We use density ridges to detect filaments.Â
The formal definition for density ridges can be found in the following papers:
Chen, Yen-Chi, Christopher R. Genovese, and Larry Wasserman. ``Asymptotic Theory for Density Ridges." The Annals of Statistics 43, no. 5 (2015): 1896-1928. arXiv: 1406.5663
Genovese, Christopher R., Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. ``Nonparametric Ridge Estimation." The Annals of Statistics 42, no. 4 (2014): 1511-1545.
In general, finding ridges from a given function is very difficult.
However, when we want to find ridges from the kernel density estimator (KDE), a well-known density estimator, there exists a very efficient algorithm--the Subspace Constrained Mean Shift (SCMS) algorithm. This algorithm is originally proposed in the following paper:
Ozertem, Umut, and Deniz Erdogmus. "Locally Defined Principal Curves and Surfaces. " The Journal of Machine Learning Research 12 (2011): 1249-1286.
The statistical consistency for the ridge estimators (the ridges from the KDE) is given in
Genovese, Christopher R., Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman. ``Nonparametric ridge estimation." The Annals of Statistics 42, no. 4 (2014): 1511-1545.
More detailed asymptotic theory and a valid procedure for constructing confidence sets for ridges are given in
Chen, Yen-Chi, Christopher R. Genovese, and Larry Wasserman. ``Asymptotic Theory for Density Ridges." The Annals of Statistics 43, no. 5 (2015): 1896-1928. arXiv: 1406.5663
Similarly to the usual KDE, there is a smoothing bandwidth needed to be selected when we apply the SCMS.
The smoothing bandwidth controls the amount we want to smooth for each data point (particle--in the language of astronomy).
A general way to select this smoothing bandwidth is given in
Chen, Yen-Chi, Christopher R. Genovese, Shirley Ho, and Larry Wasserman. ``Optimal Ridge Detection using Coverage Risk." arXiv: 1506.02278
When we apply to the SDSS data, we use another approach to select the smoothing bandwidth, see
Chen, Yen-Chi, Shirley Ho, Peter E. Freeman, Christopher R. Genovese, and Larry Wasserman. ``Cosmic Web Reconstruction through Density Ridges: Method and Algorithm." arXiv: 1501.05303
An extension for the SCMS algorithm which allows for the image data and putting different weights for different data points (particles) is described in
Chen, Yen-Chi, Christopher R. Genovese, and Larry Wasserman. ``Generalized Mode and Ridge Estimation." arXiv: 1406.1803