digital image processing lecture 11: image restoration march 30, 2005 prof. charlene tsai

Digital Image Processing Lecture 11: Image Restoration

March 30, 2005

Digital Image Processing Lecture 11: Image Restoration

March 30, 2005

Prof. Charlene TsaiProf. Charlene Tsai

Digital Image Processing Lecture 11 2

ReviewReview

In last lecture, we discussed techniques that restore images in spatial domain. Mean filtering Order-statistics filering Adaptive filering Gaussian smoothing

We’ll discuss techniques that work in the frequency domain.

In last lecture, we discussed techniques that restore images in spatial domain. Mean filtering Order-statistics filering Adaptive filering Gaussian smoothing

We’ll discuss techniques that work in the frequency domain.


Periodic Noise ReductionPeriodic Noise Reduction

We have discussed low-pass and high-pass frequency domain filters for image enhancement.

We’ll discuss 2 more filters for periodic noise reduction Bandreject Notch filter

We have discussed low-pass and high-pass frequency domain filters for image enhancement.

We’ll discuss 2 more filters for periodic noise reduction Bandreject Notch filter


Bandreject FiltersBandreject Filters

Removing a band of frequencies about the origin of the Fourier transform. Ideal filter

where D(u,v) is the distance from the center, W is the width of the band, and D0 is the radial center.

Removing a band of frequencies about the origin of the Fourier transform. Ideal filter

where D(u,v) is the distance from the center, W is the width of the band, and D0 is the radial center.

2

WD, if 1

2,

2 if 0

2

, if 1

,

0

00

0

vuD

WDvuD

WD

WDvuD

vuH


Bandreject Filters (con’d)Bandreject Filters (con’d)

Butterworth filter of order n

Gaussian filter

Butterworth filter of order n

Gaussian filter

n

DvuDWvuD

vuH 2

20

2 ,,

1

1,

220

2

,

,

2

1

1,

WvuD

DvuD

evuH


Bandreject Filters: DemoBandreject Filters: Demo

Original corrupted by sinusoidal noise

Fourier transform

Butterworth filter

Result of filtering


Notch FiltersNotch Filters

Reject in predefined neighborhoods about the center frequency.

Due to the symmetry of the Fourier transform, notch filters must appear in symmetric pairs about the origin.

Given 2 centers (u0, v0) and (-u0, -v0), we define D1(u,v) and D2(u,v) as

Reject in predefined neighborhoods about the center frequency.

Due to the symmetry of the Fourier transform, notch filters must appear in symmetric pairs about the origin.

Given 2 centers (u0, v0) and (-u0, -v0), we define D1(u,v) and D2(u,v) as

2120

201 22, vNvuMuvuD

2120

202 22, vNvuMuvuD


Notch Filters: plotsNotch Filters: plots

ideal

Butterworth Gaussian


Notch Filters (con’d)Notch Filters (con’d)

Ideal filter

Butterworth filter

Gaussian filter

Ideal filter

Butterworth filter

Gaussian filter

otherwise 1

,or , if 0, 0201 DvuDDvuDvuH

n

vuDvuDD

vuH

,,1

1,

21

20

20

22 ,,

2

1

1, D

vuDvuD

evuH


How to deal with motion blur?How to deal with motion blur?

Original Blurred by motion


Convolution Theory: ReviewConvolution Theory: Review

Knowing the degradation function H(u,v), we can, in theory, obtain the original image F(u,v).

In practice, H(u,v) is often unknow. We’ll discuss briefly one method of obtaining the

degradation functions. For interested readers, please consult Conzalez, section 5.6 for other methods.

Knowing the degradation function H(u,v), we can, in theory, obtain the original image F(u,v).

In practice, H(u,v) is often unknow. We’ll discuss briefly one method of obtaining the

degradation functions. For interested readers, please consult Conzalez, section 5.6 for other methods.

vuHvuFvuG ,,,

Filter (degradation function)

Original imageDegraded image


Estimation of H(u,v) by Experimentation Estimation of H(u,v) by Experimentation

If equipment similar to the one used to acquire the degraded image is available, it is possible, in principle, to obtain the accurate estimate of H(u,v). Step1: reproduce the degraded image by varying the

system settings. Step2: obtain the impulse response of the

degradation by imaging an impulse (small dot of light) using the same system settings.

Step3: recalling that FT of an impulse is a constant (A)

If equipment similar to the one used to acquire the degraded image is available, it is possible, in principle, to obtain the accurate estimate of H(u,v). Step1: reproduce the degraded image by varying the

system settings. Step2: obtain the impulse response of the

degradation by imaging an impulse (small dot of light) using the same system settings.

Step3: recalling that FT of an impulse is a constant (A)

A

vuGvuH

,,

What we want

Degraded impulse image

Strength of the impulse


Estimation of H(u,v) by Exp (con’d)Estimation of H(u,v) by Exp (con’d)

An impulse of light (magnified). The FT

is a constant A

G(u,v), the imaged (degraded) impulse


Undoing the DegradationUndoing the Degradation

Knowing G & H, how to obtain F? Two methods:

Inverse filtering Wiener filtering

Knowing G & H, how to obtain F? Two methods:

Inverse filtering Wiener filtering

vuHvuFvuG ,,,

Filter (degradation function)

Original image (what we’re after)

Degraded image


Inverse FilteringInverse Filtering

In the simplest form:

See any problems? Division by small values can produce very

large values that dominate the output.

In the simplest form:

See any problems? Division by small values can produce very

large values that dominate the output.

vuHvuG

vuF,

,,

Original

Inverse filtering using

Butterworth filter


Inverse Filtering (con’d)Inverse Filtering (con’d)

Solutions? There are two similar approaches:

Low-pass filtering with filter L(u,v):

Thresholding (using only filter frequencies near the origin)

Solutions? There are two similar approaches:

Low-pass filtering with filter L(u,v):

Thresholding (using only filter frequencies near the origin)

vuLvuH

vuGvuF ,

,

,,

dvuHvuG

dvuHvuH

vuGvuF

, if ,

, if ,

,,


Inverse Filtering: DemoInverse Filtering: Demo

Full filter d=40

d=70 d=80


Inverse Filtering: Weaknesses Inverse Filtering: Weaknesses

Inverse filtering is not robust enough. It is even worse if the image has been

corrupted by noise.

The noise can completely dominate the output.

Inverse filtering is not robust enough. It is even worse if the image has been

corrupted by noise.

The noise can completely dominate the output.

vuNvuHvuFvuG ,,,,

vuH

vuNvuGvuF

,

,,,


Wiener FilteringWiener Filtering

What measure can we use to say whether our restoration has done a good job?

Given the original image f and the restored version r, we would like r to be as close to f as possible.

One possible measure is the sum-squared-differences

Wiener filtering: minimum mean square error:

What measure can we use to say whether our restoration has done a good job?

Given the original image f and the restored version r, we would like r to be as close to f as possible.

One possible measure is the sum-squared-differences

Wiener filtering: minimum mean square error:

2,, jiji rf

vuG

KvuH

vuH

vuHvuF ,

,

,

,

1, 2

2

Specified constant


Comparison of Inverse and Wiener Filtering Comparison of Inverse and Wiener Filtering

Column 1: blurred image with additive Gaussian noise of variances 650, 65 and 0.0065.

Column 2: Inverse filtering

Column 3: Wiener filtering

Column 1: blurred image with additive Gaussian noise of variances 650, 65 and 0.0065.

Column 2: Inverse filtering

Column 3: Wiener filtering


SummarySummary

Removal of periodic noise: Bandreject Notch filter

Deblurring the image: Inverse filtering Wiener filtering

Removal of periodic noise: Bandreject Notch filter

Deblurring the image: Inverse filtering Wiener filtering

digital image processing lecture 11: image restoration march 30, 2005 prof. charlene tsai

Documents

image enhancement

image restorationmarch

origin inverse filtering

frequency domain filters

original image fu

weaknesses inverse filtering

degradation function

originalinverse filtering