CIS303 Advanced Forensic Computing

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CIS303Advanced Forensic Computing

• Dr Giles Oatley

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The frequency domain

• Working in the frequency domain gives many more
  opportunities for the image analyst. To convert
  an image between the spatial and the frequency
  domain we use the Fourier transform. This is
  practical because of the Fast Fourier Transform
  algorithm (FFT). Two principle areas of
  application are filtering and focus correction.
• Sub topics
• Frequency information in the image
• Fourier components in 1 and 2 dimensions
• Applying the FFT in MatLab
• Low and high pass filters
• Focus correction

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Frequency in an image
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Basic definitions
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Fourier Series

• Techniques for the analysis and manipulation of
  spatial frequency are based on the work of Jean
  Baptiste Joseph Fourier.
• The key idea is that any periodic function,
  however complex, can be represented as a sum of
  sines and cosines.
• Thus, for a function with period L

• The set of sine and cosine functions are known as
  the basis functions.
• A weighted sum of these basis functions is known
  as a Fourier Series.
• The weighting factors an and bn are the Fourier
  coefficients.

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Fourier Coefficients
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The frequency components in a square wave
function Squarewave(n) To calculate display up
to n terms of a Fourier series if ngt50 n
50 end x 0pi/1002pi total 0 for J
0n-1 K 2J1 total total
sin(Kx)/K end y 0.5 2total/pi plot(x,y
) titlestring int2str(n),' Terms' title(titl
estring)
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Square wave reconstruction
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The Fourier spectrum for the square wave
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Extension to two dimensions

• In two dimensions, the basis functions are 2D
  sine and cosine functions.
• A Fourier series of a 2D function f(x,y) can be
  written as follows

• Where u and v are the number of cycles fitting
  into one horizontal and vertical period,
  respectively.
• This Fourier series can be used to represent any
  image and we can visualise the basis functions as
  basis images.
• If u0 and v0, the basis image is constant, with
  value a00. The is the mean grey level in the
  image.
• Higher terms in u and v introduce the
  fluctuations about this mean level that are
  needed to represent changes in grey level across
  the image.

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Fourier reconstruction in two dimensions
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The Discrete Fourier Transform (DFT)

• Fourier theory provides us with a means of
  determining the contribution made by any basis
  function to the representation of some function
  f(x).
• The contribution is determined by projecting f(x)
  onto that basis function. This procedure is known
  as a Fourier transform (FT).
• The FT is a generalization of the Fourier series.
  Instead of sines and cosines, as in a Fourier
  series, the Fourier transform uses exponentials
  and complex numbers.
• When applying the procedure to images, we must
  deal explicitly with the fact that the image is
• Two-dimensional.
• Sampled.
• Of finite extent.
• These considerations give rise to the Discrete
  Fourier Transform (DFT).

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Why do we need it?

• The projection allows us to reconstruct the
  signal as a sum of exponentials!
• Why is this good? Complex exponentials are
  sinusoids. Most significantly, the operation of
  filtering is very simple once we are in the
  frequency domain - in order to change the amount
  of a certain frequency which is in a signal, we
  just multiply that coefficient by a constant.
• Thus, DFT is a convenient way of breaking down a
  signal into its frequency components. Best of
  all, efficient algorithms (called Fast Fourier
  Transforms, or FFTs) exist to compute the DFT,
  which allow us to perform efficient filtering by
  taking the DFT, multiplying the frequency
  coefficients, and then reconstructing the signal
  (i.e. taking the inverse DFT.)

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DFT of an N x N image
Or, (since cos(?)jsin(?) can be written in
exponential form)
Note that F(u,v) is a complex number we are now
dealing with a set of complex coefficients,
rather than two sets of real coefficients, as was
the case with the Fourier series shown on a
previous slide.
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Notes on the DFT

• For any particular spatial frequency specified by
  u and v, evaluating the previous equations gives
  us the contribution that the corresponding pair
  of basis images makes to the image.
• In other words, it tells us how much of that
  particular frequency is present in the image f.
• To build up a complete picture of the relevant
  importance of different frequencies, we must
  evaluate the equation for all u and v.
• There also exists an inverse Fourier Transform
  given by

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More about the DFT/inverse DFT

• The only essential difference between the DFT and
  the inverse DFT is the sign of the exponent.
• The forward transform on an NxN image yields an
  NxN array of coefficients.
• Since the inverse transform reconstructs the
  original image from this set of coefficients,
  they must constitute a complete representation of
  the information available in the image.
• When we manipulate f(x,y), we say we are
  processing the image in the spatial domain.
• When we manipulate F(u,v), we say we are
  processing the image in the frequency domain.
• The transformation from one domain to another
  does not, in itself, result in any information
  loss.

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The spectra of an image
F(u,v) is a complex number, with real and
imaginary parts. We can thus define the magnitude
and phase of F(u,v)
Thus, magnitude and phase are given by
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Amplitude and phase spectra

• The previous equations allow us to decompose an
  array of complex coefficients into arrays of
  magnitudes and phases.
• Magnitudes amplitudes of the basis images in
  the Fourier representation.
• Array of magnitudes is the amplitude spectrum.
• Array of phases is the phase spectrum.
• When we talk about the spectrum, we normally
  mean the amplitude spectrum, which is more
  significant in terms of interpretation. However,
  both are needed to reconstruct an image from the
  frequency to the spatial domain.
• Another term used is the power spectrum, or
  spectral density

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Log amplitude

• Note that the dynamic range of Fourier spectra is
  usually much higher than that of a typical
  display device, so that only the brightest parts
  of the image are visible on the display screen.
• To compensate for this we often display the
  following

Where c is a scaling constant and the logarithmic
function performs the desired compression. This
greatly facilitates visual analysis of Fourier
spectra.
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Example
Using phase only
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Fast Fourier transform (FFT)

• Calculating a single value of F(u,v) involves a
  summation over all pixels in the image.
• For an NxN image, the number of complex
  multiplications and additions for a DFT is
  proportional to N2.
• Assuming that the multiplication of a complex
  number takes 1µs, it would take 71 minutes to
  compute the FFT of a 256x256 image, 12 days for a
  1024x1024 image.
• Fortunately, a much faster method exists the
  Fast Fourier Transform, or FFT.
• The 2-D FFT algorithm takes advantage of the
  separability of the Fourier transform, which
  allows us to perform a 1-D FFT along each row of
  the image, followed by another 1-D FFT down each
  column.
• The FFT also uses the fact that a Fourier
  transform of length N can be written as the sum
  of two Fourier transforms of length N/2.

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Advantage of FFT
Overall cost of FFT procedure is Nlog2N
operations, compared to N2.
N N2 Nlog2N Ratio N2/Nlog2N
64 4,096 384 11
256 65,536 2,048 32
1024 1,048,576 10,240 102
4096 16,777,216 49,152 341
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Steps to display a Fourier transform
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Using the FFT

• The transform works most efficiently when the
  image dimensions are powers of 2 (e.g. 128, 256
  etc.). It will however work with other
  dimensional ranges.
• It generates an amplitude and phase for each
  frequency component. Both are equally important
  but usually only the amplitude component is
  displayed because the phase is difficult to
  interpret visually.
• As an alternative to amplitude and phase the
  transform can be expressed as a set of complex
  numbers Z X iY. In this case it is usually
  the intensity which displayed.
• The transform is a periodic function of which
  only one cycle is displayed. The transform also
  assumes that the original image is periodic with
  only one cycle visible. Occasionally this can
  cause unwanted artefacts on the processed image.
• Left by itself the FFT will produce a graph in
  which the zero frequency appears in the corners
  and the highest frequency in the centre. MATLAB
  provides a simple function to rectify this.

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Demonstration FFT in MatLab
function FFTDemo(SW) To demonstrate the 2D Fast
Fourier Transform SW is the width (pixels) of a
white stripe in a 256 x 256 square A
zeros(256,256) if SW gt 50 SW 50 end LL
128 - round(SW/2) UL 128
round(SW/2) A(1255,LLUL) 1 subplot(1,2,1),
imshow(A) F fft2(A) F1 fftshift(F) F2
abs(F1) subplot(1,2,2), imshow(F2,-1
8,'notruesize') colormap(jet)
colorbar improfile
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A thin line and its FFT
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A small square and its FFT
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Link between spatial and frequency domains

• Two Fourier transform relationships that
  constitute a basic link between the spatial and
  frequency domains are convolution and
  correlation.
• They are of fundamental importance to
  understanding image processing techniques based
  upon the Fourier transform.
• For any computer-based system, increasing the
  kernel size for the above operations soon means
  it is more efficient to perform the operations in
  the Fourier domain.
• i.e. the time needed to perform the FFT
  transformations from the spatial to the frequency
  domain and back is shorter than that needed to
  perform the operations in the spatial domain.

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Convolution in the frequency domain

• The equivalent to spatial domain convolution is a
  single multiplication of each pixel in the
  magnitude image by the corresponding pixel in a
  transform of the kernel.
• Thus, convolution in the spatial domain is
  exactly equivalent to multiplication in the
  frequency domain.

The main application is in filtering. Note that
in correlation, a similar relationship exists i.e.
The main application of correlation is in
template matching.
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Steps for filtering in the frequency domain
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Low pass filters
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Examples of Butterworth low pass filters
C 50 n 5
C 50 n 1
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Demonstration program part 1
function MothDemo(cutoff,order) To demonstrate
the 2D Fast Fourier Transform on a real object A
imread('moth9.gif') B double(A)/255 subplot(
2,2,1), imshow(B) Calculate and display its
Fourier transform F fft2(B) F1
fftshift(F) F2 abs(F1) subplot(2,2,2)
imshow(F2,-1 32)colormap(gray) Get the image
size row col size(B)
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Demonstration program part 2
Now define the appropriate Butterworh
Filter bwlpf zeros(row,col) centre_row
round(row/2) centre_col round(col/2) for v
1col for u 1row bwlpf(u,v) 1 / (1
(sqrt((u-centre_row)2 (v-centre_col)2)/cutoff)
(2order)) end end bwlpf fftshift(bwlpf) su
bplot(2,2,3) imshow(bwlpf) F F .
bwlpf F_lpf real(ifft2(F)) subplot(2,2,4)
imshow(F_lpf)
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Results on the moth image
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The high pass filter
C 50 n 1
C 50 n 5