Computer Vision

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Computer Vision

• Spring 2006 15-385,-685
• Instructor S. Narasimhan
• Wean 5403
• T-R 300pm 420pm

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Announcements

• Homework 1 is due today in class.
• Homework 2 will be out later this evening (due
  in 2 weeks).
• Start homeworks early.
• Post questions on bboard.

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• Image Processing and Filtering
• Lecture 5

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Image as a Function

• We can think of an image as a function, f,
• f R2 ? R
• f (x, y) gives the intensity at position (x, y)
• Realistically, we expect the image only to be
  defined over a rectangle, with a finite range
• f a,bxc,d ? 0,1
• A color image is just three functions pasted
  together. We can write this as a vector-valued
  function

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Image as a Function
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Image Processing

• Define a new image g in terms of an existing
  image f
• We can transform either the domain or the range
  of f
• Range transformation
• What kinds of operations can this perform?

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Image Processing

• Some operations preserve the range but change the
  domain of f
• What kinds of operations can this perform?
• Still other operations operate on both the domain
  and the range of f .

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Linear Shift Invariant Systems (LSIS)
Linearity
Shift invariance
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Example of LSIS
Defocused image ( g ) is a processed version of
the focused image ( f )
Ideal lens is a LSIS
Linearity Brightness variation Shift
invariance Scene movement
(not valid for lenses with non-linear distortions)
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Convolution
LSIS is doing convolution convolution is linear
and shift invariant
kernel h
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Convolution - Example
Eric Weinsteins Math World
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Convolution - Example
1
1
2
-1
-2
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Convolution Kernel Impulse Response

• What h will give us g f ?

Dirac Delta Function (Unit Impulse)
Sifting property
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Point Spread Function
Optical System
scene
image

• Ideally, the optical system should be a Dirac
  delta function.

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Point Spread Function
normal vision
myopia
hyperopia
astigmatism
Images by Richmond Eye Associates
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Properties of Convolution

• Commutative
• Associative

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How to Represent Signals?

• Option 1 Taylor series represents any function
  using polynomials.
• Polynomials are not the best - unstable and not
  very physically meaningful.
• Easier to talk about signals in terms of its
  frequencies
• (how fast/often signals change, etc).

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Jean Baptiste Joseph Fourier (1768-1830)

• Had crazy idea (1807)
• Any periodic function can be rewritten as a
  weighted sum of Sines and Cosines of different
  frequencies.
• Dont believe it?
• Neither did Lagrange, Laplace, Poisson and other
  big wigs
• Not translated into English until 1878!
• But its true!
• called Fourier Series
• Possibly the greatest tool
• used in Engineering

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A Sum of Sinusoids

• Our building block
• Add enough of them to get any signal f(x) you
  want!
• How many degrees of freedom?
• What does each control?
• Which one encodes the coarse vs. fine structure
  of the signal?

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Fourier Transform

• We want to understand the frequency w of our
  signal. So, lets reparametrize the signal by w
  instead of x

• For every w from 0 to inf, F(w) holds the
  amplitude A and phase f of the corresponding sine
  
• How can F hold both? Complex number trick!

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Time and Frequency

• example g(t) sin(2pf t) (1/3)sin(2p(3f) t)

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Time and Frequency

• example g(t) sin(2pf t) (1/3)sin(2p(3f) t)

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Frequency Spectra

• example g(t) sin(2pf t) (1/3)sin(2p(3f) t)

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Frequency Spectra

• Usually, frequency is more interesting than the
  phase

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Frequency Spectra



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Frequency Spectra



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Frequency Spectra



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Frequency Spectra



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Frequency Spectra



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Frequency Spectra

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Frequency Spectra
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FT Just a change of basis
M f(x) F(w)


. . .
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IFT Just a change of basis
M-1 F(w) f(x)


. . .
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Fourier Transform more formally
Represent the signal as an infinite weighted sum
of an infinite number of sinusoids
Note
(Frequency Spectrum F(u))
Inverse Fourier Transform (IFT)
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Fourier Transform

• Also, defined as

Note

• Inverse Fourier Transform (IFT)

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Fourier Transform Pairs (I)
Note that these are derived using
angular frequency ( )
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Fourier Transform Pairs (I)
Note that these are derived using
angular frequency ( )
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Fourier Transform and Convolution
Let
Then
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Fourier Transform and Convolution
Spatial Domain (x)
Frequency Domain (u)
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Properties of Fourier Transform
Spatial Domain (x)
Frequency Domain (u)
Note that these are derived using
frequency ( )
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Properties of Fourier Transform
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Example use Smoothing/Blurring

• We want a smoothed function of f(x)

H(u) attenuates high frequencies in F(u)
(Low-pass Filter)!
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Image as a Discrete Function
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Digital Images

• The scene is
• projected on a 2D plane,
• sampled on a regular grid, and each sample is
• quantized (rounded to the nearest integer)

Image as a matrix
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Sampling Theorem
Continuous signal
Shah function (Impulse train)
Sampled function
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Sampling Theorem
Continuous signal
Shah function (Impulse train)
Sampled function
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Sampling Theorem
Sampled function
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Nyquist Theorem
Aliasing
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Aliasing
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Announcements

• Homework 1 is due today in class.
• Homework 2 will be out later this evening.
• Start homeworks early.
• Post questions on bboard.

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Next Class

• Image Processing and Filtering (continued)
• Horn, Chapter 6