Digtial Image Processing, Spring ECES 682 Digital Image Processing Oleh Tretiak ECE Department Drexel University

Digtial Image Processing, Spring About the Course Homework 2 due today Midterm exam next week  Covers first three homeworks  90 minutes (second half of class)

Digtial Image Processing, Spring 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

Digtial Image Processing, Spring 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

Digtial Image Processing, Spring Mr. Joseph Fourier To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula

Digtial Image Processing, Spring 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

Digtial Image Processing, Spring 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 f n  Space range 0 ≤ n < N  Frequency range 0 ≤ k < N

Digtial Image Processing, Spring DFT Example

Digtial Image Processing, Spring D FT Example

Digtial Image Processing, Spring Another Example

Digtial Image Processing, Spring Examples of 2DFT a b c a b c Image Fourier transform

Digtial Image Processing, Spring Two-Dimensional Systems We would like to have a system model for vision. h x(u,v) y(u,v) Input: Image Output: Our mind’s perception

Digtial Image Processing, Spring ‘Typical’ Visual Spatial Response

low contrast high contrast

Digtial Image Processing, Spring Objective value (intensity) Subjective (perceived) value Mach Bands

Digtial Image Processing, Spring The circles have the same objective intensity.

Digtial Image Processing, Spring

Digtial Image Processing, Spring How to Filter 1.Multiply image by (-1) x+y Image dimensions MxN 2.Compute F(u, v) DFT DC at M/2, N/2. F(u, v) complex valued 3.Multiply F(u, v) by H(u, v) DC for H(u, v) at M/2, N/2. 4.Compute inverse DFT of result in (3) 5.Take real part of result in (4) 6.Multiply result in (5) by (-1) x+y

Digtial Image Processing, Spring Notch Filter

Digtial Image Processing, Spring Fourier Low- and High-Pass Filters

Digtial Image Processing, Spring High-Boost Filter

Digtial Image Processing, Spring Space and Frequency Filters

Digtial Image Processing, Spring Radial Low-Pass Filter

Digtial Image Processing, Spring Power Distribution

Digtial Image Processing, Spring 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

Digtial Image Processing, Spring Ideal vs. Butterworth

Digtial Image Processing, Spring Ideal vs. Gaussian

Digtial Image Processing, Spring ‘Morphological’ Filtering

Digtial Image Processing, Spring Sharpening Filters

Digtial Image Processing, Spring Sharpening: Ideal vs. Butterworth

Digtial Image Processing, Spring Sharpening: Ideal vs. Gaussian

Digtial Image Processing, Spring Laplacian in the Frequency Domain

Digtial Image Processing, Spring Homomorphic Filtering

Digtial Image Processing, Spring Correlation and Finding Things

Digtial Image Processing, Spring More About the Fourier Transform Shift Linearity Scaling Rotation Seperability Forward and inverse Padding and wraparound

Digtial Image Processing, Spring Wraparound: Example

Digtial Image Processing, Spring Summary Fourier methods in image processing  Filtering  Other Filtering  Space domain N 2 image, M 2 filter  Cost = cN 2 M 2  Fourier domain  Cost = kN 2 logN Other  Spectral estimation

Digtial Image Processing, Spring 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.