Digital image processing ece 09 452 ece 09 552 fall 2009
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Digital Image Processing ECE.09.452/ECE.09.552 Fall 2009. Lecture 4 October 5, 2009. Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/fall09/dip/. Plan. Image Spectrum 2-D Fourier Transform (DFT & FFT) Spectral Filtering

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Digital Image Processing ECE.09.452/ECE.09.552 Fall 2009

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Digital image processing ece 09 452 ece 09 552 fall 2009

Digital Image ProcessingECE.09.452/ECE.09.552Fall 2009

Lecture 4October 5, 2009

Shreekanth Mandayam

ECE Department

Rowan University

http://engineering.rowan.edu/~shreek/fall09/dip/


Digital image processing ece 09 452 ece 09 552 fall 2009

Plan

  • Image Spectrum

    • 2-D Fourier Transform (DFT & FFT)

    • Spectral Filtering

  • Lab 2: Spatial and Spectral Filtering


  • Dip details

    DIP: Details


    Noise models

    S

    f(x,y)

    g(x,y)

    n(x,y)

    Degradation Model: g = f + n

    Noise Models

    • SNRg = 10log10(Pf/Pn)

    • Power Variance (how?)

    • SNRg = 10log10(sf2/ sn2)


    Noise models1

    Noise Models

    • N(0,1): zero-mean, unit-variance, Gaussian RV

    • Theorem:

      • N(0,s2) = sN(0,1)

      • Use this for generating normally distributed r.v.’s of any variance

    >>imnoise

    >>nrfiltdemo

    >>filter2

    • demos/demo2spatial_filtering/lowpassdemo.m


    Image preprocessing

    Image Preprocessing

    Restoration

    Enhancement

    • Inverse filtering

    • Wiener filtering

    Spectral

    Domain

    Spatial

    Domain

    • Filtering

    • >>fft2/ifft2

    • >>fftshift

    • Point Processing

    • >>imadjust

    • >>histeq

    • Spatial filtering

    • >>filter2


    Recall 1 d cft

    Continuous Fourier Transform (CFT)

    Frequency, [Hz]

    Phase

    Spectrum

    Amplitude

    Spectrum

    Inverse Fourier Transform (IFT)

    Recall: 1-D CFT


    Recall 1 d dft

    Equal time intervals

    Recall: 1-D DFT

    • Discrete Domains

      • Discrete Time: k = 0, 1, 2, 3, …………, N-1

      • Discrete Frequency:n = 0, 1, 2, 3, …………, N-1

    • Discrete Fourier Transform

    • Inverse DFT

    Equal frequency intervals

    n = 0, 1, 2,….., N-1

    k = 0, 1, 2,….., N-1


    How to get the frequency axis in the dft

    n=0 1 2 3 4 n=N

    f=0 f = fs

    How to get the frequency axis in the DFT

    • The DFT operation just converts one set of number, x[k] into another set of numbers X[n] - there is no explicit definition of time or frequency

    • How can we relate the DFT to the CFT and obtain spectral amplitudes for discrete frequencies?

    (N-point FFT)

    Need to

    know fs


    Dft properties

    n=0 N/2 n=N

    f=0 fs/2 f = fs

    DFT Properties

    • DFT is periodic

      X[n] = X[n+N] = X[n+2N] = ………

    • I-DFT is also periodic!

      x[k] = x[k+N] = x[k+2N] = ……….

    • Where are the “low” and “high” frequencies on the DFT spectrum?


    1 d fft demo

    1-D FFT Demo

    http://engineering.rowan.edu/~shreek/spring09/ecomms/demos/dft.m

    >>fft


    2 d continuous fourier transform

    u

    x

    v

    y

    2-D Continuous Fourier Transform

    Spatial

    Frequency

    Domain

    Spatial

    Domain


    2 d discrete fourier transform

    u=0 u=N/2 u=N

    v=N v=N/2 v=0

    2-D Discrete Fourier Transform

    >>fft2

    >>ifft2


    2 d dft properties

    2-D DFT Properties

    • Conjugate symmetry

      demos/demo3dft_properties/con_symm_and_trans.m

    • Rotation

      demos/demo3dft_properties/rotation.m

    • Separability

      demos/demo3dft_properties/separability.m

    >>fftshift


    Spectral filtering radially symmetric filter

    Spectral Filtering: Radially Symmetric Filter

    u=-N/2 u=0 u=N/2

    • Low-pass Filter

      demos/demo4freq_filtering/lowpass.m

    D(u,v)

    D0

    v=N/2 v=0 v=-N/2


    Lab 2 spatial spectral filtering

    Lab 2: Spatial & Spectral Filtering

    http://engineering.rowan.edu/~shreek/fall09/dip/lab2.html


    Summary

    Summary


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