rebecca willett
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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. First principal component. Second principal component. - PowerPoint PPT PresentationTRANSCRIPT
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Commentary on “The Characterization, Subtraction, and
Addition of Astronomical Images” by Robert Lupton
Rebecca Willett
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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
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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
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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
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Simulation example
IterativeBlindDeconvolution
WeinerDeconvolution
MaximumEntropy
Deconvolution
Observations
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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.
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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