Linear Filtering – Part II - Bilkent University Computer Engineering

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Linear Filtering – Part II - Bilkent University Computer Engineering
Linear Filtering – Part II
Selim Aksoy
Department of Computer Engineering
Bilkent University
[email protected]
Fourier theory
Jean Baptiste Joseph Fourier had a crazy idea:
Don’t believe it?
Any periodic function can be written as a weighted sum
of sines and cosines of different frequencies (1807).
Neither did Lagrange, Laplace, Poisson, …
But it is true!
Fourier series
Even functions that are not periodic (but whose
area under the curve is finite) can be expressed
as the integral of sines and cosines multiplied by a
weighing function.
Fourier transform
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier theory
The Fourier theory
shows how most real
functions can be
represented in terms
of a basis of sinusoids.
The building block:
A sin( ωx + Φ )
Add enough of them
to get any signal you
want.
Adapted from Alexei Efros, CMU
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier transform
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Fourier transform
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Fourier transform
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©2010, Selim Aksoy
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Fourier transform
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©2010, Selim Aksoy
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Fourier transform
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier transform
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier transform
Adapted from Alexei Efros, CMU
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier transform
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Gonzales and Woods
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Fourier transform
Adapted from Gonzales and Woods
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier transform
CS 484, Spring 2010
©2010, Selim Aksoy
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Fourier transform
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©2010, Selim Aksoy
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Fourier transform
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Shapiro and Stockman
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Fourier transform
Example building patterns
in a satellite image and
their Fourier spectrum.
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Convolution theorem
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Frequency domain filtering
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©2010, Selim Aksoy
Adapted from Shapiro and Stockman,
and Gonzales and Woods
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Frequency domain filtering
Since the discrete Fourier transform is periodic, padding is
needed in the implementation to avoid aliasing (see section
4.6 in the Gonzales-Woods book for implementation details).
CS 484, Spring 2010
©2010, Selim Aksoy
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Frequency domain filtering
f(x,y)
|F(u,v)|
×
h(x,y)
|H(u,v)|
⇓
⇓
g(x,y)
|G(u,v)|
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Alexei Efros, CMU
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Smoothing frequency domain filters
Adapted from Gonzales and Woods
CS 484, Spring 2010
©2010, Selim Aksoy
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Smoothing frequency domain filters
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©2010, Selim Aksoy
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Smoothing frequency domain filters
The blurring and ringing caused by the ideal lowpass filter can be explained using the convolution
theorem where the spatial representation of a
filter is given below.
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Smoothing frequency domain filters
Adapted from Gonzales and Woods
CS 484, Spring 2010
©2010, Selim Aksoy
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Smoothing frequency domain filters
CS 484, Spring 2010
©2010, Selim Aksoy
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Sharpening frequency domain filters
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Sharpening frequency domain filters
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Gonzales and Woods
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Sharpening frequency domain filters
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Gonzales and Woods
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Sharpening frequency domain filters
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Gonzales and Woods
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Template matching
Correlation can also be used for matching.
If we want to determine whether an image f
contains a particular object, we let h be that
object (also called a template) and compute the
correlation between f and h.
If there is a match, the correlation will be
maximum at the location where h finds a
correspondence in f.
Preprocessing such as scaling and alignment is
necessary in most practical applications.
CS 484, Spring 2010
©2010, Selim Aksoy
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Template matching
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Gonzales and Woods
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Template matching
Face detection using template matching: face templates.
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Template matching
Face detection using template matching: detected faces.
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Resizing images
How can we generate a
half-sized version of a
large image?
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©2010, Selim Aksoy
Adapted from Steve Seitz, U of Washington
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Resizing images
1/8
1/4
Throw away every other row and column to create
a 1/2 size image (also called sub-sampling).
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Steve Seitz, U of Washington
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Resizing images
1/2
CS 484, Spring 2010
1/4 (2x zoom)
1/8 (4x zoom)
Does this look nice?Adapted from Steve Seitz, U of Washington
©2010, Selim Aksoy
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Resizing images
We cannot shrink an image by simply taking every k’th pixel.
Solution: smooth the image, then sub-sample.
Gaussian 1/8
Gaussian 1/4
Gaussian 1/2
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Steve Seitz, U of Washington
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Resizing images
Gaussian 1/2
CS 484, Spring 2010
Gaussian 1/4
(2x zoom)
©2010, Selim Aksoy
Gaussian 1/8
(4x zoom)
Adapted from Steve Seitz, U of Washington
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Sampling and aliasing
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Steve Seitz, U of Washington
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Sampling and aliasing
Errors appear if we do not sample properly.
Common phenomenon:
High spatial frequency components of the image appear
as low spatial frequency components.
Examples:
Wagon wheels rolling the wrong way in movies.
Checkerboards misrepresented in ray tracing.
Striped shirts look funny on color television.
CS 484, Spring 2010
©2010, Selim Aksoy
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Gaussian pyramids
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©2010, Selim Aksoy
Adapted from Gonzales and Woods
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Gaussian pyramids
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©2010, Selim Aksoy
Adapted from Michael Black, Brown University
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Gaussian pyramids
CS 484, Spring 2010
©2010, Selim Aksoy
Adapted from Michael Black, Brown University
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