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Warning!!
These slides do not replace text book reading
L01: about this course
Aim: to teach the elements of Computer GraphicsTraditional 3D photorealismModern use of images, and non-photorealism
Method: traditional formslecturespersonal readingpersonal practical work
Assumptions: knowledge of analytic mathematicsvectors and matricesintegration and differentiation
Lecture number / curriculum index Topic Learning objective
01 / 1 Introduction to CM30075And Compuer Graphics
Pointers to course materials given., Traditional Graphics outlined briefly.
Traditional Computer Graphics
02 / 1.1.1 Modelling – Brep The B-rep modelling scheme is introduced as just one way to build objects.
03 / 1.3.1 Rendering – cameras and projection Camera models introduced. Points are projected, so that objects models can be rendered as wire-frames.
04 / A.1 Rendering – ray-casting Lines intersected with planar polygons, result used as a basis for ray-casting. Now objects can be rendered to look solid.
05 / 1.2.1 Simple reflection models Point light sources are introduced; Lambertian and Specular reflection so objects can be shaded.
06 / 1.3.2 Rendering – ray tracing Cast shadows, reflection and refraction amongst many objects. Ray-tracing as a tree.
Milestone: material for advance practical work is now covered
07 / 1.2.2 Advance lighting and reflection The BRDF is introduced. Physical models of reflection and refraction.
08 / 1.2.3 The Lighting Equation;Radiosity
The full complexity of lighting is revealed; radiosity and ray-tracing are seen as specific solutions.
09/ 1.1.1 Modelling revisited – advanced B-rep Use of spline surfaces as B-rep models; how to define them and how to render them
10/ 1.1.2 Modelling revisited – CSG and Voxels Voxel models and Volume Rendering
11 / 1.1.3 Texture maps. Texture maps introduced as things to be rendered that relieve the burden of modelling. Aliasing issue is mentioned.
Milestone: static images now (mostly) covered
12 / 1.4.1 Animation basics The basics of animation are introduced from both technical and artistic points of view.
13 / 1.4.2 Articulated figures Modelling bodies and hierarchies of transforms; inverse kinematics/dynamics is discussed
14 / 1.4.3 Soft Objects and Fluids Elastic properties – use of physics – to model cloth, putty and such like, including fluids.
15 / 1.4.5 Fire and Fluids Use of dynamic textures and particles to model animate objects with no definitive boundary.
Milestone: Traditional Computer Graphics material covered
Modern Computer Graphics: Photograph editing and non-photorealistic rendering
16 / 2.1.1 Compositing How images layer to make a whole; simple methods for making panoramas from holiday snaps.
17 / 2.1.2 Panoramic Views How to make panoramas from holiday snaps – and from video too.
18 / 2.1.3 Intelligent scissors Cutting and pasting objects in pictures; so more on compositing
19 / 2.1.4 Texture Propagation Filling in holes in images, maybe left by cutting.
20 / 2.2.1 NPR from 3D models Why is photorealism the aim? People paint! How to paint, not photograph from models.
21 / 2.2.2 NPR from photos How can photographs be processed to look like a painting? Why is this difficult?
22 / 2.2.3 Generalised Cameras The eye of Art is often not a real camera!
Milestone: course content fully introduced
23 revision
24 revision
Reading
Lecture Slides• are NOT intended to replace text book reading.
Standard texts• Watt, 3D Computer Graphics, Addison Wesley.• Foley et al, 3D Computer Graphics, Addison Wesley.• Watt and Watt, Advanced Computer Graphics, Addison Wesley.
For the interested.• SIGGRAPH proceedings (published as journal special issue; Transactions on Graphics)
• Eurographics proceedings• IEEE Transactions on Visualization and Computer Graphics• Computer Graphics Forum
PracticalAim: You are to write a simple ray-casting / ray-tracing program.
The practical work is broken into three stages
• ray-caster with self-shading
• cast-shadows
• full ray-tracer.
You should spend no more than 15 hours on this component;
including time to prepare the documents needed for your assessment.
Assessment
Assessment will cover all material given in lectures, in assigned reading and in practical work.
75% Sat ExaminationQuestions usually comprise 4 parts:a) Basic knowledge (3rd class)b) Moderate knowledge, basic understanding (2.2nd class)c) Good knowledge / moderate understanding (2.2nd class)d) Good understanding – can solve new problems (1st class)
25% practical workAlready explained
Computer Graphics Basics
focus
window
objectcamera
point (pixel)
Traditional: What colour is this pixel?
Rendering Pipeline
Make 3D models
transform models into place
illuminate models
project models
clip invisible parts
raterization
display
The rendering pipeline shows
the flow of information uses
and the processes needed
to synthesise an image.
In fact, there are many rendering pipelines.
The order of processes can change depending, for example, on whether rendering time or rendering quality is more important.
These different pipelines differ only in details – the general flow of information from 3D model to 2D image is always the same.
Modern approaches
point (pixel)
target image source image
What colour is this pixel?
• The B-rep modelling scheme is introduced as just one way to build objects.
L02: B-rep basics
points points and lines points, lines, and polygons
B-rep basics
Brep = Boundary Representation
Start with a set of M points p = { (x, y, z)i : i = 1…M }
Make a set of N lines from points L = { (i, j)k : k = 1…N }
Make a set of m polygons from lines B = { (k1, k2, …, kn}l : l = 1…m }
A model is three-dimensional (3D),
if the points are 3D (as above).
Different informationsupports different rendering
• points onlydots (used in Chemistry, also in modern Point based Rendering)
• linesWire-frame rendering, good for quick tests in animation, say
• polygonsshaded surfaces
points points and lines points, lines, and polygons
x1 y1 z1
x2 y2 z2
xi yi zi
xM yM zM
2 i
2 k
1 3 4 6 8
2 1 7 * *
point table line table polygon table
!! ALWAYS INDEX POINTS !!
Practical Issues
• avoids repeated points, so more efficient
• avoids numerical error when animating
Different tables can be used
x1 y1 z1
x2 y2 z2
xi yi zi
xM yM zM
1 3 4
3 4 7
point table triangle table
Triangles are the most common polygon, because triangles are always flat.
But triangles are expensive – many of them; so other polygons used for modelling are often decomposed into triangles for rendering.
A Minor Complication!! POINT ORDER MATTERS !!
anti-clockwiseclockwise
The ordering gives the polygon a “front” and “back”.
eh! which side is which?
The normal of a triangular polygon
The normal of a polygon is used in lighting calculations (and in other calculations too).
Suppose a triangle has points, p, q, r, each point in 3D.
The normal direction is
n = (p-q) x (q-r)
where x is the vector cross product.
Exercise: Show that reversing the order of points also reverses the direction of the normal.
build a table of trianglesfrom these points
1
2
3
4 5
6
7
8
Exercise: The points are randomly numbered.
From the view given,point 4 is at the back of the cube,point 5 is the nearest corner.
Build a table of trianglesin which vertices are consistentlyordered clockwise,when each face of the cube is viewedfrom the outside.
B-rep basics: summary
• B-rep = boundary representation
• Models objects with points, lines, polygons
• Points are ordered around a polygon
• Best to index into tables
• The form of tables dictates rendering algorithms
Exercise: Build a 3D cube from well ordered triangles, render it as a wire-frame. See L03 for projection methods.
• Camera models introduced.
• Points are projected, so that objects models can be rendered as wire-frames.
L03: Cameras and Projection
The Linear CameraIn a linear camera
• rays of light travel in straight lines from a object
• the camera captures all rays passing through a single focus
• intersect a planar window to make the image
• the normal from the plane to the focus is the optical axis
focus
windowobject
image
ray
optical axis
The Linear Camera
focuswindow
object
image
Two variants exist:
• the “physical” model – shown in the previous slidehas the focus between the object and the window,in which the image is inverted
• the “mathematical model” – which we use, is shown belowhas the window between the object and focusleaving the image the right way up
Basic Perspective Projection
h
W
wHh
w
Uses similar triangles to compute the height of the image
object
image
focus
W
H
3D is almost as easy as 2D
),(),( yxz
fvu
v xu
y
zf
),,( zyx
),,( fvu
In the canonical camera, focal length (f) is taken as 1.
Homogeneous Points
Make projection easy and convenient
3D point (x, y, z) written as (x,y,z,1)homogeneous point (x, y, z, ) maps to real point 1/ (x, y, z)
The homogeneous points p = (x,y,z,1) and q = (sx, sy, sz, s)differ only by a scale factor s; this makes them equivalent in homogenous space.
Notice (x,y,z) and (sx,sy,sz) are two points on the same straight line passing through the origin. This makes them equivalent – they represent the same line!
In homogeneous space, lines and points are dual concepts!
object point
image point
straight line (a ray of light)
focus:at the origin
a particular line – the optical axis – passesnormally through the window and through thefocus. This line has the image of the focus –a vanishing point.
the set of all rays projectnormally onto the windowto make a pattern of “spokes”
A little homogeneous geometry
window
a scaled version of the image
a scaled version of the object
Projection with a matrix
Using homogeneous coordinates, projection can be written as a matrix
we do need to divide by homogeneous depth, , after this
Compare this to with f = 1.),(),( yxz
fvu
The camera as a matrixUsing a matrix for projection is very convenient in many ways.
1. It means we can model the camera as a matrix, C, say.Now projection of a homogenous point p is just
q = pCand we know that the homogeneous image point q is just a scale factor away from being correct – and all we need do is scale it by its depth (last element).
2. It means we can move the camera about in space just by pre-multiplying by a matrix transform
q = pMC
3. It means we can change the internals of the camera (focal length, aspect ratio, etc) by post-multiplying by a matrix
q = pMCK
4. We can do all of this at once! just set A = MCK, now A is a linear camera q = pA
Some Examples
Setting
using
rotates then translates the camera before projection.
Notice this is equivalent to applying M-1 to the point.
Another exampleTo change focal length, set
now post-multiply
All at once
Define a new linear projective camera in a new place and with a new focal length
It is easy to use this camera…
Of course, you can set M and K as you please
– not just rotate and translate and new focal length!
Exercise: See notes for an exercise
Rendering with a Projection Matrix
1. Project all points using the projection matrix
2. Keep all points that lie within the window bounds
3. Connect points in the picture that are connected in 3D
If we had a model built just of points (no lines, planes etc) then we could make a simple images using this simple technique:
This produces a “wire frame” picture – easy, and fast!
If all pixels inside a projected triangle can be identified,then they too can be coloured.
This is the basis of scan-conversion.
• Lines are intersected with planar polygons.
• The result used as a basis for ray-casting.
• Now objects can be rendered to look solid.
L04: Ray-Casting
Ray-casting algorithm
For each pixelCast a ray from the focus through the pixelCompute all intersections with all polygonsFind the nearest polygonColour pixel with polygon colour
Actually, the polygon colour is modified to create the effect of shading, as on a sphere.
Ray Casting Basics
• One ray per pixel, cast into the scene• Look for nearest intersection• Colour pixel accordingly
Expensive part: computing the intersection of a ray with a polygon
focus
pixel
window
scene of objects
camera
Line / Polygon intersection
p r
x(s)
c
n
an infinitely wide plane
an infinitely long line
compute scale factor
Given the scale factor,
the intersection is easy to get
Point Inside Triangle?inside
outside
all three “turns” are anti-clockwise some “turns” are anti-clockwise,others are clockwise
direction of turn:
p1
p2
p3
x
parallel direction of turn, sij > 0:
A practical problem
The frame buffer is in the computer.
We imagine casting rays thru’ pixel centres
i
j
The window is in “space”;
where are pixel centres in space?
Solution: recall the camera as a matrix, C = MPK
y = xC = ((xM)P)K
• external parameters M locates the camera in spaceM is a 4x4 matrix, (eg rotation)notice M maps a model point in spaceuse the inverse of M to move camera!
• a projection P maps 3D points, to the windowP is rank degeneratecan be (4x3) but usually is (4x4)
• internal parameters, maps window points to frame-buffertransform confined to the planea matrix K (a 3x3 will do!)
K
x
y
i
j
The internal parameters transform the pixels centres from “frame buffer” coordinates to “window coordinates”.
There is no single “answer” – here I’ve mapped a rectangular frame buffer to a window which is square, so pixels get stretched into rectangles – but you may keep pixels square.
pixel is typically at (j,i) point is at (x,y)
It is very common (almost universal) to “flip” coordinates so pixel (j,i) maps to location (x,y);and notices row (j) increase going DOWN - take care when designing K!
K-1
The external parameters transforms the canonical camera
in “camera space” to the desired camera in “world space”.
M-1
M
Ray-casting takes a long time
Why does ray-casting take so long?
Because every ray is compared to every polygon.
Is this necessary?
What are bounding spheres?
Homework question: what is a BSP tree?
• Point light sources are introduced.
• Lambertian and Specular reflection, so objects can be shaded.
L05: Simple Reflection
Point light sources
• A point light source is, er, a point (x,y,z) that emits light.
• Actually the point light can be at infinity (think of the sun), in which case (x,y,z) is its direction.
Most point lights sources are often at infinity in computer graphics.This makes shading/reflection efficient to compute because the direction of the light is the same for every intersection point.
Basic Reflection Model
mirror direction
light direction
surface normal
Mirror reflection - specular reflection – Phong reflectionDiffuse refection – Lambertian reflection
Diffuse/Lambertian ReflectionA beam of light spreads over an area, and is reflected equally in all directions.So here we’ve not drawn any reflected direction
The energy in the light is spread out more over the wider area.
So, the area will appear less bright.
over anarrow area over a wide area
these beamsare of equal width
Functional analogy: the sun warms the earth more near the equator than the pole
w
h
h = w/cos()
w
The light energy in the beam of width w is spread over a length h = w/cos().
Energy density in beam: I/w
Energy density on ground: I/h = Icos()/w
Lambertian reflection in Graphics
Iout = Ilightcos()The cosine we need is the dot (inner) product of the unit normal andunit vector pointing to the light.
We know that light from the source strikes a surface, to be spread equallyin all directions. So, only the angle between the light direction and thesurface is important.
Iout = Ilight n.l
BUT the surface can absorb some light, so only a fraction c is reflected
Iout = cIlight n.l
The “diffuse reflection coefficient” c depends on the surface material.
Diffuse reflection in colour
Rout = credRlight n.l
To handle colour is easy: just do the same calculation for each colour channel
Bout = cblueBlight n.l
Gout = cgreenGlight n.l
For white light, Rlight = Glight = Blight.
The coefficients c are often thought of as the colour of the material.
Specular Reflection
mirror direction, r
light direction, l
surface normal, n
eye direction, e
How much light is reflected in a particular direction?
Intuitively, the angle between the mirror and eye directions is important.Homework: The eye direction is a given, what is the mirror direction?
Phong Reflection
Phong’s specular reflection model
Iout = kIlight(r.e)n
Notice how the cosine (r.e) is used.
k is the “specular reflection coeffiecient” for the surface.
Phong reflection works for colourin just the same sort of way as Lambertian reflection
A complete but simplereflection model
Iout = aIamb+ cIlight n.l + kIlight(r.e)n
A complete model (for a single colour channel) is
We see the diffuse and specular terms are added up.
There is an additional term for “ambient” light - this is light that comes equally from every direction- ambient light allows us see the very darkest regions of a picture
Specular highlights often look white, (just look around at things to see this) so k is often given the same value in every colour channel
Ray-tracing is a global lighting method.
• Cast shadows
• Reflection and refraction amongst many objects.
• Ray-tracing as a tree.
L06: Ray-tracing
Cast Shadows
Is the box resting on the table ?
Is a point in shadow?
point light
lightpoint
A point is in shadow if the line between it and a light is blocked by an object
How can we tell?
The point to light ray is a line
Look for intersections of the this line with any object
This is almost an exact repetition of the ray-casting,
except intersections need not be ordered,
just finding one is enough
Reversibility of light
Ray tracing relies on a physical principle:a ray of light can be traced in either direction
So, if a ray of light splits into two parts then its energy is divided.
But we can “run this backwards” and add up the energy in the divided rays to get the energy in the original.
The paths of the light rays are identical, backwards or forwards.
Ray tracing extends ray casting
eye window
rayobject
reflected ray
intersection
refracted ray
Ray tracing tree
A ray/object intersection generate a new (reflected, refracted) pair of rays.These new rays also intersect to generate new rays, hence a tree
original ray from eye thru’ pixel
intersection
reflected refracted
intersection
reflected refracted
intersection
reflected refracted
Stop when a ray “leaves” the scene, or a set depth
intersection
a leaf
the root
a “parent” raywhen goingUP the tree
Using the tree in a simple wayThe ray-tracing tree is used “bottom up”, from the leaves to the root.
There are no reflected or refracted rays at a leaf.You can work out the light to the parent using the simple lighting model.But now the direction to the eye is in fact the direction of the parent ray.
The leaf contributes some energy to its parent,this parent receives light contributions from both its children.
Now work out the simple light model for the parent, as if it was a leaf,Add this to the contributions from its children.
And so on.
A detail of the tree
surface
normalto light
mirrordirection
to parent
leaf intersectionlocal light only
surface
normal
to light mirrordirection
to parent
leaf intersectionlocal light only
surface
to parent
non-leaf intersectionlocal light + child rays
Whitted ray-tracingTurner Whitted produced the first ray-racer (1980).It was the first global illumination model. It includes
• Hidden surface removal• Self shading (as in diffuse/specular reflection)• Cast shadows• Reflection and refraction (making the model “global”)
The full coursework is to write a Whitted ray-tracer.
A restriction is to not include reflection and refraction, which is “advanced” ray casting.
A further restriction include self-shading only, which is “simple” ray-casting, and the minimum required to make a picture of some kind.
• The BRDF is introduced.
• Physical models of reflection and refraction are considered.
L07: Advanced Reflectionand Refraction
Overall energy transfer
incident light Ein
reflected light, Erefl
refracted light, Erefr
schematically
Ein = Erefl + Erefr + Eabsconservation of energy
Recall the simple lighting model
c = kaIa + Ikd(n.l) + Iks(r.e)
This takes into account• ambient light• diffuse reflection• specular reflection
n
l re
The energy in the colour depends on where the viewer is, e,compared to the reflection r – and hence the light direction l.
A visualisation
Suppose we set Ia = 0, I = 1, kd = 1, ks=1, and n.l = 1Furher, fix r. Now the lighting depends only on e and .In fact
c = 1 + e.r
to get a picture of how the energy depends on point of view
A more general idea
The light energy depends on
• the direction of the incident light
• the direction of view
Diffuse/Specular lighting
diffuse to diffuse diffuse to specular
specular to diffuse specular to specular
The BDRF
BDRF = bi-directional reflectance function
f(
f the fraction of light energy transferredfrom an incident light ray at (1,2)to viewing direction (1, 2)
In the simple model this is (e.r) in which r depends on input direction l.(Recall computing r from l is a homework!)
A full BDRF is even more complicated – wavelength, polarization…!
The importance of the BDRF
The BDRF controls the kind of material an object appears to be made from, for example
• plastic
• metal
• ceramic
The simple model tends to make things look plastic.
Where do we get a BRDF from?
BRDF can be measured from the real thing; not easy!
In graphics, “micro-surfaces” can be used to estimate BRDF.
Look up Torrance Sparrow model.
see, eg www.cs.princeton.edu/~smr/cs348c-97/surveypaper.html
flat surface …under a microscope
a perfect mirroreach micro-facet a perfect mirrorbut overall surface is not!
• The full complexity of global lighting
• radiosity in brief
• radiosity and ray-tracing as specific solutions
L08: The rendering equation
See: Watt & Watt, Advanced Animation and Rendering, subsec 12.2
Global LightingEvery object reflects light
(otherwise, it would look like a black hole)
So, every object can be reflected in every other !
This makes global lighting very complicated.
what colour is the light coming out?
light in
light out
Scattering, Shadows and refraction complicate further still !
light in
light out
and real cases are still more complicated !
Kajya’s rendering equation
The light transported from point y to point x
g(x,y) is the “visibility” function, 0 if x is in shadow wrt y, 1/|y-x|2 otherwise.
(x,y) is the transfer directly from y to x
(x,y,z) is BDRF (scattered light) toward x by y given light source at z
S is the set of all points in the scene
Important: I(.,.) appears both sides.
Use of Rendering Equation
Different lighting models are special-case solutions:
The local-model
Ray-tracing
Radiosity
rewrite
as with R an integral operator
Now
hence
Solving the rendering equation
consider
first term – direct (local) lighting: x is the eye, y a point
subsequent terms account for scattering from other points
Ray-tracing
• Forward ray-tracing (from the eye to lights)
– specular to specular : bounces, Phong term– specular to diffuse: Lambertian– diffuse to diffuse (but badly!): Ambient
• Backward ray-tracing (to the eye from lights)
– diffuse to specular
Radiosity
• Diffuse to Diffuse
This is the (badly modelled) “ambient” term in “local” models
light falls onto a patch from all others
light radiates to all other patches
The light energy per unit area is called radiosity
total energy at a patch is its radiosity x its area
and energy is conserved, so at a patch
radiosity x area = emitted energy + reflected energy
In a closed environment the energy transfer betweenpatches will reach an equilibrium.
If we use a discrete environment (ie a finite model),then we can use a discrete form of the radiosity equation
The form factors are correlated:
so that, on division by Ai we get the basic equation used:
the radiosity equation in matrix form
the hard part is computing the form factors.
(See a standard text for how to do this)
Once they are at hand, solve the system for the B.
then render with a scan converter.
• Spline surfaces as B-rep models
• How to define them
• How to render them
L09: Brep with curves
Many real world objects are curved.
But so far, we have used flat modelling primitives.
Here, we learn how to use curved primitives.
We first look at curve lines in space,
and then at curved surfaces in space.
Both curves and surfaces are collections of points in space.
The points are related to one another by some function or other.
In principle the curves and surfaces contain an infinite number of points,but in practice we are forced to use finitely many points.
And we can think of surfaces as a collection of curves
There are many ways to define curves and surfaces
Implicit: “balances” a points coordinates
Unit Circle… x2 + y2 = 1
Unit sphere: x2 + y2 + z2 = 1
We will use implicit forms in CSG modelling.
Here we will use parametric forms.
Parametric forms allow you to compute points directly.
This is much better for Brep models.
Parametric forms allow you to compute points directly.
This is much better for Brep models.
Parametric curves require one parameter, u.
x(u)
is a point in 3D, on the curve.
Parametric surfaces require two parametersx(u,v)
is a point in 3D, on the surface.
Formally
x(u) is a mapping from the real line, R, to a subset of R3.
It’s as if the real line (x axis) is bent into the shape of the curve.
x(u,v) is a mapping from the real plane, R2, to a subset of R3.It’s as if the real plane (xy-plane) is bent into the shape of the curve.
u
u
v
x(u)
x(u,v)
Curves:
An easy way to specify a curve is with a polynomial;most modellers use cubics:
x(u) = a + bu + cu2 + du3
= [u3 u2 u 1][d c
b a]
The coefficients a, b, c, d specify the curve.
But this is not the most convenient way,because it’s hard to control.So the cubic is usually specified in some other way.
This is an abuse of notation!
Bezier Curves and Surfaces
are just one of the many alternatives
x(u) = UMP
U = [u3 u2 u 1]
M = [ -1 3 -3 1 3 -6 3 0 -3 3 0 0 1 0 0 0 ]
P = [p0
p1
p2
p3]
the “Bezier” matrix
parameter vector
4 control points
Examples of Bezier curves
control points
Bezier curve
Bezier surfaces
analogous issues over control of surfaces argue in favour of,
say, Bezier surfaces; again one of may controllable forms.
x(u,v) = UMPMtVt
The M is the Bezier matrix,
U and V parameter vectors,
P is now a 4x4 matrix of control points.
A Bezier surface example
Rendering curved surfaces
Can compute intersection of ray/patchbut this is difficult and expensive.
More often, the surface is broken into many small triangles;the approximation error is carefully controlled.
Homework: read and summarise (for yourself) one method for decomposing a surface into triangles.
Computing normals of surfaces
Could use normal from a triangle.Better to get the normal at a point directly.
compute partials(only one is shown)
get normal direction
Issues not considered here:
Stitching patches together to make a bigger surface.Continuity issues constrain the points around the joins.
The differential geometry of curves and surfacesFrenet framesGaussian curvature
• Construct Solid Geometry basics
• Voxel models
L10: CSG and Voxels
CSG basics
Brep is not the only way to represent objects.CSG is another common way.
CSG is models define sets of points via inequalities.
Example
A straight line, y = mx+c, divides the plane:
set f(x,y) = mx + c – y
then f(x,y) > 0 are points above the linef(x,y) = 0 are points on the linef(x,y) < 0 are point below the line
More generally
if f(x,y,z) is any scalar function of 3 spatial variablesthen it can be used to partition space points,using the sign of f(x,y,z).
More examples:
unit sphere: f(x,y,z) = x^2 + y^2 + z^2 – 1
unit cylinder: f(x,y,z) = x^2 + y^2 – 1
Let’s write f(x) in place of f(x,y,z);x has become a vector, x = (x,y,z,1).
Now suppose f(x) = x12 + x2
2 -1, a sphere
And suppose M is a transform,
then f(xM) is a mapped version of the sphere;
is an ellipse – maybe with a new centre.
Creating new shapes
Combining shapes
CSG primitives are just sets of points.
So they are easy to combine into more complex shapes.
plane(.)plane(.) plane(.) plane(.)
tetrahedron(.)
AND
sphere(.) cylinder(.)
DIF
doughnut(.)
Any set operation can be used to combine primitivesAny new shapes can be combined too.
Rendering CSG
Ray tracing could be suitable –
the “CSG” plane is already used by us!and bounding spheres are CSG models!
We just have to be clever about following the model treewhen deciding if a ray strikes an object (eg dougnut example).
And (partial) differentiating gives normals;
Homework: show the normal at a point x on the unit sphere is 2x.
CSG vs Brep
CSG B-rep
points implicit point explicit
sets easy to handle sets hard to handle
goodish to ray trace
hard for radiosity
hardish to ray-trace
good for radiosity
easy to combine hard to combine
hard to get from life easier to get from life
NB: This list is my personal view based on vast non-experience.
Voxel Models
A pixel is a picture element – an area of colour
A voxel is a volume element – a volume of colour
Voxels are often (partially) transparent.
Voxels (typcally) come from medical data:CAT - bonesMRI – soft tissue
a single voxel
many voxels
voxels are arranged into a brick;
each brick can contain hundreds – thousands – of voxels
in each direction
Typical rendering: CAT scans
CAT scan data produces voxels with a “density”,
so is a scalar field f(x,y,z).
1. Use the scalar field to assign colourand opacity
2. Compute a normal at each voxelUse finite differences for this
3. Shoot a ray through a pixel into the volume,compute lighting at each voxelintegrate colour and opacity over the ray
f
colo
ur
Example – taken from Wikipedia
• Texture maps
• Variations
• Two problems
L11: Texture maps
Texture maps: what and why
A texture map is a picture:
a carpeta label on a can of foodthe earth…many other examples
Texture maps are used where the level detail neededmake standard modelling inefficient / impossible.
The idea is to “wrap” or “warp” a flat pictureonto a 3D surface
In this example,a texture of the earth ismapped onto a sphere
+
=
Variations on a theme
• Bump mapping
• Environment mapping
• Procedural mapping
Two problems
1. Flat pictures will not map onto many (most) surfaces:toroidsbunniesetc
2. Aliasing – the texture map is made of pixelsthe pixels can show upand the texture is squashed / stretchedto fit the surfacemaking the problem worse
x
(sort of) solving problem 1
Use “intermediate” shapes, typicallyspherecylindercube
x is a point on the object
c is the object centre
y is the project of x on the im-shape
y
c
Basic algorithm has two steps:
1. Map the 2D texture (u,v) to the 3D surface (x,y,z):x = x(u,v),y = y(u,v),z = z(u,v)T(u,v) -> T(x,y,z)
This is called the S-mapping
2. Map the object to the same 3D surfacex = x’(xo,yo,zo)y = y’(xo,yo,zo)z = z’(xo,yo,zo)T(x,y,z) -> T(xo,yo,zo)
This is called the O-mapping
Other O-mappings from the object to im-surface exist
x x
x x
centroid object normal
surface normalreflected ray
this one makes reflect the environment
simple bump opacity
www.sli.unimelb.edu.au/envis/texture.html
environment
3D textures(procedural)
(sort of) solving problem 2
Recall: Aliasing of textures
solution : mip-mapping
mip = multim im parvo "many things in a small space“
The texture is scaled
and filtered
BEFORE use
1.The texture is scaled
and filtered
2.In use, the program selected the
correct section of the mipmap, according
to distance
• Animation as an art
• Animation of solid bodies
• Camera motion
L12: Animation Basics
Animation as an art
• technically, animation is motion
• artistically it means “bring to life”
Traditional animatorsuse many tricks to bringtheir characters to life –to animate them.
All tricks break Physics
streak-lines
ghosting
squash-and-stretch
rubber inertia
Animation of solid bodies
This is easy – the same transform is applied to all points
The transform is usually a matrix, but does not have to be.
The transform can be differential
x(t+dt) = f[ x(t) ]
but this often leads to numerical error;
so the transform is more often absolute
x(t) = f[ x(0) ]
The set of point {x(0)} is – conveniently – the “canonical” object
Examples
RiTiTiRi
Camera motion
Cameras often follow a path through a scene.
The path is often defined by a cubic curve.
But this just says where the camera is, we need to know:
* where it points
* and control the camera speed.
We must return to the differential geometry of a curve...
To control speed and acceleration along the curve
we need
Arc-Length Parameterisation
Recall the Brep curves; x(u) = U*P is a point on the curve
u is a “natural parameter”
parameterisation is arbitrary
define c(u) as the length of the arc
To control viewing direction along the curve
we need
Frenet Frames
• Model Specifications
• Hierarchies of transforms
• Inverse kinematics/dynamics
L13: Articulated Figures
Model specifications
Articulated models comprise
• a set of “limbs”
• joints that link the limbs
• methods to move the limbs
plus appearance information (colour etc)
Here the limbs are just rectangles;
limbs that are rigid bodies are much easier to handle
in which case rectangles are OK to use.
So we can think about the joints.
These are specified in many different ways.
Here a joint is just a point
the limbs move around the point
Often within a constrained angle
(In 3D, within a constrained solid angle)
rotation angle
parent frame
The way angles are measured
must be carefully specified.
A frame is needed.
Here the “x” axis runs between joints.
X-axis rotation is, then , angle moved.
Hierarchies of TransformsSimple articulated models have a tree-like structure
head
right upper arm
right lower armleft lower arm
left upper arm left upper leg
left lower leg
left foot
right upper leg
right lower leg
right foot
body
neck
This tree is used to locate the whole body first, then the next layer of limbs,
and so on. In this tree the feet are placed last of all.
Placing a transform at each node in the tree
is a very common method
to make the object move.
We already know how to orient the limb inside its frame,
so all we do is move the frame!
Which way around is correct?
y = xMbodyMupperMlowerMfoot
or
y = xMfootMlowerMupperMbody
or will either do?does it matter?
Problem:
in the modified video,
why does the foot not goes though the floor?
Inverse kinematics/dynamics
• Kinematics – the location of the limbs at any given time.
• Dynamics – the motion of the limbs at any given time.
IK method
Put the “puppet” into a poses at various times.
The computer works out how to move from one pose to the next,
in the time available.
The solution has to be subject to some constraints,
typically minimisation of some energy measure.
The method is hard, and tedious – too much so for these note.
See Watt & Watt, section 16.4 for details.
• Non-physical models
• Physical models; elasticity etc for cloth, putty
• Water and fluids
L14: Soft Objects and Fluids
Soft Objects and Physics
Animation of rigid bodies is easy to achieve;
Animation of articulated figures benefits from“inverse kinematics” and “inverse dynamics”.
Animation of soft objects all but requires the use of physics.
Non-physics approaches
• Parametric transforms• Spline surfaces• Free-form deformations
Parametric transforms, eg angle of turn about z depend s on z:
R( theta(z) )
Or could animate splines; eg.
Bezier surfaces are easy to move – just move the control points !
• Free form deformations
FFD’s “move the space”.
FFD’s are tricubic Bezier patches:
Q(u,v,w) = sum sum sum p(i,j,k) Bi(u) Bj(v) Bk(w)
Any point on any object and (u,v,w) maps to the point Q.
Physics models
Simple use of physics;
i) Use Newton’s second law
F = m d2x/dt2
write as a set of first order ODES
F = m du/dtu = dx/dt
ii) Integrate the differential equations over time;setting boundary conditions at time = 0;
• Elasticity is a well used model
Soft objects are stretchy !!
Force = -k * stretch
Thus given a form for F, motion is determined.
Different variants can be – and are - used
Bending forces (twisting forces)
Water and fluids
The “atom and force” model is good for “soft solids”
But liquids as best modelled in other ways
Again, though, physics is the basis of all the models
• Not given in 2008
L15: Fire and Smoke
• Why is photorealism the aim? People paint!
• What is NPR?
• NPR issues
L16: NonPhotorealistic Rendering
Why photorealism? people paint!
Why photorealism ?
• Physics provides starting point for models
• Photographs provide a foil to test against
But people paint !
• Artwork shows off important regions better
• Photography is just one depictive style amongst many
Which is the simplest to understand?
What is NPR?
• NPR is a sub-branch of Computer Graphics that studies making images that do not look photographic.
• NPR is said to have started around 1990.
• Several sub-divisions exist– paint-boxes– rendering 3D models– rendering from photographs and/or video
Paint boxes
from the simple… …to the complex
Rendering 3D models
Breslav et al, SIGGRAPH 2007
Rendering from photographs and/or video
NPR issues
• NPR depends on perception and interpretation– “drawing implies seeing”– models are much more complex– interaction is common
• Mark making– What kind of mark?– Where to mark?
• No single foil to test against– there is no “correct” drawing
NPR depends on Perception and Interpretation
No single foil to test against
Mark making: what and where?
What: Emulate real media
pencil, oil paint, chalk, …..
physical models possible
can test for quality of a given mark
What: Produce new media
Can the computer be used to make marks not made before?
Where: Should marks be made locally, globally, or both?
How to decide where to place marks?
Where: How can marks be made stable in animations?
• The problems
• Some solutions
L17: NPR from models
The problems
Consider a pencil drawing, marks are used to depict:
• object boundaries and other contours
• shadows and shading
• texture
Apparent ridges
Judd et al, SIGGRAPH 2007
For apparent ridges we need:
• differential geometry of surfaces
• projection
The curvature at a surface point S(x,y) – how curved is the surface?
Use the Hessian, H, which is a 2x2 matrix of 2nd order partials of S.
The EVD of H = UKUT
principle directions, U = [u1 u2], principle curvature, K = [k1 k2]
Ridges and valleys where k1 is an extremum in direction u1.
Project the local surface geometry (tangents, normals ‘n all),
Look for extremum in this observed geometry.
Shadows and shading
Shading lines – cross hatches etc
Can be based in “image space”………or in “object space”
Again, curvature can be used, this time to direct pencil lines.
Other methods use solid texture maps, that adapt to local lighting and geometry
Marks make textures too…
With NPR, even the marks can be animated (look at the sea)
L18: NPR from images
L18: Smart cut-and-paste
L19: Texture Propagation
L21: NPR from images
L22: Generalised Cameras
