Complex Plane. Frequency Domain. Fourier Filtering. Fourier Transform, Frequency Domain, and Fourier Filtering1. Lecture 02. See Sections 1.2 and 2.1.3 in.
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Fourier Transform, Frequency Domain, and Fourier Filtering1 Lecture 02
See Sections 1.2 and 2.1.3 in Reinhard Klette: Concise Computer Vision Springer-Verlag, London, 2014 ccv.wordpress.fos.auckland.ac.nz
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Two Images and Their Fourier Spectra
Left: Original images Fibers and Straw in the spatial domain Right: Centered and log-transformed spectra in the frequency domain
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Agenda
1
Discrete Fourier Transform
2
Inverse Fourier Transform
3
Complex Plane
4
Frequency Domain
5
Fourier Filtering
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Discrete Fourier Transform and FFT Fourier transform Named after the French mathematician J. B. J. Fourier (1768–1830) Maps real-valued function into a complex-valued function Complex number z = a + ib, with i = real part a imaginary part b
√
−1, has
Discrete Fourier Transform (DFT) transforms uniformly spaced real signals from the spatial domain into complex numbers in the frequency domain DFTs are performed basically in any device which processes signals (e.g. cell phones, CDs, DVDs and so forth), using the Fast Fourier Transform (FFT) as a time-efficient algorithm 4 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
DFT A 1-dimensional (1D) DFT maps an infinite sine wave f (x) = into the shown real part; the imaginary part is zero everywhere
1 2
· sin( x2 )
Source: madebyevan.com/dft/ . What is the DFT of f (x) = 12 · cos( x2 ) ? What are input examples with a non-zero imaginary part in their DFT?
The 2-dimensional (2D) DFT maps an Ncols × Nrows scalar image I into a complex-valued Fourier transform I 5 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Fourier Filtering – An Outlook The DFT I in the frequency domain provides insights into the image I Fourier filtering: Change data in the frequency domain (by Fourier filter operations) and map them back (by an inverse 2D DFT) into a modified image Examples: Contrast enhancement, noise removal, or smoothing of images 1D Fourier filtering: commonly used in signal theory (e.g., for audio processing in mobile phones) 2D Fourier filtering of images: follows the same principles, just in 2D instead of in 1D Note: In the context of the Fourier transform we use image coordinates from 0 to Ncols − 1 for x, and from 0 to Nrows − 1 for y 6 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
2D Discrete Fourier Transform (2D DFT)
I(u, v ) =
Ncols
NX cols −1 Nrows X−1 1 xu yv I (x, y ) · exp −i2π + · Nrows Ncols Nrows x=0
y =0
for frequencies u = 0, 1, . . . , Ncols − 1 and v = 0, 1, . . . , Nrows − 1 √ i = −1 is the imaginary unit of complex numbers Eulerian formula, for real α and e = 2.71828 . . . = exp(1): exp(iα) = e iα = cos(α) + i · sin(α) If α outside of interval [0, 2π) then taken modulo 2π Thus: Fourier transform is a weighted sum of sine and cosine functions in the complex plane 7 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Agenda
1
Discrete Fourier Transform
2
Inverse Fourier Transform
3
Complex Plane
4
Frequency Domain
5
Fourier Filtering
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Inverse Discrete Fourier Transform Maps Fourier transform I back into the spatial domain: I (x, y ) =
NX cols −1 Nrows X−1 u=0
v =0
I(u, v ) exp i2π
yv xu + Ncols Nrows
Powers exp(iα) here reversed compared to the forward transform Scaling factor 1/Ncols · Nrows in the 2D DFT and the scaling factor 1 in the inverse transform; we also could√have a split of 1/Ncols · Nrows into two scaling factors, say, for example, 1/ Ncols · Nrows in both transforms Representation of Image by Basis Functions Image I is represented above as a weighted sum of basis functions exp(iα) = cos(α) + i sin(α) being 2D combinations of cosine and sine functions in the complex plane; Fourier coefficients I(u, v ) are the weights 9 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Illustration of Basis Functions Sine functions for imaginary part; for real part a we have cosine functions
2D waves defined by sin(u + nv ), for n = 1, . . . , 5 10 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Agenda
1
Discrete Fourier Transform
2
Inverse Fourier Transform
3
Complex Plane
4
Frequency Domain
5
Fourier Filtering
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Calculus of Complex Numbers z1 = a1 + i · b1 and z2 = a2 + i · b2 z1 + z2 = (a1 + a2 ) + i · (b1 + b2 ) z1 · z2 = (a1 a2 − b1 b2 ) + i · (a1 b2 + a2 b1 ) Norm of z = a + i · b:
||z||2 =
√
a2 + b 2
Example: ||W ||2 = 1 for root of unity W = exp( i2π N ), N ≥ 1 Why? – See the Eulerian formula z? = a − i · b
is the conjugate of
If z 6= 0 then
z =a+i ·b
? z −1 = ||z||−2 2 ·z 12 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Unit Circle (r = 1) in the Complex Plane Imaginary part
W
8
6 W 7 +i W
W5 W4
W9
W3
W 10
W2
W 11
a+ib
b
W1
r α
W 12 -1
W0 +1
a
Real part
W 23
W 13 W 14
W 22
W 15
W 21 W
16
W W 17
-i W 18
20
W 19 13 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Complex Plane
Powers of W = exp( i2π N ) are all located on the unit circle (N ≥ 1) Previous page illustrates N = 24 Figure shows complex number z = a + ib having (r , α) as polar coordinates: r is the norm (the amplitude) α is the phase Complex Plane Plane of complex numbers a + i · b with points (a, b) or vectors [a, b]>
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Agenda
1
Discrete Fourier Transform
2
Inverse Fourier Transform
3
Complex Plane
4
Frequency Domain
5
Fourier Filtering
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Image Data in Matrix I in the uv Frequency Domain
Values for low frequencies u or v (i.e. close to 0) represent long wavelengths of sine or cosine components Values for large frequencies u or v (i.e. away from zero) represent short wavelengths Low frequencies (long wavelengths) represent homogeneous additive contributions to the input image I High frequencies (short wavelengths) represent local discontinuities in I such as edges or intensity outliers
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Where is ”Low” and where is ”High”? Origin in image I and Fourier transform I is in the upper left corner Imagine that values in matrix I are repeated periodically in the plane, with periods Ncols and Nrows , thus tessellating the plane
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Origin at the Center
For having the origin (i.e. the low frequencies) at the center of the Fourier transform 1
we can perform a permutation of the four quadrants of the matrix or
2
we multiply all values I (x, y ) first by (−1)x+y before performing the 2D DFT
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Two Properties of the DFT
1
2D DFT I combines two Ncols × Nrows arrays representing the real or the imaginary part; due to the symmetry property I(Ncols − u, Nrows − v ) = I(−u, −v ) = I(u, v )∗ half of the data in both arrays of I is redundant
2
Mean of I : I(0, 0) =
Ncols
NX cols −1 Nrows X−1 1 I (x, y ) · Nrows x=0
y =0
Imaginary part of I(0, 0) is always equal to zero
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Spectrum and Phase Spectrum ||I|| ||I||(u, v ) = ||I(u, v )||2
for all Ncols · Nrows frequencies (u, v ) √ Recall: amplitude ||z||2 = r = a2 + b 2
Visualisation of the spectrum ||I|| in form of a gray-level image: It would be just black with a bright dot at the origin; for better visibility, use log-transform log10 (1 + ||I(u, v )||2 ) Phase α = atan2(b, a) for z = a + ib Visualization of phase components of I is not so common; actually not corresponding to the importance of phase for representing information present in an image
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Invariance w.r.t. Histogram Equalization
Jpg-image Donkey, histogram equalization, and corresponding spectra 21 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Invariance w.r.t. Value Inversion
Jpg-image Donkey, inverted gray-levels, corresponding spectra 22 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Simple Geometric Shapes
Directional patterns in I , e.g. lines into direction β or β + π, create value distributions in I in the orthogonal direction (i.e., in direction β + π/2) 23 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Wave Pattern and a Filter
TL: Ideal wave pattern generates non-negative values only at v = 0 BL: Diagonal wave pattern is influenced by the “finite-array effect” TR: Overlaid diagonal wave pattern BR: Simple mask for blocking values in the frequency domain; inverse DFT produces a filtered image 24 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Agenda
1
Discrete Fourier Transform
2
Inverse Fourier Transform
3
Complex Plane
4
Frequency Domain
5
Fourier Filtering
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Linear Transform of Image I 1
Modify image data in the spatial domain by a linear convolution J (x, y ) = (I ∗ G ) (x, y ) =
NX cols −1 Nrows X−1 i=0
I (i, j) · G (x − i, y − j)
j=0
G is the filter kernel and J is the filtered image Example: G (0, 0) = 1 and G (x, y ) = 0 otherwise, then ... ? 2
Modify 2D DFT I of I by multiplying values in I, position by position, with corresponding complex numbers in G [i.e., I(u, v ) · G(u, v )]; the resulting complex array is transformed by the inverse 2D DFT into the modified image J
Both options lead to identical results assuming that G is the 2D DFT of G (see convolution theorem of the Fourier transform in the CCV book) 26 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Steps of Fourier Filtering In the convolution case we miss the opportunity to design frequency-dependent filter functions in the frequency domain; there is also the FFT for time-efficiency of the second way Given: Image I and complex-valued filter function G; for applying a standard FFT program, map image I at first into a larger 2n × 2n array 1
2
3
Transform image I into the frequency domain, into the complex-valued I by using the FFT program Multiply complex-valued I, element by element, with the complex-valued filter function G Transform the result back into the spatial domain by using the FFT program for the inverse DFT
Filter function G can be obtained as Fourier transform of a filter kernel G in the spatial domain, but ii is common practice to design G in the frequency domain 27 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Examples of Filter Functions in the Frequency Domain a
a
1
1
u a
u a
1
1
u
u
1D profiles of rotation-symmetric filter functions Top: A linear high-pass filter and an ideal low-pass filter Bottom: An exponential high-emphasis filter and a linear band-pass filter Attributes “linear”, “exponential”, or “ideal” of a filter function specify the way how the transition is defined from large amplitudes of the filter to low amplitudes 28 / 32
Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Design of Filter Functions: Examples
High-pass filter: e.g. for edge detection, or for visualizing details and for suppressing low frequencies Low-pass filter: e.g. for a reduction of outliers and of contrast, i.e. a smoothing effect High-emphasis filter: e.g. for enhancing contrast Band-pass filter: for allowing only a certain range of frequencies “to pass” A filter which eliminates or enhances selected frequencies (under proper consideration of the symmetry constraint)
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Example for Fourier Filtering
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Example for Fourier Filtering: Continued
Upper row, left: Intensity channel of image Emma Upper row, right: Its spectrum, centered and with log-transform Lower row, left: An ideal low-pass filtered Emma, showing a typical smoothing effect Lower row, right: An exponential-band-pass filtered Emma, already showing more higher frequencies than lower frequencies
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Fourier Transform
Inverse Fourier Transform
Complex Plane
Frequency Domain
Fourier Filtering
Copyright Information
This slide show was prepared by Reinhard Klette with kind permission from Springer Science+Business Media B.V. The slide show can be used freely for presentations. However, all the material is copyrighted. R. Klette. Concise Computer Vision. c
Springer-Verlag, London, 2014. In case of citation: just cite the book, that’s fine.
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