Texture Mapping

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Texture Mapping

• (Some Images from Rosalee Wolff)

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Texture Mapping

• Problem with shading models
• -they assume that a diffuse surface has uniform
  reflectance
• This is okay for walls or pool balls, but not
  most objects
• We could add geometric complexity
• - BAD! Too time consuming
• Alternative Texture mapping
• - Developed by Catmull (1974), Blinn and Newell
  (1976), and others

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Texture Mapping

• Goal add visual detail without adding geometric
  detail
• 8 polygons 8 polygons

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Typical Textures
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Replicating Textures
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Replicating Textures
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Textures
Any image can be used as a texture map
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Texture Mapping

• There are both 2D and 3D versions of texture
  mapping
• 2D wallpaper a 2D image onto an object
• 3D carve a 3D object out of a block

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2D Texture Mapping

• Given a 2D texture (image) and a 3D object, map
  the texture onto the object.
• Where does each point on the object map into the
  texture?

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Texture Map Shapes

• Planar Map
• Simply remove one of the objects coordinates
  (project onto that coordinate plane)

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Planar Map

• This has the unwanted side effect of keeping the
  texture constant in one direction.
• E.g., projecting along Z

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Planar Map

• Projecting along X and Y respectively

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Cylindrical Map

• Instead of a planar map, we could use a
  cylindrical map
• (x,y,z) is converted to (r, theta, height).
• For texture mapping, theta is converted into a
  u-coordinate and height is converted into a
  v-coordinate.
• This wraps the two-
• dimensional texture
• map around the object.

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Cylindrical Map

• At minimum and maximum extents of the cylinder,
  the texture gets pinched together.
• E.g., a cylindrical map parallel to Z

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Cylindrical Map

• Similarly if the map is parallel to X or Y

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Spherical Map

• Convert from (x, y, z) to spherical coordinates.
• Latitude is converted to a u-coordinate,
  longitude is converted to a v-coordinate.

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Spherical Map

• This still has the effect of pinching the texture
  at the poles, but differently than using a
  cylindrical map.

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Spherical Map

• With the poles in the X and Y directions

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Box Map

• We can use a collection of planar maps to provide
  better coverage than using a single planar map

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Box Map

• We project each plane onto its portion of the
  object.
• Use the objects normal to determine which
  texture to use.

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Box Map

• This produces

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Box Map

• Using a different set of textures

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Box Map

• We get

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Texture Mapping Applet
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Map Entity

• To texture map, we take an (x, y, z) value from
  the object and determine a (u, v) texture value.
• How we determine what we use as the (x, y, z)
  value is the map entity.
• Can use various things
• a point on the object relative to the objects
  bounding box
• the surface normal at the point being rendered
• a vector running from the objects centroid
  through the point
• the reflection vector at the current point

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Map Entity
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Map Entity

• Using the same map shape, but different map
  entity can give quite different results.
• Planar Mapping

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Map Entity

• Cylindrical Mapping

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Map Entity

• Spherical Mapping

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Map Entity

• Box Mapping

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Texture Mapped Scene
http//daemonic.xeebra.com/raytracer/texture_mappi
ng.png
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Another Example
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The Texture Mapping Process
(inverse map)
(raytrace)
Pixel
Texture Map
Surface
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• Consider surface visible at current pixel.
• Find the patch on the surface that corresponds to
  it.
• Map screen coord of pixel corners back to object
• Find texels that map to the surface patch
• If multiple texels lie on patch combine them
• weighted avg supersampling with
  postfiltering

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Mapping Parametric Surfaces

• Parametric surfaces are already parameterized by
  (s, t).
• Use the (s,t) parameters as the (u,v) texture
  parameters

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The Utah Teapot

• 32 Parametric patches

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Non-Parametric Surfaces

• If we assign values to the vertices in the range
  (0,1), we can use the same texture mapping
  approach

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Non-linear Mapping

• We can distort the texture by using a non-linear
  mapping

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Triangles

• Given a triangle defined by three points (a, b,
  c), how do we associate a texture color with a
  point on the triangle?

c
?
b
a
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Computing the Point

• Given the (x,y) point in the triangle, how do we
  transform that to a (u,v) point in the image?
• Set up a non-orthogonal coordinate system with
  origin a and basis vectors (b-a) and (c-a)

c
g2
c-a
g1
b
b-a
a
g0
b-1
b0
b1
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Barycentric coordinates

• Any point on the triangle can be defined by the
  barycentric coordinate
• p a b(b-a) g(c-a)

c
g2
p
c-a
g1
b
b-a
a
g0
b-1
b0
b1
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Barycentric coordinates

• Once we have computed the (b,g) barycentric
  coordinate for the triangle, we can determine the
  corresponding (u, v) point.
• First, establish the (u, v) system

(0, 1)
(1, 1)
(0, 0)
(1, 0)
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Computing the (u, v) coordinate

• u(b, g) ua b(ub ua) g(uc ua)
• v(b, g) va b(vb va) g(vc va)

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Performing the transformation

• We can then compute the color for that point of
  the triangle.

c
b
a
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Other Mapping Applications

• Color Mapping diffuse component of surface
  (standard texture mapping)
• Reflection Mapping specular component of surface
  to simulate reflection (environment mapping)
• Normal vector Mapping simulate 3D surface
  structure (bump mapping)
• Geometry Mapping raise/lower points to actually
  modify surface (displacement mapping)
• Transparency Mapping vary the transparency
  across the object

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Reflection Mapping

• Also called Environment Mapping
• Idea
• surround the scene with an image
• for reflective surfaces, rather than ray tracing
  them, simply see what part of the image would be
  seen on the object
• Used in Abyss, Terminator 2

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Reflection Mapping Example
From Ed Angel, Interactive Computer Graphics,
Addison-Wesley 2002
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Reflecting
N
V
R
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Reflection Mapping

• Usually done either with
• Sphere (or hemisphere)
• Cube

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Hemisphere Mapping
N
V
R
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Hemisphere Mapping

• Objects mapped onto the hemisphere will look the
  same as the objects out in space
• The hemisphere map is used as the texture map

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Cube Mapping

• Same idea as hemisphere mapping, but use a cube
  instead
• Simpler in some ways
• Have to deal with the corners and edges of the
  cube

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Hemisphere Mapping
N
V
R
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Issues with Reflection Mapping

• Must assume that the environment is very far away
  from the scene
• No self reflections (concave objects dont work
  correctly)
• No reflections between objects

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Example
from http//www.inf.ufrgs.br/7Eoliveira/Cursos/C
MP513/CMP513_trabalhos.html
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Bump Mapping

• Texture mapping adds color detail to the object,
  but the object still looks smooth
• How do you make the object look rough?
• Solution 1 Could add more polygons
• Too time intensive
• Solution 2 Bump mapping

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Bump Mapping

• Idea
• If we use fake normals, we can trick the
  renderer into thinking there are bumps on the
  surface

Flat Plane
Real Bump
Fake Bump
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Sphere with Texture and Bump Maps
Bump Map
Sphere with Texture Map
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The Bump Map

• The bump map is defined as an array of
  displacements B(u,v)
• If P(u,v) specifies positions on a parametric
  surface, the normal at the surface can be
  calculated as
• where and are partial derivatives
  of the surface at point P lying in the tangent
  plane

N
Pu
Pv
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Bump Mapping

• We can simulate bumps by adding small
  perturbations to the surface positions along the
  direction of the normal

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Bump Mapping

• So far we have

Original surface P
Bump map B
Bump map applied to surface giving P
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Bump Mapping

• Next we compute new normals for bump mapped
  surface

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• What are the partials?

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• To simplify, we approximate, using as the reason
  that the bump should be significantly smaller
  than the surface. Thus, we remove the last terms
  of the equations, giving

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• Now, since

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• And, since

then
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• This gives
• This gives a normal perturbation that is
• independent of the position of the object
• independent of the orientation of the object
• i.e., a position on the object will always get
  the same perturbation

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Bump Function

• Bump function can be
• Analytical expression
• Table lookup
• use linear interpolation to compute B
• use finite differences to approximate Bu and Bv

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Bump Mapping Examples
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Bump Mapping Examples
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Bump Mapping Examples
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Bump Mapping Examples

• from http//www.vis-sim.org/support/vp_websamples2
  .shtml

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Horizon Mapping

• A technique to try to get shadow information from
  a bump-mapped surface
• For each point on the surface, the angle to the
  horizon is computed and stored in a horizon map
• Stored for a small number of directions (8 or 16
  usually)
• To compute lighting, the horizon map is
  consulted, the height of the point being rendered
  is determined, and the angle to the light is
  computed
• If the light angle is less than the horizon map
  angle, the point is in shadow

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Horizon Mapping

• from http//www.soft.dk/directx-3d-demos.htm

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Displacement Mapping

• The actual geometry is altered in order to allow
  for a more bumpy look
• A displacement map is given, along with the
  surface
• The displacement map is applied to the surface,
  and additional geometry is introduced where
  needed
• Gives a more realistic look, particularly at the
  silhouette
• Quite a bit more costly than the other techniques

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Parallax Mapping

• Also termed view displacement mapping
• Texture coordinates are displaced as a function
  of the view angle and the height at the point
  being rendered
• Gives similar effects as displacement mapping,
  without the overhead of increased geometry

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Parallax Mapping

• Example of parallax mapping. The walls are
  textures with parallax maps. Screenshot taken
  from one of the base examples of the opensource
  Irrlicht 3d engine.

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Parallax Mapping
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Comparison
Displacement mapping
Bump mapping
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Comparison

• from http//www.chromesphere.com/Tutorials/Vue6/Op
  tics_Basics_Print.html

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Comparison
Bump Mapping
Horizon Mapping
Displacement Mapping
View Dependent Displacement Mapping