18 THE AXIS COORDINATE SYSTEM

This section describes the concepts which an NDF’s axis coordinate system represents. The following section (§19) then goes on to consider how to access an NDF’s axis components, which hold information about this coordinate system.

18.1 Pixel Coordinates

Hitherto, an NDF has been considered simply as an N-dimensional array of pixels, addressed by a set of pixel indices. Since they are integer quantities, these indices cannot represent a continuous coordinate system, although the information stored in an NDF will almost always require that positions within it be describable to sub-pixel accuracy. For example, a calculation to determine the centroid position of a star in a 2-dimensional image will inevitably give rise to a non-integer result, for which a continuous ($x,y$) coordinate system will be required.

There are a number of ways in which a continuous coordinate system can be defined for a regular array of pixels. In the absence of other information, the NDF convention is to use a pixel coordinate system in which a pixel with indices ($i,j$) has its centre at the position:

and is taken to be one unit in extent in each dimension. Pixel (1,1) would therefore be centred at the position (0.5,0.5) and would have its “lower” and “upper” corners located at positions (0.0,0.0) and (1.0,1.0) respectively, as follows:

This makes it possible to refer to fractional pixel positions—in this case within a 2-dimensional array, although the principle can obviously be extended to other numbers of dimensions.

18.2 Axis Coordinates

The pixel coordinate system described above defines how to convert pixel indices into a set of continuous coordinates and therefore introduces a coordinate axis which runs along each dimension of the NDF, as follows:

The use of the pixel size to determine the units of these axes is rather restrictive, however, and in practice we may want to use more realistic physical units. This would allow a spectrum to be calibrated in wavelength, for instance, or the output from a plate-measuring machine to be related to axes calibrated in microns.

Of course, the pixel coordinate system is only the default choice, and is intended to be used only in the absence of other information. The NDF’s axis components are designed to hold the extra information needed to define more useful coordinate systems, so that realistic axes can be associated with a NDF, along with labels and units for these axes. The method used also allows for the possibility that an NDF’s pixels may not be square and that they may not be contiguous (i.e. that they may have gaps between them, or may overlap) when their positions are expressed in axis units. Statistical uncertainty in the pixel positions may also be represented, if present.

18.3 Axis Arrays

To define the pixel coordinate system in §18.1, we specified the location of each pixel by giving its centre position and width on each axis. Thus, for a given dimension, the pixel centre position C was derived from the corresponding pixel index $i$ according to the formula:

and its width W was given by:

An NDF’s axis coordinate system extends this idea by allowing each of these centre and width functions to be determined by values stored in a 1-dimensional array. These axis arrays then act as “look-up tables” which convert pixel indices into pixel centre coordinates and width values on each axis:

This allows a wide range of possible coordinate systems to be accommodated. A third axis variance array is also provided as a look-up table to convert pixel indices into variance estimates, which can be used to represent any possible statistical uncertainty in a pixel’s centre position.

18.4 Pixel Positions and Dimensions

If C${}_n$ and W${}_n$ represent the axis centre and width arrays for the n’th dimension of an NDF, then a pixel with index i in this dimension has its centre at coordinate C${}_n$(i) and has a width of W${}_n$(i) on the corresponding axis. It therefore extends along the axis from the point:

to the point:

In two dimensions the central ($x,y$) coordinate of a pixel with indices ($i,j$) would therefore be given by:

and its size would be:

The axis variance array is used to represent any statistical uncertainty in a pixel’s centre position and hence in the position of the pixel as a whole.¹⁶ Like the NDF’s main variance component (§8.9), its values are estimates of the mean squared error in the pixel’s position, so the value which would normally be quoted as the positional uncertainly (or used to plot error bars) is the square root of this value. Axis variance arrays may also be accessed directly as standard deviation values if required (see §19.10).

18.5 Default Axis Array Values

An important feature of each axis array is a set of default values which serve to define the axis coordinate system in the absence of complete information. In the simplest case (i.e. no information), this reduces to the pixel coordinate system discussed in §18.1. The following describes how default values are obtained for each axis array:

Centre:

If values are required for an axis centre array and none have been provided, then its values are set equal to $i-\frac{1}{2}$, where $i$ is the pixel’s index in the relevant dimension. Thus, if an NDF had pixel-index bounds (3:10) in a particular dimension, the default axis centre array values for this dimension would be:

2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5

Width:

If values are required for an axis width array and none have been defined, then its values are derived from the corresponding axis centre array by forming differences between the centre coordinates of the neighbouring pixels, i.e. the default width values are obtained as follows:¹⁷

$W_n\left(i\right)=\frac{1}{2}|C_n\left(i+1\right)-C_n\left(i-1\right)|$

This means that the default pixel widths match the local average spacing between pixel centres, which is usually appropriate. Note, however, that this does not guarantee that the pixels will be contiguous (i.e. that their edges will meet exactly) except in cases where the pixel centres are uniformly spaced (see §18.6).

Variance:

If no axis variance array values have been defined, then they default to zero, implying no uncertainty in the pixel centre positions.

18.6 Contiguous and Non-Contiguous Pixels

It is important to note that the meanings attached to an NDF’s axis arrays are defined rather precisely. In particular, note that the axis centre array specifies the position of the geometrical centre of a pixel, i.e. the mid-point between its edges, so that a pixel will always extend by an equal amount on either side of this position.

The edges of adjacent pixels will therefore only meet exactly (i.e. there will be no overlap or gap) if their centre and width values are related in the correct way. To be precise, adjacent pixels with indices $i$ and $i+1$ must have centre positions separated by half the sum of their widths if they are to be contiguous along a particular axis, so that:

With contiguous pixels (the normal case), this means that the axis centre and width arrays are not independent. In fact, either could be derived from the other to within a constant, but since this constant cannot be found without additional information, it is often necessary to store both arrays. However, an important exception occurs if the pixel centres are evenly spaced, because a convenient method then exists of deriving the width array from the centre array so that contiguous pixels always result. This is the method used to generate default axis width values when necessary (§18.5).

To avoid any potential ambiguity about the interpretation of axis array values and whether an NDF’s pixels should be considered contiguous or not, the following recommendations are given about the information which should be stored in axis arrays:

• If the default NDF pixel coordinate system is satisfactory, then it should be used and no axis coordinate information should be defined. In this case the pixels will always be contiguous.
• Otherwise, if the pixel centres are evenly spaced, then...
  • If the pixels are contiguous, the axis centre array should be assigned values but the associated width array may be left undefined.
  • If the pixels are not contiguous, both the axis centre and width arrays should be assigned values.
• Otherwise, if the pixel centres are un-evenly spaced, then both the axis centre and width arrays should always be assigned values.

18.7 Processing Axis Array Values

The method by which NDF axis arrays are processed should also reflect their meanings, as defined above. By way of illustration, suppose that a transformation of axis values is to be performed, so that each axis coordinate is converted to a new value by means of some non-linear function. Rather than simply applying this function to calculate new pixel centre locations from the old ones, the correct procedure is to transform the pixel edge locations and to derive new centre positions from these, as follows:

(1)

Obtain the relevant pixel centre and width arrays, accepting their default values if necessary.

(2)

From these, calculate the positions of the edges of each pixel.

(3)

Transform the edge positions using the non-linear transformation function.

(4)

Calculate new pixel centre positions (mid-way between the new edge positions) and store them in the NDF’s axis centre array.

(5)

Calculate associated pixel width values (from the difference in the pixel edge positions). Since the pixel centres will now be non-uniformly separated, these new width values should also be stored in the NDF’s axis width array.

(6)

If axis variance array values are available, then these should be propagated through the transformation function using the usual error-propagation formulae.

This procedure is necessary to ensure that the pixels remain contiguous (or non-contiguous, if appropriate) and that the new centre positions lie mid-way between the new pixel edge locations. Furthermore, the operations above can all be reversed if necessary to recover the original axis array values.

18.8 Axis Normalisation

One aspect of the axis coordinate system which has not yet been discussed is the property of axis normalisation, which is indicated by a logical normalisation flag associated with each axis. This flag does not affect the interpretation of the axis information itself, but instead determines how the NDF’s data and variance arrays should behave when the associated axis information is modified.

If the normalisation flag for an NDF axis is set to .TRUE., then it indicates that the NDF’s data values (and by implication its variance values) are normalised to the pixel width values for that axis. To give an example, suppose that a spectrum contains data values representing energy accumulated per unit of wavelength, with each pixel having a known spread in wavelength. In this case, the sum of each pixel’s data value multiplied by its width will give the total energy in any part of the spectrum. This is an important property which may need to be retained if the axis width values are altered for any reason (e.g. to apply an instrumental correction, or to allow for red-shift).

The axis normalisation flag indicates whether this type of normalisation should be preserved. If it is set to .TRUE., and the associated axis width values are modified, then each NDF data value should be multiplied by an appropriate factor so that its data $\times$ width product remains unchanged. If present, the variance values should also be corrected by multiplying by the square of this factor. In cases where more than one axis normalisation flag is set to .TRUE., the correction factors for each axis must be applied in turn.

If all the axis normalisation flags are set to .FALSE. (the default situation), then no changes to the data or variance components will be necessary if the axis width values are modified.