Digital Imaging 20062007

1
Digital Imaging2006/2007

• Lecture 5
• 3D graphics II

Ioannis Ivrissimtzis 05-Mar-2007
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Summary of the lecture

• Geometric Modelling
• Constructive solid geometry
• Mesh processing
• Surface interrogation
• Lights and Shading
• The reflection model
• Shading
• Textures

3
Solids
A solid is a 3D object (not a surface embedded in
3D space). In Constructive Solid Geometry
(CSG), shapes are produced using Boolean
operations to combine few basic shapes (the
primitives). Combinations of very few basic
shapes and few types of operations suffice for
most purposes.
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Boolean operations

• The basic Boolean operations are
• Union
• Intersection
• Difference

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Union
B

• A U B

A
The union of A and B is denoted by A U B. A
point belongs to A U B if it belongs to A OR to B.
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Intersection
B
A

• A n B

The intersection of A and B is denoted by A n
B. A point belongs to A n B if it belongs to A
AND to B.
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Example
A n B
A U B
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Difference

• A / B

B
A
The difference of A and B is denoted by A / B. A
point belongs to A / B if it belongs to A AND
NOT to B.
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Difference
B / A
A U B
A / B

• Unlike the union and the intersection, the
  difference is not a commutative
• operation. That is, generally A / B is different
  from B / A.

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Summary of the lecture

• Geometric Modelling
• Constructive solid geometry
• Mesh processing
• Surface interrogation
• Lights and Shading
• The reflection model
• Shading
• Textures

11
Simplification
Reduce the number of vertices.
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Simplification
Having fewer vertices saves computational time.
It is useful when an object is at the background.
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Simplification
The mesh is simplified step by step. Edge
collapse the vertex b moves to the position of
vertex a, and the triangles abc and adb
disappear.
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Simplification

• A simple algorithm for mesh simplification
• Find a function computing the cost of an edge
  collapse.
• For example, a cost function can measure changes
  in the mesh volume, or changes in the tangent
  planes.
• Compute the cost of all possible edge collapses
  and perform the one with the lowest cost.
• Compute new costs after the edge collapse and
  repeat the process.
• This is not computationally expensive because
  the cost function changes only on the part of the
  mesh altered by the edge collapse.

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Subdivision
Add vertices and faces to the mesh. The mesh is
gradually refined and becomes smoother.
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Subdivision
We create new triangles by splitting the existing
ones. How do we split the triangles? It is
NOT a good idea to insert a new vertex inside a
triangle and then split it in three. The shape
of the triangles will deteriorate.
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Subdivision
Instead we do a 1 to 4 split of the triangles.
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Subdivision
Where do we place the new vertices? If we place
the new vertices exactly on the existing edges,
the result will be flat. Instead, we
compute the new vertices in a way that produces
smooth curves.
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Subdivision
Example
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Smoothing
Many models have noise. Especially those coming
from scan data.
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Smoothing
There are algorithms for removing the noise.
Such algorithms are not perfect because some of
the initial information is destroyed by the
noise and can not be retrieved. Here,
the result is too smooth, missing some of the
details of the original.
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Smoothing
Here, the smoothing algorithm preserved several
sharp features, but several unwanted artifacts
appear.
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Summary of the lecture

• Geometric Modelling
• Constructive solid geometry
• Mesh processing
• Surface interrogation
• Lights and Shading
• The reflection model
• Shading
• Textures

24
Surface Interrogation
Sometimes the quality of a surface can not be
determined by visual inspection only. Several
techniques have been developed which use
mathematical tools to analyse the surface in a
more rigorous way. One of the most fundamental
concepts used in surface interrogation is
curvature.
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Curvature
For any point of a curve, the curvature is a
number indicating how much the curves bends at
that point. If we fit a circle at a point of a
curve, the curvature at that point is the
inverse of the circles radius. Of course,
this is not the exact definition but an intuitive
geometric interpretation.
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Curvature
At this point, the surface does not bend al lot.
We can fit a large circle. The inverse of the
circles radius is small. We say that the
curvature is low.
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Curvature

At this point the surface does bend al lot. We
can only fit a small circle. The inverse of the
circles radius is large. We say that the
curvature is large.
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Curvature

The notion of curvature generalizes to
surfaces. We can visualize curvature information
with a colormap. Each colour corresponds to a
value of the curvature.
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Curvature
A curvature colourmap can reveal features of the
surface which are not visible with naked eye.
Red denotes high curvature, green average
curvature and blue low curvature.
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Summary of the lecture

• Geometric Modelling
• Constructive solid geometry
• Mesh processing
• Surface interrogation
• Lights and Shading
• The reflection model
• Shading
• Textures

31
Normals

• The normal associated with a point of a smooth
  curve is a unit vector
• (vector of length one) perpendicular to the
  tangent line.

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Normals

• Similarly, surface normals are unit vectors
  perpendicular to the tangent
• planes of a surface.

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Lighting model

• The reflection model determines the light
  reflected at the points of the
• surface.
• It computes how much light arriving from a light
  source is reflected from
• the point and leaves in the direction of the
  viewers eye point.
• This is repeated for every light source in the
  scene graph to find the
• total reflected light at the point.
• The shading model determines how the polygonal
  faces are filled.

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Reflection model

• The computation of the reflection requires three
  unit vectors
• N (the surface normal) vector
  perpendicular to the surface.
• L (the light vector) vector from point to
  light source.
• E (the eye vector) vector from point to
  eye (view) point.

N
L
E
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Reflection model

• Reflected light is considered to have three
  components
• Ambient reflection Ia - reflected ambient
  light.
• Diffuse reflection Id - matt reflection.
• Specular reflection Is - shiny highlight.
• Different combinations of these give a surface
  the appearance of being
• made from different materials.
• The Java3d lighting model also has an emissive
  component Ie which
• allows implementation of glow in the dark
  objects, although they do not
• illuminate other objects.

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Ambient light

• The ambient light models light arriving equally
  from all directions. This
• is an empirical way to account for lost light.
• We have
• Ia La Ka
• where La is the intensity of the ambient light
  source and Ka is the
• surfaces ambient reflectance.
• Notice that none of the three vectors N, L, E is
  involved in the
• computation of the ambient light.

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Diffuse light

• Diffuse reflection scatters light equally in all
  directions, giving to the
• surface a matt appearance.
• The diffuse component is computed as
• Id Lp Kd (N L)
• where Lp is the intensity of the light source and
  Kd is the surfaces
• diffuse reflectance.

N
Lp
L
E
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Diffuse light

• Diffuse reflection is uniform and regardless of
  the eye point. However, it
• depends on the position of the light source.
• Because both N,L have length 1, we have
• where a is the angle between N and L. Therefore
  the light reflected in
• direction E depends only on cosa.

N
Lp
L
E
a
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Diffuse light
Late afternoon sun. Large a gives a value of
cosa near 0, thus less diffuse light.
N
L
Lp
E
a
Lp
N
Midday sun. Small a gives a value of cosa near
1, thus more diffuse light.
L
a
E
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Specular light

• Specular reflection creates highlights on shiny
  surfaces.
• Is Lp Ks (R E)n
• where the scalar Ks is the specular reflectance.
  The mirror reflection of
• L is in direction R where a' a.
• Because R E cos?, the closer E is to R the
  brighter the highlight will
• appear.

N
Lp
R
L
E
?
a'
a
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Specular light

• A larger value of
• shininess, n, produces
• sharper highlights.

A smaller value of n produces broader
highlights.
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Putting it all together

• The intensity I of the light leaving a point in
  direction of the eye is the
• sum of the emissive, ambient, diffuse and
  specular components.
• That gives
• can be
  scalar values or colour values.
• If they are colour values this equation is
  calculated once for each R, G
• and B intensities.

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Limitations

• The main limitation of the above lighting model
  is the lack of any
• interaction between the visual objects.
• In particular, light reflected by a visual object
  can not illuminate another
• visual object.
• This gives to the scene a distinctive computer
  signature with the visual
• objects appearing plastic like and floating.
• To address this problem we can add to the
  lighting model the
• computation of shadows, or use a different
  lighting model, e.g. one
• based or ray tracing.

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Summary of the lecture

• Geometric Modelling
• Constructive solid geometry
• Mesh processing
• Surface interrogation
• Lights and Shading
• The reflection model
• Shading
• Textures

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Flat shading

• The flat shading computes one intensity value for
  one point of the
• polygon and applies this intensity value to the
  whole polygon. For that
• reason it is fast.
• Notice the discontinuities at the vertices and
  edges of the mesh.
• Flat shading is truthful to the mesh geometry,
  but this is a disadvantage
• when we use the mesh to approximate a smooth
  surface.

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Gouraud shading

• Gouraud shading first computes the intensity at
  each vertex.
• Then, it interpolates between vertices to compute
  the intensities of the
• points at the edges of the mesh.
• Finally, it interpolates between edge points to
  compute the intensities at
• the interior of a polygon.

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Gouraud shading

• Gouroud shading uses vertex normals.
• The normal at a vertex can be computed as the
  (weighted) average of
• the normals of the adjacent faces.

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Phong shading

• The Phong shading model interpolates the vertex
  normals to compute
• normals for the points at the edges and the faces
  of the mesh.
• Then applies the lighting model to compute the
  intensity of each point of
• the mesh.
• Phong shading is computationally more expensive
  than the Gouraud
• shading which uses the reflection model only for
  computing intensities
• at the mesh vertices.

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Examples
Flat
Gouraud
Phong

• http//www.dma.ufg.ac.at

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Examples
Phong
Flat
Gouraud

• http//nis-lab.is.s.u-tokyo.ac.jp

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Summary of the lecture

• Geometric Modelling
• Constructive solid geometry
• Mesh processing
• Surface interrogation
• Lights and Shading
• The reflection model
• Shading
• Textures

52
Textures

• Textures add realism on a 3D visual object,
  without increasing the
• complexity of its geometry.
• The most common texturing technique maps an image
  to the surface
• of a 3D visual object.

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Parameterization

• When the topology of the surface is simple, we
  can map the image onto
• the surface continuously. This process is called
  parameterization.
• Assuming, for simplicity, a square image,
  parameterization finds a
• continuous map f from 0,1 x 0,1 to the
  surface.
• The square 0,1 x 0,1 is called the parameter
  space.

y
1
0
1
x
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Parameterization

• The parameterization induces a polygonization of
  the parameter space.
• Assuming that the connectivity of the mesh is
  known, the parameterization
• is defined by the position of each vertex in the
  parameter space, given by
• its texture coordinates.

y
1
0
1
x
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Parameterization

• With the exception of few simple surfaces, an
  image can not be wrapped
• around a surface without distortion.
• Parameterization algorithms try to minimize
  distortion.

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Free boundary

• One way to reduce distortion is to assume a
  free-boundary parameter
• space, that is, a parameter space which is not
  necessarily a square.
• http//www.cs.technion.ac.il

3D model
Parameterization
Textured model
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Texture atlases

• Depending on the topology of the surface, it may
  not be possible to
• map the parameter space 0,1 x 0,1 onto the
  surface continuously.
• Closed surfaces like the sphere, or planar
  surfaces with more then one
• boundary are such examples.

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Texture atlases

• Even if the surface is homeomorphic to the
  parameter space, still a
• continuous texture mapping may create very large
  distortion.
• In such cases, it might be preferable to
  partition the surface into several
• pieces called charts. Putting the charts together
  creates an atlas.
• The process of partitioning a surface into
  several smaller parts is called
• segmentation.

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Segmentation

• When a surface is segmented for parameterization,
  some common
• considerations are
• Each chart can be mapped continuously onto the
  parameter space.
• Each chart can be parameterized with little
  distortion.
• The charts that can be joined seamlessly.

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Examples

• Segmentation of large complex models.
• http//web.engr.oregonstate.edu

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Examples

• Textured models and the corresponding atlases.
• http//web.engr.oregonstate.edu

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Texture atlases

• The default texturing of Java3D primitives
  involves segmentation and
• parameterization of the chart.
• The cube is segmented into six faces. Each face
  can be textured
• without any distortion. However, there are
  discontinuities along the
• edges.

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Texture atlases

• The segmentation and texturing of the sphere
  induces discontinuities
• and distortion.

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Texture atlases

• Texturing the cone and the cylinder in Java3D.

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Normal textures

• By mapping an image to a 3D surface we can convey
  the sense of
• texture (material). However, the textured
  surface reflects the light as a
• smooth surface.
• Except of the obvious solution of increasing the
  complexity of the
• model, another technique to overcome this problem
  is to modulate the
• normals of the smooth surface creating a normal
  texture.

Original normals
Modulated normals
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Normal textures

Examples of normal textures
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PovRay example
texture T_Wood20 scale 10
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PovRay example
texture T_Copper_2B
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PovRay example
texture T_Glass4 interior I_Glass
caustics 1
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PovRay example

• sphere lt0,0,0gt, 3
• texture
• pigment color Red
• normal
• bumps 1.0 // depth of bumps
• scale 0.1 // width of
  bumps

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PovRay example

• The previous bumps example

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PovRay example

• The previous example with smaller depth of bumps
  0.25