### WIENER

Applies a Wiener filter to a one- or two-dimensional array

#### Description:

This application filters the supplied one- or two-dimensional array using a Wiener filter. It takes an array holding observed data and another holding a Point-Spread Function as input and produces an output restored array with potentially higher resolution and lower noise. Generally superior results can be obtained using applications MEM2D or LUCY, but at the cost of much more processing time.

The Wiener filter attempts to minimise the mean squared difference between the undegraded image and the restored image. To do this it needs to know the power spectrum of the undegraded image (i.e. the power at each spatial frequency before the instrumental blurring and the addition of noise). Obviously, this is not usually available, and instead the power spectrum of some other image must be used (the ‘model’ image). The idea is that a model image should be chosen for which there is some a priori reason for believing it to have a power spectrum similar to the undegraded image. Many different suggestions have been made for the best way to make this choice and the literature should be consulted for a detailed discussion (for instance, see the paper Wiener Restoration of HST Images: Signal Models and Photometric Behavior by I.C. Busko in the proceedings of the first Annual Conference on Astronomical Data Analysis Software and Systems, Tucson). By default, this application uses a ‘white’ model image, i.e. one in which there is equal power at all spatial frequencies. The default value for this constant power is the mean power per pixel in the input image. There is also an option to use the power spectrum of a supplied model image.

The filter also depends on a model of the noise in the supplied image. This application assumes that the noise is ’white’ and is constant across the image. You can specify the noise power to use. If a noise power of zero is supplied, then the Wiener filter just becomes a normal inverse filter which will tend to amplify noise in the supplied image.

The filtering is done by multiplying the Fourier transform of the supplied image by the Fourier transform of the filter function. The output image is then created by taking the inverse Fourier transform of the product. The Fourier transform of the filter function is given by:

$\frac{{H}^{\ast }}{{\left|H\right|}^{2}+\frac{{P}_{n}}{{P}_{g}}}$

where $H$ is the Fourier transform of the supplied Point-Spread Function, ${P}_{n}$ is the noise power, ${P}_{g}$ is the power in the model image, and ${H}^{\ast }$ is the complex conjugate of $H$. If the supplied model includes noise (as indicated by Parameter QUIET) then ${P}_{n}$ is subtracted from ${P}_{g}$ before evaluating the above expression.

#### Usage:

wiener in psf out xcentre ycentre

#### Parameters:

The input NDF  containing the observed data. This image may contain bad values, in which case the bad values will be replaced by zero before applying the filter. The resulting filtered image is normalised by dividing each pixel value by the corresponding weight of the good input pixels. These weights are found by filtering a mask image which holds the value one at every good input pixel, and zero at every bad input pixel.
An NDF containing an image to use as the model for the power spectrum of the restored image. Any bad values in this image are replaced by the mean of the good values. If a null value is supplied then the model power spectrum is taken to be uniform with a value specified by Parameter PMODEL. [!]
##### OUT = NDF (Write)
The restored output array. An extension named WIENER is added to the output NDF to indicate that the image was created by this application (see Parameter QUIET).
The mean power per pixel in the model image. This parameter is only accessed if a null value is supplied for Parameter MODEL. If a value is obtained for PMODEL then the model image is assumed to have the specified constant power at all spatial frequencies. If a null (!) value is supplied, the value used is the mean power per pixel in the input image. [!]
The mean noise power per pixel in the observed data. For Gaussian noise this is equal to the variance. If a null (!) value is supplied, the value used is an estimate of the noise variance based on the difference between adjacent pixel values in the observed data. [!]
An NDF holding an estimate of the Point-Spread Function (PSF) of the input array. This could, for instance, be produced using the Kappa application PSF. There should be no bad pixels in the PSF otherwise an error will be reported. The PSF can be centred anywhere within the array, but the location of the centre must be specified using Parameters XCENTRE and YCENTRE. The PSF is assumed to have the value zero outside the supplied NDF.
This specifies whether or not the image given for Parameter MODEL (or the value given for Parameter PMODEL), includes noise. If the model does not include any noise then a TRUE value should be supplied for QUIET. If there is any noise in the model then QUIET should be supplied FALSE. If a null (!) value is supplied, the value used is FALSE, unless the image given for Parameter MODEL was created by a previous run of WIENER (as indicated by the presence of a WIENER extension in the NDF), in which case the run time default is TRUE (i.e. the previous run of WIENER is assumed to have removed the noise). [!]
The fraction of the PSF peak amplitude at which the extents of the PSF are determined. These extents are used to derive the size of the margins that pad the supplied input array. Lower values of THRESH will result in larger margins being used. THRESH must be positive and less than 0.5. [0.0625]
A title for the output NDF. A null (!) value means using the title of the input NDF. [!]
If the input array contains bad values, then this parameter may be used to determine the minimum weight of good input values required to create a good output value. It can be used, for example, to prevent output pixels from being generated in regions where there are relatively few good input values to contribute to the restored result. It can also be used to ‘fill in’ small areas (i.e. smaller than the PSF) of bad pixels.

The numerical value given for WLIM specifies the minimum total weight associated with the good pixels in a smoothing box required to generate a good output pixel (weights for each pixel are defined by the normalised PSF). If this specified minimum weight is not present, then a bad output pixel will result, otherwise a smoothed output value will be calculated. The value of this parameter should lie between 0.0 and 1.0. WLIM=0 causes a good output value to be created even if there is only one good input value, whereas WLIM=1 causes a good output value to be created only if all input values are good. [0.001]

The x pixel index  of the centre of the PSF within the supplied PSF array. The suggested default is the middle pixel (rounded down if there are an even number of pixels per line).
The y pixel index of the centre of the PSF within the supplied PSF array. The suggested default is the middle line (rounded down if there are an even number of lines).

#### Examples:

wiener cenA star cenA_hires 11 13
This example deconvolves the array in the NDF called cenA, putting the resulting array in the NDF called cenA_hires. The PSF is defined by the array in NDF star, and the centre of the PSF is at pixel (11, 13).
wiener cenA star cenA_hires 11 13 pnoise=0
This example performs the same function as the previous example, except that the noise power is given as zero. This causes the Wiener filter to reduce to a standard inverse filter, which will result in more high frequencies being present in the restored image.
wiener cenA star cenA_hires 11 13 model=theory quiet
This example performs the same function as the first example, except that the power spectrum of the restored image is modelled on that of NDF theory, which may for instance contain a theoretical model of the object in NDF cenA, together with a simulated star field. The Parameter QUIET is set to a TRUE value to indicate that the theoretical model contains no noise.

#### Notes:

• The convolutions required by the Wiener filter are performed by the multiplication of Fourier transforms. The supplied input array is extended by a margin along each edge to avoid problems of wrap-around between opposite edges of the array. The width of this margin is about equal to the width of the significant part of the PSF (as determined by Parameter THRESH). The application displays the width of these margins. The margins are filled by replicating the edge pixels from the supplied input NDFs.

#### Related Applications

KAPPA: FOURIER, LUCY, MEM2D.