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

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First principal component

Second principal component

Principal Components Analysis(aka KL Transform)

1. Compute sample covariance matrix (pXp)

2. Determine directions of greatest variance using eigenanalysis

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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

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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(np2)

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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 DeconvolutionAdvantages

• Not necessary to pick out “training” stars

• Potential to use prior knowledge of image structure/statistics

• Possible to estimate distended PSF features (e.g. ghosting effects)

• Potential to use information from multiple exposures

Disadvantages

• Computational complexity can be prohibitive

• Can be overkill if only PSF, and not deconvolved image, is desired

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

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Iterative error minimization

Minimize this function:

Jefferies & Christou, 1993

Simulation example

IterativeBlindDeconvolution

WeinerDeconvolution

MaximumEntropy

Deconvolution

Observations

Data from multiple exposures

H1y1 = Poisson

H2y2

H3y3

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