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Texture Mapping

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Texture Mapping Texture Mapping Texture Aliasing MIPmaps Environment Mapping Bump Mapping Displacement Mapping Shadow Maps Solid Textures Antialiasing

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The Limits of Geometric Modeling

Although graphics cards can render over 10 million polygons per second, that number is insufficient for many phenomena Clouds Grass Terrain Skin

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Modeling an Orange – 1/2 Consider the problem of modeling an orange

(the fruit) Start with an orange-colored sphere

Too simple Replace sphere with a more complex shape

Does not capture surface characteristics (small dimples)

Takes too many polygons to model all the dimples

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Modeling an Orange – 2/2 Take a picture of a real orange, scan it, and

“paste” onto simple geometric model This process is texture mapping

Still might not be sufficient because resulting surface will be smooth Need to change local shape Bump mapping

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The Quest for Visual Realism

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Three Types of Mapping Texture Mapping

Uses images to fill inside of polygons Environmental (reflection mapping)

Uses a picture of the environment for texture maps

Allows simulation of highly specular surfaces Bump mapping

Emulates altering normal vectors during the rendering process

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Texture Mapping

geometric model texture mapped

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Decal Textures

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Bump Mapping

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Where does Mapping take place? Mapping techniques are implemented at the end

of the rendering pipeline Very efficient because few polygons pass

down the geometric pipeline

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Simple Texture Mapping

s

t

x

y

z

image

geometry screen

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Is it Simple? Although the idea is simple---map an

image to a surface---there are 3 or 4 coordinate systems involved

2D image

3D surface

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Coordinate Systems Parametric coordinates

May be used to model curved surfaces Texture coordinates

Used to identify points in the image to be mapped World Coordinates

Conceptually, where the mapping takes place Screen Coordinates

Where the final image is really produced

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Texture Mapping

parametric coordinates

texture coordinatesworld coordinates

screen coordinates

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Texture Mapping = Pattern Mapping

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Mapping Functions Basic problem is how to find the maps Consider mapping from texture coordinates to

a point a surface Appear to need three functions

x = x(s,t)

y = y(s,t)

z = z(s,t) But we really want

to go the other way

s

t

(x,y,z)

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Backward Mapping We really want to go backwards

Given a pixel, we want to know to which point on an object it corresponds

Given a point on an object, we want to know to which point in the texture it correspondsNeed a map of the form

s = s(x,y,z)

t = t(x,y,z) Such functions are difficult to find in general

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Sample Mapping – 1/2 In next topic, we will study parametric surf

aces Assume we want to map a point in the text

ure map T(s, t) to a point on the surface p(u, v) by a linear map of the form:u=as + bt + cv=ds + et + fas long as aebd, this mapping is invertible

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Sample Mapping – 2/2

)(

),(

minmaxminmax

minmin

minmaxminmax

minmin

vvtt

ttvv

uuss

ssuu

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Mapping a Texture Based on parametric texture coordinates

s

t1, 1

0, 1

0, 0 1, 0

(s, t) = (0.2, 0.8)

(0.4, 0.2)

(0.8, 0.4)

A

B C

a

bc

Texture Space Object Space

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Two-part Mapping One solution to the mapping problem is to

first map the texture to a simple intermediate surface

Example: map to cylinder

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Cylindrical Mapping

parametric cylinder

x = r cos 2 uy = r sin 2uz = v/h

maps rectangle in u,v space to cylinderof radius r and height h in world coordinates

s = ut = v

maps from texture space

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Spherical MappingWe can use a parametric sphere

x = r cos 2uy = r sin 2u cos 2vz = r sin 2u sin 2v

in a similar manner to the cylinderbut have to decide where to putthe distortion

Spheres are use in environmental maps

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Box Mapping Easy to use with simple orthographic

projection Also used in environmental maps

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Second Mapping Map from intermediate object to actual object

Normals from intermediate to actual Normals from actual to intermediate Vectors from center of intermediate

intermediateactual

What if a ray is blocked?

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Texture Example

The texture (below) is a 256 x 256 image that has been mapped to a rectangular polygon which is viewed in perspective mode

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Linear Interpolation of Textures

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Texture Mapping & Polygon Rasterization

Problems of linearly interpolatingtexture coordinates in screen space. Similar problems in the interpolations of

colors for Gouraud shading and normalsfor Phong shading.

Problems in antialiasing Expanding and shrinking

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Example

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Perspective Correction Dividing the texture coordinates by w Linearly interpolating (u/w, v/w, 1/w) At each pixel, dividing (interpolated)

u/w and v/w by (interpolated) 1/w

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Aliasing Point sampling of the texture can lead to ali

asing errors

point samples in u,v (or x,y,z) space

point samples in texture space

miss blue stripes

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Area AveragingA better but slower option is to use area averaging

Note that preimage of pixel is curved

pixelpreimage

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Sampling Texture Maps

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Over-Sampling indicative of aliasing high-frequency details showing up in areas

where we expect to see low frequencies

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Spatial Filtering Prefilter the texture to remove the high fre

quencies that show up as artifacts

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Mip Mapping MIP Mapping is one popular technique for precom

puting and performing this prefiltering Computing this series of filtered images requires o

nly a small fraction of additional storage over the original texture

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Magnification and MinificationMore than one texel can cover a pixel (minification) ormore than one pixel can cover a texel (magnification)

Can use point sampling (nearest texel) or linear filtering( 2 x 2 filter) to obtain texture values

Texture Polygon

Magnification Minification

PolygonTexture

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Examples pointsampling

mipmapped pointsampling

mipmapped linear filtering

linear filtering

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Environment Mapping

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Sphere Mapping

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Box Mapping

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Bump Mapping

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Bump Mapping

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Bump Mapping

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Displacement Mapping

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Bump Mapping versus Displacement Mapping

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Texture Mapping in Quake

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Shadow Maps

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Basic Steps of Shadow Maps Render the scene from the light’s point of view, Use the light’s depth buffer as a texture (shadow map), Projectively texture the shadow map onto the scene, Use “texture color” (comparison result) in fragment shading.

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Three Dimensional or Solid Textures

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Tri-linear Interpolation

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Line Aliasing Ideal raster line is one pixel wide All line segments, other than vertical and h

orizontal segments, partially cover pixels Simple algorithms color

only whole pixels Lead to the “jaggies”

or aliasing Similar issue for polygons

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Antialiasing Can try to color a pixel by adding a fraction of its

color to the frame buffer Fraction depends on percentage of pixel

covered by fragment Fraction depends on whether there is overlap

no overlap overlap

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Area Averaging Use average area 1+2-12 as blending factor

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Image Filtering

m

mk

n

nlljkiklij

ij

ahb

aA

,

][

H: Filter(convolution) matrix of (2m+1)×(2n+1)

010

111

010

5

1H

Examples:

121

242

121

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1H

010

151

010

H

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Motion Blur Jitter an object and render it multiple times, leaving

the positions of the other objects unchanged, we get dimmer copies of the jittered object

If the object is moved along a path, rather then randomly jittered, we see the trail of the object motion blur