Wiener deconvolution

In mathematics, the Wiener deconvolution an application of the Wiener filter for noise problems in the unfolding; they tried to minimize the influence of noise in the frequency domain for the development and is therefore mostly used in low signal-to- noise ratios.

The Wiener deconvolution is widely used in development of applications in photography because the frequency spectrum of images in the visible range is relatively easy to determine.

The Wiener deconvolution is named after Norbert Wiener.

Definition

Be

Where denotes convolution and

• The (unknown ) input signal at the time.
• The well-known linear time-invariant impulse response of a system
• An unknown noise that is independent of
• The observed signal.

The goal is to determine, so that there is as follows:

With an estimate of with minimum square error is.

The Wiener filter provides such. It can be most easily described in the frequency domain:

In which

• And the Fourier transform and in frequency.
• Is the average power spectral density of the input signal
• Is the mean spectral density of the noise
• The complex conjugate respectively.

The filter operation, as above, be carried out in the time domain or in the frequency domain:

(the Fourier transform ) is. Provides an inverse Fourier transform.

It should be noted that when the images are two-dimensional and arguments; However, the result remains the same.

Interpretation

The application of the Wiener filter is shown if the above equation is rewritten as:

Here, the inverse of the output system, and the signal -to- noise ratio. Without noise (ie infinite signal-to- noise ratio) is the term in the square brackets is equal to 1, which means that the Wiener filter is simply the inverse of the system, as you would expect. If the noise increases at certain frequencies, so the signal-to - noise ratio decreases, the term within the brackets also decreases. That is, the Wiener filter attenuates the frequencies in response to its signal -to- noise ratio.

The above equation assumes that the spectral content of a typical image, and the noise is known. Most often, the two values ​​are not known but can be estimated. For example, in photos, the signal ( the original image ) is typically strong proportions of low and high frequencies and weak shares the noise components are evenly distributed over all frequencies.

Derivation

Is as described above an approximation of the original image are generated at minimizing the quadratic error. This can be extended by

Express, which is the expected value.

Will be replaced, can rewrite the expression:

The square can be developed and results:

However, it is assumed that the noise is independent of the signal, that is:

The spectral power density is defined as:

This results in:

To find the minimum error is a differentiated and set equal to zero. Because that provides a complex value is a constant.

This equation can be rewritten to obtain the Wiener filter.