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// This library implements the algorithms and procedures described inMonthly Notices of the Royal Astronomical Society,
// Volume 511, Issue 2, April 2022, Pages 2008–2020, https://doi.org/10.1093/mnras/stac189 
// ( for a preprint, see https://arxiv.org/abs/2201.12466 ).

// Copyright(c) < 2021 > 
// <Benjamin Schulz> 
// Responsible for:
// The implementation of the library in C++, 
// the development of a parallelized RANSAC inspired algorithm that can removes outlier data in the function "focusposition_Regression",
// the implementation of a repeated median regression in the function "focusposition_Regression" instead of a simple regression,
// the implementation of various known outlier detection methods within the RANSAC 
// (MAD, S, Q, and T, biweight-midvariance estimators, Grubb's test for outliers, Peirce's criterion, percentage based midvariance)
// the implementation of the image ananysis algorithm developed by the user C.Y. Tan with the help of opencv and libfitsio libraries. 
// The algorithm generates a power spectrum in fourier 
// space which can be used in the curve fitting library for determining the focus point.

// <Stephen King> 
// Responsible for:
// The Algorithm idea for the hyperbolic fit in the internal function "Regression" with the linear regression and least squares comparison, first suggested in
// https://aptforum.com/phpbb/viewtopic.php?p=25998#p25998 

// <Jim Hunt>
// Responsible for:
// Some testing of the library,
// the algorithm idea for "findbackslash_Regression", first suggested in
// https://aptforum.com/phpbb/viewtopic.php?p=26265#p26265 

// <C.Y. Tan>  
// Responsible for:
// The development of the algorithm that analyzes the image data via Fourier transform in the function image::fouriertransform. 
// Its output can be fed into a slightly modified curve fitting algorithm.
// The suggestion in https://aptforum.com/phpbb/viewtopic.php?p=26471#p26471 to throw out outlier data by comparison of the error with the Standard-deviation.
// (Note that this old idea is now supplemented by more advanced methods, since the average and standard deviation are not robust.)

// The library makes use of an imaging analysis algorithm that was developed by C. Y. Tan (main author) and B. Schulz (second author).

// The library makes use of an algorithm for student's distribution, which can be found at
// Smiley W. Cheng, James C. Fu, Statistics & Probability Letters 1 (1983), 223-227

// The library also makes use of Peirce's outlier test. An algorithm for this method was developed in
// Gould, B. A, Astronomical Journal, vol. 4, iss. 83, p. 81 - 87 (1855).

// The library also has the possibility to use MAD, S, Q and T estimators. 
// These estimators are extensively described in 

// Peter J. Rousseeuw, Christophe Croux, Alternatives to the Median-Absolute Deviation
// J. of the Amer. Statistical Assoc. (Theory and Methods), 88 (1993),p. 1273,

// Christophe Croux and Peter J.Rousseeuw, Time-effcient algorithms for two highly robust estimators of scale, 
// In: Dodge Y., Whittaker J. (eds) Computational Statistics. Physica, Heidelberg, https ://doi.org/10.1007/978-3-662-26811-7_58

// According to Rousseeuw and Croux, the S estimator has an advantange over the MAD estimator because it can also work for asymmetric distributions.
// The same authors note that the Q estimator is better optimized than the S estimator for small sample sizes.
// 
// The library also can make use of the biweight midvariance estimator that was described in 
// T. C. Beers,K. Flynn and K. Gebhardt,  Astron. J. 100 (1),32 (1990)

// Within the Ransac, the offers the possibility to fit the data using Siegel's repeated median slope from 
// Siegel, Andrew, Technical Report No. 172, Series 2 By Department of Statistics Princeton University: Robust Regression Using Repeated Medians (September 1980)

// In practice, repeated median regression is a rather slow fitting method. 
// If median regression is not used, the RANSAC will use a faster linear regression algorithm for the 
// hyperbolic fit which was provided by Stephen King at https://aptforum.com/phpbb/viewtopic.php?p=25998#p25998).

// See 
// https://aptforum.com/phpbb/viewtopic.php?p=26587#p26587 for the early development history of this software. 
//
// A recent version of this software can be downloaded at
//
// https://github.com/bschulz81/hyperbolicfitdll/ 


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