This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Medical Image Processing 1 (MIP1)
Image Signal Analysis and Processing (ISAP)
SS’09
Fourier Transform
Dr. Pierre Elbischger
School of Medical Information Technology
Carinthia University of Applied Sciences 2D harmonics  Positive vs. negative frequencies •
• 2πu 2πu f(x)
• E9 For simplicity, we consider the spatial
frequency u in one direction x only.
The negative frequency (u) is a
mathematical trick to create a real signal by
the linear combination of two complex
exponentials. Due to the negative frequency
the exponentials rotate with the same
velocity but in converse directions, thus
resulting in a conjugate complex phasor.
A sampled version of f(x) can be obtained
by sampling x with step width ∆x,
thus x(n) = n∆x. v
r
u 1 period of the continuous signal
05.05.2009 ϕ u u sampled signal
Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 2 The 2D Fourier transform
2D Fourier transform pair Frequency domain
Spatial domain
f(x,y) is a superposition of weighted complex exponentials (basis functions). The
weights F(u,v) are complex values and determine the amplitude and the phase of the
exponentials at the spatial frequency (u,v). v r
The complex exponential is directed into
direction ϕ and has frequency r. ϕ
u λx
Period vs. Frequency λy 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform λr 3 2D basic functions of the 2D Fourier transform 1) x l=1 l=0
N l=2 l=3 k=0 y M k=1 k=2 k=3
1) Basic functions (real part) with certain frequencies
05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform Suppose the axes labels
as used in the upper left
figure, ignore the others.
4 Fourier spectrum E8 The Fourier transformed F(u,v) is a complex function with a real part R(u,v) and
an imaginary part I(u,v). If f(x,y) is real, its F(u,v) is conjugate symmetric Polar representation (magnitude spectrum and phase spectrum) Note, the angle Φ denotes the phase shift of the exponential rather than the orientation
of the exponential that is given by ϕ (see previous slide). F(u,v) represents half the
amplitude of the corresponding harmonic.
The magnitude spectrum is typically logtransformed to increased the dynamic range. 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 5 Fourier domain representation – circular chirp
A pattern that reflects the Fourier domain directly in the
spatial domain is given by a 2dimensional harmonic and
radialsymmetric chirpsignal.
The frequency linearly increases in radial direction,
independent of the direction. The signals does not represent the Fourier
transform but the associated basis functions. v u whereby c is a constant factor that controls the increase of
the effective frequency r(R) when the radial distance increases
and, R denotes the spatial radial distance to the center point
of the chirp signal.
The left frequencies’ halfplane has odd symmetry in regard
to the right halfplane and, therefore, is not illustrated in the
figure.
Note, the visualized chirp signal is the corresponding spatial
pattern, mapped into the Fourier plane – this is done for
illustration purpose only. It does not represent the Fourier
transform of the chirp signal! 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 6 The phase contains significant information images 05.05.2009 logmagnitude phase Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform Images obtained by
swaping the
magnitude spectra 7 Fourier transform properties Differentiation Scaling Convolution
Shifting 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 8 Separability considerations
The twodimensional Fourier transform is separable into two nested onedimensional
Fourier transforms. This allows for a speed up of the twodimensional FFT. If the function f(x,y)=f(x)f(y) is separable, the Fourier transform can be
decomposed into a product of two onedimensional Fourier transforms. 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 9 The sampling of a continuous signal 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 10 The sampling of the frequency response 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 11 The discrete Fourier transform
• •
• The frequency spectrum of a discrete signal f(n,m) is continuous and has a periodicity of
2π. In the digital Fourier transform the continues spectrum can be sub sampled with an
arbitrary high rate The digital Fourier transform is not required to have the same size as
the transformed, discrete image!
In order to obtain the discrete values from the frequency spectrum, integration is required
(inverse transform).
Replacing the continuous integration by two discrete sums leads to the digital Fourier
transform that can be efficiently computed by the fast Fourier transform (FFT). 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 12 The 2D digital Fourier transform (DFT)
The DFT of a discrete signal f(n,m) results in a continuous and periodic frequency spectrum!
u and v can be interpreted as normalized frequencies (with respect to a sampling frequency)
and range from 0 to 1 – corresponds to one turn in the unit circle. In order to obtain a finite number of transformation values, the frequencies u and v are
sampled equally spaced around the unit circle. Although in principal an infinitely dense
sampling can be performed, one can show that a sampling of M and N points in u and v
respectively suffice to represent the full information content of the signal. Note, the final formulas of the DFT do
not contain any spatial spacing, and
thus, the samples should be interpreted
as a sequence of consecutive values
rather than as values that are spatially
related! 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 13 The 2D digital Fourier transform (DFT)
Plausibility explanation: The digital Fourier transform can be interpreted as a discretized
version of the continues Fourier transform. The discretization is performed in the spatial
domain and in the frequency domain respectively. The sampling in the frequency domain
enforces the periodicity of the spatial signal, see the previous slides. One period of the periodic discrete
signal consists of N samples, resulting
in N fundamental frequencies uk. ycomponent
equidistant sampling The factor in front of the sums is explained by the exact derivation of the transform. [Signals & Systems, Oppenheim Willsky, Nawab, 1997]
05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 14 Centered representation of the spectrum
The spectrum of a discrete
signal is a periodic
function.
As shown before, the
spectrum of a real valued
signal is symmetric about its
origin. It is thus common to
use a centered
representation of the
spectrum FFT
05.05.2009 centered FFT
Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform Matlab: fftshift
15 Orientated spatial texture/structure in the Fourier Domain
The spatial orientation of a twodimensional
cosine or sine wave with spectral coordinates
(k,l) is ¸r
l
k f(n,m) F(k,l) Conversely a twodimensional sinusoid with
effective frequency r=1/λr and spatial
orientation ϕ is represented by the spectral
coordinates
A sinosoid with spatial orientation Ψ = 45
has spectral coefficients at the diagonal of
the spectrum. Unless the image (and thus the
spectrum) is quadratic (M=N), the
angle of orientation in the image
and in the spectrum are not the
same. To obtain identical
orientations one has to scale the
spectrum to square size.
05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 16 Effects of periodicity (1)
When interpreting the 2D digital FT of
images, one must always be aware of
the fact that with any discrete Fourier
transform, the distinct function is
implicitly assumed to be periodic in
every dimension. If there is a large
intensity difference between opposing
borders of an image, then this causes
strong transitions in the resulting
periodic signal and, thus to high
bandwidth signals along the
corresponding main axis. 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 17 Effects of periodicity (2)
One solution of the problem is to multiply the image function f(x,y) by a suitable window
function w(x,y).
The window function should drop off continuously toward the image borders such that the
transition between image replicas are effectively eliminated.
Multiplying two functions in spatial domain results in a convolution of the corresponding
spectra in the spectral domain. To apply the least possible damage to the Fourier transform of the image, the ideal spectrum
of w(x,y) would be the impulse function δ(x,y), unfortunaltey resulting in a constant
window function (box car window) again. In general, we can say that a broader spectrum of the
windowing function smoothes the resulting spectrum more strongly and individual frequency
components are harder to isolate.
Taking a picture is equivalent to cutting out a finite region from an infinite image plane, which
can be interpreted as a multiplication with a rectangular (box car) window, that is far from the
ideal impulse function.
Window functions should be as wide as possible to include a maximum part of the original
function and they should also drop off to zero toward the image borders but then again not
too steeply to maintain a narrow windowing spectrum. The ideal windowing function does not
exist, it is always a trade off and its choice depends on the particular application.
05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 18 Leakage and Windowing I
A finite observation length broadens the spectrum of a sinusoid that ideally should be a Dirac
delta function in the spectrum. The spectrum of the signal is convolved with the spectrum of
the windowing function and reduces the spectral resolution Leakage. 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 19 Leakage and Windowing II
•
•
• Windowing reduces the side lopes in the spectrum.
Selecting a window function is not a simple task. Each window function has its own
characteristics and suitability for different applications.
Windowing is used to reduce the negative effects due to a limited observation length
and/or discontinuities at the range boundaries – The DFT leads to a periodic spectrum as
well as a periodic discrete signal that may lead to discontinuities at the borders of the
signal periods. W(u) w(x) 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 20 Window functions
not ideal window functions box car cosine elliptical Bartlett Gaussian Hanning elliptical Parzen 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 21 Application of windowing functions on images 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 22 Image sampling
Spatial sampling Impulse function Frequency domain Sampling Nyquist criterion 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 23 Aliasing
Without
smoothing Gaussian
smoothing The Fourier transform of the sampled signal consists of a sum
of copies of the Fourier transform of the original signal, shifted
with respect to each other by the sampling frequency.
If they do intersect (as in this figure), the intersection region is
added, and so we cannot obtain a separate copy of the Fourier
transform, and the signal has aliased.
At the top is a 256x256 pixel image showing a grid obtained
by multiplying two sinusoids with linearly increasing frequency
— one in x and one in y. The other images in the series are
obtained by resampling by factors of two.
To avoid aliasing at any orientation, the effective frequency r
must be limited to 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 24 Examples (1)
Rectangular pulse
Strong oscillating spectrum (sincfunction)
Stretching the image contracts the spectrum Repetitive pattern
three dominant orientations
Enlarging the image contracts the spectrum Superposition of image patterns Image rotation The dark beam in (b) causes broadband effects 0°, 15° and 30° 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 25 Examples (2)
Natural images with no
dominant orientation
Print pattern Natural images with
repetitive structures orientation 05.05.2009 Dr. Pierre Elbischger  MIP0'09SS  Fourier Transform 26 ...
View
Full
Document
This note was uploaded on 07/09/2009 for the course MEDIT 1 taught by Professor Pierreelschbinger during the Spring '09 term at Carinthia University of Applied Sciences.
 Spring '09
 PIERREELSCHBINGER

Click to edit the document details