Commentary on The Characterization Subtraction and Addition of

Commentary on “The Characterization, Subtraction, and Addition of Astronomical Images” by Robert Lupton Rebecca Willett

Focus of commentary • KL transform and data scarcity • Improved PSF estimation via blind deconvolution

Principal Components Analysis (aka KL Transform) 1. Compute sample covariance matrix (p. Xp) First principal component Second principal component 2. Determine directions of greatest variance using eigenanalysis

Principal Components Analysis (aka KL Transform) Key advantages: 1. Optimal linear method for dimensionality reduction 2. Model parameters computed directly from data 3. Reduction and expansion easy to compute

Data Scarcity • When using the KL to estimate the PSF, – p (dimension of data) = 120 – n (number of point sources observed) = 20 • p >> n • What effect does this have when performing PCA? – Sample covariance matrix not full rank – Need special care in implementation – Naïve computational complexity O(np 2)

Working around the data scarcity problem • Preprocess data by performing dimensionality reduction (Johnstone & Lu, 2004) • Use an EM algorithm to solve for k-term PSF; O(knp) complexity (Roweis 1998) • Balance between decorrelation and sparsity (Chennubholta & Jepson, 2001) PCA Sparse PCA

Blind Deconvolution Advantages Disadvantages • Not necessary to pick out “training” stars • Computational complexity can be prohibitive • Potential to use prior knowledge of image structure/statistics • Can be overkill if only PSF, and not deconvolved image, is desired • Possible to estimate distended PSF features (e. g. ghosting effects) • Potential to use information from multiple exposures

Example of blind deconvolution: modified Richardson-Lucy 1. Start with initial intensity image estimate and initial PSF estimate 2. R-L update of intensity given PSF 3. R-L update of PSF estimate given intensity 4. Goto 2 (depends on good initial estimates) Tsumuraya, Miura, & Baba 1993

Iterative error minimization Minimize this function: Jefferies & Christou, 1993

Simulation example Observations Weiner Deconvolution Maximum Entropy Deconvolution Iterative Blind Deconvolution

Data from multiple exposures y 1 = Poisson H 1 y 2 H 2 y 3 H 3 If Hi = H. Si, where H is the imager PSF and Si is a known shift operator, then we can use multiple exposures to more accurately estimate H.

Takeaway messages • Exercise caution when using the KL transform to estimate the PSF – Avoid computing sample covariance matrix – Consider iterative, low computational complexity methods • Blind deconvolution indirectly estimates PSF – Uses prior knowledge of image structure/statistics – Requires less arbitrary user input – Can estimate non-local PSF components