Rebecca Willett
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Transcript of Rebecca Willett
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