11-texture-mapping.pdf

(1)

RAM Memory in the graphic cards

Screen

buffer

(RGBA)

Screen

buffer 2

(RGBA)

Depth

buffer

DISPLAY

LISTs

_Texture

RAM

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tributes

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Triangles

rasterizer

set-up

Segments

rasterizer

set-up

points

rasterizer

set-up

vertex

(2)

Texture Mapping

• The fragment operations can access a specialized

RAM

– The Texture RAM

– Organized in a set of Textures

• Each texture is an array

1D, 2D o 3D

(3)

Texels

• Typical examples of texels:

– each texel is a color (components: R-G-B, or R-G-B-A)

• The texture is a "

color-map

"

– each texel is an alpha value

• the texture is an "

alpha-map

"

– each texel is a normal (components: X-Y-Z)

• the texture is a "

normal-map

" or "

bump-map

"

– each texel contains a specularity value

• the texture is a "

shininess-map

"

(4)

Typical application: mapping images on geometry

3D geometry

(quads mesh

+

RGB texture 2D

(color-map)

(5)

(6)

More examples

(7)

Texture Mapping: History

• 1974 introduced by Ed Catmull

– In its Phd Thesis

• Only in 1992 (!) we have texture mapping hardware

– Silicon Graphics RealityEngine

• 1992 on:

increasingly used and integrated in graphic cards

– First of all by low end graphic boards

• Today:

a fundamental rendering primitive

– the main image-based technique

Ed Catmull

(8)

Notation

Texture 2D

u

v

texel

Texture Space

(or "parametric space" or "u-v space")

A Texture is defined

In the region [0,1] x [0,1]

of the "parametric space"

51 2 te

xe

ls

1024 texels

(9)

Texture Space

u

v

Texture Mapping

• We associate to each

vertex

(of each triangle) its

u,v

coordinates in the

texture space

Screen Space

x

₀

,y

₀

x

₂

,y

₂

x

₁

,y

₁

u

₀

,v

₀

u

₁

,v

₁

u

₂

,v

₂

Position of the

1st vertex

Attributes of the

1st vertex

u

₀

,v

₀

u

₁

,v

₁

(10)

Texture Mapping

• More precisely, we define a

mapping

between the

3D triangle 3D and a texture triangle

(11)

Texture Mapping

• each

fragment

has its own

u,v

coordinates in the

texture space

Texture Space

Screen Space

(12)

Texture Mapping

Screen

buffer

Texture RAM

interpolation

texture

coordinates

texture

coordinates

interpolated

including:

texture

coordinates

(per vertex!)

texture

look-up

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tributes

P

ro

je

ct

ed

V

ertice

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tri

bu

te

s

Triangles

rasterizer

set-up

Segments

rasterizer

set-up

points

rasterizer

set-up

vertex

(13)

Problem: linear interpolation of texture coordinates

• Not true for perspective projection!

– It is only an approximation

– It works fine to interpolate colors, normals, ..

– Not applicable to interpolate texture coordinates...

V

₁

V

₂

V

₃

p

3 R

R

2 f(p)

f( v

₁

)

f( v

₂

)

f( v

₃

)

projection

f

p

has barycentric coordinates

a,b,c

(14)

Problem: linear interpolation of texture coordinates

• Example:

u

v

1

1 u,v= (1,0)

u

₁

,v

₁

= (1,1)

(15)

Problem: linear interpolation of texture coordinates

(16)

Solution: Perspective Correction

• p

has barycentric coordinates

c

₀

c

₁

c

₂

V

₀

V

₂

V

₁

A

₀

,B

₀

...

A

₁

,B

₁

...

A

₂

,B

₂

...

p

=

c

₀

v

₀

+

c

₁

v

₁

+

c

₂

v

₂

Attributes of

p

:

(not considering the

“perspective correction")

A

_p

=

c

₀

A

₀

+

c

₁

A

₁

+

c

₂

A

₂

B

_p

=

c

₀

B

₀

+

c

₁

B

₁

+

c

₂

B

₂

(17)

Solution: Perspective Correction

• p

has barycentric coordinates

c

₀

c

₁

c

₂

V

₀

V

₂

V

₁

A

₀

,B

₀

...

A

₁

,B

₁

...

A

₂

,B

₂

...

p

=

c

₀

v

₀

+

c

₁

v

₁

+

c

₂

v

₂

Attributes of

p

:

(not considering the

“perspective correction")

w

₀

w

₁

A

_p

=

c

₀

A

₀

0 +

c

₁

A

₁

+

c

₂

A

₂

w

₀

A

w

1 ₁

A

₂

w

₂

A

_p

=

A

_p

=

c

₀

A

1 ₀

+

c

₁

A

1 ₁

+

c

₂

A

1 ₂

w

₂

(18)

Solution: Perspective Correction

Screen

buffer

Original

attribute A

Apply

transformations

compute:

A'

= A / w

and

w'

= 1 / w

c

₀

A

_w

0 +

c

₁

+

c

₂

0 A

₁

w

₁

A

₂

w

₂

A

_p

=

1 w

₀

w

1 ₁

w

1 ₂

interpolate

A'

and

w'

Final

fragment

attribute:

A'

/

w'

_c

0 +

c

1 +

c

2

(19)

Perspective Correction

(20)

Perspective Correction

• Texture mapping with

perspective correction

– Also known as (aka):

• Perfect texture mapping

• 3 magic vectors method

(out of fashion)

(21)

Note: the texture must be loaded

Screen

buffer

Texture RAM

L

O

A

D

(22)

Note: the texture must be loaded

1. From hard disk to main RAM memory

• (in the

motherboard

)

2. From main RAM memory to Texture RAM

• (

on board

of the graphics HW)

(23)

In OpenGL

• As an example:

glEnable(GL_TEXTURE_2D);

glBindTexture (GL_TEXTURE_2D, ID);

glTexImage2D (

GL_TEXTURE_2D,

0, // mipmapping

GL_RGB, // original format

imageWidth, imageHeight,

0, // border

GL_RGB, // RAM format

GL_UNSIGNED_BYTE,

(24)

(25)

Assigning texture coordinates to vertices

• 2 possibilities:

– Computing textures coordinates on the fly

• During the rendering…

– Precomputing

• (and store them within the mesh)

(26)

Difficult problem: u-v mapping

• Associate texture coordinates to each vertex of the

mesh

– During preprocessing

u

v

(27)

Difficult problem: u-v mapping

(28)

In OpenGL

• Like any other attribute

(29)

Assigning texture coordinates to vertices

Screen

buffer

Texture RAM

interpolating

texture

coordinates

texture

coordinates

interpolated

texture

look-up

texture

coordinates

(transformed)

including:

texture

coordinates

Fr

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tributes

Pro

ject

ed

Ve

rtice

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&

co

m

pu

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d

at

tribu

te

s

Triangles

rasterizer

set-up

Segments

rasterizer

set-up

points

rasterizer

set-up

vertex

(30)

Assigning texture coordinates to vertices

• 2 possibilities:

– Computing textures coordinates on the fly

• During the rendering…

– Precomputing

(31)

(32)

Automatically computed

• Idea: from (x,y,z) to (u,v) - Linearly

• Using object or view coordinate

– (before or after the trasformation)

(33)

Automatically computed

• Even 1D

(34)

Assigning texture coordinates to vertices

• 2 possibilities:

– Computing textures coordinates on the fly

• During the rendering…

– Precomputing

(35)

Environment mapping: spherical

Environment map: a texture containing the color

of the environment “reflexed by each normal of

the half-sphere”.

(36)

Environment mapping: spherical

Simulates a mirror-like object reflecting a far-away background

(37)

Environment mapping: cube

front right back

below

above

(38)

Environment mapping: cube

Screen

buffer

Texture RAM

interpolating

3D texture

coordinates

interpolated

coordinates

3D texture

Project on the

cube, look-up the

corresponding face

compute

3D Texture

coordinates

[-1,+1] x [-1,+1] x [-1,+1]

As view ray reflexed

by the normal

(39)

Environment mapping: cube

front right back

below

above

(40)

Environment mapping: cube and spherical

• Spherical:

– one texel for each direction

in the half-sphere

• Projected on a circle

– the texture coordinate is

the normal

– It has the "headlight“ effect:

• I can only rotate the object

while the viewpoint does not

change

• Cube

– one texel for each direction

in the sphere

• Projected on the surface of

the

cube

– the texture coordinate is

the view direction reflexed

by the normal

(41)

Automated computation of texture coordinates

glEnable(GL_TEXTURE_GEN_S);

1- abilitate:

2- choice of the mode:

glTexGeni(GL_S , GL_TEXTURE_GEN_MODE ,

mode

)

S, T, R, Q

mode

=

GL_OBJECT_LINEAR

GL_EYE_LINEAR

GL_SPHERE_MAP

Computes the texture coordinates from

the position in object coordinates

(before the trasformation)

Computes the texture coordinates

from the position in view coordinates

(after the MODEL-VIEW)

(42)

Automated computation of texture coordinates

glTexGenfv(GL_S, GL_EYE_PLANE , v);

3- choice of the plane

S, T, R, Q

EYE OBJECT

or

4 elements vector

The resulting texture coordinate = v

T

_{• pos_vertex}

(43)

u

Texture Look-up out of bounds: “clamp” mode

if (u<0) u←0; if (u>1) u←1;

if (v<0) v←0; if (v>1) v←1;

1

(44)

Texture Look-up out of bounds: “repeat” mode

u

v

1

(45)

Repeated textures

• Typical use:

Very space-efficient!

(46)

In OpenGL

glTexParameteri(

GL_TEXTURE_2D,

GL_TEXTURE_WRAP_S,

GL_CLAMP );

glTexParameteri(

GL_TEXTURE_2D,

GL_TEXTURE_WRAP_S,

GL_REPEAT );

or

note:

u

and v

treated

separately

example: repeat u and clamp v

Texture parameters.

each texture loaded in

(47)

Texture Look-up

• A fragment can have non-integer coordinates

(in texels)

Texture Space

Screen Space

(48)

pixel

Texture Look-up

Texture Space

Screen Space

pixel

texel

one pixel = less than one texel

one pixel = more than one texel

(49)

Magnification

Solution 1:

Use the texel containg the pixel

(that is, the texel whose center

is closest to the u,v coordinates

of the fragment)

Equivalent to rounding up

the texel coordinates

to the nearest integer

"Nearest Filtering"

0.5 1.5 2.5 3.5 4.5 5.5 6.5

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5 u

v

(50)

Magnification

texture 128x128

Nearest Filtering:

result

(51)

Magnification

Solution 2:

Compute the average of the four

closest texels

Bilinear Interpolation

0.5 1.5 2.5 3.5 4.5 5.5 6.5

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5 u

v

(52)

Magnification

texture 128x128

(53)

Magnification

• Nearest filtering:

– Texels are visible

– Ok if texel borders are useful

– More efficient

• Bilinear Interpolation

– Usually provides better quality

– Less efficient

(54)

Minification

(55)

Minification: MIP-mapping

MIP-map

level 0

MIP-map

level 1

MIP-map

level 2

MIP-map

level 3

MIP-map

level 4

(only one texel)

(56)

Mipmap Math

• Define a

scale factor

, =texels/pixel

–

is the maximum between

_x

and

_y

– It can vary in the same triangle

– Can be derived from the transformation matrices,

computed for the

Vertices

and interpolated for the

fragments

• The

mipmap level

to use is:

log

₂

– level 0 = maximum resolution

– if level<0 what is the reason?

(57)

Minification: MIP-mapping

(58)

Minification: MIP-mapping

0

1

2

3

4

5

(59)

In OpenGL

glTexParameteri(

GL_TEXTURE_2D,

GL_TEXTURE_MAG_FILTER,

GL_NEAREST);

glTexParameteri(

GL_TEXTURE_2D,

GL_TEXTURE_MAG_FILTER,

GL_LINEAR );

or

(60)

In OpenGL

glTexParameteri(

GL_TEXTURE_2D,

GL_TEXTURE_MIN_FILTER,

mode

);

mode

= GL_NEAREST

GL_LINEAR

GL_NEAREST_MIPMAP_NEAREST

GL_LINEAR_MIPMAP_NEAREST

GL_NEAREST_MIPMAP_LINEAR

GL_LINEAR_MIPMAP_LINEAR

where

Choose the minification filter:

(61)

In OpenGL

• Load on the graphics card all the mipmapping

levels.

– One-by-one:

glTexImage2D (

GL_TEXTURE_2D,

i, // MIP-map level

GL_RGB, // original format

imageWidth, imageHeight,

0, // border

GL_RGB, // RAM format

GL_UNSIGNED_BYTE,

(62)

In OpenGL

• Load on the graphics card all the mipmapping

levels.

– All together (using the glu library):

glTexImage2D (

GL_TEXTURE_2D,

0, // MIP-map level

GL_RGB, // original format

imageWidth, imageHeight,

0, // border

GL_RGB, // RAM format

GL_UNSIGNED_BYTE,

imageData);