WIENER  FILTER

INTRODUCTION •  The Wiener filter was proposed by Norbert Wiener in

1940. •  It was published in 1949 •  Its purpose is to reduce the amount of a noise in a

signal. •  This is done by comparing the received signal with a

estimation of a desired noiseless signal. •  Wiener filter is not an adaptive filter as it assumes

input to be stationery.

DESCRIPTION •  It takes a statistical approach to solve its goal •  Goal of the filter is to remove the noise from a signal •  Before implementation of the filter it is assumed that

the user knows the spectral properties of the original signal and noise.

•  Spectral properties like the power functions for both the original signal and noise.

•  And the resultant signal required is as close to the original signal

DESCRIPTION •  Signal and noise are both linear stochastic

processes with known spectral properties. •  The aim of the process is to have minimum mean-

square error •  That is, the difference between the original signal

and the new signal should be as less as possible.

Important  Equations •  Considering we need to design a wiener filter in

frequency domain as W(u,v) •  Restored image will be given as;

Xn(u,v) = W(u,v).Y(u,v)

•  Where Y(u,v) is the received signal and Xn(u,v) is the restored image

Important  Equations •  We choose W(k,l) to minimize:

Obtained from [1]

•  Where the equation represents the mean square error.

•  The wiener filter can be represented by the equation:

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5.8 Minimum Mean Square Error (Wiener) Filtering Here we discuss an approach that incorporates both the degradation function and statistical characteristics of noise into the restoration process. Considering images and noise as random variables, the objective is to find an estimate f̂ of the uncorrupted image f such that the mean square error between them is minimized. The error measure is given by

{ }2 2ˆ( )e E f f= − (5.8-1)

where { }E i is the expected value of the argument. By assuming that

1. the noise and the image are uncorrelated;

2. one or the other has zero mean;

3. the intensity levels in the estimate are a linear function of the levels in the degraded image.

Then, the minimum of the error function in (5.8-1) is given in the frequency domain by the expression

2

( , ) ( , )ˆ( , ) ( , )( , ) ( , ) ( , )

f

f

H u v S u vF u v G u v

S u v H u v S u vη

∗ = +

2

( , )( , )

( , ) ( , )/ ( , )f

H u vG u v

H u v S u v S u vη

∗ = +

(5.8-2) 2

2

( , )1( , )

( , ) ( , ) ( , )/ ( , )f

H u vG u v

H u v H u v S u v S u vη

= +

Important  Equations

•  Obtained from [1]

Important  Equations •  H(u,v) = degradation function •  |H(u,v)|^2 = H*(u,v)H(u,v) •  H*(u,v) = complex conjugate of H(u,v) •  Sn(u,v) = |N(u,v)|^2 power spectrum of noise •  Sf(u,v) = |F(u,v)|^2 power spectrum of

undegraded image . G(u,v) is the transform of the degraded image.

Important  Equations •  The signal to noise ration can be approximated

using the following equation:

Obtained from [1]

•  Low noise gives high SNR and High noise gives Low SNR. The value is a good metric used in characterizing the performance of restoration algorithm

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The terms in (5.8-2) are as follows:

ˆ( , ) F u v is the frequency domain estimate

( , ) G u v is the transform of the degraded image

( , )H u v is the transform of the degradation function

( , )H u v∗ is complex conjugate of ( , )H u v

2( , ) ( , ) ( , )H u v H u v H u v∗=

2( , ) ( , )S u v N u vη = = power spectrum of the noise

2( , ) ( , )fS u v F u v= = power spectrum of the undegraded image This result is known as the Wiener filter, which also is commonly referred to as the minimum mean square error filter or the least square error filter. The Wiener filter does not have the same problem as the inverse filter with zeros in the degradation function, unless the entire denominator is zero for the same value(s) of u and v . If the noise is zero, then the Wiener filter reduces to the inverse filter. One of the most important measures is the signal-to-noise ratio, approximated using frequency domain quantities such as

1 12

0 01 1

2

0 0

( , )

( , )

M N

u vM N

u v

F u vSNR

N u v

− −

= =− −

= =

=∑ ∑

∑ ∑ (5.8-3)

Important  Equations •  The MSE in statistical form can be calculated as:

Obtained from [1]

•  If restored signal is considered as signal and difference between the restored and degraded as the noise, then we can obtain SNR in spatial domain

Obtained from [1]

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The mean square error given in statistical form in (5.8-1) can be approximated also in terms a summation involving the original and restored images:

1 12

0 0

1 ˆ( , ) ( , )M N

x y

MSE f x y f x yMN

− −

= =

= − ∑ ∑ (5.8-4) If one considers the restored image to be signal and the difference between this image and the original to be noise, we can define a signal-to-noise ratio in the spatial domain as

1 12

0 01 1

2

0 0

(̂ , )

ˆ( , ) ( , )

M N

x yM N

x y

f x y

SNR

f x y f x y

− −

= =− −

= =

= −

∑ ∑

∑ ∑ (5.8-5)

The closer f and f̂ are, the larger this ratio will be.

If we are dealing with white noise, the spectrum 2( , )N u v is a constant, which simplifies things considerably. However,

2( , )F u v is usually unknown. An approach is used frequently when these quantities are not known or cannot be estimated:

2

2

( , )1ˆ( , ) ( , )( , ) ( , )

H u vF u v G u v

H u v H u v K

= +

(5.8-6) where K is a specified constant that is added to all terms of

2( , )H u v .

Note: White noise is a random signal (or process) with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency.

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The mean square error given in statistical form in (5.8-1) can be approximated also in terms a summation involving the original and restored images:

1 12

0 0

1 ˆ( , ) ( , )M N

x y

MSE f x y f x yMN

− −

= =

= − ∑ ∑ (5.8-4) If one considers the restored image to be signal and the difference between this image and the original to be noise, we can define a signal-to-noise ratio in the spatial domain as

1 12

0 01 1

2

0 0

(̂ , )

ˆ( , ) ( , )

M N

x yM N

x y

f x y

SNR

f x y f x y

− −

= =− −

= =

= −

∑ ∑

∑ ∑ (5.8-5)

The closer f and f̂ are, the larger this ratio will be.

If we are dealing with white noise, the spectrum 2( , )N u v is a constant, which simplifies things considerably. However,

2( , )F u v is usually unknown. An approach is used frequently when these quantities are not known or cannot be estimated:

2

2

( , )1ˆ( , ) ( , )( , ) ( , )

H u vF u v G u v

H u v H u v K

= +

(5.8-6) where K is a specified constant that is added to all terms of

2( , )H u v .

Note: White noise is a random signal (or process) with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency.

Important  Equations •  But it is sometimes hard to estimate the power

spectrum of either the un-degraded image or the noise.

•  In that case we assume a constant K, that is then added to all terms of H|(u,v)|^2

•  The new equation in that case becomes:

Obtained from [1]

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The mean square error given in statistical form in (5.8-1) can be approximated also in terms a summation involving the original and restored images:

1 12

0 0

1 ˆ( , ) ( , )M N

x y

MSE f x y f x yMN

− −

= =

= − ∑ ∑ (5.8-4) If one considers the restored image to be signal and the difference between this image and the original to be noise, we can define a signal-to-noise ratio in the spatial domain as

1 12

0 01 1

2

0 0

(̂ , )

ˆ( , ) ( , )

M N

x yM N

x y

f x y

SNR

f x y f x y

− −

= =− −

= =

= −

∑ ∑

∑ ∑ (5.8-5)

The closer f and f̂ are, the larger this ratio will be.

If we are dealing with white noise, the spectrum 2( , )N u v is a constant, which simplifies things considerably. However,

2( , )F u v is usually unknown. An approach is used frequently when these quantities are not known or cannot be estimated:

2

2

( , )1ˆ( , ) ( , )( , ) ( , )

H u vF u v G u v

H u v H u v K

= +

(5.8-6) where K is a specified constant that is added to all terms of

2( , )H u v .

Note: White noise is a random signal (or process) with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency.

Working  Example  1 •  We apply the filter to the following set of images

1 obtained from [1] 2 Obtained from [1]

•  We reduce the noise variance (noise power):

3 obtained from[1] 4 obtained from [1]

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Example 5.13: Further comparisons of Wiener filtering

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Example 5.13: Further comparisons of Wiener filtering

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Example 5.13: Further comparisons of Wiener filtering

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Example 5.13: Further comparisons of Wiener filtering

Working  Example  1 •  We decrease the noise variance even further:

5 obtained from [1] 6 obtained from [1]

•  As we can see A wiener filter does a very good job at deblurring of an image and reducing the noise.

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Example 5.13: Further comparisons of Wiener filtering

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Example 5.13: Further comparisons of Wiener filtering

Example  2 •  The problem is to estimate the power spectrum of

noise and even more difficult is to estimate the power spectrum of the image.

•  We know that most of the images have similar power spectrum.

•  We take two images and calculate their individual power spectrum

•  The images derived are obtained from [2]

Example  2

Obtained from [2]

Example  2 •  We calculate the power spectrum of each image:

Obtained from [2]

Example  2 •  If we restore the cameraman image using its own

power spectrum, the image will look like this:

Obtained from [2]

Example  2 •  But we use the power spectrum obtained from the

house image, the restored image will look like this:

Obtained from [2]

Example  2 •  Now if we consider a large set of images and

calculate the power spectrum for them and find a mean, that could then be used as the power spectrum input for the wiener filter, we are likely to get better results.

•  Hence, it is important to have a large data set, to calculate power spectrum for.

•  In the previous scenario a user can derive the noise power spectrum from previous knowledge or can calculate it by observing the variance within an image’s smoother part.

How  to  use  Wiener  filter? •  Implementation of wiener filter are available both in

Matlab and Python. •  These implementations can be used to perform

analysis on images.

Conclusion •  Wiener filter is an excellent filter when it comes to

noise reduction or deblluring of images. •  A user can test the performance of a wiener filter

for different parameters to get the desired results. •  It is also used in steganography processes. •  It considers both the degradation function and

noise as part of analysis of an image.

References •  [1] R. Gonzalez and W. RE, Digital Image

Processing, Third Edit. Pearson Prentice Hall, 2008, pp. 352–357.

•  [2] S. Eddins, “Matlab Central Steve on Image Processing.” [Online]. Available: http://blogs.mathworks.com/steve/2007/11/02/image-deblurring-wiener-filter/. [Accessed: 25-Aug-2012].