10 Image Processing II

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ENGG 167

MEDICAL IMAGING

Lecture 10: Oct. 13

Image Processing II

Frequency Domain & Transform Processing

References: Chapter 10, The Essential Physics of Medical Imaging, Bushberg

Radiation Detection and Measurement, Knoll, 2nd ed.

Intermediate Physics for Medicine and Biology, Hobbie, 3rd ed.

Principles of Computerized Tomographic Imaging, Kak and Slaney.

(http://rvl4.ecn.purdue.edu/~malcolm/pct/pct-toc.html)

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Preparation -

Review Imaging Processing Toolbox Help Manual(on your computer)

Download NIHImage (Mac) / ImageJ (Windows)Create a Macro which will analyze images and save a

processed version of the images

(http://rsbweb.nih.gov/ij/developer/macro/macros.html)

Complete Image Processing Assignment


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The Fourier Theorum

Ref: Gonzalez et al, Text

All signals can be decomposed into pure sinusoidal signals

+

+

+

=

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The Fourier transform


Spatial signalCorresponding

Frequency-domain signal

All signals can be decomposed into pure sinusoidal signals

This theorum is especially appropriate for periodic signals, but

can be used for discrete signals if enough frequencies are used

to capture the relevant information.


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1) Basics 1-dimensional sinusoidal representation of signals

Ref: Rizzoni

Where magnitude and phase of the coefficients are given by:

or:

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1) Basics complex numbers

Sinusoids are related to complex exponential expressions.

A complex number is one which is of the form:

z = a + ib, where I is the square root of -1, an imaginary number

Recall that the magnitude and phase of z can be calculated by:

magnitude => I = [a2 + b2]

phase => = tan-1(b/a)

So that another way to express z is :

z = I ei

Now, we can make use of a definition called Eulers formula:

ei = cos() + i sin()And

e-i = cos() - i sin()

Or written for the sinusoid signals:

cos() = [ei + e-i]/2 and sin() = [ei - e-i]/(2i)


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1) Basics complex numbers

So that equivalent expressions for a time domain signal are :

x(t) = an cos(nt) + bn sin (nt)

where an and bn are the magnitudes of the signal at each frequency n, and the summationis carried out over all values of n.

x(t) = In exp(int+n)

where In and n are the amplitude and phase at each frequency n.

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1) Basics fourier transform of a signal

X() = x(tn) exp(-itn)

where the signal X() is now in the frequency domain (recall=2f = 2/T), whereas theoriginal signal x(t) was a time resolved signal with N total data points. Summation is from

n=1 to n=N.

Alternatively a spatial data set can be transformed to spatial frequency data set by the same

approach:

F(k) = f(xn) exp(-i 2kxn)

where the signal F(k) is now in spatial frequency, k, and describes the exact discritized

shape of the original signal f(xn).

1

N

1

N


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1) Basics 1-D Fourier Transforms f(x) F(kx)

Ref: Rizzoni

Input analytic function Fourier Transformed function

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Fourier Transform Summary



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2D Fourier Transforms

Where is the information in (u,v) space?

Where are the low frequencies?

Where are the high frequencies?Ref: Gonzalez et al, Text


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The Fourier space image (k-space)

Lowest frequency


Highest positive

kx frequency

Highest negative

kx frequency

Highest positive

ky frequency

Highest negative

ky frequency

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Fourier space filtering

Spatial frequency changes in the Fourier domain are simply done

with linear mathematics!



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Spatial frequency changes in the Fourier domain are simply done

with linear mathematics!

Edge enhancement example (what is the shape of this filter)


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Try the online DEMO

http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertransform/

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Special filters Butterworth & Gaussian


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Noise



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Noise


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Noise: simple linear filtering can help



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Noise: frequency domain filtering


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Noise: frequency domain filtering



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Online resources for more info about image processing

http://micro.magnet.fsu.edu/primer/digitalimaging/imageprocessingintro.html

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Geometric Linear Transformations


Affine transform


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Geometric Linear Transformations


Affine transform

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Image compression : changing the binary

code used to reduce # of bits required


Use fewer bits to encode

Information that occurs a lot and

then use more bits to encode

information that occurs little in

the image

p(r) is the probability of

occurrence

I2 is a more efficient coding of

the bits than I1


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Lossy Image compression :transforms to fewer bits across the entire image


8 bits 4 bits

2 bits

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Minimum Lossy Image compression :

bit reduction at the local level, not globally!



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Minimum Lossy Image compression :bit reduction at the local level, not globally!


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Minimum Lossy Image compression :

bit reduction at the local level, 8x8 blocks -JPEG


predicted

error

compressed

image

close up

view

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10 Image Processing II