Computer Vision

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Computer Vision

• Spring 2012 15-385,-685
• Instructor S. Narasimhan
• Wean Hall 5409
• T-R 1030am 1150am

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• Image Processing and Filtering
• Lecture 3

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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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Point Operations
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Point Processing
Nonlinear Lower Contrast
Original
Darken
Lower Contrast
Nonlinear Raise Contrast
Invert
Lighten
Raise Contrast
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Point Processing
Nonlinear Lower Contrast
Original
Darken
Lower Contrast
Nonlinear Raise Contrast
Invert
Lighten
Raise Contrast
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Neighborhood Operations
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Neighborhood operations
Image
Edge detection
Blur
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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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Face Morphing
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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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Images are Discrete and Finite
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Averaging
Lets think about averaging pixel values
Which is faster?
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Averaging
The convolution kernel
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Gaussian Smoothing
Gaussian kernel
(truncate, if necessary)
Use two 1D Gaussian filters
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Gaussian Smoothing

• A Gaussian kernel gives less weight to pixels
  further from the center of the window (5x5,
  sigma0.5)
• 0.0000 0.0000 0.0002 0.0000 0.0000
• 0.0000 0.0113 0.0837 0.0113 0.0000
• 0.0002 0.0837 0.6187 0.0837 0.0002
• 0.0000 0.0113 0.0837 0.0113 0.0000
• 0.0000 0.0000 0.0002 0.0000 0.0000
• This kernel is an approximation of a Gaussian
  function

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Gaussian Smoothing
original
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Mean vs. Gaussian filtering
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Gaussian Smoothing
by Charles Allen Gillbert
by Harmon Julesz
http//www.michaelbach.de/ot/cog_blureffects/index
.html
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Gaussian Smoothing
http//www.michaelbach.de/ot/cog_blureffects/index
.html
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Border Problem
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Border Problem

• Ignore
• Output image will be smaller than original
• Pad with constant values
• Can introduce substantial 1st order derivative
  values
• Pad with reflection
• Can introduce substantial 2nd order derivative
  values

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Median Filter

• Smoothing is averaging
• (a) Blurs edges
• (b) Sensitive to outliers

(a)
(b)

• Sort values around the pixel
• Select middle value (median)
• Non-linear (Cannot be implemented with
  convolution)

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Salt and pepper noise
Gaussian noise
3x3
5x5
7x7
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Correlation
template
How do we locate the template in the image?
Cross-correlation
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Cross-correlation
Note
Auto-correlation
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Normalized Correlation

• Account for energy differences

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Announcements

• Project 1 will be out later this evening.
• WARNING! Start projects early.

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Next Class

• Image Processing and Filtering (continued)
• Fourier or frequency domain analysis
• Horn, Chapter 6