ray tracing(week 4a)

7/28/2019 Ray Tracing(Week 4a)

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Komputer Grafik 2 (AK045206)

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Ray Tracing


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Outline Review Illumination

Render dengan ray tracing

Pembentukan sinar Interseksi sinar-obyek

Interseksi world space

Vektor normal pada titik interseksi Transformasi vektor normal

Ray tracing rekursif

Bayangan

Permukaan tembus (transparan)


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Review Illumination

Global, physically-based illumination models

describe energy transport and radiation

subject to the properties of light and materials

subject to the geometry of light, objects andviewer

For each surface, there is a distribution that

characterizes its absorption and reflection at

each wavelength

All illumination models are, by definition,

approximate

based on sampling geometry, light distribution

and material properties, and taking rendering

shortcuts


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Ray Tracing 4/38

Why Trace Rays?

More elegant than polygon

rasterization, especially for

sophisticated physics Testbed for techniques and

effects

modeling (reflectance, transport) rendering (e.g. Monte Carlo)

texturing (e.g. hypertexture)

Easiest photorealistic renderer

to implement


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Ray Tracing

Ray tracing allows to add more

realism to the rendered scene

by allowing effects such as shadows

transparency

reflections


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Raytracing, a simple idea: instead of forward

mapping an infinite number of rays from light

source to object to viewer, backmap finite number

of rays from viewer through each pixel to object to

light source (or other object)

Raytracing: shoot rays from eye through sample

point (e.g., a pixel center) of a virtual photo

compute what color/intensity you see at the

corresponding sample point on the object out there;paint this on photo at the ray-photo intersection

point

What you see out there combination of the color of the material of the

surface element you hit first and the colors of lightthat hit it and reflect towards you

Ray Tracing


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Rendering with Raytracing

Subproblems to solve Generate primary (eye) ray

Find closest object along ray path

find the first intersection of a ray that goes out fromthe eye through a pixel center (or any other sample

point on image plane) into the scene

Light sample

use illumination model to determine light at closest

surface element

generate secondary rays from object


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Ray origin

Lets look at the geometry of the problem inuntransformedworld-space

We start a ray from an eye point: P

We send it out in some direction dfrom your eye

toward a typical point on the film plane whose colorwe want to know

Points along the ray have the form P+ td where

- Pis the base point of the ray: the cameras eyepoint

- d is the unit vector direction of the ray

- tis a nonnegative real number

The eye point is the center of projection in theperspective view volume (we use perspectivecoords to avoid having to deal with the inverse of aperspective transformation later)

Generating Rays (1/3)


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We start raycasting with screen-space points (pixels).Need to find a point in 3-D that lies on thecorresponding ray we will use ray to intersect with the original objects in the

original, untransformed world coordinate system

Transform 2D screen-space points into points on thecameras 3D film plane

Any planes orthogonal to the look vector is aconvenient film plane: constant z in canonical viewvolume

Choose a plane to be the film plane and then create afunction that maps screen-space points onto it whats a convenient plane? Try the far plane :-)

to convert, we just have to scale integer screen-spacecoordinates into floating point values between -1 and 1

Generating Rays (2/3)Ray direction


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Transform film plane point into world-space point - we can make the direction vector between the

eye and this world-space point

- we need direction to be in world space so that wecan intersect with the original object in the original,untransformed world coordinate system

Normalizing transformation takes world-spacepoints to points in the canonical view volume - translate to the origin, orient with the axes, scalex

and yto adjust viewing angles, scale so that z: [-1,0];x,y: [-1, 1]

Apply the inverse of the normalizingtransformation: the Viewing Transformation

Ray direction (cont.)

Generating Rays (3/3)


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If an object is defined implicitly by a function fsuch that f(Q) = 0 if and only ifQ is a point on thesurface of the object, then ray-object intersectionis comparatively easy

we can define many objects implicitly implicit functions give you potentially infiniteresolution

tessellating implicit functions is more difficult thanusing them directly

For example, a circle of radius Ris an implicitobject in the plane, and its equation is

points where f(x, y) = 0 are points on the circle

An infinite plane is defined by the function:

A sphere of radius Rin 3-space:

Implicit objects

Ray Object Intersection (1/5)

(A,B,C

(0,0,0)

-D

222),( Ryxyxf

DCzByAxzyxf ),,(

2222),,( Rzyxzyxf


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Ray Object Intersection (2/5)

At what points (if any) does the ray intersect theobject?

Points on the ray have the form P+ td tis any nonnegative real

A point Q lying on the object has the property thatf(Q) = 0

Combining, we want to know For which values oftis f(P + td)= 0 ? (if any)

We are solving a system of simultaneousequations inx, y(in 2D) orx, y, z(in 3D)

Implicit objects(cont.)


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An Explicit Example (1/3)

Consider the eye-point P= (-3, 1), thedirection vectord = (.8, -.6) and the unitcircle given by:

A typical point of the ray is:

Plugging this into the equation of thecircle:

Expanding, we get:

Setting this to zero, we get:

1),( 22 yxyxf

)6.1,8.3()6.,8(.)1,3( ttttPQ d

1)6.1()8.3()6.1,8.3()( 22 ttttfQf

136.2.1164.8.49 22 tttt

0962 tt

2D ray-circle intersection example


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2D ray-circle intersection example (cont.) Using the quadratic formula:

We get:

Because we have a root of multiplicity 2, ourray intersects the circle at one point (i.e., its

tangent to the circle) We can use the discriminant D = b2- 4acto

quickly determine if a ray intersects a curve ornot

- ifD < 0, imaginary roots; no intersection

- ifD = 0, double root; ray is tangent

- ifD > 0, two real roots; ray intersects circle at twopoints

Smallest non-negative real trepresentsintersection nearest to eye-point

a

acbbroots

2

42

3,3,2

36366

tt


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2D ray-circle intersection example (cont.)

Generalizing: our approach will be to take anarbitrary implicit surface with an equation f(Q)= 0 and a ray P+ td, and plug the latter intothe former:

This results, after some algebra, in anequation with tas unknown

We then solve fort, analytically or numerically

0)( dtPf


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Ray Object Intersection (3/5)Implicit objects-multiple conditions

For objects like cylinders, the equation

in 3-space defines an infinite cylinder of unitradius, running along the y-axis Usually, its moreuseful to work with finite objects, e.g. such a unitcylinder truncated with the limits

But how do we do the caps?

The cap is the inside of the cylinder at the yextrema of the cylinder

0122 zx

1

1

y

y

1,0122 yzx


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Ray Object Intersection (4/5)Multiple conditions (cont.)

Picture

We want intersections satisfying the cylinder:

or top cap:

or bottom cap:

11

0122

y

zx

1

0122

y

zx

1

0122

y

zx


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Turn an implicit surface equation into an equationfortand solve

The thing you see first from eye point is at thesmallest non-negative t-value

For complicated objects (not defined by a singleequation), write out a set of equalities andinequalities and then code as cases

Latter approach can be generalized cleverly to

handle all sorts of complex combinations ofobjects- Constructive Solid Geometry (CSG), where objects

are stored as a hierarchy of primitives and 3-D setoperations (union, intersection, difference)

- blobby objects, which are implicit surfaces definedby sums of implicit equations

Ray Object Intersection (5/5)Implicit surface strategy summary


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World-Space IntersectionWorld space - a global view

One way to do intersection is to compute theequation of your object in world-space

Example: a sphere translated to (3, 4, 5) after itwas scaled by 2 in the x-direction has equation

One can then take the ray P+td and plug it intothe equation

Solve the resulting equation fort Always start with the untransformed object

definition in its own coordinate system

We then try to derive what the transformedversion should be, given the CTM (call it M) this is not easy for general transformations, and

furthermore, the transformed version of theequation is always more complicated and thusmore expensive to compute

can we just work with the object in its owncoordinate system? Yes, as follows:

1)5()4(2

)3(),,( 22

2

2 zy

xzyxf

0)( dtPf


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Object-Space IntersectionTransform ray into object-space

Express world-space point of intersection as MQ,where Q is some point in object-space:

If is the equation of the untransformedobject, we just have to solve

- note d is probably not a unit vector

- the parametertalong this vector and its worldspace counterpart always have the same value.Normalizing d would alter this relationship.Do NOT normalize d .

dd

d

d

d

11

11

1

~,

~Let

)(

MPMP

QtMPM

QtPM

MQtP

),,(~

zyxf

0)~~

(~

dtPf

(0,0,0)

MQ

dtP dd 11~~ tMPMtP

0),,( zyxf

Q

0),,(

~

zyxf


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World-Space vs. Object-Space

Use object-space if objects defined there

To compute world-space intersections, you have totransform the implicit equation of your canonical objectdefined in object space - often difficult

But to compute intersections in object-space, you needonly apply a matrix (M-1) to Pand d, which is much

simpler. does M-1 exist?

Mwas composed from two parts: the cumulative modelingtransformation that positions the object in world-space,and the cameras normalizing transformation

the modeling transformations are just translations,rotations, and scales (all invertible)

the normalizing transformation also consists oftranslations, rotations and scales (also invertible); noperspective! (Now you see why we used the hingedperspective view volume)

When youre done, you get a t-value

This tcan be used in two ways: P+td is the world-space location of the intersection of the

ray with the transformed object

P + td is the corresponding point on theuntransformed object (in object space)


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Normal Vectors at

Intersection Points (1/4)Normal vector to implicit surfaces

For illumination we need to find a way, given apoint on the object in object-space, to determinethe normal vector to the object at that point so thatwe can calculate angles between the normal andother vectors

If a surface bounds a solid whose interior is givenby

then we can find the normal vector at a point(x, y,z) via the gradientat that point:

Recall that the gradient is a vector with threecomponents:

0),,( zyxf

),,( zyxf n

),,(),,,(),,,(),,( zyx

z

fzyx

y

fzyx

x

fzyxf


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Normal Vectors at

Intersection Points (2/4)Sphere normal vector example

For the sphere, the equation is

The partial derivatives are

So the gradient is

Normalize n before using in dot products! In some degenerate cases, the gradient may be

zero, and this method failsuse nearby gradientas a cheap hack

1),,( 222 zyxzyxf

zzyxz

f

yzyxy

f

xzyxx

f

2),,(

2),,(

2),,(

)2,2,2(),,( zyxzyxf n


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Normal Vectors at

Intersection Points (3/4)Transforming back to world-space

We have an object-space normal vector

We want a world-space normal vector

To transform the object to world coordinates, wejust multiplied its vertices by M, the objects CTM

Can we treat the normal vector the same way? Answer: NO

Example: say Mscales inxby .5 and yby 2

See the normal scaling applets inAppletsLighting and Shading

objectworld Mnn

nobject

Mnobject

Wrong!Normal must be

perpendicular


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Normal Vectors at

Intersection Points (4/4) Why doesnt multiplying the normal work?

For translation and rotation, which are rigid bodytransformations, it actually does work

Scale, however, distorts the normal in exactly the

opposite sense of the scale applied to the surface,not surprising given that the normal isperpendicular scaling y by 2 causes normal to scale by .5:

Well see this algebraically in the next slides

xy2

1:normal

xy 2

xy4

1:normal

xy 4


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Transforming Normals (1/4)

Object-space to world-space

Take the polygonal case, for example

Lets compute the relationship between theobject-space normal nobjto a polygon Hand the

world-space normaln

worldto a transformedversion ofH, say MH

Ifv is a vector in world space that lies in thepolygon (e.g., one of the edge vectors), then thenormal is perpendicular to the vectorv:

But vworldis just a transformed version of somevector in object space, vobj. So we could write

But wed be wrong, because v is a tangent vector its transformed by the derivative

which includes a 3D->4D->3D mapping

0 worldworld vn

0 objworld Mvn

03

objworld M vn

M Mof3x3leftupper3


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Transforming Normals (2/4)Object-space to world-space(cont.)

So we want a vector nworld such that

We will show in a moment that this equation canbe rewritten as

We also already have

Therefore

Left-multiplying by (Mt)-1,

03 objworld M vn

objworld

tM nn 3

03 objworldt

M vn

0 objobj vn

objtworld M nn-1

3


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Transforming Normals (3/4)Object-space to world-space(cont.)

So how did we rewrite this:

As this:

Recall that if we think of vector as nx1 matrices,then switching notation,

Rewriting the formula above, we have

Writing M = Mtt, we get

Recalling that (AB)t = BtAt, we can write this:

Switching back to dot product notation:0)(3

obj

t

world

t

M vn

03 objworld M vn

baba t

03 objt

worldMvn

03 objttt

worldM vn

0)( 3 objworldt

M vn

0)( 3 objworldt

M vn


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Transforming Normals (4/4)Applying inverse-transpose of M to normals So we ended up with

Fortunately, invert and transpose can be swapped,

to get our final form:

Why do we do this? Its easier!

A hand-waving interpretation of(M3-1)t M3 is a composition of rotations and scales, Rand S (why

no translates?). Therefore

so were applying the transformed (inverted, transposed)versions of each individual matrix in their original order

for a rotation matrix, this transformed version is equal tothe original rotation, i.e., normal rotates with object

(R-1)t=R; the inverse reverses the rotation, and the transposereverses it back

for a scale matrix, the inverse reverses the scale, while

the transpose does nothing: (S(x,y,z)-1)t = S(x,y,z)-1=S(1/x,1/y,1/z)

obj

t

world M nn1

3 )(

obj

t

world M nn )(1

3

...))()(()(...)...)(( 11111 tttt SRRSRS


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Summary :

Non-Recursive Ray TracerP = eyePt

For each sample of image

Compute d

for each object

Intersect ray P+td with object

Select object with smallest non-negative t-value(this is

the visible object)

For this object, find the object-space intersection point

Compute the normal at that point

Transform the normal to world space

Use the world-space normal for lighting computations


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Recursive RaytracingSimulating global lighting effects (Whitted, 1979)

By recursively casting new rays into

the scene, we can look for more

information Start from the point of intersection

Wed like to send rays in all

directions

Send rays in directions we predict

will contribute the most:

toward lights (shadow)

bounce off objects (specular reflection) through the object (transparency)


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Shadows

Each light in scene makes contribution to thecolor and intensity of a surface element

Iff it reaches the object!

- could be occluded by other objects in the scene

- could be occluded by another part of the same

object Construct a ray from the surface to each light

Check to see if ray intersects other objects beforeit gets to the light

- if ray does NOT intersect that same object oranother object on its way to the light, then the full

contribution of the light can be counted - if ray does intersect an object on its way to the

light, then no contribution of that light should becounted

- what about transparent objects?

numlights

light

specular[diffusesitylightIntennattenuatio

ambientnsityobjectInte

1


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Transparent Surfaces (1/2)Nonrefractive transparency

For a partially transparent polygon

2polygonforcalculatedintensity

1polygonforcalculatedintensity

1polygonofncetransmitta

2

1

1

2111)1(

I

I

k

IkIkI

t

tt

I2

I1

polygon 1

polygon 2


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Transparent Surfaces (2/2)Refractive transparency

We model the way light bends at interfaces with

Snells Law

2mediumofrefractionofindex

1mediumofrefractionofindex

t

i

t

iit

sinsin


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Recursive Raytracing (1/2)

Trace secondary rays at intersections: - light: trace ray to each light source. If light source

blocked by opaque object, it does not contribute tolighting, and object in shadow of blocked light at thatpoint

- specular reflection: trace ray in mirror direction byreflecting about N

- refraction/transparency: trace ray in refractiondirection by Snells law

- recursively spawn new light, reflection, refractionrays at each intersection until contribution negligible

Your new lighting equation

note: intensity from recursed rays calcd with same eqn

light sources contribute specular and diffuse lighting

Limitations - recursive interobject reflection is strictly specular:

along ray reflected about N - diffuse interobject reflection handled by radiosity

Informative videos: silent movie on raytracing, ALong Rays Journey into Light

recursive

refracted

ttt

reflected

ssm

specular

nss

diffuse

ddpatt

ambient

dkaa IOkOkVROkLNOkIfOII ])([


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Recursive Raytracing (2/2)Light-ray Trees


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Raytracing Pipeline

Raytracer produces visible samples from

model

samples convolved with filter to form pixel image

Additional pre-processing

pre-process database to speed up per-sample

calculations (done once, sometimes must be

redone if objects transformed (resize, translate))


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Referensi F.S.Hill, Jr., COMPUTER GRAPHICS

Using Open GL, Second Edition,Prentice Hall, 2001

Andries van Dam, Introduction toComputer Graphics, Slide-Presentation, Brown University, 2003,(folder : brownUni)

______, Interactive Computer Graphic,Slide-Presentation, (folder :Lect_IC_AC_UK)

ray tracing(week 4a)

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