### MAKEMOS

Make a mosaic by combining and (optionally) normalising a set of images

#### Description:

This is a comprehensive application for combining a set of images (normally representing overlapping coverage of an object) into a single mosaic. It addresses the problems of (a) combining a sequence of separate data sets into a single image and (b) optionally normalising each image so that they match each other in regions where they overlap. Mutual alignment of the separate images is not performed by this application and must be addressed beforehand (although images may be aligned to the nearest pixel simply by shifting their pixel origin).

MAKEMOS registers the set of images supplied by matching their pixel indices and then forms a mosaic by combining the separate input pixel values at each location using a nominated data-combination method (by default, it takes the median). The resulting mosaic is of sufficient extent to accommodate all the input data, with any output data pixels which do not receive values from the input being set to the "bad" pixel value. Account is taken of variance information associated with the input images, and all calculations are optimally weighted to minimise the output noise. Output variance estimates for the final mosaic may also be produced.

Forming a mosaic in this way will normally be successful only so long as the input data are mutually consistent. Unfortunately, this is often not the case, since data frequently have differing effective exposure times and background levels which give discontinuities in the final mosaic. Thus, MAKEMOS also addresses the problem of normalising the input images to make them mutually consistent. It does this by optionally applying optimised multiplicative and/or additive corrections (termed scale-factor and zero-point corrections) to each image before forming the mosaic. These optimised corrections are determined by inter-comparing the input images in pairs, using the regions where they overlap to determine the relative scale-factor and/or zero-point difference between each pair. A self-consistent set of corrections is then found which, when applied to each input image, will best eliminate these observed differences and give a smooth mosaic.

makemos in out

#### Parameters:

The fraction of extreme values to remove before combining input data if the "trimmed mean" data combination method is selected for producing the output mosaic (see the METHOD parameter). A fraction alpha (approximately) of the available values is removed from each extreme. This may take values in the range 0 to 0.5. [0.2]
This parameter controls the use of statistical error (variance) information contained in the input images when they are inter-compared in pairs to derive scale-factor or zero-point corrections. It is only used if either SCALE or ZERO is set to TRUE and if two or more of the input images contain variance information (a "reference image" also counts, if supplied). In this case, if CMPVAR is set to TRUE, then variance information is used to correctly weight the input data whenever a pair of input images are inter-compared and both have variance information available.

The default behaviour is to use variance information during inter-comparisons. This may be suppressed by setting CMPVAR to FALSE, which sometimes gives faster execution without greatly affecting the result (also see the "Algorithms Used" section). However, if input data with similar values have widely differing variance values within the same input image, then use of input variance information is recommended (this could happen, for instance, if an input image is the result of a previous mosaic-ing process). [TRUE]

The name of the file used to output the scale and zero-point corrections (see SCALE and ZERO parameters). This file can be read by the DRIZZLE task. If the file already exists, it is overwritten. If a null (!) value is supplied, or if SCALE and ZERO are both set to FALSE, no file is written. [!]
If GENVAR is set to TRUE and all the input images supplied contain statistical error (variance) information, then variance information will also be calculated for the output mosaic image, provided that USEVAR is also TRUE.

Otherwise if GENVAR is TRUE and either USEVAR is FALSE or some of the input images do not contain error information, then output variances will be generated using the natural variations in the input data. Obviously this method should only be used if there are many input datasets, which also provide good coverage of the output area. If this option is chosen any regions of the output image that have only one input value will have their associated variances set bad.

The default for this parameter depends on the presence of error information in the input images. If all have error information then the default is TRUE, otherwise it is FALSE.

[DYNAMIC]

##### IN = LITERAL (Read and [optionally] Write)
A list of the names of the input images which are to be combined into a mosaic. The image names should be separated by commas and may include wildcards.

The input images are normally accessed only for reading. However, if the MODIFY parameter is set to TRUE (and scale-factor or zero-point corrections are being calculated) then each of the "input" images will be modified by applying the calculated corrections.

If a TRUE value is given for this parameter (the default), then the names of all the images supplied as input will be listed (and will be recorded in the logfile if this is enabled). Otherwise, this listing will be omitted. [TRUE]
Name of the CCDPACK logfile. If a null (!) value is given for this parameter, then no logfile will be written, regardless of the value of the LOGTO parameter.

If the logging system has been initialised using CCDSETUP, then the value specified there will be used. Otherwise, the default is "CCDPACK.LOG". [CCDPACK.LOG]

Every CCDPACK application has the ability to log its output for future reference as well as for display on the terminal. This parameter controls this process, and may be set to any unique abbreviation of the following:
• TERMINAL – Send output to the terminal only

• LOGFILE – Send output to the logfile only (see the LOGFILE parameter)

• BOTH – Send output to both the terminal and the logfile

• NEITHER – Produce no output at all

If the logging system has been initialised using CCDSETUP, then the value specified there will be used. Otherwise, the default is "BOTH". [BOTH]

Upper limit for input data values which may contribute to the output mosaic if the "threshold" data combination method is selected (see the METHOD parameter). [Maximum real value]
This parameter specifies the maximum number of iterations to be used when inter-comparing pairs of input image data arrays to determine their relative scale-factor and/or zero-point. It is only used if (a) both the SCALE and ZERO parameters have been set to TRUE, or (b) SCALE has been set to TRUE and statistical error (variance) information obtained from the input images is being used to weight the data during the inter-comparison. In other cases the inter-comparison operation is not iterative.

If the specified number of iterations is exceeded without achieving the accuracy required by the settings of the TOLS and TOLZ parameters, then a warning message will be issued, but the results will still be used. The value given for MAXIT must be at least one. [20]

The method to be used to combine the input images’ data values to form the output mosaic. This may be set to any unique abbreviation of the following:
• MEAN – Mean of the input data values

• MEDIAN – Weighted median of the input data values

• TRIMMED – An "alpha trimmed mean" in which a fraction alpha of the values are removed from each extreme

• MODE – An iteratively "sigma clipped" mean which approximates to the modal value

• SIGMA – A sigma clipped mean

• THRESHOLD – Mean with values above and below given limits removed

• MINMAX – Mean with the highest and lowest values removed

• BROADENED – A broadened median (the mean of a small number of central values)

• CLIPMED – A sigma clipped median
• FASTMED – Unweighted median of the input data values

[MEDIAN]

Lower limit for input data values which may contribute to the output mosaic if the "threshold" data combination method is selected (see the METHOD parameter). [Minimum real value]
By default, the images supplied via the IN parameter are regarded as "input" images and will not be modified. However, if scale-factor or zero-point corrections are being calculated (see the SCALE and ZERO parameters), then giving a TRUE value for MODIFY indicates that these images are themselves to be modified by applying the calculated corrections before the output mosaic is formed.

This facility provides a means of applying corrections to individual images (e.g. to mutually normalise them) without necessarily also combining them into a mosaic. It may also be useful if several invocations of MAKEMOS are to be made with different parameter settings; by specifying MODIFY=TRUE for the first invocation, scale-factor or zero-point corrections may be applied to normalise the input data so that this need not be repeated on each invocation.

WARNING: Caution should be exercised if setting MODIFY to TRUE, as information about the uncorrected data values of the "input" images will not be retained. [FALSE]

Maximum number of refining iterations used if the "mode" data combination method is selected (see the METHOD parameter). [7]
This parameter specifies the "optimum number of overlaps" which an image should have with its neighbours and controls the number of inter-comparisons made between pairs of overlapping images when determining scale-factor or zero-point corrections (see the SCALE and ZERO parameters).

The need for this parameter arises because when multiple input images are supplied there may be a large number of potential pair-wise overlaps between them. To prevent them all being used, which may take far longer than is justified, this set of potential overlaps is reduced by elimination, starting with the smallest ones (as measured by the number of overlapping pixels) and continuing until no more overlaps can be removed without reducing the number of overlaps of any image below the value given for OPTOV. In practice, this means that each image will end up with about (although not exactly) OPTOV overlaps with its neighbours, with the largest overlaps being preferred.

Note that although this algorithm is effective in reducing the number of overlaps, it is not guaranteed always to result in a set of overlaps which allow the optimum set of corrections to be calculated. In practice, problems from this cause are unlikely unless unusual patterns of image overlap are involved, but they may be solved by increasing the value of OVOPT and/or constructing the required mosaic in pieces by running MAKEMOS several times on different sets of input images.

In some cases, reducing the value of OVOPT may reduce the number of inter-comparisons made, and hence reduce the execution time, but if too few inter-comparisons are made, there is a risk that the corrections obtained may not be the best possible.

This parameter is only used if SCALE or ZERO is set to TRUE. [3]

##### OUT = image (Write)
Name of the image to contain the output mosaic. This is normally mandatory. However, if the "input" images are being modified (by setting the MODIFY parameter to TRUE), then it may optionally be omitted by supplying a null value (!). In this case, no output mosaic will be formed.
If a TRUE value is given for this parameter (the default), then the data type of the output mosaic image will be derived from that of the input image with the highest precision, so that the input data type will be "preserved" in the output image. Alternatively, if a FALSE value is given, then the output image will be given an appropriate floating point data type.

When using integer input data, the former option is useful for minimising the storage space required for large mosaics, while the latter typically permits a wider output dynamic range when necessary. A wide dynamic range is particularly important if a large range of scale factor corrections are being applied (as when combining images with a wide range of exposure times).

If a global value has been set up for this parameter using CCDSETUP, then that value will be used. [TRUE]

If scale-factor and/or zero-point corrections are being applied (see the SCALE and ZERO parameters) then, by default, these are normalised so that the median corrections are unity and zero respectively. However, if an image is given via the REF parameter (so as to over-ride its default null value), then scale-factor and zero-point corrections will instead be adjusted so that the corrected data are normalised to the "reference image" supplied.

This provides a means of retaining the calibration of a set of data, even when corrections are being applied, by nominating a reference image which is to remain unchanged. It also allows the output mosaic to be normalised to any externally-calibrated image with which it overlaps, and hence allows a calibration to be transferred from one set of data to another.

If the image supplied via the REF parameter is one of those supplied as input via the IN parameter, then this serves to identify which of the input images should be used as a reference, to which the others will be adjusted. In this case, the scale-factor and/or zero-point corrections applied to the nominated input image will be set to one and zero, and the corrections for the others will be adjusted accordingly.

Alternatively, if the reference image does not appear as one of the input images, then it will be included as an additional set of data in the inter-comparisons made between overlapping images and will be used to normalise the corrections obtained (so that the output mosaic is normalised to it). However, it will not itself contribute to the output mosaic in this case. [!]

This parameter specifies whether MAKEMOS should attempt to adjust the input data values by applying scale-factor (i.e. multiplicative) corrections before combining them into a mosaic. This would be appropriate, for instance, if a series of images had been obtained with differing exposure times; to combine them without correction would yield a mosaic with discontinuities at the image edges where the data values differ.

If SCALE is set to TRUE, then MAKEMOS will inter-compare the images supplied as input and will estimate the relative scale-factor between selected pairs of input data arrays where they overlap. From this information, a global set of multiplicative corrections will be derived which make the input data as mutually consistent as possible. These corrections will be applied to the input data before combining them into a mosaic.

Calculation of scale-factor corrections may also be combined with the use of zero-point corrections (see the ZERO parameter). By default, no scale-factor corrections are applied. [FALSE]

Number of standard deviations at which to reject values if the "mode", "sigma" or "clipmed" data combination methods are selected (see the METHOD parameter). This value must be positive. [4.0]
A positive "sky noise suppression factor" used to control the effects of sky noise when pairs of input images are inter-compared to determine their relative scale-factor. It is intended to prevent the resulting scale-factor estimate being biased by the many similar values present in the "sky background" of typical astronomical data. SKYSUP controls an algorithm which reduces the weight given to data where there is a high density of points with the same value, in order to suppress this effect. It is only used if a scale factor is being estimated (i.e. if SCALE is TRUE).

A SKYSUP value of unity can often be effective, but a value set by the approximate ratio of sky pixels to useful object pixels (i.e. those containing non-sky signal) in a "typical" image overlap region will usually be better. The precise value is not critical. A value of zero disables the sky noise suppression algorithm completely. The default value for SKYSUP is 10$\ast$$\ast$(n/2.0), where n is the number of significant dimensions in the output mosaic. Hence, for a 2-dimensional image, it will default to 10 which is normally reasonable for CCD frames of extended objects such as galaxies (a larger value, say 100, may give slightly better results for star fields). [10$\ast$$\ast$(n/2.0)]

Title for the output mosaic image. [Output from MAKEMOS]
This parameter defines the accuracy tolerance to be achieved when inter-comparing pairs of input image data arrays to determine their relative scale-factor. It is only used if the inter-comparison is to be performed iteratively, which will be the case if (a) both the SCALE and ZERO parameters have been set to TRUE, or (b) SCALE has been set to TRUE and statistical error (variance) information obtained from the input images is being used to weight the data during the inter-comparison.

The value given for TOLS specifies the tolerable fractional error in the estimation of the relative scale-factor between any pair of input images. This value must be positive. [0.001]

This parameter defines the accuracy tolerance to be achieved when inter-comparing pairs of input image data arrays to determine their relative zero-points. It is only used if the inter-comparison is to be performed iteratively, which will be the case if both the SCALE and ZERO parameters have been set to TRUE.

The value given for TOLZ specifies the tolerable absolute error in the estimation of the relative zero-point between any pair of input images whose relative scale-factor is unity. If the relative scale-factor is also being estimated, then the value used is multiplied by this relative scale-factor estimate (which reflects the fact that an image with a larger data range can tolerate a larger error in estimating its zero-point). The TOLS value supplied must be positive. [0.05]

The value of this parameter specifies whether statistical error (variance) information contained in the input images should be used to weight the input data when they are combined to produce the output mosaic. This parameter is only used if all the input images contain variance information, in which case the default behaviour is to use this information to correctly weight the data values being combined. If output variances are to be generated (specified by the GENVAR parameter) then this parameter (and GENVAR) should be set TRUE.

If insufficient input variance information is available, or if USEVAR is set to FALSE, then weights are instead derived from the scale-factor corrections applied to each image (see the WEIGHTS parameter for details); unit weight is used if no scale-factor corrections are being applied. Alternatively, explicit weights may be given for each input image via the WEIGHTS parameter.

If you want to add estimated variances to the output image (based on the natural variations of the input images) and all your input images contain variances then you will need to set this parameter FALSE (see GENVAR).

[TRUE]

##### WEIGHTS( ) = _REAL (Read)
A set of positive weighting factors to be used to weight the input images when they are combined. If this parameter is used, then one value should be given for each input image and the values should be supplied in the same order as the input images. If a null (!) value is given (the default) then a set of weights will be generated internally - these will normally all be unity unless scale-factor corrections are being applied (see the SCALE parameter), in which case the reciprocal of the scale factor correction for each input image is used as its weight. This corresponds to the assumption that variance is proportional to data value in each input image.

This parameter is only used if the USEVAR parameter is set to FALSE or if one or more of the input images does not contain variance information. Otherwise, the input variance values are used to weight the input data when they are combined. [!]

This parameter specifies whether MAKEMOS should attempt to adjust the input data values by applying zero-point (i.e. additive) corrections before combining them into a mosaic. This would be appropriate, for instance, if a series of images had been obtained with differing background (sky) values; to combine them without correction would yield a mosaic with discontinuities at the image edges where the data values differ.

If ZERO is set to TRUE, then MAKEMOS will inter-compare the images supplied as input and will estimate the relative zero-point difference between selected pairs of input data arrays where they overlap. From this information, a global set of additive corrections will be derived which make the input data as mutually consistent as possible. These corrections will be applied to the input data before they are combined into a mosaic.

Calculation of zero-point corrections may also be combined with the use of scale-factor corrections (see the SCALE parameter). By default, no zero-point corrections are applied. [FALSE]

#### Examples:

makemos ’$\ast$’ mymos
Combines the set of images matching the wild-card "$\ast$" into a single mosaic called mymos. By default, no normalisation corrections are applied to the input data, which are combined by taking the median in regions where several input images overlap.
makemos in=’"a,b,c,d"’ out=combined zero
Combines the four overlapping input images a, b, c and d into a single mosaic called combined. Optimised zero-point corrections are derived and applied to the data before combining them so as to make them as mutually consistent as possible. This helps to eliminate unwanted discontinuities in the output mosaic.
makemos ’"a,b,c,d"’ out=combined scale
Combines the four images a, b, c and d as above, but makes optimised corrections to the scale factor of each (i.e. multiplies each by an appropriate constant) before they are combined. This would be appropriate if, for instance, the input data were CCD frames acquired using different exposure times and had subsequently had their sky background removed.
makemos in=’frame$\ast$’ out=result scale zero
Combines the set of input images matching the wild-card "frame$\ast$" into a single mosaic called result. Optimised scale factor and zero point corrections are applied before combining the data. This would be appropriate if, for instance, the input data had been acquired using different exposure times and also had different levels of sky background.
makemos in="frame$\ast$" out=result scale zero modify
This is identical to the previous example, except that in addition to forming the output result, the MODIFY parameter causes all the input images to be modified using the same optimised corrections as are applied when forming the mosaic, thus mutually normalising all the separate images. Note that this feature should be used with care, as information about the original normalisation of the input data will be lost. When MODIFY is specified, a null value "!" may be given for the OUT parameter if an output mosaic is not actually required.
makemos ’"a,b,c,d"’ result scale zero ref=b
This example merges the four input images a, b, c and d into a mosaic called result. In calculating the optimised scale factor and zero point corrections to apply, b is regarded as a "reference image" and the other images are normalised to it. This means that if b has previously been calibrated, then the output mosaic will inherit this calibration.
makemos ’"a,b,c,d"’ result scale zero ref=e
This example is identical to that above, except that the "reference image" e is not one of the input images and will not form part of the output mosaic. Nevertheless, the scale factor and zero point corrections applied will be such that all the input images are normalised to it (the reference image must overlap with at least one of the input images). Thus, if e has been calibrated, this calibration will be transferred to the output mosaic (note that if MODIFY is specified, then the calibration could also be transferred to each of the input images).
makemos "frame$\ast$" mosaic nopreserve nogenvar method=minmax skysup=0
This example illustrates some of the less commonly used MAKEMOS options. nopreserve causes the output data type to be a floating point type rather than preserving the input data type, nogenvar prevents generation of an output variance array (possibly to save space with a large mosaic), method=minmax indicates that output pixels are to be calculated by taking the mean of input pixels after discarding the lowest and highest values, and skysup=0 is used to disable the sky noise suppression algorithm (perhaps for data which contain few sky pixels).

#### Algorithms Used

Some of the algorithms used by MAKEMOS require a little explanation. The first of these is used to inter-compare overlapping regions of the input images to determine their relative scale-factor and zero-point difference (in the most general case). In effect, this algorithm has to fit a straight line to a scatter plot representing the pixel values in the two overlapping images.

Rather than use a conventional least-squares fit for this purpose, which would be sensitive to spurious data, a fit based on minimisation of the sum of the absolute values of the residuals is used instead. This is considerably more robust. It also allows the residuals to be defined by the perpendicular distance of each point from the fitted line, rather than the vertical distance used in conventional least squares. In turn, this removes the distinction between dependent and independent variables and allows the statistical uncertainty on both axes (described by an error ellipse) to be properly taken into account along with other weighting factors used to implement sky noise suppression.

In general, this fitting algorithm is iterative and is controlled via the MAXIT, TOLS and TOLZ parameters which specify the convergence criteria. However, in some important cases the fit can be obtained in a single pass, with consequent savings in execution time. This occurs if:

• Only zero-point corrections are being determined, or

• Only scale-factor corrections are being determined and no input variance information is being used to weight the inter-comparison process (see the CMPVAR parameter).

The second stage of normalisation involves a global optimisation process which seeks to determine the best corrections to be applied to each input image. The algorithm which performs this task makes a guess at the best corrections to apply and then calculates the scale-factor and/or zero-point differences which would remain between each pair of overlapping images if they were corrected in this way. These corrections are then adjusted until the weighted sum of squares of the remaining differences is minimised. The weights used in this process are derived from error estimates produced by the earlier (inter-comparison) algorithm. This allows information about the required corrections to be optimally combined from many overlaps, even in cases where individual overlaps may be small and contain inadequate information on their own.

The algorithm used for combining the separate input images into a mosaic requires no special explanation, except to note that it is designed to operate on large mosaics without making excessive demands on system resources such as memory. It does this by partitioning the mosaic into small regions for processing.

#### Behaviour of parameters

Most parameters retain their current value as default. The "current" value is the value assigned on the last run of the application. If the application has not been run then the "intrinsic" defaults, as shown in the parameter help, apply. The exceptions to this rule are:
• SKYSUP – dynamically defaulted

• GENVAR – dynamically defaulted
• SCALE – always FALSE
• ZERO – always FALSE
• MODIFY – always FALSE
• TITLE – always "Output from MAKEMOS"

Retaining parameter values has the advantage of allowing you to define the default behaviour of the application but does mean that additional care needs to be taken when using the application on new datasets/different devices, or after a break of sometime. The intrinsic default behaviour of the application may be restored by using the RESET keyword on the command line.

Certain parameters (LOGTO, LOGFILE and PRESERVE) have global values. These global values will always take precedence, except when an assignment is made on the command line. Global values may be set and reset using the CCDSETUP and CCDCLEAR commands.

#### Implementation Status:

• MAKEMOS supports "bad" pixel values and all non-complex data types, with arithmetic being performed using the appropriate floating point type. It can process images with any number of dimensions. The DATA, TITLE and VARIANCE components of an image are directly supported, with the AXIS, HISTORY, LABEL and UNITS components and all extensions being propagated from the first input image supplied (note that AXIS values, if present, will normally be extrapolated as a result of propagation to the output mosaic, which will typically have a larger extent than any of the input images).