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DREAM PLAN IDEA IMPLEMENTATION 1
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3 Introduction to Image Processing Dr. Kourosh Kiani Email: kkiani2004@yahoo.comkkiani2004@yahoo.com Email: Kourosh.kiani@aut.ac.irKourosh.kiani@aut.ac.ir Email: Kourosh.kiani@semnan.ac.irKourosh.kiani@semnan.ac.ir Web: www.kouroshkiani.comwww.kouroshkiani.com Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University
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Lecture 10 4 2D Discrete Fourier Transform (DFT)
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The two-dimensional Fourier transform and its inverse Fourier transform (discrete case) DFt Inverse Fourier transform: u, v : the transform or frequency variables x, y : the spatial or image variables
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DFT -6.80 + 7.89i 17.8 - 12.76i17.8 -7.89i -6.8 - 7.89i103.00 19.28 + 9.2i 2.05+ 20.34i5.07 - 13.76i -5.09+- 13.76i9.28-10.82i 8.43 + 1.31i9.22 + 14.89i 6.09 + 3.25i-3.55 +13.88i-0.78+ 8i -3.55 – 13.88i6.09 - 3.25i9.22 - 14.89i8.43 - 1.31i-0.78+ 8i -5.09 + 13.76i5.07 + 2.13i2.05 - 1.83i19.28 - 9.2i 9.28-10.82i
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103.0010.4121.90 10.41 14.2614.675.5020.4421.36 8.0414.336.9017.518.53 8.048.5317.516.9014.33 14.2621.3620.445.5014.67 |F|=abs(F)= 2.011.021.34 1.02 1.151.170.741.311.33 0.911.160.841.240.93 0.910.931.240.841.16 1.151.331.310.741.17 log 10 |F|=log 10 (abs(F)) =
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0.00- 2.28-0.620.622.28 -0.861.93-0.401.470.45 -1.671.820.491.020.15 1.67-0.15-1.02 -0.49-1.82 0.86-0.45-1.470.40-1.93 angleF = 0.00- 130.76- 5.6435.64130.76 -49.39110.30- 2.7484.2625.51 - 95.57104.3328.0858.238.86 95.57- 8.8658.23--28.08- 04.33 49.39 -25.51 - 84.2622.74110.30- radToDegF=
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Inverse Fourier transform -6.80 + 7.89i 17.8 - 12.76i17.8 -7.89i -6.8 - 7.89i103.00 19.28 + 9.2i 2.05+ 20.34i5.07 - 13.76i -5.09+- 13.76i9.28-10.82i 8.43 + 1.31i9.22 + 14.89i 6.09 + 3.25i-3.55 +13.88i-0.78+ 8i -3.55 – 13.88i6.09 - 3.25i9.22 - 14.89i8.43 - 1.31i-0.78+ 8i -5.09 + 13.76i5.07 + 2.13i2.05 - 1.83i19.28 - 9.2i 9.28-10.82i
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Inverse Fourier transform
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Magnitude and Phase of DFT What is more important? magnitude phase
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Magnitude and Phase of DFT Reconstructed image using magnitude only Reconstructed image using phase only
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original amplitude phase Magnitude and Phase of DFT
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Example: DFT of 2D rectangle function Fourier spectrum Input function Spectrum displayed as an intensity function
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Extending DFT to 2D 2D cos/sin functions
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2D - DFT Base-functions are waves u v
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Why is DFT Useful? Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.
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Properties in the frequency domain Fourier transform works globally – No direct relationship between a specific components in an image and frequencies Intuition about frequency – Frequency content – Rate of change of gray levels in an image
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Example: Removing undesirable frequencies remove high frequencies reconstructed signal frequencies noisy signal To remove certain frequencies, set their corresponding F(u) coefficients to zero!
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How do frequencies show up in an Signal? Low frequencies correspond to slowly varying information High frequencies correspond to quickly varying information
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How do frequencies show up in an image? Low frequencies correspond to slowly varying information (e.g., continuous surface). High frequencies correspond to quickly varying information (e.g., edges)
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How do frequencies show up in an image?
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The 2D DFT and its inverse Centered spectrum for display
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2-D Fourier transform Frequency axis x y u v u v Fshift 0
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Low and high frequencies Low High Low High Low High Frequencies of the 2D DFT
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Periodicity of 2-D DFT For an image of size NxM pixels, its 2-D DFT repeats itself every N points in x- direction and every M points in y-direction. We display only in this range 0N2N-N 0 M 2M -M 2-D DFT: f(x,y)f(x,y)
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Conventional Display for 2-D DFT High frequency area Low frequency area F(u,v) has low frequency areas at corners of the image while high frequency areas are at the center of the image which is inconvenient to interpret.
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2-D FFT Shift : Better Display of 2-D DFT 2D FFTSHIFT 2-D FFT Shift is a MATLAB function: Shift the zero frequency of F(u,v) to the center of an image. High frequency areaLow frequency area
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Original display of 2D DFT 0N2N-N 0 M 2M -M Display of 2D DFT After FFT Shift 2-D FFT Shift : How it works
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Example of 2-D DFT Original image 2D DFT 2D FFT Shift
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Example of 2-D DFT Original image 2D DFT 2D FFT Shift
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Spectrum shift Original image Log enhanced transform Shifted log enhanced transform
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Computing DFT – Use the two-dimensional FFT command E=fft2(A) – Puts center of the beam in the corners – Use the fftshift command to put it into the center A=imread(‘slit’, ‘gif’) fft2( A ) fftshift( fft2( A ) )
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DFT Properties: Rotation Rotating f(x,y) by θ rotates F(u,v) by θ
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DFT DFT Properties: Rotation
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The Property of Two-Dimensional DFT Linear Combination The Property of Two-Dimensional DFT Linear Combination DFT A B 0.25 * A + 0.75 * B
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Two-Dimensional DFT with Different Functions Sine wave Rectangle Its DFT
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Two-Dimensional DFT with Different Functions 2D Gaussian function Impulses Its DFT
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Two-Dimensional DFT with Different Functions 2D Gaussian function Impulses Its DFT
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Relation Between Spatial and Frequency Resolutions where x = spatial resolution in x direction y = spatial resolution in y direction u = frequency resolution in x direction v = frequency resolution in y direction N,M = image width and height x and y are pixel width and height. )
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How to Perform 2-D DFT by Using 1-D DFT f(x,y)f(x,y) 1-D DFT by row F(u,y)F(u,y) 1-D DFT by column F(u,v)F(u,v)
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How to Perform 2-D DFT by Using 1-D DFT f(x,y)f(x,y) 1-D DFT by row F(x,v)F(x,v) 1-D DFT by column F(u,v)F(u,v) Alternative method
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Filtering in the Frequency Domain with FFT shift f(x,y)f(x,y) 2D FFT FFT shift F(u,v)F(u,v) 2D IFFTX H(u,v) (User defined) G(u,v)G(u,v) g(x,y)g(x,y) In this case, F(u,v) and H(u,v) must have the same size and have the zero frequency at the center.
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Smoothing filters: Gaussian The weights are samples of the Gaussian function mask size: σ = 1.4
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Smoothing filters: Gaussian As σ increases, more samples must be obtained to represent the Gaussian function accurately. Therefore, σ controls the amount of smoothing σ = 3
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Smoothing filters: Gaussian
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Fourier Low Pass Filtering
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Gaussian Low Pass Filter
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Low pass filtering
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Fourier High Pass Filtering
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Linear filtering and convolution DFT IDFT
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High pass filtering
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Example of noise reduction using DFT
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Questions? Discussion? Suggestions ?
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