### MEM2D

Performs a Maximum-Entropy deconvolution of a two-dimensional NDF

#### Description:

MEM2D is based on the Gull and Skilling Maximum Entropy package MEMSYS3. It takes an image and a Point-Spread Function as input and produces an equal-sized image as output with higher resolution. Facilities are provided to ‘analyse’ the resulting deconvolved image, i.e. to calculate an integrated flux in some area of the deconvolved image and also an estimate of the uncertainty in the integrated flux. This allows the significance of structure visible in the deconvolution to be checked.

For a detailed description of the algorithm, and further references, see the MEMSYS users manual, and SUN/117.

#### Usage:

mem2d in out mask=? $\left\{\right\$
fwhmpsf=? psf=?

psftype

#### Parameters:

ANALYSE should be given a TRUE value if an analysis of a previously generated deconvolution is to be performed, instead of a whole new deconvolution being started. An analysis returns the integrated flux in some area of the deconvolved image you specify, together with the standard deviation on the integrated flux value. The area to be integrated over is specified by an image associated with Parameter MASK. This facility can, for instance, be used to assess the significance of structure seen in the deconvolution. An analysis can only be performed if the input NDF (see Parameter IN) contains a MEM2D extension  (see Parameter EXTEND). If the input does contain such an extension, and if the extension shows that the deconvolution was completed, then ANALYSE is defaulted to TRUE, otherwise it is defaulted to FALSE. []
This is the value to which the output image will default in areas for which there is no valid data in the input. The ‘zero entropy’ image is defined to be a flat surface with value given by Parameter DEF. Any deviation of the output image away from this image will cause its entropy to become negative. Thus a maximum-entropy criterion causes the output image to be as similar as possible to a flat surface with value DEF (within the constraints of the data). DEF is defaulted to the mean data value in the input image and must always be strictly positive. []
If EXTEND has a TRUE value, then the output NDF will contain an extension called MEM2D which will contain all the information required to either restart or analyse the deconvolution. Note, including this extension makes the output file much bigger (by about a factor of seven). [TRUE]
This is the Full Width at Half Maximum (in pixels) of a Gaussian Intrinsic Correlation Function (ICF) to be used in the deconvolution. The ICF can be used to encode prior knowledge of pixel-to-pixel correlations in the output image. A value of 0 for FWHMICF causes no ICF to be used, and so no correlations are expected in the output. Larger values encourage smoothness in the output on the scale of the ICF. If a non-zero ICF is used, the image entropy which is maximised is not the output image, but a ‘hidden’ image. This hidden image is the deconvolution of the output image with the ICF, and is assumed to have no pixel-to-pixel correlations. [2]
This is the Full Width at Half Maximum (in pixels) of a Gaussian Point Spread Function (PSF). This PSF is used to deconvolve the input only if Parameter PSFTYPE has the value "Gaussian".
ILEVEL controls the amount of information displayed as MEM2D runs. If set to 0 then no information is displayed. Larger values up to a maximum of 3, give larger amounts of information. A value of 3 gives full MEMSYS3 diagnostics after each iteration. [1]
The input NDF . This can either contain an image to be deconvolved, or the output from a previous run of MEM2D. The NDF is considered to be an output from MEM2D if it contains an extension called MEM2D (see Parameter EXTEND). If such an extension is found, a check is made to see if the NDF contains a completed deconvolution or a partial deconvolution. If the deconvolution is complete, the ANALYSE parameter is defaulted to TRUE, and unless you override this default, an analysis of the deconvolution contained in the input NDF is performed. If the input deconvolution is not complete, then the deconvolution process is restarted from where it left off. If no MEM2D extension is found, then a new deconvolution is started from scratch.
An image to use as a mask to define the areas to be integrated when performing an analysis (see Parameter ANALYSE). The integrated-flux value calculated by the analysis is actually the total data sum in the product of the mask and the deconvolved image. Mask pixel values can be positive or negative (or zero) and so, for instance, masks can be arranged which subtract off a background brightness from a source before returning the integrated source flux.
An image to use as the default model for the reconstruction. If a null value is given, then a constant value given by the Parameter DEF is used to define a flat default model. The section of the given image which matches the bounds of the input image is used. Any bad pixels in the image cause the corresponding pixels in the input image to be ignored. Such pixels are set bad in the output. The model image should contain no pixels with a value of zero or less. The default model is defined to have zero entropy. The hidden image will tend to the default model in the absence of data. It should be noted that this model applies to the ‘hidden’ image, not the actually required reconstructed image. The reconstructed image is obtained from the hidden image by blurring the hidden image with the ICF. [!]
##### MODELOUT = NDF (Write)
An image which can be used for the default model in a further run of MEM2D. Each pixel value in the created image is a linear combination of the model value at the corresponding pixel in the current reconstruction, and the hidden image pixel value. Pixels for which the hidden image is well away from the current model, tend towards the value of the hidden image; pixels for which the hidden image is close to the current model tend towards the model. Running MEM2D several times, using the new model created on the previous run as the model for the current run, can reduce the ‘mottling’ often seen in MEM2D reconstructions. [!]
The maximum number of maximum-entropy iterations to perform. MEM2D continues the deconvolution until either MEMSYS3 indicates that the termination criterion ($\Omega =1.0$) has been reached, or the maximum number of iterations is reached. If a deconvolution requires more iterations than was allowed by NITER, then you can choose to continue the deconvolution by giving the prematurely terminated output from MEM2D as the input to another run of MEM2D, specifying a larger value for NITER. [50]
NOISE defines the noise statistics within the input image. It can take the value "Gaussian" or "Poisson". If Gaussian noise is selected, the data variances are set initially to the values stored in the VARIANCE  component of the input NDF. If no such component exists, then the data variances are set to a constant value equal to the RMS difference between adjacent pixels in the x direction. MEMSYS3 scales these initial noise estimates to maximise the data ‘evidence’. The evidence is displayed as "LOG(PROB)" and the noise scaling factor as "SIGMA", if Parameter ILEVEL is set to 2 or more. If Poisson statistics are selected the uncertainty in each data value is, as usual, of the order of the square root of the data value. When using Poisson statistics, there is no equivalent to the noise scaling performed when using Gaussian statistics. Any input VARIANCE component is ignored. ["Gaussian"]
##### OUT = NDF (Write)
The output image in a ‘primitive’ NDF. The output is the same size as the input. Any pixels which were flagged as bad in the input will also be bad in the output. If Parameter EXTEND is TRUE, then the output NDF contains an extension called MEM2D containing information which allows the deconvolution to be either continued or analysed. There is no VARIANCE component in the output, but any QUALITY values are propagated from the input to the output. If Parameter UPDATE is TRUE, then the output NDF is created after the first iteration and is updated after each subsequent iteration.
An NDF holding an estimate of the Point Spread Function (PSF) of the input image. This PSF is used to deconvolve the input only if Parameter PSFTYPE has the value "NDF". The PSF can be centred anywhere within the image, the location of the centre is specified using Parameters XCENTRE and YCENTRE. The extent of the PSF actually used is controlled by Parameter THRESH.
PSFTYPE determines if the Point Spread Function used in the deconvolution is to be Gaussian (if PSFTYPE="Gaussian"), or is to be defined by an image you supply (if PSFTYPE="NDF"). ["NDF"]
This is the value to use for the MEMSYS3 RATE parameter. It determines the rate at which the convergence is allowed to proceed. If RATE is high, each maximum-entropy iteration is allowed to make a big change to the current reconstruction. This can cause numeric problems within MEMSYS3 resulting in MEM2D crashing with a "floating overflow" error. If this happens, try reducing RATE. Useful values will normally be of the order of unity, and must lie in the interval 0.0001 to 100. [0.5]
The fraction of the PSF peak amplitude at which the extents of the NDF PSF are determined. It must be positive and less than 0.5. This parameter is only used when PSFTYPE="NDF". An error will result if the input PSF is truncated above this threshold. [0.0625]
A title for the output NDF. A null (!) value means using the title of the input NDF. [!]
If UPDATE is given a TRUE value, then the output NDF will be created after the first iteration, and will then be updated after each subsequent iteration. This means that the current reconstruction can be examined without aborting the application. Also, if Parameter EXTEND is TRUE, then if the job aborts for any reason, it can be restarted from the last completed iteration (see Parameter IN). [TRUE]
The x pixel index of the centre of the PSF within the supplied PSF image. This is only required if PSFTYPE is "NDF". XCENTRE is defaulted to the middle pixel (rounded down if there are an even number of pixels per line). []
The y pixel index (line number) of the centre of the PSF within the supplied PSF image. This is only required if PSFTYPE is "NDF". YCENTRE is defaulted to the middle line (rounded down if there are an even number of lines). []

#### Results Parameters

##### DSUM = _REAL (Write)
The standard deviation of the integrated-flux value calculated if an analysis is performed (see Parameter ANALYSE).
##### SUM = _REAL (Write)
The integrated-flux value calculated if an analysis is performed (see Parameter ANALYSE).

#### Examples:

mem2d m51 m51_hires psftype=gaussian fwhmpsf=3
This example deconvolves the data array in the NDF called m51, putting the resulting image in the data array of the NDF called m51_hires. A circular Gaussian Point-Spread Function is used with a Full Width at Half Maximum of 3 pixels.
mem2d m51 m51_hires psf=star xcentre=20 ycentre=20
This example performs the same function as the previous example, but the PSF is defined by the data array of the NDF called star, instead of being defined to be Gaussian. This allows the PSF to be any arbitrary two-dimensional function. NDF star could be produced for example, by the Kappaapplication called PSF. Parameters XCENTRE and YCENTRE give the pixel indices of the centre of the beam defined by the PSF in star. The PSF is truncated to one sixteenth of its peak amplitude.
mem2d m51_hires m51_hires niter=70 psf=star
If the previous example failed to converge within the default 50 iterations, the deconvolution can be started again from its current state, rather than having to start again from scratch. Here NITER gives the upper limit on the total number of iterations which can be performed (including those performed in the previous run of MEM2D), not just the number performed in this single run of MEM2D. This facility can also be used if a MEM2D run is interrupted for any reason, such as the host computer going down, or a batch-queue CPU limit being reached. To use this facility the Parameters EXTEND and UPDATE should have the default values of TRUE.
Once a deconvolved image has been produced, the significance of features seen in the deconvolution can be assessed. This example takes in the NDF m51_hires produced by a previous run of MEM2D. If this is a completed deconvolution then the Parameter ANALYSE will be defaulted to TRUE, and an analysis will be performed. This effectively results in the deconvolution being multiplied by the data array of the NDF called nucleus, and the total data sum in the resulting image being displayed, together with the standard deviation on the total data sum. The image in m51_hires is the most probable deconvolution, but there may be other deconvolutions only slightly less probable than m51_hires. The standard deviation produced by an analysis takes account of the spread between such deconvolutions. If the total data sum is not significantly greater than the standard deviation, then the feature selected by the mask image (called nucleus in this case) may well be spurious. The mask image itself may for instance consist of an area of uniform value $+$1 covering some feature of interest, and the bad value (or equivalently the value zero) everywhere else. The analysis would then give the integrated flux in the feature, assuming that the background is known to be zero. If the background is not zero, then the mask may contain a background region containing the value $-$1, of equal area to the region containing the value $+$1. The resulting integrated flux would then be the total flux in the source minus the flux in a background region of equal area.

#### Notes:

• MEM2D requires a large quantity of memory–-almost as much as the rest of Kappa. In order for the Kappa monolith to load without you having to increase your memory or datasize resources, and because MEM2D is batch oriented (see Timing) it is only available as a separate application.

• Memory is required to store several intermediate images while the deconvolution is in progress. If the input image is small enough, these images are stored in a statically declared, internal array. Otherwise, they are stored in dynamically mapped external arrays. There is no limit on the size of image which can be processed by MEM2D (other than those imposed by limited resources on the host computer).

• It is sometimes desirable for the pixels in the output image to be smaller than those in the input image. For instance, if the input data are critically sampled (two samples per PSF), the output image may not be a very good deconvolution. In such cases sub-dividing the output pixels would give better results. At the moment MEM2D cannot do this. Be warned that sub-dividing the input pixels and then running the current version of MEM2D will not have the same effect, since the noise in the input image will then have pixel-to-pixel correlations, and be interpreted as real structure.

#### Timing

MEM deconvolution is extremely CPU intensive. The total CPU time taken depends partly on the size of the image, and partly on the complexity of the structure within the image. As a typical example, a 100$×$100 image containing 20 Gaussians on a flat background took about 34 minutes of elapsed time on an unloaded DEC Alpha 2000. Deconvolution jobs should therefore always be done in batch. To perform an analysis on a deconvolution takes about the same length of time as a single deconvolution iteration.

#### Related Applications

KAPPA: FOURIER, LUCY, WIENER.