Computer Graphics

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Computer Graphics
School of Computing Staffordshire University
Mapping Techniques
Dr. Claude C. Chibelushi
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Outline

• Introduction
• Texture Mapping
• Coordinate transformation
• Distortions
• Relief Mapping
• Reflection Mapping
• Summary

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Introduction

• Most objects have patterns on their surface
• colour pattern, surface bumps, reflections, ...
• e.g. brick wall, gravel, hair, wood, orange skin
• Improved appearance of surface details enhances
  photo-realism
• e.g. surface roughness, reflectivity, ...

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Introduction

• Costly or ineffective to produce intricate
  surface pattern by using
• smooth shading (i.e. shades of single colour)
  colour details
• high number of small polygons, or curves
  roughness details
• ray tracing of reflective surfaces
  reflection details

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Introduction

• Mapping can produce realistic surface details for
  little geometric complexity
• for colour details
• copy picture of pattern onto surface
• process known as texture mapping
• for roughness details
• apply pattern of bumps onto flat surface
• process known as bump mapping
• for reflection details
• apply reflected pattern (of objects in
  environment) onto surface
• process known as environment mapping
• for other details lighting,

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Introduction
Illustration of smooth shading, and texture
mapping
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Texture Mapping

• Texture mapping
• Variety of techniques for adding surface details
• enhanced realism but higher computational and
  storage cost
• typically maps 2D texture onto 3D object (or vice
  versa)
• assigns texel from texture to point on object
  surface
• can be viewed as coordinate transformation /
  mapping

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Texture Mapping

• Texture pattern can be obtained from
• camera real image (rectangular grid of pixels
• popular approach, particularly for intricate or
  fine surface details
• procedure that synthesises desired texture
• e.g. pixel-value undulations (sinusoids coupled
  with some randomness) for producing wood-grain
  pattern

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Texture Mapping

• Texture stretched / shrunk like rubber to fit
  onto object
• pattern typically mapped onto planar object faces
• sometimes (e.g. when no texel-vertex
  correspondence is available -- see environment
  mapping) pattern
• first warped onto (symmetrical) dummy with
  curved surfaces (e.g. cylinder, sphere)
• then transferred from dummy onto target object

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Texture Mapping

• Mapping process
• transformation of coordinates from texture space
  to object / image space
• 2D texel array mapped onto 3D surface
• but, how good is mapping of flat texture onto
  curved 3D surface?
• also, desired object appearance may require 3D
  texture (e.g. for semi-transparent object)
• 3D surface projected onto 2D view window

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Texture Mapping

• Coordinate transformation

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Texture Mapping

• Coordinate transformation
• Image-order (or pixel-order) scanning more
  popular than texture scanning
• avoids possible gaps in target image
• some approaches require inversion of forward
  transformation matrix

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Texture Mapping

• Coordinate transformation

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Texture Mapping

• Coordinate transformation
• Part of texture stretched or contracted to fit
  target polygon
• based on specified location of vertices on
  texture
• pre-calculated or manual attachment of texture to
  object vertices
• implementation
• rendering approaches
• interpolation incremental computation of
  coordinates
• homogeneous matrices for transformations of
  coordinates
• data structures for polygonal model with texture
• polygon vertex coordinates, texture coordinates,
  normal(s)

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Texture Mapping

• Coordinate transformation
• Naïve (linear) interpolation method
• only 2D coordinates are considered
• options
• discard depth
• map texture onto projection of facet
• easy implementation and low computational cost
• may give poor results (see distortions)
• lacks perspective foreshortening

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Texture Mapping

• Coordinate transformation
• Naïve (linear) interpolation method

(x1, y1) -gt (s1, t1) (x2, y2) -gt (s2, t2) (x3,
y3) -gt (s3, t3)

• Replace t by s to obtain second coordinate of
  pixel
• Efficiency use incremental calculations

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Texture Mapping

• Coordinate transformation
• Naïve (linear) interpolation method
• use incremental computation
• along polygon edge
• tcurrent tprevious (t1 - t2) / (y1 - y2)
  (along edge between 1 and 2)
• tcurrent tprevious (t1 - t3) / (y1 - y3)
  (along edge between 1 and 3)
• along scan line
• tcurrent tprevious (t5 - t4) / (x5 - x4)
  (along scan line from 4 to 5)

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Texture Mapping

• Coordinate transformation
• Texture mapping which incorporates perspective
  effect
• shrink texture map as corresponding z increases
• modify naive interpolation equations by
  interpolating homogeneous coordinates (s, t, q)
• s s / q, t t / q, q 1 / z
• i.e linear interpolation between (s, t , q) and
  (x, y), but non linear interpolation between (s,
  t) and (x, y)

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Texture Mapping

• Coordinate transformation
• Other texture mapping approach
• rational linear transformation two mapping
  equations in form of ratios of polynomials
• x as function of s and t and y as function of s
  and t
• 8 unknown coefficients in total
• solution requires association between 4 texels
  (texture space) and four vertices (screen space)
• homogeneous matrices often used

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Texture Mapping

• Coordinate transformation
• Rational linear transformation using homogeneous
  texture- and screen-space coordinates

i can be set 1 hence 8 unknowns determined using
4 (2D) points
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Texture Mapping

• Coordinate transformation
• Rational linear transformation

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Texture Mapping

• Pixel lighting
• Two options
• texel directly used as colour of pixel
• no illumination model applied
• texel modulates reflection coefficients
• hence varies amount of reflected light

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Texture Mapping

• Distortions
• Shape and aliasing distortion due to
• curvature of target surface
• texture image generally flat but object surface
  may be curved
• mapping rectangular texture onto two triangles
• solutions bilinear interpolation onto quad use
  of perspective correction or reduction of
  polygon size
• size differences between texture and target
  images
• many-to-one (or one-to-many) mapping results in
  aliasing
• solution filtering / blending (averaging,
  multi-resolution texture, )

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Texture Mapping
Shape distortion
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Texture Mapping
Shape distortion
Original texture map
Linear transformation
Bilinear transformation
Perspective transformation
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Texture Mapping
Aliasing distortion
Pixel popping (or crawlies) pixels appear
and disappear during animation
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Texture Mapping

• Aliasing distortion
• Texture shrunk ? / stretched ? onto screen
  polygon
• ? minification problem pixel corresponds to many
  texels
• abrupt changes in appearance (e.g. scintillation)
• solution texture blurring and map size reduction
  
• ? magnification texel corresponds to many pixels
• blocky appearance
• solutions
• use larger texture map (may be precluded by
  storage limitations)
• combine texture map with approximation of edges,
  and increase map size

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Texture Mapping

• Anti-aliasing techniques
• Mip-mapping
• multiple texture maps at different resolutions
• obtained by blurring higher-resolution map and
  reducing size
• e.g. averaging of groups of 4 texels
• selection of texture map determined by e.g.
  screen size of object being mapped
• Reduces aliasing but increases computation and
  storage cost

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Relief Mapping

• Bump mapping
• distortion of smooth surface to give illusion of
  wrinkles or dimples
• i.e. change of curvature
• small perturbations applied to orientation of
  local surface normals
• perturbation derived from height field specified
  in bump map
• perturbed normals used in intensity calculation
• result in variation of diffuse and specular
  components of reflected light

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Relief Mapping

• Bump mapping

surface normal
perturbation related to local curvature of
surface and of height map
perturbed surface normal
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Relief Mapping
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Reflection Mapping

• Reflection of environment wrapped onto object
  surface
• object shows reflections of surrounding world
• rough / blurry reflection (chrome mapping)
• recognisable image of world (environment mapping)
• can also be used to simulate specular highlights
• map depicting bright spot(s) representing
  reflection of light source

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Reflection Mapping
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Reflection Mapping

• Environment mapping
• two-part mapping
• pre-computation real or synthetic image of
  environment wrapped onto enclosing volume
  (sphere, cube, )
• computational simplification object centred
  within and much smaller than enclosing volume
• ray reflected by object approximated by ray from
  centre of enclosure

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Reflection Mapping

• two-part mapping (ctd.)
• run-time rendering
• calculate reflected ray for ray incident from
  eye onto each object vertex
• (formula refer to specular reflection model)
• reflected ray use to index into reflection map
• interpolate across polygon

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Suggested Reading

• Relevant parts of Ch. 14, D. Hearn, M.P. Baker,
  Computer Graphics, 2nd Ed. in C, Prentice-Hall,
  1996.
• Relevant parts of Ch. 12, A. LaMothe, Black Art
  of 3D Game Programming, Waite Group Press, 1995.
• A. LaMothe, Affine Texture Mapping, Dr. Dobbs
  Journal, pp. 58 - 63, July 1998.
• P.S. Heckbert, Survey of Texture Mapping, IEEE
  Computer Graphics and Applications, pp. 56 - 67,
  Nov. 1986.

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Summary

• Mapping techniques can produce realistic surface
  details for little geometric complexity
• e.g. colour, bump, or environment mapping
• Texture mapping
• stretch / shrink texture onto surface
• requires coordinate transformation
• image-order scanning is popular
• enhanced realism but may suffer from aliasing and
  shape distortion

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Summary

• Bump mapping
• perturbation of surface smoothness
• Environment mapping
• generation of reflection of environment on object
  surface