive in the scene DO

IF there is a ray/primitive intersection

test if this is the closest intersection computed so far.

If this is the case,

keep this primitive as the closest primitive for the next tests.

• Basic Complexity: – O(number of rays*number of primitives). – Optimized algorithms can do much better (See the next lectures)!

Mathematics for Ray-Tracing intersections


Swansea

Parametric Ray/Plane Formulation


Swansea

• A ray can be described with a parametric formulation from a (starting) point and a direction

• A parametric plane can be described from one point and two vectors:

• A plane can also be defined from a point and a normal vector (here P0 and Vz).

directionVtPtP 0

yx VyVxPyxP 0,

P0

Vx

VyVz

Ray/Plane Intersection


Swansea

• We are looking for the point p(t) such that p(t) belong to the plane h.

• h is defined with a point o and a normal n.

• If vdir perpendicular to n, then either 0 solution, or an infinite number of solutions if p0 belongs to h (the vector p0o perpendicular to n).

p0

h

n

op(t)

vdir

Ray/Plane Intersection


Swansea

• Otherwise, there is only one point that is solution. We have to find out t.

• The solution comes from the fact that the vector p(t)o is perpendicular to n:

(6)

(5)

(4)

(3)

(2) 0.

(1) 0)(

0

00

0

nv

nopt

nopnpnonvt

nonvtp

nontp

ntpo

notp

dir

dir

dir

p0

h

n

op(t)

vdir

13Benjamin Mora

University of Wales Swansea

The ray intersects the cell only if max(txmin, tymin)<min(txmax, tymax).

Finding out the exit face consists of finding out the minimum of (txmax, tymax).

txmin txmax

tymin

tymax

t

tymin tymax

txmin txmax

p(t) = p0 +v•t

p0 v

Ray/Cube Intersection

Ray/Cube Intersection


Swansea

• A cube can be defined with 6 planes (faces).• 3 pairs of opposite faces (let’s reference them as x,y,z) .• A parametric ray p(t) intersects 2 opposite faces at 2

locations, with 2 respective parameters tmin and tmax. (let’s suppose tmin<=tmax).

• The 6 faces are intersected with 6 different parameters (txmin,txmax, tymin,tymax, tzmin,tzmax).

• The ray intersects the cube only if:Max (txmin,tymin,tzmin)< Min(txmax,tymax,tzmax).

Ray/Sphere Intersection


Swansea

• A sphere can be defined by a center o and a radius r.

• Looking for the points p(t) such as: norm(p(t)o)=r.

p0r

o

vdir p(t)

02 (3)

(2)

)( (1)

2200

22

220

20

22

ropvoptvt

rvtopvtpo

rotp

dirdir

dirdir

Ray/Triangle Intersections


Swansea

• Several ways to achieve this goal.

– Badouel’s Algorithm.– Moller’s Algorithm (not shown).– Other Algorithm (close to Segura et al. 98).

p0 vdir

p(t)

Badouel’s Algorithm


Swansea

• First, find the intersection p(t) between the ray and the plane containing the triangle.

• Then use the barycentric coordinates of the point to sort the intersection out (2D algorithm).

p1

p(t)

p0

p2 20100 .. pppptpp 1,0,0

)(

)(

)(

02010

02010

02010

zzzzzz

yyyyyy

xxxxxx

t

t

t

Other Ray/Triangle Algorithms


Swansea

• The edges of the triangles can be considered as oriented.

• The Ray should always be “on the same side” of the edges

• => Three triple products with the same sign.

Segura, R.J., Feito, F.R. Segura, R.J., Feito, F.R.: An algorithm for determining intersection segment-polygon in 3D. Computer & Graphics, Vol.22, No.5, pp.587-592, 1998.

p vdir

p(t)

p0p1

p2

Other Ray/Triangle Algorithms


Swansea

• There is an intersection Ray/Triangle only if:

• Fast algorithm.– Can take advantage of SIMD instructions on processor.

• But:– Only in order to detect an intersection.– Extra work must be done in order to get the tri-linear interpolating

weights (including a div) and the real intersection location if needed (but not always required).

p vdir

p(t)

p0p1

p2

vppppSignvppppSignvppppSign 022110

Triangle/Box Intersection


Swansea

• Why?

– Bounding boxes are used much to speed-up intersections.

• Standalone boxes.

• Spatial subdivision techniques.

– Therefore needed to construct such acceleration structures.



Swansea

• Solution 1: Working in the triangle plane.

• For every face of the cube:

– Find the face/triangle plane intersection.• A line!

– Clip the triangle/polygon with this line. (Sutherland ).

• Intersection occurs if result not NULL!

• Complex and Costly.



Swansea

• Solution 2: Use the Separating Axis Theorem. (Akenine-Moller)

• No intersection occurs between 2 convex objects if there exists a separating plane.

Triangle/Box Intersection (2D)


Swansea



Swansea



Swansea

• 13 planes to be tested in 3D.

• For each plane:– Compute distances to the triangle vertices

• Using the plane equation.

– Compute distances to the box vertices.– No intersection if MIN of one distance set greater

than MAX of the other distance set.

• If condition is false for all 13 cases, an intersection exists.



Swansea

• Plane directions are determined from:–

Where Ei are the directions of the edges of the box, and Tj the directions of the edges of the triangle. (9 cases)

– The faces of the box. (3 cases)– The triangle plane normal. (1 case)

Ei X Tj

Simple Ray-Tracing Effects


Swansea



Swansea

• The main operation is to find an intersection from a given ray.

• This operation can be used (by casting secondary rays) to compute simple effects that are more

difficult to implement with OpenGL like (but fog):– Reflection.

– Refraction.

– Shadows.

– Fog. Source: Min Chen



Swansea

• Basic Lighting model:

objectcolour

result oftracing

reflective ray

result oftracing

refractive ray++

Rin

Rfl

Rfr

refractiont

reflectionm

object

Ik

Ik

II

kmkt



Swansea

• Shading can be done in the same way as to OpenGL.– But better methods exist.

Source: Min Chen

NV (-Rin)

L

nsdpaa RVkNLkIdfkII

R



Swansea

• Reflection:



Swansea

• Reflection:

N

Vincident Vreflected

ii

1,2 NVNNVV incidentincidentreflected

L



Swansea

• Refraction:

NV (-Rin) Rfl

Lii

VNR riririfr )coscos(

R

Rfr

Snell’s Law

rirr

i

i

r

sinsin



Swansea

• Total Internal Reflection:

NV (-Rin)

Rfr

criticalanglecritical



Swansea

• Taking into account reflection and refraction several times (recursively) requires an exponential number of rays!

• The algorithm should stop after reaching a maximum number of consecutive reflections/refractions.

• Only reduced to some specific lighting effects.



Swansea



Swansea

• Fog effect (also possible with OpenGL).– Depth Cue:

– Fog:

max

maxminminmax

min

min

1

0

Dd

DdDDD

DdDd

f

D

df

e

11

fIfII foginout 1

or



Swansea

• Super Sampling can be used in order to improve Anti-Aliasing.

non-adaptivesupersampling

stochasticsampling

adaptivesupersampling



Swansea



Swansea

• The final color is the average of all the sub-sampled rays.

• Uniform Super-Sampling:– Nice but not always needed.

• Adaptive Super-Sampling: – Useful for the regions with large color transitions (high frequencies),

like object boundaries.

– Trade-off between quality and speed.

• Stochastic Super-Sampling:– Take advantage of the visual system limitations.



Swansea

• Stochastic Super-Sampling:

– Proposed by Robert L. Cook.

• Stochastic Sampling in Computer Graphics, ACM transaction on Graphics No. 5(1), January 1986, pp. 51-72.

– The ray location(s) are randomly chosen inside a pixel.

– Our visual system is less sensitive to noise than to contours.

Other Ray-Tracing Algorithms.


Swansea

Other Ray-Tracing Algorithms.


Swansea

• Pyramidal Ray-Tracing.

• Cone Tracing.

• Beam Tracing.

Pyramidal Ray-Tracing.


Swansea

• Proposed by:

• And improved by:

• Main Idea: Replacing Rays by Pyramids to improve aliasing.

Whitted, Turner. An Improved Illumination Model for Shaded Display. C.A.C.M. vol. 23, no. 6, June 1980, pp. 343-3~19.

J. Genetti, D. Gordon and G. Williams. Adaptive supersampling in object space using pyramidal rays. Computer Graphics Forum 17 (1998) 29-54.

viewlocation

image plane

primaryray



Swansea

J. Genetti, D. Gordon and G. Williams. Adaptive supersampling in object space using pyramidal rays. Computer Graphics Forum 17 (1998) 29-54.

The triangle will contribute here to 7/64 of the final pixel colour.



Swansea

• If a pyramid only partially intersects a graphics primitive (e.g., triangle), then the pyramid is subdivided until a given depth is reached.– The final pixel value is the (weighted) average of all the

samples inside the triangle.

– Non-intersecting pyramids keep on computing intersections with the rest of the scene.

Cone Tracing.


Swansea

• Proposed by:John Amanatides. Ray Tracing with Cones. Siggraph 1984, pp. 129-135.

viewlocation

image plane

primaryray

Partial intersection

Cone Tracing.


Swansea

• Similar to pyramidal ray-tracing, but the pyramid is now replaced by a cone.– Allows easier intersection computations.

– 1 cone per pixel, no subdivision.

• For a ray-triangle intersection:– The cone is projected onto the triangle plane.

• Circular projection.

– If there is a partial intersection detected, the colour of the triangle is weighted by the corresponding covering percentage, and computing intersections will carry on.

Cone Tracing: Result


Swansea

Beam Tracing.


Swansea

• Proposed by:

• Takes advantages of spatial coherency between rays to group them and save computations.

• Rays are distributed after Reflections and

Refractions according to the

reflection/refraction coefficients.

• Constructs a beam tree.

• Outdated.

Paul S. Heckbert and Pat Hanrahan. BEAM TRACING POLYGONAL OBJECTS. Siggraph 1984, pp. 119-127.

Current Ray-Tracing Technology


Swansea



Swansea

• Ray-Tracing is now used a lot for realistic rendering of scenes. – Not just Reflections/Refractions.

• Has a better complexity than OpenGL based renderings (when optimized).– Scalable.

• A simple ray representation (no cone or pyramid) is often used.

• Therefore, researchers have recently tried to hardware accelerate it.



Swansea

• 3 different approaches:– Cluster based CPU approach.

• Use SIMD instructions with packed rays.• A lot of threads in parallel.

– GPU Based.• Not very efficient because not really designed for RT.

– Hardware (FPGA) accelerated ray-tracing.• Seems to be efficient.• Need to compute a spatial subdivision tree first, which is not

really appropriate to dynamic scenes.

Tim Foley, Jeremy Sugerman. KD-Tree Acceleration Structures for a GPU Raytracer. In Proceedings of Graphics Hardware 2005.

Ingo Wald, Timothy J. Purcell, Jörg Schmittler, Carsten Benthin and Philipp Slusallek. Realtime Ray Tracing and its use for Interactive Global Illumination. Eurographics 2003 State of the Art Report.



Swansea

• The RPU unit. See Video !!!

Sven Woop, Jorg Schmittler, and Philipp Slusallek. RPU: A Programmable Ray Processing Unit for Realtime Ray Tracing. Siggraph 2005.

Documents

Computer Graphics 2 Lecture 13: Ray-Tracing Techniques