10 Image Processing II
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Transcript of 10 Image Processing II
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ENGG 167
MEDICAL IMAGING
Lecture 10: Oct. 13
Image Processing II
Frequency Domain & Transform Processing
References: Chapter 10, The Essential Physics of Medical Imaging, Bushberg
Radiation Detection and Measurement, Knoll, 2nd ed.
Intermediate Physics for Medicine and Biology, Hobbie, 3rd ed.
Principles of Computerized Tomographic Imaging, Kak and Slaney.
(http://rvl4.ecn.purdue.edu/~malcolm/pct/pct-toc.html)
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Preparation -
Review Imaging Processing Toolbox Help Manual(on your computer)
Download NIHImage (Mac) / ImageJ (Windows)Create a Macro which will analyze images and save a
processed version of the images
(http://rsbweb.nih.gov/ij/developer/macro/macros.html)
Complete Image Processing Assignment
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The Fourier Theorum
Ref: Gonzalez et al, Text
All signals can be decomposed into pure sinusoidal signals
+
+
+
=
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The Fourier transform
Ref: Gonzalez et al, Text
Spatial signalCorresponding
Frequency-domain signal
All signals can be decomposed into pure sinusoidal signals
This theorum is especially appropriate for periodic signals, but
can be used for discrete signals if enough frequencies are used
to capture the relevant information.
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1) Basics 1-dimensional sinusoidal representation of signals
Ref: Rizzoni
Where magnitude and phase of the coefficients are given by:
or:
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1) Basics complex numbers
Sinusoids are related to complex exponential expressions.
A complex number is one which is of the form:
z = a + ib, where I is the square root of -1, an imaginary number
Recall that the magnitude and phase of z can be calculated by:
magnitude => I = [a2 + b2]
phase => = tan-1(b/a)
So that another way to express z is :
z = I ei
Now, we can make use of a definition called Eulers formula:
ei = cos() + i sin()And
e-i = cos() - i sin()
Or written for the sinusoid signals:
cos() = [ei + e-i]/2 and sin() = [ei - e-i]/(2i)
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1) Basics complex numbers
So that equivalent expressions for a time domain signal are :
x(t) = an cos(nt) + bn sin (nt)
where an and bn are the magnitudes of the signal at each frequency n, and the summationis carried out over all values of n.
x(t) = In exp(int+n)
where In and n are the amplitude and phase at each frequency n.
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1) Basics fourier transform of a signal
X() = x(tn) exp(-itn)
where the signal X() is now in the frequency domain (recall=2f = 2/T), whereas theoriginal signal x(t) was a time resolved signal with N total data points. Summation is from
n=1 to n=N.
Alternatively a spatial data set can be transformed to spatial frequency data set by the same
approach:
F(k) = f(xn) exp(-i 2kxn)
where the signal F(k) is now in spatial frequency, k, and describes the exact discritized
shape of the original signal f(xn).
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N
1
N
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1) Basics 1-D Fourier Transforms f(x) F(kx)
Ref: Rizzoni
Input analytic function Fourier Transformed function
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Fourier Transform Summary
Ref: Gonzalez et al, Text
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Fourier Transform Summary
Ref: Gonzalez et al, Text
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Fourier Transform Summary
Ref: Gonzalez et al, Text
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Fourier Transform Summary
Ref: Gonzalez et al, Text
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2D Fourier Transforms
Where is the information in (u,v) space?
Where are the low frequencies?
Where are the high frequencies?Ref: Gonzalez et al, Text
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The Fourier space image (k-space)
Lowest frequency
Ref: Gonzalez et al, Text
Highest positive
kx frequency
Highest negative
kx frequency
Highest positive
ky frequency
Highest negative
ky frequency
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Fourier space filtering
Spatial frequency changes in the Fourier domain are simply done
with linear mathematics!
Ref: Gonzalez et al, Text
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Fourier space filtering
Spatial frequency changes in the Fourier domain are simply done
with linear mathematics!
Edge enhancement example (what is the shape of this filter)
Ref: Gonzalez et al, Text
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Fourier space filtering
Ref: Gonzalez et al, Text
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Try the online DEMO
http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertransform/
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Fourier space filtering
Ref: Gonzalez et al, Text
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Special filters Butterworth & Gaussian
Ref: Gonzalez et al, Text
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Special filters Butterworth & Gaussian
Ref: Gonzalez et al, Text
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Special filters Butterworth & Gaussian
Ref: Gonzalez et al, Text
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Noise
Ref: Gonzalez et al, Text
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Noise
Ref: Gonzalez et al, Text
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Noise: simple linear filtering can help
Ref: Gonzalez et al, Text
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Noise: frequency domain filtering
Ref: Gonzalez et al, Text
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Noise: frequency domain filtering
Ref: Gonzalez et al, Text
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Online resources for more info about image processing
http://micro.magnet.fsu.edu/primer/digitalimaging/imageprocessingintro.html
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Geometric Linear Transformations
Ref: Gonzalez et al, Text
Affine transform
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Geometric Linear Transformations
Ref: Gonzalez et al, Text
Affine transform
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Image compression : changing the binary
code used to reduce # of bits required
Ref: Gonzalez et al, Text
Use fewer bits to encode
Information that occurs a lot and
then use more bits to encode
information that occurs little in
the image
p(r) is the probability of
occurrence
I2 is a more efficient coding of
the bits than I1
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Lossy Image compression :transforms to fewer bits across the entire image
Ref: Gonzalez et al, Text
8 bits 4 bits
2 bits
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Minimum Lossy Image compression :
bit reduction at the local level, not globally!
Ref: Gonzalez et al, Text
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Minimum Lossy Image compression :bit reduction at the local level, not globally!
Ref: Gonzalez et al, Text
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Minimum Lossy Image compression :
bit reduction at the local level, 8x8 blocks -JPEG
Ref: Gonzalez et al, Text
predicted
error
compressed
image
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