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Since its inception in the eighties and early nineties, research in this field has grown enormously. Large telescopes all around the world are now equipped with integral field units, and two instruments of the future James Webb Space Telescope will have integral field spectroscopic capabilities. Nowadays, more effort is dedicated to refining techniques for reducing, analyzing and interpreting the data obtained with 3D spectrographs.

If the replicas do overlap, on the other hand, the function cannot be determined uniquely. Sometimes these terms are interchanged, however, or used to refer to uc. Thus, by the Convolution Theorem, f x is reconstructed in the spatial domain by a summation of regularly spaced sinc functions Figure 2. Observational procedures and data reduction 59 Figure 2.

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Reconstruction of a continuous function from regularly spaced samples, by convolution with a sinc function. Each grey curve is a sinc function centred at the appropriate sample point and normalized to the corresponding sample value cf. The thicker black curve shows the overall sum as a function of x. Equation 2. In the Fourier spectrum of the sampled signal, F u then overlaps its neighbouring replicas Figure 2. Turner fn x x a Figure 2. An undersampled frequency component and its lower frequency alias have the same values at regular sampling intervals, making them indistinguishable.

The Nyquist, or critical sampling, criterion requires the sampling rate to be high enough with respect to the signal bandwidth that such ambiguities cannot occur. The aliasing can be thought of as beating between highfrequency components of the signal and the sampling period itself. The only way to distinguish between these aliases is to disallow one of them the higher frequency in our case ; hence the requirement for band limitation.

Where undersampling occurs, the transfer of power from higher to lower frequencies creates artifacts in the reconstruction of f x from samples. Any super-Nyquist frequencies are not only unmeasurable, but contaminate otherwise good data. With irregular sampling patterns, the similarity between alias frequencies is broken — as can be seen from Figure 2.

Nevertheless, information is still lost if the sampling rate is too low, which unavoidably leads to reconstruction error. In general, the theory of reconstruction from irregular samples is considerably more involved than that discussed here see references in Section 2. Further information on regular sampling, including proofs of the theorems relied on here, can be found in Bracewell , or earlier editions. The image produced on a screen or detector by the resulting optical interference pattern depends on distance from the aperture.

For a rectangular aperture described by a 2D version of the box function discussed in Section 2. For a circular telescope mirror, it is the closely related Airy pattern, shown in Figure 2. A derivation can be found in Goodman or Hecht Turner a b Figure 2.

Theoretically, we would therefore need a FWHM of something like 20 times the pixel size to reach the Nyquist rate. Fortunately, in practice, we can approximate critical sampling with a less stringent criterion. Super-Nyquist frequencies are therefore heavily suppressed in amplitude, to Observational procedures and data reduction 63 2. This zero location corresponds to twice the Nyquist frequency divided by n i. Convolving F u with something much narrower than itself does not extend the frequency band appreciably, so the same sampling criterion applies. To second order, things are a bit more complicated.

This frequency extension is manifested as reconstruction error towards the edges of the data, similar to the oscillation seen at the ends of a Fourier series when approximating a sharp edge. The measured sample value is therefore the convolution of the telescope image at the relevant point with the pixel shape. Poisson noise is incoherent and spread over all frequencies, rather than contributing power at any particular frequency. The question is therefore how the noise level propagates through the resampling process.

When interpolating images, the value at a particular x is a weighted sum of sinc functions, one per sample see Equation 2. Since noise terms add in quadrature, each of the original sample values contributes to the total variance at x according to the square of its weight. Since the values of sinc2 x at unit intervals add up to one Bracewell, , the noise level in the interpolated image is the same as the original.

Noise therefore gets carried through 64 James E. In practice, of course, the noise level will vary over the image and the sinc function may allow bright peaks to dominate further out. Tapering the sinc function with an envelope such as a Gaussian or cosine multiplier provides a more robust and computationally feasible solution. This type of interpolant is good for well-sampled data, albeit relatively slow to compute unless look-up tables are used.

Nearest-neighbour zero-degree spline interpolation is the simplest and fastest to compute method.

Beyond that, sinc au goes negative and therefore reverses the phase of some frequencies. In terms of reproducing the analogue telescope image, the nearest-neighbour algorithm is about the least accurate method. Nevertheless, the values are guaranteed to be reasonable, since they are the same as the original samples: blockiness is an artifact, but a well-behaved and understood one. It should also be noted that this interpolant does not smooth the data, as each value comes from a single sample. In this example, the dark line shows the summation of three individual triangle functions, one per sample.

In each of the three cases above, the height at each point represents the relative contribution to the interpolated value at the position of the arrow. The interpolated values themselves are continuous, however, and follow the original curve more closely than in the nearest-neighbour case. Resampling at the original points clearly just reproduces the original values, with no smoothing at all.

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Resampling at the mid-points, on the other hand, is the same as averaging pairs of samples and gives maximal smoothing; this is exactly equivalent to sampling with pixels that are twice as large to begin with5 , so the values are guaranteed to be sensible even when undersampled. Finally, resampling at some arbitrary position is equivalent to initial sampling with pixels that have a spatial step in sensitivity from one half to the other Figure 2.

Its 5 Since the sampling rate is unchanged, the larger pixels would have to overlap, as for multiple dither positions.

In fact, the Central Limit Theorem Bracewell, asserts that the result of repeated convolution tends towards a Gaussian curve, so higher-order b-splines and their transforms are all Gaussian-like. Convolving sampled data with a cubic b-spline gives a well-behaved result, which is more accurate than nearest-neighbour or linear interpolants but which causes strong smoothing of the data.

It is also not strictly an interpolation because it does not pass through the original samples. Better results can be achieved using a cubic spline curve that does interpolate the data. Lehmann et al. It is therefore the best compromise for many applications involving reasonably well-sampled data. Polynomial interpolants can also be constructed that directly approximate the form of a sinc function, for use as convolution kernels.

These can again be simple and relatively compact, producing sinc-like results with less computation and less oscillation than a summation of sinc functions. Some other possible basis functions include wavelets, Fourier series, Taylor series and Gaussian summations. Irregular sampling It is possible to formulate more general theories for irregular sampling, but the relatively advanced mathematics is beyond the scope of this lecture for an overview, see Marvasti, Beutler, However, the vectors describing sample weights in the space of the basis functions are no longer orthogonal for nonuniform spacing, so reconstructing the signal can become an ill-conditioned problem.

For guaranteed stability, some methods require that the maximum separation between samples is no greater than the Nyquist interval. Reconstruction from irregular samples can be approached in more than one way. Strohmer, The spatial pixels of an image slicer form a grid that appears irregular in two or more dimensions, but which can be separated into a series of regular grids in 1D. For datasets that are only slightly non-uniform on large scales due to geometric distortions, it may be acceptable just to treat the sampling as regular, as long as the Nyquist criterion is still met, and then resample the interpolant at variable intervals corresponding to a regular grid in the appropriate system.

Observational procedures and data reduction 67 2. For accurate reconstruction with minimal smoothing, a tapered sinc function or smooth piecewise interpolant such as a cubic spline is likely to be the best option. Smooth spline-like interpolants with compact kernels are better behaved, but still prone to some oscillation. Convolution kernels that are positive everywhere are insensitive to such issues, but can cause strong smoothing, especially near the Nyquist limit. The ideal strategy is therefore to avoid interpolating altogether.

It may be necessary to interpolate in order to combine datasets or work with a square grid otherwise analysis would require special end-to-end treatment, incorporating basic processing. Unfortunately, sinc-like interpolation of undersampled data would introduce serious aliasing artifacts, whilst smooth, positive convolution kernels could cause excessive loss of detail.

The best compromise may therefore be to use simple nearest-neighbour or linear methods, or the related Drizzle algorithm Fruchter and Hook, These are relatively inaccurate in terms of reconstructing a smooth image, but ensure that values remain meaningful. They can, however, introduce their own peculiarities in the presence of image distortions, such as skipping nearest neighbours or variable smoothing.

## Development of compact integral field unit for spaceborne solar spectro-polarimeter

Whilst the above discussion considers sampling telescope images, the same principles apply to sampling a dispersed spectrum. The spectral PSF of a slit spectrograph or image slicer is a rectangular slit for extended sources or a narrow section of the telescope PSF, truncated by the slit for stars. This is less of an issue for spectra with intrinsically broad features, where the sampling may be good even though the instrumental PSF for narrow lines would be undersampled.

At the same time, we would like to preserve the integrity of raw data, which, amongst other things, means avoiding degradation through excessive 68 James E. Key IFU data reduction steps, in no particular order. The whole process should be optimized, depending on the project, to meet suitable signal-to-noise or resolution criteria. In the remainder of this lecture, I consider each operation in turn, roughly in the order of application.

Although details vary between instruments, the concepts are fairly generic. In the next lecture, I shall give some examples of how the steps can be combined in a complete reduction process. If the detector is read out in separate quadrants, those might need pasting together. Any overscan columns can be trimmed after subtraction, since they do not contain real data. If a standard bad-pixel mask is available for the detector, one may want to store that information with each exposure as part of a data quality array see Section 2.

A variance array can be created from the almost-raw pixel values if one intends to propagate error information Section 2.