Map in object space

In texture space for quadrilaterals

solve for u and v.

In general case, add P_ij's u,v coordinates into equation.

Before do this,

• undo viewport map (x_i, y_i, z_i)
• undo perspective and normalizing transform (object -- world)
• perspective project points

Now what do you do ?

• take closest point -- alias prone
• bilinear interpolate 4 closest values -- solve some problems
• map pixel corners back and filter projected area
• disadvantage: expensive

Solution: Antialiased texture mapping

• mip mapping [Lance Williams 83] (square filter)
• Store texture map at multiple resolutions (power of 2)
  Each one integrated (square filter) on higher resolution table
  (e.g., 512x512, 256x256, 128x128, 64x64, 32x32, 16x16, 8x8, 4x4, 2x2,1x1)
• Interpolate between value from map i and map i-1
• How do you choose map i ?

  Use square root of texture area subtended by pixel

• simple method is to map back (x+1/2, y+1/2), (x-1/2, y-1/2)
• d = sqrt(UVarea);
• choose i such that d < (1 / (i res)) and d > (1 / (i-1) res)
• now average the u and v values of the 4 pixel corners to get [u][v] (or do this procedure for each of the pixel corners).
• then get value of texture[i][u][v] and texture[i-1][u][v] and linearly interpolate them.
• To calculate texture[i][u][v] and texture[i-1][u][v] bilinearly interpolate the texture map.
• Williams used pyramid storage

  

• Example Mip-maps

• Summed Area Tables -- precalculated single table
• each point is the sum of texture value at that point plus the sum of intensities at lower left hand corners.
  

  
  

• Advantages:
• less storage
• handles rectangular filters
• Disadvantages: - Why doesn't anybody use it?
• Can you do it with 8 bits per entry?
• Since uses rectangular filter, still not good for edge on viewing
• Must use rectangular filter

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Object Space vs. World Space for Mapping

• In world space, object will move through the texture. -- The texture is not stuck to the object.

• Example:

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Other Mapping Functions

• Spherical Mapping Example
• Cylindrical Mapping Examples
• Box Mapping Examples

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Bump Mapping [Blinn 78]

• Texture map contains displacement to the surface.

Mathematical Derivation:

Decompose N into partial surface derivatives in the u and v directions: P_u P_v.

N = P_u X P_v

F(u,v) is displacement of surface,

P' = P + F(u,v) * N /||N||

Assuming F is negligibly small,

N' = P'_u X P'_v

P'_u = P_u + F_u * N /||N||

P'_v = P_v + F_v * N /||N||

N' = N + D

D = [F_u(N X P_v) - F_v(N X P_u)] / ||N||

How do we calculate F_u, F_v?

Given F(u,v) function, ep = 1 / (TextureMapResolution)

F_u = (F(u+ep, v) - F(u - ep, v)) / (2*ep)

F_v = (F(u, v+ep) - F(u, v - ep)) / (2*ep)

How do we calculate P_u, P_v?

• Displace world space point in +u, -u direction and +v, -v direction by small amount and calculate approximated partial derivative as :
  P_u = (P(p+u_epsilon) - P(p-u_epsilon)) / (2 * epsilon)
  P_v = (P(p+v_epsilon) - P(p-v_epsilon)) / (2 * epsilon)

  If smooth shading, need to adjust this because of interpolated normals:

  P_Vtemp = P_u X N

  P_u = N X P_Vtemp

  P_Utemp = P_v X N

  P_v = N X P_Utemp

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Dependence on scale:

If scale object, need to adjust displacement map accordingly

D' = alpha * D * |N|/|D| ,

alpha = sqrt ( F_u² + F_v²)

N'' = N + D'

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Displacement Mapping [Cook 84]

• Use texture map to displace actual surface geometry. (vertices, control vertices)

• Done as a pre-process to geometry before the actual rendering

• Requires fine resolution in geometry or will get unexpected holes.

• Alternative is for program to add geometry when displacement variation can't smoothly be represented in geometry.

• Example of Bump Mapping vs. Displacement Mapping
  
• How do you tell the difference between the two?

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Environment Mapping (Reflection Mapping) [Newell 76]

• Used to simulate ray traced reflections.

• Use reflection vector as index into texture map.

• Texture map is either hand drawn or precomputed rendering of scene viewed from that location (often times 6 directional rendering ).

• Normally use box projection or spherical.

• Examples of box projection:
  

• Notice the differences in the ray traced vs. environmental mapped images:
  

• Key to tell which is which? Look for object reflecting itself.

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Return to CMSC 635 Notes Page

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Last modified: Fri Mar 26 10:56:17 EST 1999