Astronomical Techniques - Imaging and the Inverse Problem

We may take an image to be some mapping between properties of radiation from the sky and one, two, or three (sometimes even more) dimensions of a data structure. In most normal cases, we refer to a mapping of sky intensity onto an (x,y) grid, but many of the processes we discuss apply just as well to spectral images and more exotic data forms. This includes indirect images, where there is no optical formation of an image but one is constructed from interferometer measurements, photon counts through moving masks, fiber-array spectra, and so forth.

Quite generally, imaging takes the form of an inverse problem. Some true source distribution S is observed by a system with some response pattern (different from a δ-function) R and in the presence of noise N to give an observed image I. We may express this as a convolution equation I(x,y) = ∫ S(v,w) R(x-v,y-w) dv dw + N which we must sometimes consider to be further integrated within the area of each pixel, perhaps with some nonuniform internal weighting.

Often the convolution is understood so intuitively that it is not explicitly treated. We expect photographs to have large round images of bright stars, though the source distributions of these stars will be no larger than for faint ones. Our eyes accustom us to this, and it is second nature to consider what an object will look like at a given resolution. Complexities enter when the response R has lots of structure or wings very extended compared to its core profile. Such systems have introduced the need for deconvolution - inverting the convolution equation to estimate S given I and R. This may not be mathematically well-conditioned, so different approaches may be needed for various regimes. In principle, the problem lends itself to a Fourier approach, since convolution in the function domain becomes multiplication in the Fourier (spatial frequency) domain. Denoting the Fourier transform by lower case, we have (neglecting the noise term) i = sr and thus (in principle, again) s = i / r from which the inverse Fourier transform gives the source distribution. No matter what the textbook says, this is almost never useful in imaging applications. The reasons lie first in the noise behavior, and second in the fact that real response function have zeros, or near zeros, in their Fourier transforms. Thus certain spatial frequencies are almost pure noise in the data, and Fourier deconvolution will amplify this noise with no attendant signal.

There are many distinct algorithms used to approach this problem. Modified Fourier methods, such as the Wiener filter, use a smoothed or filtered Fourier transform to suppress noise amplification. Methods working directly in the Fourier domain are known collectively as linear. Radio astronomers have widely used the CLEAN algorithm, in which point sources are iteraftively removed from the image until the residuals are "adequately" close to the expected noise level. The Lucy-Richardson technique has found wide use in HST data reduction; this takes a guess image, compares its observed counterpart to the image, and uses the differences to improve the guess. Various more sophisticated techniques are being tested, most notably maximum-entropy and Bayesian methods. If we have extensive a priori knowledge about an image, we can do a better job of inferring the source distribution. For example, if we know that a globular cluster contains only stars whose images are identical except for a scale factor, we can derive the brightnesses of 105 stars per cluster much more accurately than we could without such knowledge. If we know (say from a short pre-refurbishment HST exposure) exactly where they are, we may do better yet. In general, the fewer parameters we need to determine at once, the more accurately they can be measured from a given set of data. Another example is optimal extraction from two-dimensional spectra - one can impose a constant spatial profile and smoothly varying location along the spectrograph slit to increase the signal-to-noise ratio of a spectrum by at least √ 2.

For use in many of these techniques, and in further analysis, it can be crucial to understand the noise. We have already discussed noise models for CCDs. Other detectors, such as Vidicon TV systems and photographic plates, may have their own behaviors, such as noise constant with intensity or slowly varying with intensity. It is often useful to propogate the error per pixel through subsequent analysis, always watching whether the pixels are still statistically independent (an assumption that breaks down during deconvolution or any rebinning operation, for example).

Image assessment and analysis
In approaching image data, we need ways of assessing its content - if you don't know what's there, you can't measure it. Overall statistics may be helpful in getting the sky background level (say from a histogram of intensity values) and range of useful data. For data of high dynamic range, multiple ways of display may all give a different view - positive and negative contrast, linear and logarithmic intensity mapping, histogram equalization mapping, grayscale and pseudocolor display. These are needed because many displays (and eyes) have less dynamic range than the detectors now do, and details can easily hide from view even when clearly measurable. Further techniques can reveal much in a decent image: we consider median filters and median windows, gradient operations, convolutions for optimal detection, or model subtraction.

It is also important to recognize artifacts - features present in the data but produced by the instrument, not the sky. Some frequent artifacts include:
• Interferometer aliasing and deconvolution artifacts
• Reflections
• Saturation and charge bleeding
• Spectral aliasing in undersampled data
• Diffraction structures (mostly radial spikes)
• Photographic blemishes (Eastman nebulae)
• Calibration (flat-field) flaws
• Interference fringes
• Cosmic-ray events
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