Computer Graphics

1
Computer Graphics

• Lecture Eight
• Photorealistic Computer Graphics
• Shadows
• Ray Tracing Algorithm
• LecturerHeather Sayers
• E-mailhm.sayers_at_ulster.ac.uk
• URLhttp//www.infm.ulst.ac.uk/heather

2
Content

• Shadows the z-buffer algorithm
• Object/image space rendering
• The Ray Tracing Algorithm
• Ray sphere intersections
• Ray polygon intersections
• Efficiencies
• Bounding volumes
• Spatial subdivision

3
Depth Sorting

• Used for hidden surface removal
• The basic problem in hidden surface removal is to
  find the closest object to the viewer at that
  pixel
• Soevery object in the scene can be sorted in
  depth (z) and then scan converted in order,
  starting with the surface of greatest depth
• Such algorithms (eg Painters Algorithm) work well
  for surfaces that do not overlap in depth, which
  require further processing (and since this is the
  case for most scenes these are rarely used today)

4
Z-buffer

• A common image-based approach is to compare
  surface depths (z-values) at each pixel position
  on the projection plane

y
x
P(x,y)
S1
S2
z
ZS1
ZS2
5
Z-buffer

• Commonly implemented in hardware by an additional
  block of memory, storing the current z-value for
  the surface currently rendered at that pixel
• Initially the z-buffer locations are initialised
  to -infinity
• During scan conversion, each surface listed in
  the polygon table is processed scanline by
  scanline (top to bottom), calculating the depth
  (z) at each pixel position
• The algorithm is shown on the next slide

6
Z-buffer algorithm

• // Initialise z-buffer and FB for every location
• depth(x,y) -infinity
• FB(x,y) RGB background
• // For every (x,y) position on every projected
  polygon surface,
• // compare depth value to previously stored value
  in z-buffer
• zcurrent getDepthValue(current_surface, x,
  y)
• if zcurrent gt depth(x,y) then
• depth(x,y) zcurrent
• FB(x,y) RGBcurrent_surface(x,y)

7
Z-buffer and shadows

• The classic z-buffer algorithm can be augmented
  to support shadows
• The technique requires a separate shadow Z-buffer
  for each light source, and is a 2-step process
• Firstly the scene is rendered and depth
  information stored in the shadow Z-buffer using
  the light source as the viewpoint
• This computes a depth image from the light
  source of the polygons that are visible to the
  light

8
Shadow Z-buffer

• Step 2 - render the scene normally using a
  Z-buffer algorithm
• The normal process is enhanced as followsif a
  point (x, y, z) is visible, a transformation maps
  the point from 3D screen space to (x, y, z) in
  screen space with the light source as origin.
• The transformed points x and y are used to
  index the shadow buffer, and the corresponding
  depth value is compared with z.

9
Shadow Z-buffer

• If z is greater than the stored value in the
  shadow z-buffer for that point, then another
  surface is nearer to the light source than the
  point under consideration and the point is in
  shadowor else the point is not in shadow and is
  rendered normally
• With this algorithm there are obviously high
  memory requirements
• Also inefficient in that calculations are
  performed for surfaces that are subsequently
  overwritten

10
3D Rendering

• Can be classified into 2 types image space and
  object space
• Image space rendering algorithms are most
  common.
• the polygons are first clipped to the viewing
  frustum,
• an illumination algorithm is applied to light the
  surfaces,
• the surfaces are then transformed into image
  space,
• and finally rasterised

11
Image-based Rendering

• Clipping involved removing surfaces which lie
  outside the viewing frustum
• An illumination algorithm takes account of any
  ambient light, and the position of the light
  sources (and the viewing position), the surface
  characteristics and calculates a colour at the
  vertices of each triangle
• Transformation matrices are applied to each lit
  3D triangle to project the triangle onto the 2D
  image plane
• Each triangle is then rasterised.colours for
  each pixel covering the triangle are interpolated
  from the vertex colours, and written to the
  z-buffer

12
Object-based rendering

• Object-based rendering operates in object (or
  scene) coordinate spacei.e. 3D space
• Algorithms such as ray tracing and radiosity are
  object-based algorithmsthese techniques do not
  involve projections of the 3D objects onto the 2D
  image plane, are therefore more accurate and
  produce more realistic images
• The downside is that since they operate in 3D
  (i.e. floating point) space, they are not (yet)
  suitable for implementation in hardware, and are
  therefore very much slower than image-space
  techniques.

13
The Ray Tracing Algorithm

• Attempts to model the physics of light
  energylight energy is modelled as a number of
  light rays emanating from the light source and
  illuminating the scene.
• The light rays are absorbed by some objects,
  reflect off some objects and pass through some
  objects
• Some of these light rays then pass out of the
  scene through the camera to the viewer
• In practice, backward ray tracing is employed,
  where rays emanate from the viewer, and are
  traced through the centre of each pixel into the
  scene

14
Ray Tracing

• First set up a coordinate system with the pixel
  positions designated in the xy plane
• From the centre of projection, determine a ray
  path that passes through the centre of each
  screen-pixel position
• Illumination effects accumulated along this ray
  path are then assigned to the pixel
• Based on the principle of geometric optics
  light rays from the surfaces in a scene emanate
  in all directions, and some will pass through the
  pixel positions in the projection plane
• Since there is an infinite number of ray paths,
  the contributions to a particular pixel are found
  by tracing a light path backward from the pixel
  to the scene

15
Backward Ray Tracing
Cube B (opaque)
Light source
Reflection ray
Image plane
Shadow rays
Cube A (reflective)
Primary ray
16
Ray Tracing

• For each pixel ray, we test each surface in the
  scene to determine if it is intersected by the
  ray
• If a surface is intersected, we calculate the
  distance from the pixel to the surface
  intersection point
• The closest surface (smallest distance)
  intersected is found (hidden surface removal)
  it is the visible surface for that pixel
• Depending on the surface characteristics, more
  rays are traced from that intersection point to
  the light sources, and in the reflection and
  transparent directions

17
Ray Tracing

• We then reflect the ray off the visible surface
  along a specular path (angle of reflection
  angle of incidence)
• If the surface is transparent, we also send a ray
  through the surface in the refraction direction
• Reflection and refraction rays secondary rays
• This is a recursive process which is repeated for
  each secondary ray until a predetermined maximum
  recursion level is reached (set by user /
  determined by amount of storage space available)
  or if the ray strikes a light source.

18
Ray Tracing

• The result of this process is an intersection
  tree, which is initialised with the intersection
  produced by the primary ray, and a node is added
  to the tree for each object intersected by the
  secondary rays (left branches for reflection
  paths, right branches for transmission paths)
• The intensity assigned to a pixel is determined
  by accumulating the intensity contributions,
  starting at the surfaces that are intersected at
  the bottom of the ray-tracing tree
• If no surfaces are intersected by the pixel ray,
  the tree is empty and the pixel is assigned the
  background colour
• The contribution of each node to the final
  intensity of the pixel represented by the tree is
  determined as described in Whitteds 1980 paper

19
Ray Tracing

• The final colour at a pixel depends on
• the properties of the intersected surfaces,
• the relative positions of the surfaces, and their
  orientation with respect to the light sources.
• The intensity of light at an each intersection
  point P is a combination of the direct
  illumination at P, and the indirect illumination
  due to reflection and refraction, and is given
  by

20
Local Illumination

• According to Whitteds local illumination model,
  the colour at intersection point P is given by

where Iambi is the ambient light component, Id is
the diffuse reflection component, Is is the
specular reflection component, and It is the
specular transmission component
21
Ambient light

• This component models light that arrives on a
  surface indirectly. Ambient light does not
  originate from a particular source but is rather
  the product of light from all the surfaces in the
  environment. The ambient light term has the
  form

where Ia is the ambient light, assumed to be
constant for all objects, and ka is the ambient
reflection coefficient, which determines the
amount of ambient light reflected from a surface.
Ambient light is a uniform illumination, product
of multiple reflections of light on other objects
in the environment
22
Diffuse Reflection (Lambertian)

• Diffuse reflection, also called Lambertian
  reflection, models light reflection of dull
  surfaces. The intensity of light at a point on
  the surface is proportional to the angle ?
  between the normal to the surface at that point,
  N, and the direction to the light source, L

Id Ilkd cos ?
L
N
Il is the intensity of the light source kd is
the diffuse reflection coefficient
q
23
Diffuse Reflection

• Note this can be re-written in terms of the
  normal vector and the light vector as

Id Ilkd(L.N)
Diffuse reflection does not depend on the view
point. These surfaces appear equally bright
regardless of the viewing angle.
24
Specular Reflection

• Specular reflection refers to the amount of light
  reflected due to the shininess of a surface. In
  contrast to diffuse reflection, specular
  reflection is highly dependent on the viewing
  angle.

R
L
N
V
q
q
a
25
Specular Reflection

• Ideally, in the case of perfectly specular
  surfaces, the light is reflected back in one
  direction, R, which is exactly opposite the
  normal from the direction that light hits the
  surface.
• Specular reflection is proportional to the cosine
  of the angle ?, between the direction R, and the
  viewer direction, V.
• In nature, shiny surfaces are not perfectly
  specular, but they reflect light in different
  directions around the direction of the perfect
  specular reflection, R.

26
Specular Transmission

• Transmission refers to the amount of light
  refracted due to the transparency of a surface
• Each medium (surface) has a refractive index
  associated with it, that describes the speed of
  light through the medium compared to the speed of
  light in vacuum

N
L
qi
ni
nt
qt
T
27
Specular Transmission

• If ni and nt are the refraction indices of the
  mediums, the direction of transmission, T, can be
  computed as

28
Ray-object intersections

• The idea behind the intersection test between a
  ray and an object is to put the mathematical
  equation of the ray into the object equation and
  look for a real solution.
• If there is a real solution, then there is an
  intersection between the ray and the object, and
  the actual intersection point can be calculated.
• A ray in ray tracing is defined in terms of the
  ray origin, O(x0, y0, z0), and the ray direction
  vector, D(xd, yd, zd). The set of points on the
  line defined by O and D are given by the
  parametric equation of a ray...

29
Ray-object intersections

• t is the distance of the point from the origin,
  if the ray direction vector is normalised

30
Ray-object intersections

• The x, y, z coordinates of a point on the path of
  the ray are given by

31
Ray-object intersections

• We will now look at ray-object intersections for
  some common
• primitivesspheres and polygons
• Ray-sphere intersections
• Ray-polygon intersections

32
Ray-sphere intersections

• The mathematical equation for a sphere at centre
  C(xc, yc, zc) and radius r is given by

• By substituting the ray equation, into the above
  equation we get
• (x0xd.t-xc)2 (y0yd.t-yc)2 (z0zd.t-zc)2
  r2

• which gives a quadratic equation in t of the
  form
• ax2 bx c 0

33
Ray-sphere intersection

• If the equation does not have any real roots,
  the ray does not intersect the sphere.
• If there is only one real root, the ray is
  tangential to the sphere.
• Finally, if there are two real roots, the
  smallest positive root gives the ray entry point
  to the sphere, and the other root gives the ray
  exit point.
• The roots of this equation are substituted back
  to the parametric equation of the ray to give the
  actual intersection points

34
Ray-sphere intersection
35
Ray-polygon intersection

• The first step is to test for intersection with
  the ray and the plane defined by the points of
  the polygon.
• The mathematical equation for a plane through
  point P(xp, yp, zp), with normal N(a, b, c) is
  given by

By substituting the ray equation into the above
equation we get
which gives a linear equation in t...
36
Ray-polygon intersection
If the denominator of this equation is equal to
zero, the ray is parallel to the plane, so an
intersection does not occur. If the ray
intersects the plane, the intersection point
needs also to be tested against the polygon for
containment.
IP
37
Ray-polygon intersection
Ray1
Ray2
IP1
IP2
38
Ray-polygon intersection

• Point in polygon test
• extend a line to infinity from the IP in the
  plane,
• count the number of edges crossed.if even, IP
  does not lie inside the polygon,
• if odd, IP lies inside the polygon.

39
Illumination

• Once a valid intersection point has been found,
  this is stored.
• Once all the polygons have been tested, the
  closest valid intersection point is
  retained.this represents the closest visible
  surface to the viewer, and the illumination
  algorithm is then performed on this point.

40
Ray tracing efficiencies - 1

• Back facing surface removal takes place before
  illumination, after transformation into scene
  space
• Recall that solid objects are made up of a
  number of surfaces which describe the boundary of
  the object (b-rep models)

G
H
E
F
Surfaces ABFE, BCGF and FGHE are
visible, surfaces AEHD, CDHG and ADBC are back
facing from this viewing position, and can be
removed prior to illumination and rendering
D
C
B
A
41
Back face surface removal

• In order to determine whether or not a surface is
  back facing, take the dot product of the viewing
  vector with the surface normal (viewed from
  outside the object)

Remember v.n v n cos(theta) If theta lt 90,
the surface is visible, otherwise it is back
facing If theta lt 90, then cos(theta) is gt
0 Therefore, if v.n gt 0 the surface is visible,
otherwise the surface is back facing, and is
culled.
42
Efficiencies - 2

• Since every ray must be intersected with every
  surface, which is a very high computational load
  on the system (up to 90 of the work), a number
  of efficiencies have been introduced to
  accelerate the naïve ray tracing algorithm
• bounding volume hierarchies
• spatial data structures

43
Bounding volumes

• By analysing ray-sphere and ray-polygon
  intersection tests it is apparent that it is much
  easier to perform a ray-sphere intersection test
  than a ray-polygon intersection test.
• Also, for any single ray, it is likely that only
  a small number of surfaces actually intersect
  with the ray
• So.one efficiency is to enclose every polygon
  inside a tightly fitting bounding volume (spheres
  are commonly used)
• The ray is then intersected with the bounding
  volume first.and if no intersection is found,
  the surface need not be subjected to the more
  expensive ray-polygon test
• If an intersection is found, of course, the real
  ray-polygon test must then be performed as usual

44
Bounding Volumes

• The trade-off is then the extra computation
  required to generate the bounding volumes and the
  extra storage required (in the case of a sphere
  this is an (x,y,z) centre and a real radius)
  versus the reduction in computation for the
  ray-intersection tests
• For a complex scene this is almost always worth
  doing.

45
Bounding Volumes

• In some cases, bounding volume hierarchies are
  employedhere the whole scene and all the objects
  in it are enclosed by the bounding volume
• If the ray intersects the root bounding volume,
  then the child bounding volumes are intersected,
  and so on until individual polygons are
  intersected.

46
Bounding Volumes
C
A
B
Ray
Ray does not intersect with the bounding volume
surrounding A or B, and therefore need not be
intersected with the real geometry of A or B. Ray
does intersect with the bounding volume of C, and
is then intersected with Cs geometry as usual.
47
Space-Subdivision Methods

• Another way to reduce intersection calculations
  is to use space-subdivision methods
• We can enclose a scene within a cube and then
  successively subdivide the cube until each
  subregion (cell) contains no more than a preset
  maximum number of surfaces
• We then trace rays through the individual cells
  of the cube and only perform intersection tests
  within those cells containing surfaces
• The first object surface intersected by a ray is
  the visible surface for that ray
• Trade-off cell size vs number of surfaces per
  cell. If maximum number of surfaces per cell is
  too low, cell size can become so small that much
  of the savings in reduced intersection tests goes
  into cell-traversal processing

48
Spatial subdivision (2D)
49
Summary

• Z buffer shadow algorithm
• image-space techniques
• object-space techniques
• ray tracing
• ray generation
• illumination
• ray-sphere intersections
• ray-polygon intersections
• efficiencies
• bounding volumes
• spatial subdivision