Lecture 2: 2D Fourier transforms and applications
Lecture 2: 2D Fourier transforms and applications Fourier transforms and spatial frequencies in 2D ... Filtering vs convolution in 2D in Matlab.
CS425 Lab: Frequency Domain Processing
Fourier transform. MATLAB has three functions to compute the DFT: 1. fft -for one dimension (useful for audio). 2. fft2 -for two dimensions (useful for
Discrete Two Dimensional Fourier Transform in Polar Coordinates
16 juil. 2019 Sample Matlab code is included in the appendix of the paper. 2 Definition of the Discrete 2D Fourier Transform in Polar Coordinates.
Discrete two dimensional Fourier transform in polar coordinates part II
2 mars 2020 Specifically we demonstrate how the decomposition of the 2D DFT as a DFT
Signalprocessing Background
11 avr. 2016 Extension of the Fourier Transform to 2D Functions ... Beware that the scaling term in MATLAB is with the inverse rather than the transform ...
Image Processing
28 juin 2010 General 2D Spatial Transformations. • A three-step process in MATLAB. 1. Define the transformation parameters. 2. Create a transformation ...
Exercise 1: Fouriertransform
You are asked to write a MATLAB program to perform the following tasks: 1. Load the sample image HeadCT.jpg calculate its 2D DFT and plot the. Fourier spectrum
2D Discrete Fourier Transform (DFT)
2D DFT can be regarded as a sampled version of 2D DTFT. a-periodic signal periodic transform periodized signal periodic and sampled transform
The Image Processing Handbook Fourth Edition
The Fourier transform and other frequency space transforms are applied to two-dimensional im- ages for many different reasons. Some of these have little to
Two-Dimensional Fourier Transform and Linear Filtering
Continuous Space Fourier Transform (CSFT) Each color component of an image is a 2D real signal with finite support ... Using MATLAB freqz2: f=[12
2-D fast Fourier transform - MATLAB fft2 - MathWorks
This MATLAB function returns the two-dimensional Fourier transform of a matrix using a fast Fourier transform algorithm which is equivalent to computing
2-D Fourier Transforms - MATLAB & Simulink - MathWorks
The fft2 function transforms 2-D data into frequency space For example you can transform a 2-D optical mask to reveal its diffraction pattern Two-Dimensional
[PDF] Lecture 2: 2D Fourier transforms and applications
Lecture 2: 2D Fourier transforms and applications B14 Image Analysis Michaelmas 2014 A Zisserman • Fourier transforms and spatial frequencies in 2D
Two dimensional Fourier transform using MATLAB - IEEE Xplore
This paper describes the use of two dimensional Fourier transform in the MATLAB environment to compute the length of a wave and the angle between the line that
Two-Dimensional Fourier Analysis in MAtlAB
Lecture 10 Notes Two-Dimensional Fourier Analysis in MAtlAB The MAtlAB functions fft2 and ifft2 are used to compute the discrete Fourier transform (DFT)
Two dimensional Fourier transform using MATLAB Request PDF
Request PDF Two dimensional Fourier transform using MATLAB Two dimensional Fourier transform is a powerful tool for image processing especially in the
[PDF] 2-D Fourier Transforms - Electrical and Computer Engineering
2D FT • Fourier Transform for Discrete Time Sequence In MATLAB frequency scaling is such that 1 represents maximum freq uv=1/2
[PDF] 1 Preliminaries 2 Exercise 1 – 2-D Fourier Transforms - UCSB ECE
Image Processing in MATLAB – Fourier Analysis and Filtering of Images the image representation of the 2-D DFT magnitude of the image being studied in
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In light of its importance this article presents a tutorial for 2-D DFT utilizing MATLAB® for both 2-D signals and images The analysis of the discrete signals
[PDF] 2D • Fourier Properties • Convolution Theorem • FFT • Examples
Discrete Fourier Transform - 1D • Discrete Fourier Transform - 2D • Fourier Properties • Convolution Theorem • FFT • Examples Matlab: F=fft(f);
How to do a 2D Fourier transform in MATLAB?
Y = fft2( X ) returns the two-dimensional Fourier transform of a matrix using a fast Fourier transform algorithm, which is equivalent to computing fft(fft(X). '). ' . If X is a multidimensional array, then fft2 takes the 2-D transform of each dimension higher than 2.What is 2 dimensional Fourier transform?
The Fourier Transform ( in this case, the 2D Fourier Transform ) is the series expansion of an image function ( over the 2D space domain ) in terms of "cosine" image (orthonormal) basis functions.How to use MATLAB for DFT?
To plot the magnitude and phase in degrees, type the following commands: f = (0:length(y)-1)*100/length(y); % Frequency vector subplot(2,1,1) plot(f,m) title('Magnitude') ax = gca; ax. XTick = [15 40 60 85]; subplot(2,1,2) plot(f,p*180/pi) title('Phase') ax = gca; ax. XTick = [15 40 60 85];- As known, FFT(x) performs 1D-FFT transformation, column wise. However, FFT2(x), performs 2D-FFT transformation.
Lecture 2: 2D Fourier transforms and applications
B14 Image Analysis Michaelmas 2014 A. Zisserman • Fourier transforms and spatial frequencies in 2D • Definition and meaning • The Convolution Theorem • Applications to spatial filtering • The Sampling Theorem and Aliasing Much of this material is a straightforward generalization of the 1D Fourier analysis with which you are familiar.Reminder: 1D Fourier Series
Spatial frequency analysis of a step edge
Fourier decomposition
xFourier series reminder
f(x)=sinx+13sin3x+...
Fourier series for a square wave
f(x)= X n=1,3,5,...1nsinnx
Fourier series: just a change of basis
M f(x)= F(
Inverse FT: Just a change of basis
M -1 F( )= f(x)1D Fourier TransformReminder transform pair - definitionExample
x u2D Fourier transforms
2D Fourier transform Definition
Sinusoidal Waves
To get some sense of what
basis elements look like, we plot a basis element --- or rather, its real part --- as a function of x,y for some fixed u, v. We get a function that is constant when (ux+vy) is constant. The magnitude of the vector (u, v) gives a frequency, and its direction gives an orientation. The function is a sinusoid with this frequency along the direction, and constant perpendicular to the direction. uv slide: B. FreemanHere u and v are larger than
in the previous slide. uvAnd larger still...
uvSome important Fourier Transform Pairs
FT pair example 1
rectangle centred at origin with sides of length Xand Y |F(u,v)| separabilityf(x,y) |F(u,v)| v u FT pair example 2Gaussian centred on origin•FT of a Gaussian is a Gaussian • Note inverse scale relation f(x,y)F(u,v)
FT pair example 3Circular disk unit height and
radius a centred on origin • rotational symmetry • a '2D' version of a sincf(x,y)F(u,v)
FT pairs example 4
f(x,y)F(u,v) =+++ ...f(x,y)Summary
Example: action of filters on a real image
f(x,y) |F(u,v)| low pass high passoriginal Example 2D Fourier transformImage with periodic structure f(x,y) |F(u,v)| FT has peaks at spatial frequencies of repeated textureExample - Forensic application
Periodic background removed
|F(u,v)| remove peaksExample - Image processing
Lunar orbital image (1966)
|F(u,v)| remove peaks join lines removedMagnitude vs Phase
f(x,y)|F(u,v)| • |f(u,v)| generally decreases with higher spatial frequencies • phase appears less informativephase F(u,v) cross-sectionThe importance of phase
magnitude phase phaseA second example
magnitude phase phase TransformationsAs in the 1D case FTs have the following properties • Linearity • Similarity •Shift f(x,y) |F(u,v)| ExampleHow does F(u,v) transform if f(x,y) is rotated by 45 degrees?In 2D can also rotate, shear etcUnder an affine transformation:
The convolution theorem
Filtering vs convolution in 1D
100 | 200 | 100 | 200 | 90 | 80 | 80 | 100 | 100
f(x)1/4 | 1/2 | 1/4
h(x) g(x) | 150 | | | | | | | molecule/template/kernel filtering f(x) with h(x) g(x)= Z f(u)h(xu)du Z f(x+u 0 )h(u 0 )du 0 X i f(x+i)h(i) convolution of f(x) and h(x) after change of variable note negative sign (which is a reflection in x) in convolution •h(x)is often symmetric (even/odd), and then (e.g. for even)Filtering vs convolution in 2D
image f(x,y) filter / kernel h(x,y) g(x,y) = convolution filtering for convolution, reflect filter in x and y axesConvolution
• Convolution: - Flip the filter in both dimensions (bottom to top, right to left) h f slide: K. Grauman h filtering with hconvolution with h Filtering vs convolution in 2D in Matlab2D filtering • g=filter2(h,f);2D convolution
g=conv2(h,f); lnkmflkhnmg lk f=image h=filter lnkmflkhnmg lk In words: the Fourier transform of the convolution of two functions is the product of their individual Fourier transformsSpace convolution = frequency multiplication
Proof: exercise
Convolution theorem
Why is this so important?
Because linear filtering operations can be carried out by simple multiplications in the Fourier domainThe importance of the convolution theorem
Example smooth an image with a Gaussian spatial filterGaussian
scale=20 pixels It establishes the link between operations in the frequency domain and the action of linear spatial filters1. Compute FT of image and FT of Gaussian
2. Multiply FT's
3. Compute inverse FT of the result.
f(x,y) xFourier transform
Gaussian
scale=3 pixels |F(u,v)| g(x,y) |G(u,v)|Inverse Fourier
transform f(x,y) xFourier transform
Gaussian scale=3 pixels
|F(u,v)| g(x,y) |G(u,v)|Inverse Fourier
transform There are two equivalent ways of carrying out linear spatial filtering operations:1. Spatial domain: convolution with a spatial operator
2. Frequency domain: multiply FT of signal and filter, and compute
inverse FT of productWhy choose one over the other ?
• The filter may be simpler to specify or compute in one of the domains • Computational costExerciseWhat is the FT of ...
2 small disks
The sampling theorem
Discrete Images - Sampling
x X f(x) xxFourier transform pairs
Sampling Theorem in 1D
spatial domain frequency domain replicated copies of F(u) F(u)x uApply a box filter
The original continuous function f(x) is completely recovered from the samples provided the sampling frequency (1/X) exceeds twice the greatest frequency of the band-limited signal. (Nyquist sampling limit) u 1/X F(u) f(x) xThe Sampling Theorem and Aliasing
if sampling frequency is reduced ... spatial domain frequency domainFrequencies above the Nyquist limit are
'folded back' corrupting the signal in the acceptable range.The information in these frequencies is
not correctly reconstructed. x uSampling Theorem in 2D
frequency domain 1/YF(u,v)
1/X frequencies beyond u b and v b ,i.e. if the Fourier transform is completely reconstructed from its samples as long as the sampling distances w and h along the x and y directions are such that and b uw21 b vh21The sampling theorem in 2D
Aliasing
Insufficient samples to distinguish the high and low frequency aliasing: signals "travelling in disguise" as other frequenciesAliasing : 1D example
If the signal has frequencies above the Nyquist limit ...Aliasing in video
Slide by Steve Seitz
Aliasing in 2D - under sampling example
originalreconstruction signal has frequencies above Nyquist limitAliasing in images
What's happening?
Input signal:
x = 0:.05:5; imagesc(sin((2.^x).*x))Plot as image:
Aliasing
Not enough samples
Anti-Aliasing • Increase sampling frequency
• e.g. in graphics rendering cast 4 rays per pixel • Reduce maximum frequency to below Nyquist limit • e.g. low pass filter before samplingExample
convolve withGaussian
down sample by factor of 4 down sample by factor of 44 x zoomHybridImages
FrequencyDomainandPerception
Campbell-Robson contrast sensitivity curve
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