Lecture #10 - Case Western Reserve Universityengr.case.edu/.../eecs490f07/Lectures/Lecture10.pdf · 2012-02-16 · Lecture #10 • Wraparound and padding • Image Correlation •

EECS490: Digital Image Processing

Lecture #10

• Wraparound and padding

• Image Correlation

• Image Processing in the frequency domain

• A simple frequency domain filter

• Frequency domain filters

– High-pass, low-pass

– Apodization

• Zero-phase filtering

• Frequency domain filters

– ideal, Butterworth, Gaussian

– Ringing


Wraparound and Padding

f x( ) h x( ) =1

Mf m( )

m=0

h x m( )

DFT makes f & h periodic

If f and g are 400 points f★ggoes out to 800 points

Mirror h to get h(-m)

Now shift h(-m) to get h(x-m)

Wraparound comes from overlap of the periodic functions

© 2002 R. C. Gonzalez & R. E. Woods

We must pad h with 399 points ofzero to get the correct result withthis periodic function

This was nothere before




We also haveto padimages but intwo-dimensionsto get thecorrectconvolutionresults


No blurring at edges — no padding



1

16

1 2 1

2 4 2

1 2 1



Wraparound and PaddingPadding with zeros





Image Correlation

Original images

Padded images

Correlation output

f x, y( )*h x, y( ) =1

MNf * m,n( )h x + m, y + n( )

n=0

N 1

m=0

M 1



Image Processing in theFrequency Domain


a. Pad original image to PxQb. Multiply padded image by (-1)x+y

d. Fourier transform F(u,v)e. Construct real, symmetric filter H(u,v) of size PxQf. Compute F(u,v)H(u,v)g. Compute Re[F-1 {F(u,v)H(u,v)}] and multiply by (-1)x+y

h. Crop to remove padding in result


Fourier Transform


Due to the shortextrusion


Simple Frequency Domain Filter

What if we simplymultiply F(0,0) byzero?

This is the minimalhigh-pass filter



Simple Frequency DomainFiltering

Multiplying F(0,0) by zeromakes the average imagevalue zero— these arenegative image values whichcannot be displayed, but getcropped to zero!




Gaussian Filters

Frequency domain filters

Corresponding spatialdomain filters

Low-pass filters High-pass filters

This is a Difference ofGaussians, called a DoG


Gaussian Filters

Low-pass filter

High-pass filters

Blurs image Enhances detail;lowers contrast

Addition of apreserves DC term



Frequency Domain Filters


1. Original frequencydomain filter H(f)

2. Spatial domainfilter h(x)= F-1{H(f)}

3. Padded spatialdomain filter hp(x)

4. Fourier transform ofpassed spatial domain filterhp(x)

Discontinuitiescaused by padding



Phase Filters

Results of multiplying ONLY thephase of an image by a constant

MOST image processing filters operate ONLY on themagnitude and are called zero-phase-shift filters



Frequency Domain FilteringExample

a. Original imageb. Padded imagec. Multiply by (-1)x+y

d. Fourier transform F(u,v)e. Centered Gaussian LPFH(u,v)f. Compute F(u,v)H(u,v)g. Compute F-1 {F(u,v)H(u,v)}and multiply by (-1)x+y

h. Crop to remove padding

Padded imagesoften show darkedges



Fourier Transform of an Image

600x600 pixels F(u,v)



Sobel spatial/frequency domainfilters

Frequency domain Sobel

Spatial domain Sobel,3x3

Frequency domainfiltering whereh(x,y) was paddedto 602x602 andcentered prior tomultiplying byF(u,v) in thefrequency domain

Spatial domainfiltering



Ideal Low-pass Filter

Do is picked based upon the fraction of image power to be removed



Test Pattern

Magnitude transformof padded test image

u =1

M xIf M=688 and, say, there are 688 pixels/incheach pixel in the Fourier transformscorresponds to u=1/inch



Filtering with Ideal LPF

Filtering the paddedtest image with anILPF in the frequencydomain



Computing the inverse Fourier transform of theILPF shows ringing in the spatial domain whichis clearly shown in Figure 4.42

Ringing



Butterworth LPF

H u,v( ) =1

1+D u,v( )D0

2n

For n=1 there is no ringing. Theringing will increase as n increases.


Filtering with Butterworth filter,n=2



Spatial Butterworth filters



Gaussian LPF

H u,v( ) = eD2 u ,v( )

2D02



Low-pass Filters



Filtering with Gaussian LPF



Filtering for OCR

No filtering. Brokencharacters are difficult torecognize.

Filtered with a GLPF withD0=80. Characters are fullerand filled in.


Documents

Lecture #10 - Case Western Reserve Universityengr.case.edu/.../eecs490f07/Lectures/Lecture10.pdf · 2012-02-16 · Lecture #10 • Wraparound and padding • Image Correlation •