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GR2 Advanced Computer Graphics AGRPowerPoint Presentation

GR2 Advanced Computer Graphics AGR

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object space

V

texture

space

U

X

W

Z

Solid 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 solid, texturing
- texture now occupies a volume
- can imagine object being carved out of the texture volume

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

Defining 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 (w)

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

- ie as a function that can be evaluated, such as:

texture

space

U

W

Example: 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,

then

texture (u,v,w) = dark

else

texture (u,v,w) = light

looking down:

cross section view

Example: 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

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) )

- basic pattern:
- with turbulence:
- marble (x) = marble_colour (sin (x + turbulence (x) ) )

Bump 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 trick
- surface stays the same
- but the true normal is perturbed, or jittered, to give the illusion of surface ‘bumps’

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 normal

N’(u) for reflection

model

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

- eg cylinder of fixed radius r is defined by parameters (s,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)

How 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

How it Works - The Maths

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

or

N’ = 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

- Ps x Pt = N and N x N = 0, this simplifies to:

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) = 0

yt (s,t) = 0

zt (s,t) = 1

N = Ps x Pt :

Nx = r cos (s)

Ny = r sin (s)

Nz = 0

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

Bump MappingA Bump Map

Bump MappingResulting Image

Bump MappingAnother Example

Bump MappingProcedurally Defined Bump Map

Environment 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?

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 map

projection of scene

on face of cube -

environment map

camera

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

environment

map

eye

point

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 mapping
- Can use sphere rather than cube

Jim 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

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