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GR2Advanced Computer

GraphicsAGR

GR2Advanced Computer

GraphicsAGR

Lecture 12Solid TexturesBump Mapping

Environment Mapping

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Marble TextureMarble Texture

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Solid TextureSolid Texture

A difficulty with 2D textures is the mapping from the object surface to the texture image– ie constructing fu(x,y,z) and fv(x,y,z)

This is avoided in 3D, or solidsolid, texturing– texture now occupies a volume– can imagine object being carved out of

the texture volumeU

V

W

texturespace X

Y

Z

object space

Mapping functions trivial: u = x; v = y; w = z

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Defining the TextureDefining the Texture

The texture volume itself is usually defined procedurally– ie as a function that can be

evaluated, such as:

texture (u, v, w) = sin (u) sin (v) sin texture (u, v, w) = sin (u) sin (v) sin (w) (w)

– this is because of the vast amount of storage required if it were defined by data values

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Example: Wood TextureExample: Wood Texture

Wood grain texture can be modelled by a set of concentric cylinders – cylinders coloured dark, gaps

between adjacent cylinders coloured light

radius r = sqrt(u*u + w*w)

if radius r = r1, r2, r3,

thentexture (u,v,w) = darkelsetexture (u,v,w) = light

looking down:cross section view

U

V

W

texturespace

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Example: Wood TextureExample: Wood Texture

It is a bit more interesting to apply a sinusoidal perturbation– radius:= radius + 2 * sin( 20*) , with

0<<2 .. and a twist along the axis of the

cylinder– radius:= radius + 2 * sin( 20* + v/150

) This gives a realistic wood texture

effect

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Wood TextureWood Texture

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How to do Marble?How to do Marble?

First create noise function (in 1D):– noise [i] = random numbers on lattice

of points Next create turbulence:

– turbulence (x) = noise(x) + 0.5*noise(2x) + 0.25*noise(4x) + …

Marble created by:– basic pattern:

• marble (x) = marble_colour (sin (x) )

– with turbulence:• marble (x) = marble_colour (sin (x + turbulence

(x) ) )

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Marble TextureMarble Texture

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Bump MappingBump Mapping

This is another texturing technique Aims to simulate a dimpled or

wrinkled surface– for example, surface of an orange

Like Gouraud and Phong shading, it is a tricktrick– surface stays the same– but the true normal is perturbed, or

jittered, to give the illusion of surface ‘bumps’

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Bump MappingBump Mapping

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How Does It Work?How Does It Work?

Looking at it in 1D:

original surface P(u)

bump map b(u)

add b(u) to P(u)in surface normal direction, N(u)

new surface normalN’(u) for reflectionmodel

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How It Works - The Maths!How It Works - The Maths!

Any 3D surface can be described in terms of 2 parameters– eg cylinder of fixed radius r is defined by

parameters (s,t)

x=rcos(s); y=rsin(s); z=t Thus a point P on surface can be written

P(s,t) where s,t are the parameters The vectors:

Ps = dP(s,t)/ds and Pt = dP(s,t)/dt

are tangential to the surface at (s,t)

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How it Works - The MathsHow it Works - The Maths

Thus the normal at (s,t) is:N = Ps x Pt

Now add a bump map to surface in direction of N:

P’(s,t) = P(s,t) + b(s,t)N To get the new normal we need to

calculate P’s and P’t

P’s = Ps + bsN + bNs

approx P’s = Ps + bsN - because b small

P’t similar– P’t = Pt + btN

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How it Works - The MathsHow it Works - The Maths

Thus the perturbed surface normal is:N’ = P’s x P’t

orN’ = Ps x Pt + bt(Ps x N) + bs(N x Pt) + bsbt(N x N)

But since– Ps x Pt = N and N x N = 0, this simplifies to:

N’ = N + D

– where D = bt(Ps x N) + bs(N x Pt)

= bs(N x Pt) - bt(N x Ps )

= A - B

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Worked Example for a Cylinder

Worked Example for a Cylinder

P has co-ordinates:

Thus:

and then

x (s,t) = r cos (s)y (s,t) = r sin (s)z (s,t) = t

Ps : xs (s,t) = -r sin (s)ys (s,t) = r cos (s)zs (s,t) = 0

Pt : xt (s,t) = 0yt (s,t) = 0zt (s,t) = 1

N = Ps x Pt : Nx = r cos (s)Ny = r sin (s)Nz = 0

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Worked Example for a Cylinder

Worked Example for a Cylinder

Then: D = bt(Ps x N) + bs(N x Pt) becomes:

and perturbed normal N’ = N + D is:

D : bt *0 + bs*r sin (s) = bs*r sin (s)bt *0 - bs*r cos (s) = - bs*r cos (s)bt*(-r2) + bs*0 = - bt*(r2)

N’ : r cos (s) + bs*r sin (s)r sin (s) - bs*r cos (s)-bt*r2

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Bump MappingA Bump Map

Bump MappingA Bump Map

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Bump MappingResulting ImageBump Mapping

Resulting Image

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Bump Mapping - Another Example

Bump Mapping - Another Example

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Bump MappingAnother ExampleBump Mapping

Another Example

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Bump MappingProcedurally Defined Bump

Map

Bump MappingProcedurally Defined Bump

Map

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Environment MappingEnvironment Mapping

This is another famous piece of trickery in computer graphics

Look at a highly reflective surface– what do you see?– does the Phong reflection model predict

this? Phong reflection is a local illumination

model– does not convey inter-object reflection– global illumination methods such as ray

tracing and radiosity provide this .. but can we cheat?

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Environment Mapping - Recipe

Environment Mapping - Recipe

Place a large cube around the scene with a camera at the centre

Project six camera views onto faces of cube - known as an environment mapenvironment map

camera

projection of sceneon face of cube -environment map

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Environment Mapping - Rendering

Environment Mapping - Rendering

When rendering a shiny object, calculate the reflected viewing direction (called R earlier)

This points to a colour on the surrounding cube which we can use as a texture when rendering

eyepoint

environmentmap

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Environment Mapping - Limitations

Environment Mapping - Limitations

Obviously this gives far from perfect results - but it is much quicker than the true global illumination methods (ray tracing and radiosity)

It can be improved by multiple environment maps (why?) - one per key object

Also known as reflection mappingreflection mapping Can use sphere rather than cube

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Environment MappingEnvironment Mapping

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Environment MappingEnvironment Mapping

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Jim BlinnJim Blinn

Both bump mapping and environment mapping concepts are due to Jim Blinn

Pioneer figure in computer graphics

www.research.microsoft.com/~blinn

www.siggraph.org/s98/conference/keynote/slides.html