# References¶

This section of the documentation is devoted to providing references for the algorithms implemented in scikit-guess. Each paper comes with a link, a PDF where permitted, and any additional materials.

## Régressions et équations intégrales¶

The ideas proposed in this paper by Jean Jacquelin were the seed for this scikit.

### Concept¶

The concept behind this paper is that integrals and derivatives can be estimated through differentials and cumulative sums. The goal is to set up an integral or differential equation whose solution is the model function. If the right-hand side of such an equation can be expressed as a linear combination of integrals and derivatives of itself multiplied by some other pre-determined functions, we can estimate the numerical values of the equation’s terms. While the coefficients that make the equation work depend on the fitting parameters, the functions themselves do not. It is therefore possible to set up a simple linear regression for the coefficients based on the numerical approximations of the integrals and derivatives. The numerical approximation of integrals by cumulative sums tend to be more robust than the approximations of derivatives by differentials, so integral equations are generally preferred.

### Citation¶

 [Jacquelin] Jacquelin, “Régressions et équations intégrales,” Scribd, pp. 1–85, Jan. 2014.

Available online at https://www.scribd.com/doc/14674814/Regressions-et-equations-integrales.

A PDF is available with this documentation: Régressions et équations intégrales.

## Circle Fitting by Linear and Nonlinear Least Squares¶

This paper by Ian Coope demonstrates a way to linearize a non-linear least squares problem.

### Concept¶

Rather than solving the traditional non-linear least squares problem for n-dimensional circles, this paper proposes a change of variable that reduces the problem to a simple linear least squares. The change appears to yield more robust results in some cases. This is one of the multidimensional optimizations offered in the scikit.

### Citation¶

 [Coope] I. D. Coope, “Circle fitting by linear and nonlinear least squares,” Journal of Optimization Theory and Applications, vol. 76, no. 2, pp. 381–388, 1993.

Preprint available online at https://ir.canterbury.ac.nz/bitstream/handle/10092/11104/coope_report_no69_1992.pdf.

## Image Analysis with Rapid and Accurate Two-Dimensional Gaussian Fitting¶

This is one of two papers used to seed the idea for the n-dimensional Gaussian estimators.

### Concept¶

This paper introduces the idea of linearizing least squares fit to a the a two dimensional Gaussian function. It suggests using a weighted fit and performing thresholding on the image. The supplemental materials show a suggested asymmetrical weighting of

where is the standard deviation of the noise.

The paper also implies, but does not show, a cross-coupling term for rotated elliptical Gaussians.

### Citation¶

 [Anthony-Granick] Anthony and S. Granick, “Image Analysis with Rapid and Accurate Two-Dimensional Gaussian Fitting,” Langmuir, vol. 25, no. 14, pp. 8152–8160, 2009.

Supplemental materials (MATLAB implementation) available at https://pubs.acs.org/doi/10.1021/la900393v

### Supplement¶

A supplement deriving the linearized regression for the N-Dimensional case is provided for this paper and Star Centroiding Based on Fast Gaussian Fitting for Star Sensors. See the corresponding Star Centroiding Based on Fast Gaussian Fitting for Star Sensors section.

## Star Centroiding Based on Fast Gaussian Fitting for Star Sensors¶

This paper uses a similar linearized regression to Image Analysis with Rapid and Accurate Two-Dimensional Gaussian Fitting, but with materially different suggestions for weighting and thresholding, as well as a somewhat different intended application.

### Concept¶

This paper suggests a two-pass approach using a similar regression to Image Analysis with Rapid and Accurate Two-Dimensional Gaussian Fitting, but with an initial pass to find high-SNR pixels followed by a second pass to fine tune the results.

The suggested weighting derived here is the value of the pixel itself. This is the weighting used by the scikit by default.

This paper too only deals with ellipses without cross coupling and does not generalize to more than two dimensions.

### Citation¶

 [Wan-Wang-Wei-Li-Zhang] Wan, G. Wang, X. Wei, J. Li, and G. Zhang, “Star Centroiding Based on Fast Gaussian Fitting for Star Sensors,” Sensors, vol. 18, no. 9, p. 2836, 2018.

Available online at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6163372/

### Supplement¶

The two papers on Gaussian centroiding provide the basic idea for N-dimensional Gaussian fitting without quite getting there. The writeup below fills in a couple of the missing steps. The math is fairly rudimentary, with the main attraction being the vectorized implementation provided by skg.ngauss.