Last Time

1
Last Time

• The end of Sampling and Filtering
• Compositing

2
Today

• Compositing Summary
• Intro to 3D (and 2D) Graphics
• Homework 2 due

3
Compositing Operations

• F and G describe how much of each input image
  survives, and cf and cg are pre-multiplied
  pixels, and all four channels are calculated

4
Unary Operators

• Darken Makes an image darker (or lighter)
  without affecting its opacity
• Dissolve Makes an image transparent without
  affecting its color

5
PLUS Operator

• Computes composite by simply adding f and g, with
  no overlap rules
• Useful for defining cross-dissolve in terms of
  compositing

6
Obtaining ? Values

• Hand generate (paint a grayscale image)
• Automatically create by segmenting an image into
  foreground background
• Blue-screening is the analog method
• Remarkably complex to get right
• Lasso is the Photoshop operation
• With synthetic imagery, use a special background
  color that does not occur in the foreground
• Brightest blue or green is common

7
Compositing With Depth

• Can store pixel depth instead of alpha
• Then, compositing can truly take into account
  foreground and background
• Generally only possible with synthetic imagery
• Image Based Rendering is an area of graphics
  that, in part, tries to composite photographs
  taking into account depth

8
Where to now

• We are now done with images
• We will spend several weeks on the mechanics of
  3D graphics
• Coordinate systems and Viewing
• Clipping
• Drawing lines and polygons
• Lighting and shading
• We will finish the semester with modeling and
  some additional topics

9
Graphics Toolkits

• Graphics toolkits typically take care of the
  details of producing images from geometry
• Input (via API functions)
• Where the objects are located and what they look
  like
• Where the camera is and how it behaves
• Parameters for controlling the rendering
• Functions (via API)
• Perform well defined operations based on the
  input environment
• Output Pixel data in a framebuffer an image in
  a special part of memory
• Data can be put on the screen
• Data can be read back for processing (part of
  toolkit)

10
OpenGL

• OpenGL is an open standard graphics toolkit
• Derived from SGIs GL toolkit
• Provides a range of functions for modeling,
  rendering and manipulating the framebuffer
• What makes a good toolkit?
• Alternatives Direct3D, Java3D - more complex and
  less well supported

11
Coordinate Systems

• The use of coordinate systems is fundamental to
  computer graphics
• Coordinate systems are used to describe the
  locations of points in space
• Multiple coordinate systems make graphics
  algorithms easier to understand and implement

12
Coordinate Systems (2)

• Different coordinate systems represent the same
  point in different ways
• Some operations are easier in one coordinate
  system than in another
• We saw this in filtering Its easier to define
  the filter in one space, but transform it to
  another for use

(2,3)
(1,2)
v
v
y
y
u
u
x
x
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Transformations

• Transformations convert points between coordinate
  systems

(2,3)
(1,2)
ux-1 vy-1
v
v
y
y
u
u
xu1 yv1
x
x
14
Transformations(Alternate Interpretation)

• Transformations modify an objects shape and
  location in one coordinate system
• The previous interpretation is better for some
  problems, this one is better for others

(2,3)
xx-1 yy-1
y
(1,2)
y
xx1 yy1
x
x
15
2D Affine Transformations

• An affine transformation is one that can be
  written in the form

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Why Affine Transformations?

• Affine transformations are linear
• Transforming all the individual points on a line
  gives the same set of points as transforming the
  endpoints and joining them
• Interpolation is the same in either space Find
  the halfway point in one space, and transform it.
  Will get the same result if the endpoints are
  transformed and then find the halfway point

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Composition of Affine Transforms

• Any affine transformation can be composed as a
  sequence of simple transformations
• Translation
• Scaling (possibly with negative values)
• Rotation
• See Shirley 1.3.6

18
2D Translation

• Moves an object

y
y
by
x
x
bx
19
2D Translation

• Moves an object

y
y
by
x
x
bx
20
2D Scaling

• Resizes an object in each dimension

y
y
syy
y
x
x
sxx
x
21
2D Scaling

• Resizes an object in each dimension

y
y
syy
y
x
x
sxx
x
22
2D Rotation

• Rotate counter-clockwise about the origin by an
  angle ?

y
y
?
x
x
23
2D Rotation

• Rotate counter-clockwise about the origin by an
  angle ?

y
y
?
x
x
24
X-Axis Shear

• Shear along x axis (What is the matrix for y axis
  shear?)

y
y
x
x
25
X-Axis Shear

• Shear along x axis (What is the matrix for y axis
  shear?)

y
y
x
x
26
Reflect About X Axis
x
x

• What is the matrix for reflect about Y axis?

27
Reflect About X Axis
x
x

• What is the matrix for reflect about Y axis?

28
Rotating About An Arbitrary Point

• What happens when you apply a rotation
  transformation to an object that is not at the
  origin?

y
?
x
29
Rotating About An Arbitrary Point

• What happens when you apply a rotation
  transformation to an object that is not at the
  origin?
• It translates as well

y
x
x
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How Do We Fix it?

• How do we rotate an about an arbitrary point?
• Hint we know how to rotate about the origin of a
  coordinate system

31
Rotating About An Arbitrary Point
y
y
x
x
y
y
x
x
32
Rotate About Arbitrary Point

• Say you wish to rotate about the point (a,b)
• You know how to rotate about (0,0)
• Translate so that (a,b) is at (0,0)
• xxa, yyb
• Rotate
• x(x-a)cos?-(y-b)sin?, y(x-a)sin?(y-b)cos?
• Translate back again
• xfxa, yfyb

33
Scaling an Object not at the Origin

• What also happens if you apply the scaling
  transformation to an object not at the origin?
• Based on the rotating about a point composition,
  what should you do to resize an object about its
  own center?

34
Back to Rotation About a Pt

• Say R is the rotation matrix to apply, and p is
  the point about which to rotate
• Translation to Origin
• Rotation
• Translate back
• The translation component of the composite
  transformation involves the rotation matrix. What
  a mess!

35
Homogeneous Coordinates

• Use three numbers to represent a point
• (x,y)(wx,wy,w) for any constant w?0
• Typically, (x,y) becomes (x,y,1)
• Translation can now be done with matrix
  multiplication!

36
Basic Transformations

• Translation Rotation
• Scaling

37
Homogeneous Transform Advantages

• Unified view of transformation as matrix
  multiplication
• Easier in hardware and software
• To compose transformations, simply multiply
  matrices
• Order matters AB is generally not the same as BA
• Allows for non-affine transformations
• Perspective projections!
• Bends, tapers, many others