Digtial Image Processing, Spring 2006

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ECES 682 Digital Image Processing

Oleh TretiakECE Department

Drexel University

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About the Course

• Homework 2 due today• Midterm exam next week

Covers first three homeworks 90 minutes (second half of class)

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Last Week’s Lecture

• Image Enhancement in the Spatial Domain Gray level transformations Histogram processing Arithmetic/Logic operations Spatial filtering

Smoothing Sharpening

• Matlab image processing Image datatypes Image display

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This Week’s Lecture

• Chapter 4, Image enhancement in the frequency domain Fourier transform and the frequency domain Filtering with Fourier methods Spatial vs. Fourier filtering Smoothing filters Sharpening filters Laplacian Unsharp masking, homomorphic filtering Funny stuff with the FFT Convolution and correlation

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Mr. Joseph Fourier

• To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.f (x) =A0 + Ak cos(2πkx/ L)+ Bk sin(2πkx/ L)

k=1

∞

∑k=1

∞

∑

0 ≤x≤L

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Fourier Methods

Continuous time, real function, finite interval

Sine/cosine Fourier series

Continuous time, complex function, finite interval

Fourier series, complex exponentials

Discrete time, complex function, infinite interval

Fourier transform, finite interval in frequency

Discrete time, complex function, finite interval

Discrete Fourier transform (DFT)

Two dimensional complex function, infinite intervals

2-D Fourier transform

Two dimensional complex function, polar coordinates

Fourier-Bessel transform, angular harmonics

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FT and FFT

• We normally deal with low-pass functions centered at the origin f(x) <—> F(u) Space range -X/2 < x < X/2 Frequency range -W< u <W

• Natural coordinates for DFT are fn

Space range 0 ≤ n < N Frequency range 0 ≤ k < N

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DFT Example

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2D FT Example

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Another Example

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Examples of 2DFT

a

b

c

a

bc

Image Fouriertransform

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Two-Dimensional Systems

• We would like to have a system model for vision.

hx(u,v) y(u,v)

• Input: Image• Output: Our mind’s perception

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‘Typical’ Visual Spatial Response

low contrast

high contrast

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Objective value

(intensity)

Subjective (perceived)

value

Mach Bands

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The circles have the same objective intensity.

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How to Filter

1. Multiply image by (-1)x+y

Image dimensions MxN

2. Compute F(u, v) DFTDC at M/2, N/2.F(u, v) complex valued

• Multiply F(u, v) by H(u, v)DC for H(u, v) at M/2, N/2.

• Compute inverse DFT of result in (3)• Take real part of result in (4)• Multiply result in (5) by (-1)x+y

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Notch Filter

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Fourier Low- and High-Pass Filters

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High-Boost Filter

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Space and Frequency Filters

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Radial Low-Pass Filter

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Power Distribution

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Power Removal

(a) Original image, (b) 8% power removal, (c) 5.4% power removal, (d) 4.3%, (e) 2%, (f) 0.5%. Radii are 5, 15, 30, 80, and 230. Max frequency is 250

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Ideal vs. Butterworth

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Ideal vs. Gaussian

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‘Morphological’ Filtering

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Sharpening Filters

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Sharpening: Ideal vs. Butterworth

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Sharpening: Ideal vs. Gaussian

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Laplacian in the Frequency Domain

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Homomorphic Filtering

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Correlation and Finding Things

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More About the Fourier Transform

• Shift• Linearity• Scaling• Rotation• Seperability• Forward and inverse• Padding and wraparound

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Wraparound: Example

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Summary

• Fourier methods in image processing Filtering Other

• Filtering Space domain N2 image, M2 filter

Cost = cN2M2

Fourier domain Cost = kN2logN

• Other Spectral estimation

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References on the FT

• Ron Bracewell, The Fourier Transform and its Applications, McGraw-Hill, 2000

• About Josef Fourier www-groups.dcs.st-and.ac.uk

(University of Saint Andrews MacTutor history of mathematics web site). The image on the right is from that site.