Image Transforms

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Image Transforms

• Basic idea

Input Image, I(x,y) (spatial domain)
Mathematical Transformation F( )
Transformed Image F(u, v)
Processing F(u,v)
Inverse Transformation F-1( )
Output Image, I(x,y) (spatial domain)
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Image Transforms

• Fourier Transform
• There are many different transformations, Fourier
  Transform (FT) or its fast implementation (FFT)
  is the most well-known.
• For the purpose of this course, we will treat FFT
  as a black box, and will not go through the
  detail mathematics (i.e., not required).
• Instead of formal mathematics, which will be more
  elegant, we will try to explain the essential
  idea of FFT informally.

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Image Transforms

• Fourier Transform Essential idea
• Any given function (an image is a 2D function)
  can be approximated by a weighted sum of sines
  and conses.

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Image Transforms

• Fourier Transform
• Basic idea

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5
Image Transforms
f(t)
Cos(2?t)
f(t)
f(t) Cos(?t/2)Cos(?t)Cos(2?t)
Cos(?t)
Cos(?t/2)
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Image Transforms
f(t)

• In general,
• The weight, Fi, indicates the importance of
  cos(uit)
• ui represents the frequency of the cosine signal
• A larger ui, cos(uit) changes faster -gt higher
  frequency component of f(t)
• A smaller ui, cos(uit) changes slower -gt lower
  frequency component of f(t)

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Image Transforms
f(t)
Cos(2?t)
f(t) Cos(?t/2)Cos(?t)Cos(2?t) F1 1 u1
?/2 F2 1 u2 ? F3 1 u3 2?
Cos(?t)
Cos(?t/2)
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Image Transforms
f(t)
Cos(2?t)
f(t) 5Cos(?t/2)2Cos(?t)Cos(2?t) F1
5 u1 ?/2 F2 2 u2 ? F3 1 u3 2?
Cos(?t)
Cos(?t/2)
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Image Transforms
f(t)
Cos(2?t)
f(t) Cos(?t/2)2Cos(?t)5Cos(2?t) F1
1 u1 ?/2 F2 2 u2 ? F3 5 u3 2?
Cos(?t)
Cos(?t/2)
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Image Transforms
f(t)
F1 1 u1 ?/2 F2 2 u2 ? F3 5 u3 2?
F1 5 u1 ?/2 F2 2 u2 ? F3 1 u3 2?
F1 1 u1 ?/2 F2 1 u2 ? F3 1 u3 2?
f(t) F1Cos(?t/2)F3Cos(?t)F3Cos(2?t)
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Image Transforms
f(t)
F1 1 u1 ?/2 F2 2 u2 ? F3 5 u3 2?
F1 5 u1 ?/2 F2 2 u2 ? F3 1 u3 2?
F1 1 u1 ?/2 F2 1 u2 ? F3 1 u3 2?
f(t) F1Cos(?t/2)F3Cos(?t)F3Cos(2?t)
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Image Transforms
f(t)
F1 1 u1 ?/2 F2 2 u2 ? F3 5 u3 2?
F1 5 u1 ?/2 F2 2 u2 ? F3 1 u3 2?
F1 1 u1 ?/2 F2 1 u2 ? F3 1 u3 2?
Power spectrum or Frequency Distribution
f(t) F1Cos(?t/2)F3Cos(?t)F3Cos(2?t)
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Image Transforms
f(t)
F1 1 u1 ?/2 F2 2 u2 ? F3 5 u3 2?
F1 5 u1 ?/2 F2 2 u2 ? F3 1 u3 2?
F1 1 u1 ?/2 F2 1 u2 ? F3 1 u3 2?
What can we tell about the function (image) from
its frequency distribution?
f(t) F1Cos(?t/2)F3Cos(?t)F3Cos(2?t)
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Image Transforms
f(t)

• Fourier Transform so far our informal
  illustration

f(t)
FFT
F(u)
FFT
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Image Transforms
f(t)

• Fourier Transform Actual

A continuous function rather than discrete
f(t)
FFT
F(u)
F(u)
FFT
u
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Image Transforms
f(t)

• Fourier Transform Actual

F(u)
u
Similar to the discrete case, from F(u), we can
tell something about the signal f(t)
F(u)
u
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Image Transforms
f(t)

• Fourier Transform To summaries
• From F(u), we can tell something about its
  spatial signal, whether it contains fast/slow
  changing features

f(t)
FFT
F(u)
F(u)
FFT
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Image Transforms

• Now, come back to the idea of frequency domain
  processing
• F( )

Input Image, I(x,y) (spatial domain)
Mathematical Transformation F( )
Transformed Image F(u, v)
F
Processing F(u,v)
Inverse Transformation F( )
Output Image, I(x,y) (spatial domain)
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Image Transforms

• Now, come back to the idea of frequency domain
  processing
• F( )

To achieve smoothing, low-pass filtering, we
attenuate the higher frequency part of F(u)
Input Image, I(x,y) (spatial domain)
Mathematical Transformation F( )
Transformed Image F(u, v)
F
Processing F(u,v)
Inverse Transformation F( )
Output Image, I(x,y) (spatial domain)
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Image Transforms

• Now, come back to the idea of frequency domain
  processing
• F( )

To achieve sharpening, low-pass filtering, we
attenuate the lower frequency part of F(u)
Input Image, I(x,y) (spatial domain)
Mathematical Transformation F( )
Transformed Image F(u, v)
F
Processing F(u,v)
Inverse Transformation F( )
Output Image, I(x,y) (spatial domain)
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Image Transforms

• Band limiting signals A signals Fourier
  transform equal to zero above a certain finite
  frequency
• All images (natural signals) are band limiting
  signals

F(u)
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Frequency Domain Processing
im

• In practice
• Matlab
• f2fft2(im)
• f2fftshift(f2)
• fabs2FH_abs(f2)
• (calculate magnitude,
• FFT are complex number)
• imshow(fabs2)

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Frequency Domain Processing

• Another
• example

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Convolution and Spatial Filtering

• Spatial filtering is the convolution between the
  input image and the filtering mask

f(x,y)
w(x,y)
f(x,y)w(x,y)
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Frequency Domain Processing

• The foundation of frequency domain techniques is
  the convolution theorem

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Frequency Domain Processing
H(u, v) is called the transfer function
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Frequency Domain Processing

• Typical lowpass filters and their transfer
  functions

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Frequency Domain Processing

• Typical lowpass filters and their transfer
  functions

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Frequency Domain Processing

• Example

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Frequency Domain Processing

• Example

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Frequency Domain Processing

• Typical lowpass filters and their transfer
  functions

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Frequency Domain Processing

• Example

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Frequency Domain Processing

• Typical lowpass filters and their transfer
  functions

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Frequency Domain Processing

• Example

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Frequency Domain Processing

• Example

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Frequency Domain Processing

• Example

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Frequency Domain Processing

• Typical highpass filters and their transfer
  functions

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Frequency Domain Processing

• Typical highpass filters and their transfer
  functions

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Frequency Domain Processing

• Typical highpass filters and their transfer
  functions

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Frequency Domain Processing

• Examples

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Frequency Domain Processing

• Examples

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Frequency Domain Processing

• Examples

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Frequency Domain Processing

• More examples

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Frequency Domain Processing

• Examples

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Frequency Domain Processing

• Examples

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Frequency Domain Processing

• Spatial vs frequency domain

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Frequency Domain Processing

• Spatial vs frequency domain

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Frequency Domain Processing

• Examples