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A 2-D generalization of the Haar Transform which is used for the compression of astronomical images. The algorithm consists of dividing the $2^N\times 2^N$ image into blocks of $2\times 2$ pixels, calling the pixels in the block $a_{00}$, $a_{10}$, $a_{01}$, and $a_{11}$. For each block, compute the four coefficients

$\displaystyle h_0$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(a_{11}+a_{10}+a_{01}+a_{00})$  
$\displaystyle h_x$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(a_{11}+a_{10}-a_{01}-a_{00})$  
$\displaystyle h_y$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(a_{11}-a_{10}+a_{01}-a_{00})$  
$\displaystyle h_c$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(a_{11}-a_{10}-a_{01}+a_{00}).$  

Construct a $2^{N-1}\times 2^{N-1}$ image from the $h_0$ values, and repeat until only one $h_0$ value remains. The H-transform can be performed in place and requires about $16N^2/3$ additions for an $N\times N$ image.

See also Haar Transform


Capaccioli, M.; Held, E. V.; Lorenz, H.; Richter, G. M.; and Ziener, R. ``Application of an Adaptive Filtering Technique to Surface Photometry of Galaxies. I. The Method Tested on NGC 3379.'' Astron. Nachr. 309, 69-80, 1988.

Fritze, K.; Lange, M.; Möstle, G.; Oleak, H.; and Richter, G. M. ``A Scanning Microphotometer with an On-Line Data Reduction for Large Field Schmidt Plates.'' Astron. Nachr. 298, 189-196, 1977.

Richter, G. M. ``The Evaluation of Astronomical Photographs with the Automatic Area Photometer.'' Astron. Nachr. 299, 283-303, 1978.

White, R. L.; Postman, M.; and Lattanzi, M. G. ``Compression of the Guide Star Digitised Schmidt Plates.'' In Digitised Optical Sky Surveys: Proceedings of the Conference on ``Digitised Optical Sky Surveys'' held in Edinburgh, Scotland, 18-21 June 1991 (Ed. H. T. MacGillivray and E. B. Thompson). Dordrecht, Netherlands: Kluwer, pp. 167-175, 1992.

© 1996-9 Eric W. Weisstein