Part IX Lighting

1
Part IX Lighting Shading
2
Lighting and its Diffuse Component

• Rasterization stage has to compute a color for an
  object point through lighting or illumination
  which describes the interaction between light
  sources and objects.
• We need a model for lighting, and a lighting
  model is usually divided into diffuse, specular
  and ambient components.
• The diffuse component is based on Lamberts law,
  which states that reflections from the ideally
  diffuse (totally matte) surface (e.g. chalk) are
  scattered with equal intensity in all directions,
  independently of the viewing direction. The
  reflected light is determined by the cosine of
  the angle ? between the surface normal n and the
  light vector l.

E
n
l
(n and l are unit vectors.)
3
Diffuse Component

• If ? gt90o, cos? nl lt 0.
• In reality, however, the reflected light should
  become 0, not a negative value, because the
  object point does not receive light at all.
• So, instead of nl, use max(0, nl).

n
n
l
l
4
Interaction between Colored Lights and Surfaces

• A colored light has RGB components.
• Example 1 Suppose a white light sdiff (sdiff-R,
  sdiff-G, sdiff-B)(1,1,1). If an object lit by
  the light appears yellow, it means that the
  object reflects R and G and absorbs B. We can
  easily implement this kind of filtering through
  material parameter mdiff, i.e. if (mdiff-R,
  mdiff-G, mdiff-B)(1,1,0), then
  (1,1,1)?(1,1,0)(1,1,0) where ? is component-wise
  multiplication.
• Example 2 If (sdiff-R sdiff-G, sdiff-B)(
  0.5,0.5,0.5) and (mdiff-R, mdiff-G, mdiff-B) (
  0.5,1,0.5), then (0.5,0.5,0.5)?(0.5,1,0.5)
  (0.25,0.5,0.25).

G
R
B
G
R
B
intensity .5
.5
.5
intensity 1
1
1
.5x.5.25
.5x1.5
1x11
1x00
.5x.5.25
1x11
(.5,1,.5)
(1,1,0)
where ? is component-wise multiplication.
5
Specular Component

• In order to make a surface look shiny by making
  highlight appear, we need specular component.
• For a perfectly shiny surface, the highlight is
  only where ?0.
• For non-perfect objects, the highlight occurs at
  ?0 but falls off sharply as ? increases. Such
  rapid fall-off is approximated by cosm?, where m
  describes the shininess of the surface
• For a perfectly shiny surface, m?. (For example,
  a perfect mirror.)
• Objects have both diffuse and specular
  properties. The diffuse component is
  view-independent while the specular component is
  highly view-dependent.
• Wrt the light sources, sdiff and sspec should be
  the same.

cos2?
cos?
cos64?
n
r
l
?
v
6
Ambient Component

• The diffuse and specular components deal with
  direct rays from the light source.
• In real world, however, some rays emanating from
  a light source may bounce off a wall and then
  reach an object when the light is turned on. This
  bounced light (called ambient light) is assumed
  to be so scattered that there is no way to tell
  its origin direction, but it disappears if a
  particular light source is turned off.
• The ambient component catches such indirect rays.
• The amount of ambient light incident on each
  object is a constant for all surfaces and over
  all directions, and reflections from the surfaces
  are scattered with equal intensity in all
  directions.

7
.
Colored Lights and Surfaces Summary

• Every light source has three parameters ambient
  color samb, diffuse color sdiff and specular
  color sspec.
• Some systems such as OpenGL allow different sdiff
  and sspec, which might look unnatural but
  provides us with flexibility. For a realistic
  effect, however, sdiff and sspec should be
  identical.
• samb can be different. Imagine a white light in a
  room with red walls. The scattered light tends to
  be red.
• A material consists of ambient, diffuse,
  specular, shininess and emissive parameters
  mamb, mdiff, mspec, m (shininess) and memi.
• memi describes how much light a surface itself
  (e.g. headlight) emits.
• mamb and mdiff define the materials color, and
  are typically similar if not identical.
• In contrast, mspec is in the gray level (e.g.
  white) so that highlights end up being the color
  of the light source. Imagine a while light
  shining on a shiny red plastic sphere. Most of
  the sphere appears red, but the shiny highlight
  is white.

8
Lighting Equation

• The lighting is the sum of ambient, diffuse and
  specular components.
• The intensity of light decreases as distance from
  the light increases.

9
Light Sources

• So far, weve implicitly assumed a point light.
  However, we may have various types of light
  sources.
• A directional light is positioned infinitely far
  away (e.g. sun) and so all rays from the source
  are parallel when arriving at an object.
• A point light is at a certain position, and
  radiates in all directions.
• A spot light is similar to a point light, but
  emits light only within the volume of a cone,
  defined by a cut-off angle and a light direction.
  (We may say that both the point and spot lights
  are positional.)

10
Directional Light vs. Positional Light

• Directional lights are described by their
  direction vectors, and positional lights (point
  spot lights) are by their positions.
• Recall that, for attenuation, we introduced spos
  (the light source position), which is used for
  point spot lights.
• Lets also store the directional lights
  direction vector into spos.
• We are in homogeneous coordinates. The 4th
  component of spos is often used to distinguish
  between the two cases (e.g. OpenGL). Note that
  w0 implies a vector and w?0 implies a point.
• Also note that attenuation is disabled for a
  directional light.
• So far, parameters for defining light sources are
  samb, sdiff, sspec and spos.

11
Spot Light

• The spot light is another positional light, and
  spos is used to define its position.
• In addition to samb, sdiff, sspec and spos, a
  spot light needs more parameters.
• a direction vector sdir
• a cut-off angle scut
• a spot exponent sexp for fall-off control
• A factor denoted cspot can describe the spot
  light effect.
• If the point to be lit is inside the spot light
  cone defined by scut,
• If outside,
• cspot 0

12
Phong Lighting Equation

• The lighting equation is extended as follows.
• itot cspot( d( iamb idiff ispec ) )
  where cspot1 if not a spot light source
• The parameter memi describes how much light a
  surface emits.
• itot memi cspot( d( iamb idiff ispec ) )
  
• There is also a global ambient light source
  parameter aglob.
• There can be multiple light sources
• After lighting computations, resulting color
  values outside the range 0,1 are clamped to the
  range 0,1.
• The above is called Phong lighting model.

13
Example Model of A Scene
14
Example of Ambient Reflection Only
15
Application of Lighting Model for Shading

• Shading of a polygon mesh
• The polygon mesh could be shaded by applying the
  Phong lighting model for every (pixel)
  position.
• Two problems
• expensive
• may not be able to avoid faceted appearance
• Before talking about the faceted appearance,
  lets recall that a polygon mesh is an
  approximation of a smooth surface.

16
Why Faceted Appearance?

• Without loss of generality, suppose only the
  diffuse reflection of a directional light source
  where a uniform light vector l is used for
  lighting all polygons.
• Lets consider 3 adjacent polygons which have 3
  points p1, p2 and p3, respectively.
• Because the intensity is proportional to cos?
  nl, p1 is brighter than p2, and p2 is brighter
  than p3.
• All points in a polygon have the same normal n,
  and so same intensity.
• Therefore, faceted appearance is inevitable.

n2
n3
n1
l
l
l
medium
light
dark
17
Flat Shading

• If we use a single normal for a face (as in the
  previous example), its called flat/constant
  shading.
• Flat shading is fast, and in fact many of old
  games such as Virtual Fighter I used flat shading
  for rendering speedup.

polygon mesh faceted image
smooth image
current state what we
want
18
Example of Flat Shading with Diffuse Reflection
19
Normals of the Polygon Mesh

• Noting again that a polygon mesh is an
  approximation of a smooth surface, consider the
  triangle marked by bold lines.
• The triangle corresponds to a patch of the sphere
  surface, and so all points of the triangle should
  have different normals, which would lead to
  different intensities/colors. However, the flat
  shading assumes a uniform normal for the entire
  triangle and leads to a faceted image.
• On the other hand, consider a vertex shared by
  multiple triangles. It also corresponds to a
  point of the sphere, and so its normal should be
  set accordingly.
• Interpolation shading techniques take the points
  into account.

20
Gouraud Shading

• Gouraud Shading is an color interpolation
  shading.
• Determine the normal of each vertex.
• Its typically given at the modeling stage.
• Otherwise, it can be obtained by averaging the
  surface normals of all polygons sharing the
  vertex.
• Compute colors of all vertices (with Phong
  lighting model).
• In WdC, linearly interpolate vertex colors along
  each edge of polygon.
• Then, linearly interpolate pixel colors along
  each scan line.
• This is a magic to get smooth images from polygon
  meshes.

n1
average of n1, n2, n3, n4 and n5
n5
n2
n4
n3
21
Linear Interpolation of Colors

• RGB colors I1, I2 and I3 (computed at vertices)
  are interpolated to determine the colors Ia at
  xleft and Ib at xright for a scan line.
• Then, Ia and Ib are interpolated to determine a
  color Ip for each pixel for the scan line.

22
Why Smooth Appearance?

• Color interpolation results in smooth transition
  of lightness.

n2
n3
n1
becoming lighter along this direction
polygon mesh flat shading
Gouraud shading
23
Example of Gouraud Shading with Diffuse Reflection
24
X-intersection Computation (revisited)

• Recall x-intersection interpolation for polygon
  filling.
• Pre-computed delta value per a unit move is
  repeatedly added.
• Initial/starting value is computed (using similar
  triangles).

i1
1
m
i
ymxB
1
1/m
new-x old-x 1/m
old-x
8 7 6 5 4 3 2 1
y
(5.4,7.3)
(1.5?,2)
2 1
0.4
(1.5,1.6)
1
?
(1.5,1.6)
1/m
x 1 2 3 4 5 6 7
25
Color Interpolation

• Color interpolation is achieved basically through
  the same mechanism.
• Z-value interpolation for z-buffering is also
  done in the same manner.

(r,g,b)(245,87,32) at (5.4,7.3)
8 7 6 5 4 3 2 1
y
(r,g,b)(1880.410,,)(192,,) at (,2)
r20210
2 1
r19210
0.4
(r,g,b)(188,23,99)at (1.5,1.6)
x 1 2 3 4 5 6 7
(r,g,b)(188,23,99) at (1.5,1.6)
?y 7.3-1.6 5.7 ?r 245-18857 ?r/?y 57/5.7
10
26
Z-buffer Algorithm with Gouraud Shading

• for all x, y
• z-bufferx,y background depth
• frame-bufferx,y background color
• for each polygon T
• if T is a front face, i.e. neither a back face
  nor a silhouette
• obtain vertex colors (I1, I2 I3 for ?) and
  z-values (z1, z2 z3 for ?)
• compute Ts y-range ylow,yhigh
• ymin ?? ylow? ?
• ymax ? yhigh ? ? -1
• for y ymin to ymax
• compute xleft,xright x-intersections
  of T along y
• xmin ?? xleft? ?
• xmax ? ?xright ? ? -1
• compute ltIa, Ibgt and ltza, zbgt by interpolation
• for x xmin to xmax
• compute Ip and zp at (x,y) by interpolation
• if zp gt z-bufferx,y

27
Problems in Gouraud Shading

• Recall cosm?(rv)m for highlighting in specular
  reflection.
• Suppose that A and B are two vertices of a
  triangle, and m is a big value. Because m is big,
  highlight falls sharply as ? increases.
• Then, we should have the triangle shown at the
  right.

vertex A
r1
l
l
v2
r2
v1
n1
n2
vertex B
B small ?
A large ?
28
Problems in Gouraud Shading (contd)

• However, Gouraud shading is color
  interpolation, and therefore the colors at
  vertices are linearly/evenly interpolated along
  the edges and then along scan lines.
• Worse still, highlights inside a triangle cannot
  be caught.

Gouraud shading what we should have
small ?
scan line
large ?
To avoid the second problem, break up the (large)
polygon into smaller ones.
29
Phong Shading

• The problems of Gouraud shading can be overcome
  by Phong shading, which is a normal interpolation
  shading.
• Determine the normal of each vertex. (Same as in
  Gouraud shading.)
• Linearly interpolate vertex normals along each
  edge.
• Linearly interpolate normals along each scan
  line.
• Compute colors at each pixel (with Phong lighting
  model).

interpolation along edges
interpolation along a scan line
30
Example of Gouraud Shading with Specular
Reflection
31
Example of Phong Shading with Specular Reflection
32
Example with Curved Surfaces, not Polygon Meshes
33
Shading Summary

• Three types of shading
• Flat shading Lighting computation per polygon
• Gouraud shading Lighting computation per vertex
• Phong shading Lighting computation per pixel
• Most graphics hardware implementations have
  adopted Gouraud shading.
• It is as fast as flat shading and produces
  improved quality.
• It is highly dependent on the LOD of the mesh.
  For a coarse mesh, highlights are often missing,
  the spot-light effect cannot be caught, etc.
• Thanks to the advance of hardware technology, the
  trends start to move from Gouraud shading to
  Phong shading.

Gouraud shading
Phong shading
Flat shading

34
Discussions Lighting in Eye Coordinates

• Recall that, in the geometry stage, projection
  transformation incurs distortion, which will
  distorts the configuration of l, n, r and v. So,
  lighting computation should be completed before
  the projection transformation.
• Lighting is computed at EC.
• Think about why not in LC or WC.

modeling viewing transformation
projection transformation
in EC
in EC
in CC
lighting
in LC
perspective division
viewport transformation
in CC
clipping
in WdC
in NDC
35
Discussions Normal

• When a series of transformations is applied to a
  model, the models vertex coordinates are
  transformed. How about the normals? Can we apply
  the same transformations to the normals?
• With 2D cross-section of a polygon, consider
  applying a scaling to both the polygons vertices
  and normal.
• As discussed earlier, we need vertex normals at
  EC for lighting, and we just found that the
  modelview transformation Mmodelview which
  transforms the model from LC to EC should not be
  directly applied to normals.

sx0.5 sy1
36
Discussions Normal (contd)

• Suppose the normal n which should be
  perpendicular to the edge connecting p and q.
• Apply the transpose of the inverse of Mmodelview
  to normal vectors.

n
q
p
to vertices
to normals
37
Discussions Normal (contd)

• We have to apply (Mmodelview-1)t to normals, but
  .
• What if Mmodelview is simply a rotation?
• Recall that every rotation is an orthogonal
  matrix, whose inverse is equal to its transpose.
• So, we can simply use Mmodelview instead of
  (Mmodelview-1)t.
• How about translations?
• Translations do never affect vector direction.
• So, if Mmodelview is a combination of rotations
  and translations, i.e. a rigid-body motion, we
  can use Mmodelview instead of (Mmodelview-1)t.
• The scaling we took as an example was a
  non-uniform scaling.
• How about uniform scaling where sxsysz?
• It affects only the length of the transformed
  normal, not its direction.
• We just need the normalization step to make a
  unit vector.
• Also note that, for (Mmodelview-1)t, we can
  compute only the transpose of the adjoint of the
  upper-left 3x3 portion of Mmodelview.

38
Discussions Reflection Halfway Vectors

• In the Phong lighting model,
• l is either computed by connecting the vertex
  with the light source position (for positional
  light) or given as a constant (for directional
  light),
• n is either computed or given, and
• v is either computed by connecting the vertex
  with COP (for perspective projection) or given as
  (0,0,1) (for orthographic projection).
• The only remaining is r, and it has to be
  computed as follows.

n
r
l
?
v
s
s
39
Discussions Reflection Halfway Vectors (contd)

• Note that r2n(nl)-l requires 7
  multiplications and 5 additions or subtractions
  for every vertex.
• Suppose that we use parallel projection and
  directional light, where
• l is a constant, and
• v is also a constant (0,0,1).
• Now, lets take halfway vector h(lv)/lv,
  which is also a constant for every vertex.
• If l, n, r, and v are in a plane, b?/2.
  Otherwise, bgt?/2.
• cos?nh and cos?rv are essentially
  proportional nh is maximized when rv is
  maximized. Approximately, (rv)m (nh)4m.
• Lets replace (cosm?) by (cosm?) and (rv)m by
  (nh)m in the specular term ispec(cosm?)mspec?ss
  pec(rv)mmspec?sspec.

?
?
40
Discussions Reflection Halfway Vectors (contd)

• ispec (cosm?)mspec?sspec (nh)mmspec?sspec
  has been proposed.
• Is it safe? Yes, it is even though (rv)m and
  (nh)m are different.
• Such a difference does not matter because m is
  set to a specific value through trial and error.
  So is m.
• In fact, many systems such as OpenGL recommend
  infinite viewer even though the perspective
  projection is chosen.
• With the infinite viewer, we can avoid computing
  v for every vertex.
• Combined with the directional light which makes l
  constant, the halfway vector h is also made
  constant.
• Pretty efficient computation is possible with
  infinite viewer only for lighting, but the
  rendered image is less realistic.

?
?