Simple Mathematics for graphics

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Simple Mathematics for graphics
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Volumes from Silhouettes

• Start with collection of calibrated images

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Volumes from Silhouettes

• Cup on turntable example

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3D Curves from Edges

• Feature-based stereo matching

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3D Curves from Edges

• Extract extremal and internal edges

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3D Curves from Edges

• Match curves along epipolar lines

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What is Computer Graphics?

• Using a computer to generate an image from a
  representation.

computer
Model
Image
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Representations

• How do we represent an object?
• Points
• p x y z
• Mathematical Functions
• X2 Y2R2
• Polygons (most commonly used)
• Points
• Connectivity

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Representing Points and Vectors

• A 3D point p x y z
• Represents a location with respect to some
  coordinate system
• A 3D vector v x y z
• Represents a displacement from a position

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Linear Algebra

• Why study Linear Algebra?
• Deals with the representation and operations
  commonly used in Computer Graphics

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Vector Spaces

• Consists of a set of elements, called vectors and
  two operations that are defined on them, addition
  and multiplication

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Vector Addition

• Given V X Y Z and W A B C
• VW XA YB ZC
• Properties of Vector addition
• Commutative VWWV
• Associative (UV)W U(VW)
• Additive Identity V0 V
• Additive Inverse VW 0, W-V

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Parallelogram Rule

• To visualize what a vector addition is doing,
  here is a 2D example

VW
V
W
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Vector Multiplication

• Given V X Y Z and a Scalar s and t
• sV sX sY sZ
• Properties of Vector multiplication
• Associative (st)V s(tV)
• Multiplicative Identity 1V V
• Scalar Distribution (st)V sVtV
• Vector Distribution s(VW) sVsW

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Dot Product and Distances

• Given u x y z and v a b c
• vu axbycz
• The Euclidean distance of u from the origin is
  sqrt( x2y2z2) and is denoted by u
• Notice that u sqrt(uu)
• The Euclidean distance between u and v is sqrt(
  (x-a) 2(y-b) 2(z-c) 2) ) and is denoted by
  u-v

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Properties of the Dot Product

• Given a vector u, v, w and scalar s
• The result of a dot product is a SCALAR value
• Commutative vw wv
• Non-degenerate vv0 only when v0
• Bilinear v(usw)vus(vw)

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Angles and Projection

• Alternative view of the dot product
• vwv w cos(?) where ? is the angle
  between v and w
• If v is a unit vector (v 1) then if we
  perpendicularly project w onto v can call this
  newly projected vector u then u vw

w
v
u
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Matrices

• A compact way of representing operations on
  points and vectors
• 3x3 Matrix A looks like
• a(i,j) refers to the element of matrix A

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Matrix Multiplication

• If A is an n?k matrix and B is a k?p then AB is a
  n?p matrix with entries c(i,j)where
• c(i,j)?a(i,s)b(s,j)
• Alternatively, if we took the rows of A and
  columns of B as individual vectors then
• c(i,j)AiBj where the subscript refers to the
  row and column, respectively

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Matrix Multiplication Properties

• Associative (AB)C A(BC)
• Distributive A(BC) ABAC
• Multiplicative Identity I diag(1) (square
  matrix
• NOT commutative AB?BA

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Determinant of a Matrix

• Defined on a square matrix (nxn)
• Where A1i determinant of (n-1)x(n-1) submatrix A
  gotten by deleting the first row and the ith
  column

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Recursive Definition!!

• The basis case
• det of a 2x2 matrix is defined to be ad-bc where

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Uses of the Determinant?

• Linear Independence of columns in a matrix
• Cross Product
• Given 2 vectors vv1 v2 v3, ww1 w2 w3, the
  cross product is defined to be the determinant of

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Cross Product Properties

• The Cross Product of v and w is denoted by v?w
• Is a VECTOR, perpendicular to the plane defined
  by v and w
• v?wv w sin?
• ? is the angle between v and w
• v?w-(w?v)

v?w
w
v
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Matrix Transpose and Inverse

• The Transpose of a matrix A, denoted by AT is
  defined as aijaji (exchanging the rows and
  columns)
• If A and B are nxn matrices and ABBAI then B is
  the inverse of A, denoted by A-1
• (AB)-1B-1A-1 same applies for transpose

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Methods for finding the Inverse

• Explicit Methods
• Gaussian-Jordan Elimination
• Create the Augmented matrix AI and reduce the
  left side to the identity using elementary row
  operations and the right hand side will be the
  inverse. ie. IA-1
• Cramers Rule
• Solve for A where aijdet(submatrix(Aij))
• A-1(1/det(A))(A)T

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Implicit Methods

• Instead of explicitly calculating A-1, there are
  techniques that solve equations of the form Axb
  (system of linear equations).
• Clearly xA-1b but we do not need to explicitly
  calculate A-1 to calculate x.
• LU Decomposition
• QR Factorization
• Singular Value Decomposition (SVD)
• Conjugate Gradient if sparse

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Transformations

• Why use transformations?
• Create object in convenient coordinates
• Reuse basic shape multiple times
• Hierarchical modeling
• System independent
• Virtual cameras

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Translation




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Properties of Translation




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Rotations (2D)
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Rotations 2D

• So in matrix notation

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Rotations (3D)
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Properties of Rotations
order matters!
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Combining Translation Rotation
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Combining Translation Rotation
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Scaling
Uniform scaling iff
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Homogeneous Coordinates
can be represented as
where
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Translation Revisited
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Rotation Scaling Revisited
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Combining Transformations
where
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Transforming Tangents
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Transforming Normals
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Rotations about an arbitrary axis
Rotate by around a unit axis
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• We can view the rotation around an arbitrary axis
  as a set of simpler steps
• We know how to rotate and translate around the
  world coordinate system
• Can we use this knowledge to perform the rotation?

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Rotation about an arbitrary axis

• Translate the space so that the origin of the
  unit vector is on the world origin
• Rotate such that the extremity of the vector now
  lies in the xz plane (x-axis rotation)
• Rotate such that the point lies in the z-axis
  (y-axis rotation)
• Perform the rotation around the z-axis
• Undo the previous transformations

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Rotation about an arbitrary axis

• Step 1
• Rotate x-axis

y
(a,b,c)
x
x
z
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Closer Look at Y-Z Plane

• Need to rotate ? degrees around the x-axis

y
?
z
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Equations for ?
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Rotation about the Y-axis

• Using the same analysis as before, we need to
  rotate ? degrees around the Y-axis

y
x
z
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Rotation about the Z-axis

• Now, it is aligned with the Z-axis, thus we can
  simply rotate ? degrees around the Z-axis.
• Then undo all the transformations we just did

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Equation summary
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Deformations

• Transformations that do not preserve shape
• Non-uniform scaling
• Shearing
• Tapering
• Twisting
• Bending

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Shearing
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Tapering
Image courtesy of Watt, 3D Computer Graphics
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Twisting
Image courtesy of Watt, 3D Computer Graphics
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Bending
Image courtesy of Watt, 3D Computer Graphics
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What have we seen so far?

• Basic representations (point, vector)
• Basic operations on points and vectors (dot
  product, cross products, etc.)
• Transformation manipulative operators on the
  basic representation (translate, rotate,
  deformations) 4x4 matrices to encode all
  these.

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Why do we need this?

• In order to generate a picture from a model, we
  need to be able to not only specify a model
  (representation) but also manipulate the model in
  order to create more interesting images.

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Overview

• The next set of slides will deal with the other
  half of the process (at least in a simplistic
  fashion)
• From a model, how do we generate an image

computer
Model
Image
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Accumulation of Transformations
Example Robot arm
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Orthonormal Coordinates
Iff u and v are orthonormal
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Coordinate Systems

• Object coordinates
• World coordinates
• Camera coordinates
• Normalized device coordinates
• Window coordinates

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Object Coordinates

• Convenient place to model the object

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World Coordinates

• Common coordinates for the scene

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Camera Coordinates

• Coordinate system with the camera in a convenient
  pose

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Normalized Device Coordinates

• Device independent coordinates
• Visible coordinate usually range from

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Window Coordinates

• Adjusting the NDC to fit the window

is the lower left of the window
height
width
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Perspective Projection

• Taking the camera coordinates to NDC

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Perspective Projection
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Rasterization

• Array of pixels

3
2
1
0
0
1
2
3
4
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Rasterizing Lines
Given two endpoints, find the pixels that make
up the line.
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Rasterizing Lines

• Requirements
• No gaps
• Minimize error (distance to line)

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Rasterizing Lines
Assume 1 lt m lt 1, x0 lt x1
Line(int x0, int y0, int x1, int y1) float dx
x1 x0 float dy y1 y0 float m
dy/dx float x x0, y y0 for(x x0 x lt
x1 x) setPixel(x,round(y)) y ym
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Rasterizing Lines

• Problems with previous algorithm
• round takes time
• uses floating point arithmetic

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Midpoint Algorithm
NE
Q
M
E
P(x,y)
If Q lt M, choose E. If Q gt M, choose NE
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Implicit Form of a Line
Implicit form
Explicit form
Positive below the line Negative above the
line Zero on the line
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Decision Function
Choose NE if d gt 0 Choose E if d lt 0
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Incrementing d
If choosing E
But
So
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Incrementing d
If choosing NE
But
So
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Initializing d
Multiply everything by 2 to remove fractions
(doesnt change the sign)
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Midpoint Algorithm
Assume 0 lt m lt 1, x0 lt x1
Line(int x0, int y0, int x1, int y1) int dx
x1 x0, dy y1 y0 int d 2dy-dx int
delE 2dy, delNE 2(dy-dx) int x x0, y
y0 setPixel(x,y) while(x lt x1)
if(dlt0) d delE x x1 else
d delNE x x1 y y1
setPixel(x,y)
See Bresenham Algorithm
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Limitations?

• The midpoint line algorithm assumes that the
  slope (m) is between 0 and 1
• This implies that this algorithm only applies to
  lines in region 1
• Extending to other regions left as an a
  programming assignment

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3
1
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5
8
7
6
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Anti-aliasing Lines
Lines appear jaggy
Sampling is inadequate
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Anti-aliasing Lines
Trade intensity resolution for spatial resolution
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Anti-aliasing Lines
Assume 0 lt m lt 1, x0 lt x1
Line(int x0, int y0, int x1, int y1) float dx
x1 x0 float dy y1 y0 float m
dy/dx float x x0, y y0 for(x x0 x lt
x1 x) int yi floor(y) float f y
yi setPixel(x,yi, 1-f)
setPixel(x,yi1, f) y ym
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Putting it all together!!

• Take your representation (points) and transform
  it from Object Space to World Space
• Take your World Space point and transform it to
  Camera Space
• Perform the remapping and projection onto the
  image plane in Normalized Device Coordinates
• Perform this set of transformations on each point
  of the polygonal object
• Connect the dots through line rasterization

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Intuitively
World Space
Object Space
Camera Space
Rasterization