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Astronomical Data Analysis II
In the first article about Astronomical data analysis, we discussed about, “What is data analysis”, “What is astronomical data” and “Tools for astronomical data analysis”. We are going to present further details about Astronomical data analysis from this article.
Image and Signal Processing.
The major areas of application of image and signal processing include the following.
 Visualization: Seeing our data and signals in a different light is very often a revealing and fruitful thing to do.
 Filtering: A signal in the physical sciences rarely exists independently of noise, and noise removal is therefore a useful preliminary to data interpretation. More generally, data cleaning is needed, to bypass instrumental measurement artifacts, and even the inherent complexity of the data.
 Deconvolution: Signal “deblurring” is used for reasons similar to filtering, as a preliminary to signal interpretation. Motion deblurring is rarely important in astronomy, but removing the effects of atmospheric blurring, or quality of seeing, certainly is of importance.
 Compression: Longterm storage of astronomical data is important. A current trend is towards detectors accommodating everlarger image sizes. Research in astronomy is a cohesive but geographically distributed activity. All three facts point to the importance of effective and efficient compression technology.
 Mathematical morphology: Combinations of dilation and erosion operators, giving rise to opening and closing operations, in Boolean images and in greyscale images, allow for a truly very esthetic and immediately practical processing framework. The median function plays its role too in the context of these order and rank functions. Multiple scale mathematical morphology is an immediate generalization.
 Edge detection: Gradient information is not often of central importance in astronomical image analysis.
 Segmentation and pattern recognition: Dealing with object detection. In areas outside astronomy, the term feature selection is more normal than object detection.
 Multidimensional pattern recognition: General multidimensional spaces are analyzed by clustering methods, and by dimensionality mapping methods. Multiband images can be taken as a particular case.
 Hough and Radon transforms (leading to 3D tomography and other applications): Detection of alignments and curves is necessary for many classes of segmentation and feature analysis, and for the building of 3D representations of data. Gravitational lensing presents one area of potential application in astronomy imaging, although the problem of faint signal and strong noise is usually the most critical one.
In this article we will discuss about visualization (Transformation and Data Representation) of Astronomical data.
Visualization (Transformation and Data Representation)
Many different transforms are used in data processing. The goal of these transformations is to obtain a sparse representation of the data, and to pack most information into a small number of samples. Wavelets and related multiscale representations pervade all areas of signal processing. The reason for the success of wavelets is due to the fact that wavelet bases represent well a large class of signals. Therefore this allows us to detect roughly isotropic elements occurring at all spatial scales and locations.
This broad success of the wavelet transform is due to the fact that astronomical data generally gives rise to complex hierarchical structures, often described as fractals. Using multiscale approaches such as the wavelet transform, an image can be decomposed into components at different scales, and the wavelet transform is therefore welladapted to the study of
astronomical data.
Here are some existing transformation types.
 Fourier Analysis
 TimeFrequency Representation
 TimeScale Representation: The Wavelet Transform
 The Radon Transform
Fourier Analysis
The Fourier transform of a continuous function f (t) is defined by:
And the inverse Fourier transform is:
The discrete Fourier transform is given by:
And the inverse discrete Fourier transform is:
In the case of images (two variables), this is:
Since 𝑓̂(𝑢,𝑣) is generally complex, this can be written using its real and imaginary parts:
With:
It can also be written using its modulus and argument:
𝑓̂(𝑢,𝑣)^{2} is called the power spectrum, and
Two other related transforms are the cosine and the sine transforms. The discrete cosine transform is defined by:
With 𝑐(𝑖) = 1√2 when 𝑖 = 0 and 1 otherwise.
Other types of data transformation will be discussed in upcoming data analysis article series.
References:
 Handbook of Astronomical Data Analysis by JeanLuc Starck and Fionn Murtagh
~Pasan Muthugala~