ENG4BF3 Medical Image Processing

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ENG4BF3Medical Image Processing

• Image Enhancement in Frequency Domain

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Image Enhancement
Original image
Enhanced image
Enhancement to process an image for more
suitable output for a specific application.
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Image Enhancement

• Image enhancement techniques
• Spatial domain methods
• Frequency domain methods
• Spatial (time) domain techniques are techniques
  that operate directly on pixels.
• Frequency domain techniques are based on
  modifying the Fourier transform of an image.

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Fourier Transform a review

• Basic ideas
• A periodic function can be represented by the sum
  of sines/cosines functions of different
  frequencies, multiplied by a different
  coefficient.
• Non-periodic functions can also be represented as
  the integral of sines/cosines multiplied by
  weighing function.

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Joseph Fourier(1768-1830)
Fourier was obsessed with the physics of heat and
developed the Fourier transform theory to model
heat-flow problems.
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Fourier transformbasis functions
Approximating a square wave as the sum of sine
waves.
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Any function can be written as thesum of an even
and an odd function
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Fourier Cosine Series

• Because cos(mt) is an even function, we can write
  an even function, f(t), as
•  
•  
• where series Fm is computed as
•  
• Here we suppose f(t) is over the interval (p,p).

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Fourier Sine Series

• Because sin(mt) is an odd function, we can write
  any odd function, f(t), as
•  
•  
• where the series Fm is computed as
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•  

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Fourier Series
So if f(t) is a general function, neither even
nor odd, it can be written
Even component
Odd component
where the Fourier series is
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The Fourier Transform

• Let F(m) incorporates both cosine and sine series
  coefficients, with the sine series distinguished
  by making it the imaginary component
• Lets now allow f(t) range from ? to ?, we
  rewrite
• F(u) is called the Fourier Transform of f(t). We
  say that f(t) lives in the time domain, and
  F(u) lives in the frequency domain. u is called
  the frequency variable.

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The Fourier Transform
?u?x ? 1/M
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a
d
time domain frequency domain
time domain frequency domain
b
d a b c
c
time domain frequency domain
time domain frequency domain
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The Inverse Fourier Transform
We go from f(t) to F(u) by
Fourier Transform

• Given F(u), f(t) can be obtained by the inverse
  Fourier transform

Inverse Fourier Transform
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Discrete Fourier Transform (DFT)

• A continuous function f(x) is discretized as

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Discrete Fourier Transform (DFT)
Let x denote the discrete values (x0,1,2,,M-1),
i.e.
then
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Discrete Fourier Transform (DFT)

• The discrete Fourier transform pair that applies
  to sampled functions is given by

u0,1,2,,M-1
and
x0,1,2,,M-1
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2-D Discrete Fourier Transform

• In 2-D case, the DFT pair is

u0,1,2,,M-1 and v0,1,2,,N-1
and
x0,1,2,,M-1 and y0,1,2,,N-1
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Polar Coordinate Representation of FT

• The Fourier transform of a real function is
  generally complex and we use polar coordinates

Polar coordinate
Magnitude
Phase
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Fourier Transform shift

• It is common to multiply input image by (-1)xy
  prior to computing the FT. This shift the center
  of the FT to (M/2,N/2).
  
• 
  

Shift
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Symmetry of FT

• For real image f(x,y), FT is conjugate symmetric
  
• 
  

• The magnitude of FT is symmetric
  
• 
  

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FT
IFT
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IFT
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The central part of FT, i.e. the low frequency
components are responsible for the general
gray-level appearance of an image.
The high frequency components of FT are
responsible for the detail information of an
image.
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Image
Frequency Domain (log magnitude)
Detail
General appearance
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10
5
20
50
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Frequency Domain Filtering
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Frequency Domain Filtering

• Edges and sharp transitions (e.g., noise) in an
  image contribute significantly to high-frequency
  content of FT.
• Low frequency contents in the FT are responsible
  to the general appearance of the image over
  smooth areas.
• Blurring (smoothing) is achieved by attenuating
  range of high frequency components of FT.

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Convolution in Time Domain
g(x,y)h(x,y)?f(x,y)

• f(x,y) is the input image
• g(x,y) is the filtered
• h(x,y) impulse response

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Convolution Theorem
Multiplication in Frequency Domain
G(u,v)F(u,v)?H(u,v)
Convolution in Time Domain
g(x,y)h(x,y)?f(x,y)

• Filtering in Frequency Domain with H(u,v) is
  equivalent to filtering in Spatial Domain with
  f(x,y).

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blue line sum of 3 sinusoids (20, 50, and 80
Hz) random noise red line sum of 3 sinusoids
without noise
blue line sum of 3 sinusoids after filtering in
time domain 1x average 1 1 1 1 1 / 5 blue line
sum of 3 sinusoids after filtering in
frequency domain cut-off 90 Hz
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Algorithm Complexity

• We can compute the DFT directly using the formula
• An N point DFT would require N2 floating point
  multiplications per output point
• Since there are N2 output points , the
  computational complexity of the DFT is N4
• N44x109 for N256
• Bad news! Many hours on a workstation

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Algorithm Complexity

• The FFT algorithm was developed in the 60s for
  seismic exploration
• Reduced the DFT complexity to 2N2log2N
• 2N2log2N106 for N256
• A few seconds on a workstation

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Examples of Filters
Frequency domain
Spatial domain
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Ideal low-pass filter (ILPF)
(M/2,N/2) center in frequency domain
D0 is called the cutoff frequency.
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Shape of ILPF
Frequency domain
h(x,y)
Spatial domain
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FT
ringing and blurring
Ideal in frequency domain means non-ideal in
spatial domain, vice versa.
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Butterworth Lowpass Filters (BLPF)

• Smooth transfer function, no sharp discontinuity,
  no clear cutoff frequency.

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Butterworth Lowpass Filters (BLPF)
n1
n2
n5
n20
h(x)
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No serious ringing artifacts
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Gaussian Lowpass Filters (GLPF)

• Smooth transfer function, smooth impulse
  response, no ringing

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GLPF
Frequency domain
Spatial domain
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No ringing artifacts
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Examples of Lowpass Filtering
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Examples of Lowpass Filtering
Low-pass filter H(u,v)
Original image and its FT
Filtered image and its FT
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High-pass Filters

• Hhp(u,v)1-Hlp(u,v)
• Ideal
• Butterworth
• Gaussian

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High-pass Filters
h(x,y)
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Ideal High-pass Filtering
ringing artifacts
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Butterworth High-pass Filtering
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Gaussian High-pass Filtering
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Gaussian High-pass Filtering
Gaussian filter H(u,v)
Original image
Filtered image and its FT
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Laplacian in Frequency Domain
Frequency domain
Spatial domain
Laplacian operator
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Subtract Laplacian from the Original Image to
Enhance It
Original image
enhanced image
Laplacian output
Spatial domain
Frequency domain
new operator
Laplacian
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Unsharp Masking, High-boost Filtering

• Unsharp masking fhp(x,y)f(x,y)-flp(x,y)
• Hhp(u,v)1-Hlp(u,v)
• High-boost filtering
• fhb(x,y)Af(x,y)-flp(x,y)
• fhb(x,y)(A-1)f(x,y)fhp(x,y)
• Hhb(u,v)(A-1)Hhp(u,v)

One more parameter to adjust the enhancement
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End of Lecture