SEE_ALSO

## NAME

ils -- Iterative Least Squares image restoration.

## USAGE

`ils input psf output`

## DESCRIPTION

This task restores images by the Least Squares (or minimum residual norm) criterion. The implemented algorithm has as its main features:

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- Smoothness constraint (Miller) regularization through a 2-D Laplacian operator and fixed regularization parameter.

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- Either spatially-invariant regularization, or spatially-adaptive regularization using an "eye-model" weight function.

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- Projection into the set of meaningful solutions, which takes the form of a positivity constraint.

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- Multiple input images and PSFs.

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- Bad pixel masking.

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- Edge extension.

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- Iterative solution by the fast "conjugate gradients" algorithm.

The basic input consists of one degraded image to be restored, and one image which contains the (isolated) PSF. Their sizes in x and y may be different, and the output image may have either the same size as the degraded image, or a (larger) size specified by the `xsizeout' and `ysizeout' parameters. In this way, edge effects that come from the Fourier transforms used internally by the algorithm can be overcome.

The PSF does not have to be centered in the PSF image; the task will look for the brightest pixel in that image and take it as the PSF center. Alternatively, the PSF may come from an image populated by several stars, including the degraded image itself. In this case parameters `px0', `py0' and `mask' are used to isolate the desired PSF star from the full data frame. `px0' and `py0' must point to the central PSF pixel, and `mask' defines the radius (in pixels) of a circular area extracted around that pixel and apodized by a smooth taper function. If the PSF comes from an actual, noisy, non-synthetic image, parameter `nlpsf' must be set to "no". These PSF-related parameters are resident in the `ilspars' pset.

In the case where more than one degraded version of the same scene is available, the algorithm can combine simultaneously all degraded images into one single restored image. In this case the solution will benefit from complementary information available in each single input image and PSF. The task may accept either one single PSF image, which will be applyed to all input images, or one separate PSF image for each input degraded image. In this case, the input lists for the `input' and `psf' task parameters must present a one-to-one relationship. Also, parameters `px0' and `py0' should be set to -1, and each input PSF should be shifted according to its corresponding input image.

Parameter `mask' does not apply in this case. All degraded images must have the same size; the PSF images however may be of any size. No provision exists to date to include variable weighting among the input images; all of them will contribute with the same weight to the final result.

Iterations usually depart from a constant, zeroed, first-guess image generated internally, and proceed up to the number specified by task parameter `niter', or until current chi-square becomes smaller than task parameter `limchisq'. If however a previously computed image estimate is already available, this image may be input to the algorithm via task parameter `model'. In this way the algorithm may be run in an incremental way. If edge extension is not enabled (xsizeout = ysizeout = 0), the model image must have the same size as the input image(s). If edge extension is enabled, the model image must have the same size as specified by task parameters `xsizeout' and `ysizeout'.

Notice, however, that since the conjugate direction information is not saved in the output file, when restarting iterations from a previously computed solution, the algorithm may follow a slightly different path to the minimum. Thus, a solution from a total of N iterations obtained in M incremental runs may look slightly different than one obtained in a single N-iteration run. This effect can be minimized if, at each incremental run, a minimum of about 5 iterations is performed; in this way the algorithm may have enough time at each run to catch up with the same path to the minimum. The fastest way to the minimum is always by a single, N-iteration run.

The task may optionally perform bad pixel masking. Masks are images with same size as input image(s), which contain pixel values to be used as weights in restoration (do not confuse with the weights used in regularization !). Weights must be in the range of zero to one. The larger the weight, the more the associated pixel will contribute to the final result. A zero weight totally masks out the corresponding pixel's contribution. Notice that this definition is NOT the same as used in HST data quality files. To convert a data quality image to the format used by this task, task `imcalc' may be used as, for example, with command string

equals = "if im1 .gt. 0. then 0. else 1.".

The mask image(s) is(are) specified by task parameter `wmask'. Either one single mask image, which will be applyed to all input images, or one separate mask image for each input degraded image, are acceptable. In this case, the lists for the `input' and `wmask' task parameters must present a one-to-one relationship.

When using edge extension, task parameter `center' may be used to specify where in the output array the restored image will be put. If center=yes, the restored image will be centered. If center=no, it will be put with its [1,1] corner coincident with the [1,1] corner of the output array, and the (extended) edges will be laid out in a wrap-around configuration in the output array's remaining space. This makes it easier to trim out the edges with a imcopy operation.

An image with the current estimate may be output at each `nsave'-th iteration, updating the contents of the file specified by task parameter `output'.

Algorithm control and regularization details are specified by parameters inside pset `ilspars'. The remaining paragraphs explain these details.

The conjugate gradients algorithm works by computing, at each iteration, the conjugate direction based on the previous iteration's direction and the current (steepest-descent) gradient. A line minimization routine is used next to compute the "step" (or gain) `beta' to be used over the conjugate direction to find a minimum. This operation mode, the default, is specified by task parameter `auto' set to "yes", and in this case the value of task parameter `beta' is disregarded.

For certain types of images, typically of low dynamic range but rich in spatial structures of many scales, it was found empirically that the line minimization routine can be skipped, and a fixed gain used instead. The number of iterations must be larger than when using line minimization, but nevertheless results in substantial savings in CPU time. This mode is enabled by setting task parameter `auto' as "no", and in this case parameter `beta' will be used as the (fixed) step. Too large values (>> 1) may lead to divergency, however. When auto = no, chi-square computation is disabled.

Noise regularization is available through two techniques: standard Miller regularization and projection into the set of physically meaningful solutions. Miller regularization usually works better in scenes with complex and intrinsically smooth structures, but tends to impair convergency on images where most of the signal energy is concentrated in stellar images. The regularization parameter `alpha' describes the ratio between the observation noise and the maximum allowable power present in the solution's spatial high frequencies. For typical images, its value may be in the range of a few 1.E-3. A value of zero disables regularization by this mechanism. If `alpha' is set to zero and simultaneously the step `beta' is fixed as 1., the resulting algorithm is identical to the "modified VanCittert" or "Landweber" iteration, except that it uses conjugate directions instead of steepest-descent.

The inverse behavior is observed in what regards the projection technique, which in the current implementation takes the form of a positivity constraint. It has little effect on terrestrial images with strong signals, but is very effective in controlling sky background noise buildup in astronomical images with dark sky.

Standard regularization by smoothness constraint tends to impair restoration quality near edges (regions were there is a large intensity gradient), with generation of artifacts (ringing). This behavior may be minimized by weighting the regularization term, with weights that allow full regularization in smooth image regions, but damp regularization near edges. The edge information ("spatial activity") is obtained by comparing the measured local variance at each pixel, with the expected variance derived from a noise model. Local variance information can be derived either from the input degraded image (the first in the list), or from an external, independent image.

Task parameter `adap' control the spatial adaptivity mechanism. If set as "none", standard space-invariant smoothing will be performed. If set as "fixed", local variance information will be computed before the first iteration, and kept fixed along the iteration sequence. If set as "update", local variance information will be updated at each iteration, based on the current estimate.

The optional external image from which to derive local variance information is defined by task parameter `activity'. This image must have the same size as the input image(s). If using an external activity image, the usual procedure is to set adap="fixed". If not using an external image, use adap="update" instead.

The sensitivity of regularization weights to the local variance may be adjusted by task parameter `atune'. Larger values will increase sensitivity, smaller values decrease it. In the limit atune=0, no spatial adaptivity is performed at all.

The noise model used in spatial activity computations includes both Poisson and Gaussian terms, derived from task parameters `adu' and `noise'. These are also used to compute chi-squared. If no Poisson noise is present, set adu=0 and `noise' in DN instead of electrons. If no Gaussian noise is present, set noise=0. If the input image has no noise, set both parameters to zero. In this case, the chi-square value printed at output will be the average of squared residuals.

Some caution must be exercised in specifying the output image size. The FFT algorithm used by the task is faster when the axis sizes are composite numbers, faster yet when rich in factors of 2, and even faster with exact powers of two. So it may be worth to extend the edges of a, say, 200 X 200 image to 256 X 256 output size. For efficiency reasons, odd-sized output sizes will be trimmed off to even size.

Memory usage may be of concern, in particular when working with multiple input images. The maximum data memory M (in bytes) needed by the task can be estimated by

M = Nin + Npsf + 10 if (alpha > 0.) M = M + 2 if (spatial adaptivity) M = M + 1 if (edge extension) M = M + 1 if (masks) M = M + Nmask if (write intermediate result) M = M + 1 M = M * 4 * Npixwhere Nin is the number of input degraded images, Npsf the number of PSF images, Nmask the number of mask images, and Npix the number of pixels in the output image (assuming real numbers use 4 bytes each).

## PARAMETERS

- input [file name template/list]
- 2-d image section(s) to be restored.

- psf [file name template/list]
- 2-d PSF(s) image section(s).

- output [file name]
- Output restored image. Always type real, regardless of input image type.

- (model = "") [file name]
- Model image to start iterations. If left as a null string (""), no model image is read.

- (wmask = "") [file name template/list]
- Bad pixel mask(s) / weight image(s).

- (ilspars) [pset]
- Pset with regularization controls, algorithm controls and PSF specs.

- (niter = 10) [int, min=1]
- Number of iterations.

- (limchisq = 1.) [real]
- Chi-square for stopping iterations.

- (xsizeout,ysizeout = 0) [int, min=0]
- The size of the output image along each axis. A (default) value of 0 means that the output image will have the same size as the input image. If a non-zero value is specified, it must be larger than the size of the input image.

- (center = no) [boolean]
- Center the output image in the output array ?

- (nsave = 0) [int, min=0]
- The interval with which to write out intermediate results. If set to zero, no output of intermediate results is performed.

- (verbosity = 2) [int, min=0, max=2]
- Verbosity level. If set to 2, detailed information from the conjugate gradient computation is output at STDOUT. If set to 1, only current iteration number is output. Zero disables all output.

- (auto = yes) [boolean]
- Turn on conjugate gradients algorithm ? If set to "no", steepest-descent is used, if set to "yes", full conjugate gradient with line minimization is used instead.

- (beta = 1.0) [real, min=0.]
- Iteration gain. Only used in steepest-descent mode.

- (alpha = 0.0) [real, min=0.]
- Miller regularization parameter. If set to zero, disables this form of regularization.

- (positivity = yes) [bool]
- Turn on positivity constraint ?

- (adap = "update") [string]
- Allowed values: none | fixed | update. Type of spatial adaptivity. "none": no spatial adaptivity; "fixed": compute regularization weights only once, at start of iteration sequence; "update": update regularization weights at each iteration, based on current estimate.

- (activity = "") [file name]
- Image from which to derive spatial adaptivity information (local variance). If left as a null string (""), no activity image is read and the needed information is derived from the (first) input image.

- (atune = 1.0) [real]
- "Eye-model" tuning parameter.

- (adu = 1.0) [real, min=0.]
- Conversion constant from data numbers to electrons. Typical values are 7.5 for WF/PC and 1.0 for FOC. If only Gaussian noise is present, set adu=0.

- (noise = 0.) [real, min=0.]
- Additive Gaussian noise, or read-out noise, given in electrons. Typical values are 13 for WF/PC and 0 for FOC. If only Gaussian noise is present, noise must be specified in data number units.

- (nlpsf = yes) [boolean]
- Is(are) the PSF image(s) noiseless ? If PSFs are taken from an observed image, this parameter must be set to "no", in which case, a "pruning" noise filter will be used in their Fourier transforms. If the PSFs are synthetic and without noise, this parameter must be set to "yes".

- (px0, py= INDEF) [real, min=1.]
- Center coordinates, in pixels, of PSF in
`psf`image section(s). If either one, or both, are left as INDEF, the task will locate the maximum pixel value in the PSF image section(s), and use its coordinates instead. If set to -1, this instructs the task to not change the PSF image(s) in any way. This should be used when restoring multiple data sets that have relative shifts between them. Each PSF image should then be shifted to match its associated input image shifts.

- (mask = INDEF) [real, min=1.]
- PSF masking radius, in pixels. If INDEF, no masking is performed.

## EXAMPLES

## BUGS

## REFERENCES

This task was written by I.Busko

## SEE ALSO