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Ch4 Frequency Filtering 1

Overview Common Fourier Transforms Ideal Lowpass Butterworth Lowpass Gaussian Lowpass Ideal Highpass Butterworth Highpass Gaussian Highpass Conclusions

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Overview

Forlinearfrequency domain filters, we pointmultiply the FT of our image by a filter H(u,v)and do an IFT to obtain the output image:

F(u,v) = FT{ f(x,y) }

G(u,v) = F(u,v).H(u,v)

g(x,y) = IFT{ G(,u,v) }

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Overview

By varying the filter H(u,v) we can implementa wide variety of image processing operations

Lowpass filterssmooth an image by keepingonly the low frequencies (the big features)

Highpass filterssharpen an image by keepingonly the high frequencies (the small features)

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Common Fourier Transforms

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Ideal Lowpass

An ideal lowpass filter keeps low frequenciesin an image and discards the high frequencies

This is accomplished by multiplying F(u,v) bya filter H(u,v) that equals 1 in desired rangeand 0 outside this range

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Ideal Lowpass

H(u,v) is a disk of radius D0 centered at origin

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Ideal Lowpass

Filter radii: 10, 30, 60, 160, 460

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Ideal Lowpass

To understand the ringing artifacts above, weneed to consider what convolution maskcorresponds to the ideal lowpass filter

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Ideal Lowpass

void im_complex::IdealLP

(float freq)

{

// Perform FFT

FastFT();

// Perform filtering

int ydim = Re.Ydim;

int xdim = Re.Xdim;

for (int v = 0; v < ydim; v++)

for (int u = 0; u < xdim; u++)

{ // Calculate distance to origin

int dv = (v < ydim / 2) ? v : v - ydim;

int du = (u < xdim / 2) ? u : u - xdim;

float dist = (float)(dv * dv + du * du);

// Apply filter

if (dist > freq * freq)

{

Re.Data2D[v][u] = 0;

Im.Data2D[v][u] = 0;

}}

// Perform inverse FFT

FastIFT();

}

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Butterworth Lowpass

A Butterworth lowpass keeps frequenciesinside radius D0 and discards values outside

It introduces a gradual transition from 1 to 0 toreduce ringing artifacts

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Butterworth Lowpass

A family of filters can be created by varying nto increase or decrease the slope at D0

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Butterworth Lowpass

Cutoff frequencies: 10, 30, 60, 160, 460

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Butterworth Lowpass

Ringing artifact increases as n increases

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Butterworth Lowpass

void im_complex::ButterworthLP

(float freq, float power)

{

// Perform FFT

FastFT();


int ydim = Re.Ydim;

int xdim = Re.Xdim;



{

// Calculate distance to origin




// Apply filter

float filter = 1 / (1 +

pow(dist / (freq*freq), power));

Re.Data2D[v][u] *= filter;

Im.Data2D[v][u] *= filter;

}


FastIFT();

}

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Gaussian Lowpass

A Gaussian lowpass filter has the most naturaland well behaved filter shape (blurring with a

Gaussian filter corresponds to heat diffusion)

The FT of a Gaussian function is a Gaussianso no ringing artifacts are introduced

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Gaussian Lowpass

A family of filters can be created by varyingthe standard deviation D0 to increase ordecrease filter width

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Gaussian Lowpass

Cutoff frequencies: 10, 30, 60, 160, 460

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Gaussian Lowpass

void im_complex::GaussLP

(float freq)

{

// Perform FFT

FastFT();


int ydim = Re.Ydim;

int xdim = Re.Xdim;



{





// Apply filter

float filter =

exp(dist / (-2*freq*freq));

Re.Data2D[v][u] *= filter;

Im.Data2D[v][u] *= filter;

}


FastIFT();

}

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Ideal Highpass

An ideal highpass filter keeps the highfrequencies and discards the low frequencies

This is accomplished by multiplying F(u,v) bya filter H(u,v) that equals 1 in desired rangeand 0 outside this range

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Ideal Highpass

H(u,v) is a disk of radius D0 centered at origin

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Ideal Highpass

Cutoff frequencies: 30, 60, 160

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Ideal Highpass

void im_complex::IdealHP

(float freq)

{

// Perform FFT

FastFT();


int ydim = Re.Ydim;

int xdim = Re.Xdim;



{





// Apply filter

if (dist < freq * freq)

{

Re.Data2D[v][u] = 0;

Im.Data2D[v][u] = 0;

}}


FastIFT();

}

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Butterworth Highpass

A Butterworth highpass keeps frequenciesoutside radius D0 and discards values inside

It has a gradual transition from 0 to 1 toreduce ringing artifacts

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A family of filters can be created by varying nto increase or decrease the slope at D0

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void im_complex::ButterworthHP

(float freq, float power)

{

// Perform FFT

FastFT();


int ydim = Re.Ydim;

int xdim = Re.Xdim;



{





// Apply filter

float filter = 1 / (1 +

pow(dist / (freq*freq), power));

Re.Data2D[v][u] *= (1-filter);

Im.Data2D[v][u] *= (1-filter);

}


FastIFT();

}

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Gaussian Highpass

The Gaussian highpass filter discards lowfrequencies and keeps high frequencies

Since Gaussian HP = (1 Gaussian LP) itdoes not introduce any ringing artifacts

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Gaussian Highpass

A family of filters can be created by varyingthe standard deviation D0 to increase ordecrease filter width

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Gaussian Highpass

Cutoff frequencies: 30, 60, 160

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Gaussian Highpass

void im_complex::GaussHP

(float freq)

{

// Perform FFT

FastFT();


int ydim = Re.Ydim;

int xdim = Re.Xdim;



{





// Apply filter

float filter =

exp(dist / (-2*freq*freq));

Re.Data2D[v][u] *= (1-filter);

Im.Data2D[v][u] *= (1-filter);

}


FastIFT();

}

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Conclusion

In this section we have seen three types oflowpass filters and three types of highpassfilters

In the next section we will see that frequencyfiltering can do much more to an image

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