### WCSALIGN

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

#### Description:

This application resamples or rebins a group of input NDFs , producing corresponding output NDFs which are aligned pixel-for-pixel with a specified reference NDF, or POLPACK catalogue (see Parameter REFCAT).

If an input NDF has more pixel axes than the reference NDF, then the extra pixel axes are retained unchanged in the output NDF. Thus, for instance, if an input RA/Dec/velocity cube is aligned with a reference two-dimensional galactic-longitude/latitude image, the output NDF will be a galactic-longitude/latitude/velocity cube.

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.

Two algorithms are available for determining the output pixel values: resampling and rebinning (the method used is determined by the REBIN parameter).

Two methods exist for determining the bounds of the output NDFs. First you can give values for Parameters LBND and UBND which are then used as the pixel index bounds for all output NDFs. Second, if a null value is given for LBND or UBND, default values are generated separately for each output NDF so that the output NDF just encloses the entire area covered by the corresponding input NDF. Using the first method will ensure that all output NDFs have the same pixel origin, and so the resulting NDFs can be directly compared. However, this may result in the output NDFs being larger than necessary. In general, the second method results in smaller NDFs being produced, in less time. However, the output NDFs will have differing pixel origins which need to be taken into account when comparing the aligned NDFs.

#### Usage:

wcsalign in out lbnd ubnd ref

#### Parameters:

This controls what happens if an error occurs whilst processing one of the input NDFs. If a FALSE value is supplied for ABORT, then the error message will be displayed, but the application will attempt to process any remaining input NDFs. If a TRUE value is supplied for ABORT, then the error message will be displayed, and the application will abort. [FALSE]
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 larger run times. [0.5]
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 aligning 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 aligning 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 aligning 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 differecen 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. The dynamic default is TRUE if rebinning, and FALSE if resampling (see Parameter REBIN). []
A group of input NDFs (of any dimensionality). This should be given as a comma-separated list, in which each list element can be:
• 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 minus sign "-", 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.

If INSITU is set to TRUE, then no output NDFs are created. Instead, the pixel origin of each input NDF is modified in order to align the input NDFs with the reference NDF (which is a much faster operation than a full resampling). This can only be done if the mapping from input pixel co-ordinates to reference pixel co-ordinates is a simple integer pixel shift of origin. If this is not the case an error will be reported when the input is processed (what happens then is controlled by the ABORT parameter). Also, in-situ alignment is only possible if null values are supplied for LBND and UBND. [FALSE]
An array of values giving the lower pixel-index bound on each axis for the output NDFs. The number of values supplied should equal the number of axes in the reference NDF. The given values are used for all output NDFs. If a null value (!) is given for this parameter or for Parameter UBND, then separate default values are calculated for each output NDF which result in the output NDF just encompassing the corresponding input NDF. The suggested defaults are the lower pixel-index bounds from the reference NDF, if supplied (see Parameter REF).
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 any 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 sampling the input pixel values (if resampling), or dividing an input pixel value up between a group of neighbouring output pixels (if rebinning). For details of these schemes, see the descriptions of routines AST_RESAMPLEx and AST_REBINx in SUN/210. METHOD can take the following values.
• "Bilinear" –- When resampling, the output pixel values are calculated by bi-linear interpolation among the four nearest pixels values in the input NDF. When rebinning, the input pixel value is divided up bi-linearly between the four nearest output pixels. Produces smoother output NDFs than the nearest-neighbour scheme, but is marginally slower.

• "Nearest" –- When resampling, the output pixel values are assigned the value of the single nearest input pixel. When rebinning, 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 interpolation point (resampling) or transformed input pixel centre (rebinning), 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. A valuable general-purpose scheme, intermediate in its visual effect on NDFs between the bi-linear and nearest-neighbour schemes.

• "SincCos" –- Uses the $\mathrm{\text{sinc}}\left(\pi x\right)cos\left(k\pi x\right)$ kernel. 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 $x$ is the pixel offset from the interpolation point (resampling) or transformed input pixel centre (rebinning), and $\mathrm{\text{somb}}\left(z\right)=2\ast {J}_{1}\left(z\right)/z$. ${J}_{1}$ 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 interpolation 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]

##### OUT = NDF (Write)
A group of output NDFs corresponding one-for-one with the list of input NDFs given for Parameter IN. This should be given as a comma-separated list, in which each list element can be:
• an NDF name. If the name contains an asterisk character "$\ast$", the name of the corresponding input NDF (without directory or file suffix) is substituted for the asterisk (for instance, "$\ast$_al" causes the output NDF name to be formed by appending the string "_al" to the corresponding input NDF name). Input NDF names can also be edited by including original and replacement strings between vertical bars after the NDF name (for instance, $\ast$_al$|$b4$|$B1$|$ causes any occurrence of the string "B4" in the input NDF name to be replaced by the string "B1" before appending the string "_al" to the result).

• 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 editing, 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 hyphen. All the NDFs given in this way are concatenated into a single group.

This parameter is only accessed if the INSITU parameter is FALSE.

##### 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 pixels are to contribute to the interpolated result on either side of the interpolation or binning point in each dimension. 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 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 schemes, 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. On astronomical NDFs and spectra, good results are often obtained by approximately matching the FWHM of the envelope function, given by PARAMS(2), to the point-spread function of the input data. []

Determines the algorithm used to calculate the output pixel values. If a TRUE value is given, a rebinning algorithm is used. Otherwise, a resampling algorithm is used. See the “Choice of Algorithm” topic below. The initial default is FALSE. [current value]
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. This parameter is only used if no catalogue is supplied for Parameter REFCAT.
A POLPACK catalogue defining the WCS to which all the input NDFs are to be aligned. If a null value is supplied for this parameter, the WCS will be obtained from an NDF using Parameter REF. [!]
An array of values giving the upper pixel-index bound on each axis for the output NDFs. The number of values supplied should equal the number of axes in the reference NDF. The given values are used for all output NDFs. If a null value (!) is given for this parameter or for Parameter LBND, then separate default values are calculated for each output NDF which result in the output NDF just encompassing the corresponding input NDF. The suggested defaults are the upper pixel-index bounds from the reference NDF, if supplied (see Parameter REF).
This parameter is only used if REBIN is set TRUE. It specifies the minimum number of good pixels which must contribute to an output pixel for the output pixel to be valid. Note, fractional values are allowed. A null (!) value causes a very small positive value to be used resulting in output pixels being set bad only if they receive no significant contribution from any input pixel. [!]

#### Examples:

wcsalign image1 image1_al ref=image2 accept
This example resamples the NDF called image1 so that it is aligned with the NDF call image2, putting the output in image1_al. The output image has the same pixel-index bounds as image2 and inherits WCS information from image2.
wcsalign m51$\ast$ $\ast$_al lbnd=! accept
This example resamples all the NDFs with names starting with the string "m51" in the current directory so that they are aligned with the first input NDF. The output NDFs have the same names as the input NDFs, but extended with the string "_al". Each output NDF is just big enough to contain all the pixels in the corresponding input NDF.
wcsalign ^in.lis ^out.lis lbnd=! accept
This example is like the previous example, except that the names of the input NDFs are read from the text file in.lis, and the names of the corresponding output NDFs are read from text file out.lis.

#### Choice of Algorithm

The algorithm used to produce the output image is determined by the REBIN parameter, and is based either on resampling the output image or rebinning the input image.

The resampling algorithm steps through every pixel in the output image, sampling the input image at the corresponding position and storing the sampled input value in the output pixel. The method used for sampling the input image is determined by the METHOD parameter. The rebinning algorithm steps through every pixel in the input image, dividing the input pixel value between a group of neighbouring output pixels, incrementing these output pixel values by their allocated share of the input pixel value, and finally normalising each output value by the total number of contributing input values. The way in which the input sample is divided between the output pixels is determined by the METHOD parameter.

Both algorithms produce an output in which the each pixel value is the weighted mean of the near-by input values, and so do not alter the mean pixel values associated with a source, even if the pixel size changes. Thus the total data sum in a source will change if the input and output pixel sizes differ. However, if the CONSERVE parameter is set TRUE, the output values are scaled by the ratio of the output to input pixel size, so that the total data sum in a source is preserved.

A difference between resampling and rebinning is that resampling guarantees to fill the output image with good pixel values (assuming the input image is filled with good input pixel values), whereas holes can be left by the rebinning algorithm if the output image has smaller pixels than the input image. Such holes occur at output pixels which receive no contributions from any input pixels, and will be filled with the value zero in the output image. If this problem occurs the solution is probably to change the width of the pixel spreading function by assigning a larger value to PARAMS(1) and/or PARAMS(2) (depending on the specific METHOD value being used).

Both algorithms have the capability to introduce artefacts into the output image. These have various causes described below.

• Particularly sharp features in the input can cause rings around the corresponding features in the output image. This can be minimised by suitable settings for the METHOD and PARAMS parameters. In general such rings can be minimised by using a wider interpolation kernel (if resampling) or spreading function (if rebinning), at the cost of degraded resolution.

• The approximation of the Mapping using a piece-wise linear transformation (controlled by Parameter ACC) can produce artefacts at the joints between the panels of the approximation. They are caused by the discontinuities between the adjacent panels of the approximation, and can be minimised by reducing the value assigned to the ACC parameter.

#### Notes:

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

• QUALITY is propagated from input to output only if Parameter METHOD is set to "Nearest" and REBIN is set to FALSE.

#### Related Applications

KAPPA: WCSFRAME, REGRID; CCDPACK: TRANNDF.