### WCSMOSAIC

Tiles a group of NDFs using World Co-ordinate System information

#### Description:

This application aligns and rebins a group of input NDFs into a single output NDF. It differs from WCSALIGN in both the algorithm used, and in the requirements placed on the input NDFs. WCSMOSAIC requires that the transformation from pixel to WCS co-ordinates be defined in each input NDF, but (unlike WCSALIGN) the inverse transformation from WCS to pixel co-ordinates need not be defined. For instance, this means that WCSMOSAIC can process data in which the WCS position of each input pixel is defined via a look-up table rather than an analytical expression. Note however, that the WCS information in the reference NDF (see Parameter REF) must have a defined inverse transformation.

The WCSMOSAIC algorithm proceeds as follows. First, the output NDF is filled with zeros. An associated array of weights (one weight for each output pixel) is created and is also filled with zeros. Each input NDF is then processed in turn. For each pixel in the current input NDF, the corresponding transformed position in the output NDF is found (based on the WCS information in both NDFs). The input pixel value is then divided up between a small group of output pixels centred on this central output position. The method used for choosing the fraction of the input pixel value assigned to each output pixel is determined by the METHOD and PARAMS parameters. Each of the affected output pixel values is then incremented by its allocated fraction of the input pixel value. The corresponding weight values are incremented by the fractions used (that is, if 0.25 of an input pixel is assigned to an output pixel, the weight for the output pixel is incremented by 0.25). Once all pixels in the current input NDF have been rebinned into the output NDF in this way, the algorithm proceeds to rebin the next input NDF in the same way. Once all input NDFs have been processed, output pixels which have a weight less than the value given by Parameter WLIM are set bad. The output NDF may then optionally (see Parameter NORM) be normalised by dividing it by the weights array. This normalisation of the output NDF takes account of any difference in the number of pixels contributing to each output pixel, and also removes artefacts which may be produced by aliasing between the input and output pixel grids. Thus each output pixel value is a weighted mean of the input pixel values from which it receives contributions. This means that the units of the output NDF are the same as the input NDF. In particular, any difference between the input and output pixel sizes is ignored, resulting in the total input data sum being preserved only if the input and output NDFs have equal pixel sizes. However, an option exists to scale the input values before use so that the total data sum in each input NDF is preserved even if the input and output pixel sizes differ (see Parameter CONSERVE).

If the input NDFs contain variances, then these are propagated to the output. Alternatively, output variances can be generated from the spread of input values contributing to each output pixel (see Parameter GENVAR). Any input variances can also be used to weight the input data (see Parameter VARIANCE). By default, all input data are given equal weight. An additional weight for each NDF can be specified using Parameter WEIGHTS.

The transformations needed to produce alignment are derived from the co-ordinate system information stored in the WCS components of the supplied NDFs. For each input NDF, alignment is first attempted in the current co-ordinate Frame  of the reference NDF. If this fails, alignment is attempted in the current co-ordinate Frame of the input NDF. If this fails, alignment occurs in the pixel co-ordinate Frame. A message indicating which Frame alignment was achieved in is displayed.

#### Usage:

wcsmosaic in out lbnd ubnd ref

#### Parameters:

The positional accuracy required, as a number of pixels. For highly non-linear projections, a recursive algorithm is used in which successively smaller regions of the projection are fitted with a least-squares linear transformation. If such a transformation results in a maximum positional error greater than the value supplied for ACC (in pixels), then a smaller region is used. High accuracy is paid for by longer run times. [0.05]
Determines the co-ordinate system in which each input NDF is aligned with the reference NDF. If TRUE, alignment is performed in the co-ordinate system described by the current Frame of the WCS FrameSet in the reference NDF. If FALSE, alignment is performed in the co-ordinate system specified by the following set of WCS attributes in the reference NDF: AlignSystem AlignStdOfRest, AlignOffset, AlignSpecOffset, AlignSideBand, AlignTimeScale. The AST library provides fixed defaults for all these. So for instance, AlignSystem defaults to ICRS for celestial axes and Wavelength for spectral axes, meaning that celestial axes will be aligned in ICRS and spectral axes in wavelength, by default. Similarly, AlignStdOfRest defaults to Heliocentric, meaning that by default spectral axes will be aligned in the Heliocentric rest frame.

As an example, if you are mosaicing two spectra which both use radio velocity as the current WCS, but which have different rest frequencies, then setting ALIGNREF to TRUE will cause alignment to be performed in radio velocity, meaning that the differences in rest frequency are ignored. That is, a channel with 10 Km/s in the input is mapping onto the channel with 10 km/s in the output. If ALIGNREF is FALSE (and no value has been set for the AlignSystem attribute in the reference WCS), then alignment will be performed in wavelength, meaning that the different rest frequencies cause an additional shift. That is, a channel with 10 Km/s in the input will be mapping onto which ever output channel has the same wavelength, taking into account the different rest frequencies.

As another example, consider mosaicing two maps which both have (azimuth,elevation) axes. If ALIGNREF is TRUE, then any given (az,el) values in one image will be mapped onto the exact same (az,el) values in the other image, regardless of whether the two images were taken at the same time. But if ALIGNREF is FALSE, then a given (az,el) value in one image will be mapped onto pixel that has the same ICRS co-ordinates in the other image (since AlignSystem default to ICRS for celestial axes). Thus any different in the observation time of the two images will result in an additional shift.

As yet another example, consider mosaicking two spectra which are both in frequency with respect to the LSRK, but which refer to different points on the sky. If ALIGNREF is TRUE, then a given LSRK frequency in one spectrum will be mapped onto the exact same LSRK frequency in the other image, regardless of the different sky positions. But if ALIGNREF is FALSE, then a given input frequency will first be converted to Heliocentric frequency (the default value for AlignStdOfRest is “Heliocentric”), and will be mapped onto the output channel that has the same Heliocentric frequency. Thus the difference in sky positions will result in an additional shift. [FALSE]

If set TRUE, then the output pixel values will be scaled in such a way as to preserve the total data value in a feature on the sky. The scaling factor is the ratio of the output pixel size to the input pixel size. This option can only be used if the Mapping is successfully approximated by one or more linear transformations. Thus an error will be reported if it used when the ACC parameter is set to zero (which stops the use of linear approximations), or if the Mapping is too non-linear to be approximated by a piece-wise linear transformation. The ratio of output to input pixel size is evaluated once for each panel of the piece-wise linear approximation to the Mapping, and is assumed to be constant for all output pixels in the panel. This parameter is ignored if the NORM parameter is set FALSE. [TRUE]
If TRUE, output variances are generated based on the spread of input pixel values contributing to each output pixel. Any input variances then have no effect on the output variances (although input variances will still be used to weight the input data if the VARIANCE parameter is set TRUE). If GENVAR is set FALSE, the output variances are based on the variances in the input NDFs, so long as all input NDFs contain variances (otherwise the output NDF will not contain any variances). If a null (!) value is supplied, then a value of FALSE is adopted if and only if all the input NDFs have variance components (TRUE is used otherwise). [FALSE]
A group of input NDFs (of any dimensionality). This should be given as a comma-separated list, in which each list element can be one of the following options.
• An NDF name, optionally containing wild-cards and/or regular expressions ("$\ast$", "?", "[a-z]" etc.).

• The name of a text file, preceded by an up-arrow character "^". Each line in the text file should contain a comma-separated list of elements, each of which can in turn be an NDF name (with optional wild-cards, etc.), or another file specification (preceded by an up-arrow). Comments can be included in the file by commencing lines with a hash character "#".

If the value supplied for this parameter ends with a hyphen, then you are re-prompted for further input until a value is given which does not end with a hyphen. All the NDFs given in this way are concatenated into a single group.

An array of values giving the lower pixel-index bound on each axis for the output NDF. The suggested default values just encompass all the input data. A null value (!) also results in these same defaults being used. [!]
A value which specifies an initial scale size in pixels for the adaptive algorithm which approximates non-linear Mappings  with piece-wise linear transformations. If MAXPIX is larger than sny dimension of the region of the output grid being used, a first attempt will be made to approximate the Mapping by a linear transformation over the entire output region. If a smaller value is used, the output region will first be divided into subregions whose size does not exceed MAXPIX pixels in any dimension, and then attempts will be made at approximation. [1000]
The method to use when dividing an input pixel value between a group of neighbouring output pixels. For details on these schemes, see the description of AST_REBINx in SUN/210. METHOD can take the following values.
• "Bilinear" –- The input pixel value is divided bi-linearly between the four nearest output pixels. This produces smoother output NDFs than the nearest-neighbour scheme, but is marginally slower.

• "Nearest" –- The input pixel value is assigned completely to the single nearest output pixel.

• "Sinc" –- Uses the $\mathrm{\text{sinc}}\left(\pi x\right)$ kernel, where $x$ is the pixel offset from the transformed input pixel centre, and $\mathrm{\text{sinc}}\left(z\right)=sin\left(z\right)/z$. Use of this scheme is not recommended.

• "SincSinc" –- Uses the $\mathrm{\text{sinc}}\left(\pi x\right)\mathrm{\text{sinc}}\left(k\pi x\right)$ kernel. This is a valuable general-purpose scheme, intermediate in its visual effect on NDFs between the bilinear and nearest-neighbour schemes.

• "SincCos" –- Uses the $\mathrm{\text{sinc}}\left(\pi x\right)cos\left(k\pi x\right)$ kernel. It gives similar results to the "Sincsinc" scheme.

• "SincGauss" –- Uses the $\mathrm{\text{sinc}}\left(\pi x\right){e}^{-k{x}^{2}}$ kernel. Good results can be obtained by matching the FWHM of the envelope function to the point-spread function of the input data (see Parameter PARAMS).

• "Somb" –- Uses the $\mathrm{\text{somb}}\left(\pi x\right)$ kernel, where $\mathrm{\text{somb}}\left(z\right)=2\ast {J}_{1}\left(z\right)/z$. ${J}_{1}$ is the first-order Bessel function of the first kind. This scheme is similar to the "Sinc" scheme.

• "SombCos" –- Uses the $\mathrm{\text{somb}}\left(\pi x\right)cos\left(k\pi x\right)$ kernel. This scheme is similar to the "SincCos" scheme.

• "Gauss" –- Uses the ${e}^{-k{x}^{2}}$ kernel. The FWHM of the Gaussian is given by Parameter PARAMS(2), and the point at which to truncate the Gaussian to zero is given by Parameter PARAMS(1).

All methods propagate variances from input to output, but the variance estimates produced by schemes other than nearest neighbour need to be treated with care since the spatial smoothing produced by these methods introduces correlations in the variance estimates. Also, the degree of smoothing produced varies across the NDF. This is because a sample taken at a pixel centre will have no contributions from the neighbouring pixels, whereas a sample taken at the corner of a pixel will have equal contributions from all four neighbouring pixels, resulting in greater smoothing and lower noise. This effect can produce complex Moiré patterns in the output variance estimates, resulting from the interference of the spatial frequencies in the sample positions and in the pixel-centre positions. For these reasons, if you want to use the output variances, you are generally safer using nearest-neighbour interpolation. The initial default is "SincSinc". [current value]

In general, each output pixel contains contributions from multiple input pixel values, and the number of input pixels contributing to each output pixel will vary from pixel to pixel. If NORM is set TRUE (the default), then each output value is normalised by dividing it by the number of contributing input pixels, resulting in each output value being the weighted mean of the contibuting input values. However, if NORM is set FALSE, this normalisation is not applied. See also Parameter CONSERVE. Setting NORM to FALSE and VARIANCE to TRUE results in an error being reported. [TRUE]
##### OUT = NDF (Write)
The output NDF. If a null (!) value is supplied, WCSMOSAIC will terminate early without creating an output cube, but without reporting an error. Note, the pixel bounds which the output cube would have had will still be written to output Parameters LBOUND and UBOUND, even if a null value is supplied for OUT.
##### PARAMS( 2 ) = _DOUBLE (Read)
An optional array which consists of additional parameters required by the Sinc, SincSinc, SincCos, SincGauss, Somb, SombCos and Gauss methods.

PARAMS(1) is required by all the above schemes. It is used to specify how many output pixels on either side of the central output pixel are to receive contribution from the corresponding input pixel. Typically, a value of 2 is appropriate and the minimum allowed value is 1 (i.e. one pixel on each side). A value of zero or fewer indicates that a suitable number of pixels should be calculated automatically. [0]

PARAMS(2) is required only by the Gauss, SombCos, SincSinc, SincCos, and SincGauss schemes. For the SombCos, SincSinc and SincCos schemes, it specifies the number of output pixels at which the envelope of the function goes to zero. The minimum value is 1.0, and the run-time default value is  2.0. For the Gauss and SincGauss scheme, it specifies the full-width at half-maximum (FWHM) of the Gaussian envelope measured in output pixels. The minimum value is 0.1, and the run-time default is 1.0. []

The NDF to which all the input NDFs are to be aligned. If a null value is supplied for this parameter, the first NDF supplied for Parameter IN is used. The WCS information in this NDF must have a defined inverse transformation (from WCS co-ordinates to pixel co-ordinates). [!]
An array of values giving the upper pixel-index bound on each axis for the output NDF. The suggested default values just encompass all the input data. A null value (!) also results in these same defaults being used. [!]
If TRUE, then any input VARIANCE components in the input NDFs are used to weight the input data (the weight used for each data value is the reciprocal of the variance). If FALSE, all input data is given equal weight. Note, some applications (such as CCDPACK:MAKEMOS) use a parameter named USEVAR to determine both whether input variances are used to weights input data values, and also how to calculate output variances. However, WCSMOSAIC uses the VARIANCE parameter only for the first of these purposes (determining whether to weight the input data). The second purpose (determining how to create output variances) is fulfilled by the GENVAR parameter. [FALSE]
An optional group of numerical weights, one for each of the input NDFs specified by parameter IN. If VARIANCE is TRUE, the weight assigned to each input pixel is the value supplied in this group correspoinding to the appropriate input NDF, divided by the variance of the pixel value. An error is reported if the number of supplied weights does not equal the number of supplied input NDFs. [!]
This parameter specifies the minimum number of good pixels that must contribute to an output pixel for the output pixel to be valid. Note, fractional values are allowed. If a value less than 1.0E-10 is supplied, a value of 1.0E-10 is used. [1.0E-10]

#### Results Parameters

##### FLBND( ) = _DOUBLE (Write)
The lower bounds of the bounding box enclosing the output NDF in the current WCS Frame. The number of elements in this parameter is equal to the number of axes in the current WCS Frame. Celestial axis values will be in units of radians.
##### FUBND( ) = _DOUBLE (Write)
The upper bounds of the bounding box enclosing the output NDF in the current WCS Frame. The number of elements in this parameter is equal to the number of axes in the current WCS Frame. Celestial axis values will be in units of radians.
##### LBOUND() = _INTEGER (Write)
The lower pixel bounds of the output NDF. Note, values will be written to this output parameter even if a null value is supplied for Parameter OUT.
##### UBOUND() = _INTEGER (Write)
The upper pixel bounds of the output NDF. Note, values will be written to this output parameter even if a null value is supplied for Parameter OUT.

#### Examples:

wcsmosaic m51$\ast$ mosaic lbnd=! accept
This example rebins all the NDFs with names starting with the string "m51" in the current directory so that they are aligned with the first input NDF, and combines them all into a single output NDF called mosaic. The output NDF is just big enough to contain all the pixels in all the input NDFs.

#### Notes:

• WCS information (including the current co-ordinate Frame) is propagated from the reference NDF to the output NDF.

• QUALITY is not propagated from input to output.

• There are different facts reported, their verbosity depending on the current message-reporting level set by environment variable MSG_FILTER. If this is set to QUIET, no information will be displayed while the command is executing. When the filtering level is at least as verbose as NORMAL, the interpolation method being used will be displayed. If set to VERBOSE, the name of each input NDF will also be displayed as it is processed.

#### Related Applications

KAPPA: WCSFRAME, WCSALIGN, REGRID; CCDPACK: TRANNDF.