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Extracting information from images and uv-data
European Radio Interferometry School Bonn, 13/09/2007
Andrei Lobanov
MPIfR Bonn
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Noise and measurement A. Lobanov
Assessing image quality
Noise in interferometry -- “irreducible” factors-- “controllable” factors
Practical estimates of the expected noise in an image
Quantifying the brightness distribution in an image-- information and its limits in interferometric images-- representing the observed structure; model fit-- error estimates for model fits
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Assessing image quality A. Lobanov
Fit to baseline visibilities:Fit to baseline visibilities:-- should represent adequately amplitudes, phases, and phase closures.
Residual flux:Residual flux:-- should be distributed smoothly, with a nearly zero mean and comparable positive and negative amplitudes.
Noise in the final image:Noise in the final image:-- should have a Gaussian distribution;-- should be approaching the thermal noise level, corrected for the bandwidth and time-average smearingand modified by self-calibration.
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Additive errors, -- system noise, interference, cross-talk, baseline-dependent errors
Multiplicative errors, -- uv-coverage, gain calibration errors, atmospheric and ionospheric errors
Convolution errors,
Position dependent errors-- pointing errors, bandwidth and time-averaging smearing
Noise in interferometric images A. Lobanov
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Noise in interferometric images A. Lobanov
Gridding and convolution
Averaging in frequency and time
Editing
Tapering and weighting
Deconvolution (CLEAN-ing)
Self-calibration
During data processing, the noise is modified by
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Thermal noise A. Lobanov
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Bandwidth smearing A. Lobanov
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Bandwidth smearing A. Lobanov
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Time average smearing A. Lobanov
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Time average smearing A. Lobanov
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Self-calibration A. Lobanov
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Gridding and convolution A. Lobanov
uv-coverage is incomplete and irregular: needs to be projected onto a regular (rectangular) grid
convolution: is used for interpolating between the observed distribution (u0, v0) and gridded distribution (uk, vk).
convolution function is chosen so that it suppresses the responses to sources outside the image (a suppression factor of ~100 is typically reached).
Simple recipe: avoid undersampling and bright sources near the edges of the image-- then most of the noise due to convolution can be
removed during deconvolution, and remaining (irreducible) addition to the image noise should be within
~0.01ΔIm
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Effect of uv-sampling A. Lobanov
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Image dimensions A. Lobanov
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Weighting and tapering A. Lobanov
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Deconvolution A. Lobanov
has to be used because the visibility data are incomplete...which leads to diffraction patterns in the image plane that cannot be removed either by direct Fourier inversion,or by linear methods
non-linear deconvolution has to be applied to correct for the diffraction patterns.
a number of complementary deconvolution algorithms exist (CLEAN, MEM, etc.) that can be applied to specific obser-vational setups and particular brightness distributions.
a successful deconvolution does not make a significantcontribution to the noise level
problems may arise with the distribution of noise.
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Excessive use of CLEAN A. Lobanov
Image “A”: properD. Range = 190
Image “B”: overCLEANedD. Range = 750
Estimated thermalnoise: 16mJy/beam
Noise in “A”: Gaussian
Noise in “B”: non-Gaussian
“B” vs. “A pixel flux
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Excessive use of CLEAN A. Lobanov
Sneg = 11mJy/beam Sneg = 4 mJy/beam Sneg = 4 mJy/beam
Sneg = 4 mJy/beam Sneg = 4 mJy/beam Sneg = 4 mJy/beam
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Gaussian noise A. Lobanov
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Expected noise in image A. Lobanov
Suppose that-- data are properly sampled and image dimensions are
properly set-- natural weighting is chosen-- no strong uv-tapering is applied -- no significant errors are introduced during self-
calibration and deconvolution
The resulting noise in the image should approach
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Practical suggestions A. Lobanov
Imaging doctrine in thirteen (sixteen) words:
The less you have to do to your data,the better you (and the data!) are.
In other words, one should try to minimize the number of operations required to achieve the expected noise level in an image.
The shortest path is usually the right one.
If the reduction process does not converge, some problem has most likely occurred early on.
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Information in images A. Lobanov
Ideally, information can be extracted from each pixel... but:-- pixel fidelity?-- physical significance?
Fitting by a set of a priori definedshapes (gaussians, discs, etc.) isa common remedy:
-- it provides a viable descriptionof the structure observed
-- but reduces the number ofdegrees of freedom of the description
200 pixels, 600 DF
7 Gaussians, 40 DF
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Information in images A. Lobanov
Examples of pixel-based extraction of information:
-- spectral index imaging
-- turnover frequency imaging
-- 2D correlations between pixels in different images
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Fitting a priori defined patterns A. Lobanov
Fitting measurements with a priori defined patterns is a general approach to inverse problems:-- design a model with a number of adjustable parameters-- use the model to predict measurements-- choose a figure-of-merit function to quantify deviation
between model predictions and measurements-- adjust the parameters to minimize the merit function
Goals:-- best-fit values for the parameters and their uncertainties -- a measure of the goodness-of-fit of the optimized model
Types of fitting:-- fitting in the image plane (IMFIT)-- fitting in the Fourier domain – model fitting
(UVFIT, MODELFIT, DIFMAP)
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Model fitting A. Lobanov
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Problems with least squares A. Lobanov
Chance of finding a local minimum instead of the global one
Slow convergence along the axes representing poorly-constrained parameters
Choosing the right number of parameters
-- a tedious task; requiresproper statistical modelling
-- F-test can be used, butthis depends on estimatingaccurately the DOF number of the dataset (number of independent samples)
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Errors are determined by a boundary of constant χ2 (taken at a desired confidence level) in the multidimensional space of all parameters fit
-- approximate method: Fisher matrix
Error estimation A. Lobanov
-- Fisher matrix often becomes degenerate-- Monte Carlo methods can be used generallly-- correlation between model fit parameters (e.g. flux density
and size) may cause problems
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Error estimates A. Lobanov
An analytical (first order) approximation can be given to relate uncertainties of the fit parameters to SNR of detection of a given model fit component
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Noise and resolution A. Lobanov
Noise and SNR play a fundamental role.
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Summary A. Lobanov
Noise and information in interferometric images are affected by a number of „irreducible“ and „controllable“factors (bandwidth and time-average smearing, gridding, convolution, tapering and weghting, deconvolution and self-calibration).
The net effect of these factors must be evaluated and understood, in order to be able to produce and analyze high-quality images.
Information can be extracted from every pixel of an image, but this sets extremely high requirements on image fidelity.
Analytical and numerical methods are available to quantify the information in images by fitting a priori defined patterns of brightness distribution.
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Further reading A. Lobanov
Most of the material covered in this lecture can be found inthe latest version of the „NRAO Summerschool Book“:
Check individual chapters for further references