V. MARCO TEORICO
5.5. ANTISÉPTICOS Y DESINFECTANTES
Before the imaging process, the data will have undergone calibration, modifying the in- terferometric equation to account for a number of issues, explained briefly in Section 1.1.1 and in more detail later in Chapter 3 (Radio Interferometric Calibration Pipeline for CO- BRaS). Now a description on imaging with deconvolution is given.
The interferometric equation (Equation 1.8) from Section 1.1.1 is three dimensional, but can be approximated to two dimensions when the 2πi wn component is much less than unity. This is the coplanar baseline assumption, and is valid when the baseline vectors trace out a concentric circle with the Earth’s rotation axis (Thompson 1999). This is appli- cable to East-West interferometers. However, one-dimensional East-West interferometers suffer a lack of visibilities in the v direction. For two-dimensional interferometers, certain conditions will allow for a two dimensional representation of the visibility equation. If the |l| and |m| terms in Equation 1.8 are small enough i.e. a small field of view (as l and m describe the source structure), the n term can be treated as approximately zero (Equation 1.14 is from Thompson 1999)
p
1 − l2 − m2− 1w ≈ −1
2 l
2 + m2 w ≈ 0. (1.14)
Ignoring the w term can induce errors from aberration into the observations, similar to field curvature in an optical telescope (Anita Richards, MERLIN user guide). Removing the w dependence reduces the measurement equation to
Vν(u, v) = Z l Z m Bν(l, m) e−2πi (lu + mv)dl dm. (1.15)
where Vν(u, v) and Bν(l, m) are now functions of frequency, i.e. not assuming monochro-
matic radiation as in Equation 1.8 for the simplified two element interferometer example. Reducing the interferometric equation to 2D has a significant benefit in aperture syn- thesis. The computational cost of the 2D Fourier transform is more feasible than the transform of the 3D relation. This is applicable for observations with a small field of view, or where the source of interest is located at the phase centre. However, for COBRaS this is not true, with wide-field imaging necessary for the science goals (see Sections 1.3.3 and 1.3.4), the problems and solutions to wide-field and wide-band imaging are discussed later in Chapter 3, but involve the 2D relation.
The u, v coverage is the response of the interferometer, whose Fourier transform is the complete sky brightness only if the u, v coverage is completely filled. In practice this is not true as is evident in Figure 1.5. Therefore a sampling function S (u, v) is introduced with the visibility function in Equation 1.15 where S (u, v) is zero where no data exists in the u, v plane. Directly inverting Equation 1.15 to find the sky brightness, or as is the new case, the dirty image BD(l, m), produces
BνD(l, m) = Z
u
Z
v
Vν(u, v) S (u, v) e2πi (ul + vm)du dv. (1.16)
It is possible to apply convolution theory to the variables in Equation 1.16. Convolution theory states that for the Fourier transform of two functions f and g,
f ∗ g = F−1{F (f ) · F (g)} , (1.17)
where ∗ denotes convolution and F represents the Fourier transform. This relation is used for the terms on the right hand side of Equation 1.16 giving
BνD = F−1{F (Vν) · F (S)}
BνD = Bν ∗ P , (1.18)
where the inverse Fourier transform of Equation 1.15 is used for Bν and
P (l, m) = Z
u
Z
v
S (u, v) e2πi (ul + vm)du dv (1.19)
is the point spread function (PSF), also known as the synthesised beam or dirty beam. To calulcate the Fourier transform for Equation 1.18, Fast Fourier Transforms (FFT) are used with the data distributed over a grid. This is preferred to the direct Fourier transform which in O formalism (denoting processing time or number of operations i.e. performance) is O N4 compared to O N2 for the FFT (Briggs et al. 1999). There-
fore considering FFTs and the incomplete u, v coverage, gridding methods or non-linear methods are necessary for imaging in radio interferometry.
The gridding of the visibility data for the FFT enables the synthesised beam shape to be manipulated by different weighting schemes. These weighting schemes extrapolate over the grid positions where no visibility data exists in different ways, maximising different
properties in the final image.
Natural weighting is one of the density weighting schemes which gives equal weight to all visibilities, thereby emphasising grid cells containing many visibilities. This increases sensitivity at the cost of producing a synthesised beam with large side-lobes, decreasing the resolution.
Uniform weighting is another density weighting scheme which gives equal weight to all grid cells, regardless of visibility distribution. This maximises the effect of long baselines on the beam shape, creating a narrow beam profile with lower side-lobe levels thereby increasing resolution at the cost of sensitivity.
Robust weighting (Briggs 1995) is a density weighting scheme which is a hybrid of natural and uniform weighting. Robust weighting attempts to find a balance between the two schemes by creating a PSF that smoothly varies from one scheme to the other depending on a single tunable parameter.
Tapering is a weighting scheme which multiplies the weights by a Gaussian. Tapering can be combined with any other weighting scheme to compromise between sensitivity and resolution. Gaussian tapering is the optimum weighting for detecting Gaussian sources and increases the detectability of an extended source (SMA user guide).
CLEAN Algorithm
The most common traditional imaging algorithms used in aperture synthesis are the CLEAN and Maximum Entropy Method (MEM) algorithms. More recent and sophis- ticated imaging algorithms derived from these two are discussed in the wide-field and wide-band sections in Chapter 3.
The original CLEAN algorithm devised by H¨ogbom (1974) solves the convolution equa- tion (Equation 1.16) by representing radio sources as a number of point sources. The peak strengths and positions of these point sources is found iteratively and the final image is the sum of these components convolved with the CLEAN beam, usually represented by a Gaussian (Cornwell et al. 1999). The algorithm proceeds as follows:
1. Search for the strength and position of the highest intensity peak in the dirty image BD.
2. Subtract the dirty beam P from the dirty image at the position of the point source found in step 1. The beam P is multiplied by a loop gain γ ≤ 1.
3. The position and strength of the point source subtracted is recorded in a model. 4. Go back to step 1, unless all remaining peaks are below a specified user level. The
remaining dirty image (minus subtracted peaks) is the residual image.
5. Convolve the model of point sources with a CLEAN beam, which is usually an elliptical Gaussian fitted to the central lobe of the dirty beam.
6. Add the residual image to the CLEANed image.
A FFT-based CLEAN algorithm by Clark (1980) performs in a similar fashion to H¨ogbom’s algorithm, but finds the positions and strengths of point sources by only using a fraction of the dirty beam profile. The Clark algorithm operates in two cycles, the major and minor cycles and proceeds as follows:
1. Minor Cycle - A segment of the beam is selected with the highest exterior side-lobe i.e. the central portion of the dirty beam.
2. Minor Cycle - Peaks are selected from the dirty image if the strength of the source is greater than the highest side-lobe of the beam from step 1.
3. Minor Cycle - A H¨ogbom CLEAN is performed on all of the selected points from step 2 using the segment of the beam from step 1. This continues for this list of points until all sources selected are weaker than the side-lobe.
4. Major Cycle - The model created in step 3 is then transformed via a FFT, multi- plied by the sampling function (the inverse transform of the PSF) transformed back and subtracted from the dirty image. Also any errors in the residual images from previous minor cycles are corrected by subsequent minor cycles.
This algorithm is sufficient to find CLEAN components for dirty beams which have fairly good side-lobe patterns. One further modification to the CLEAN algorithm is the Cotton-Schwab algorithm (Schwab 1984) where the major cycle subtracts CLEAN components from the un-gridded visibility data. This enables the removal of aliasing noise and gridding errors if the inverse Fourier transform of these components to each u, v sample is accurate enough (Cornwell et al. 1999). The algorithm also decides whether to use the direct Fourier transform for a small number of CLEAN components to increase accuracy, or to use the FFT for large numbers of CLEAN components.
A major advantage of the Cotton-Schwab algorithm is its ability to CLEAN many separate fields independently in the minor cycles and remove all of the CLEAN components together in the major cycle. Calculating the residual for each small field allows for the w- term in the full 3D measurement equation (Equation 1.8) to be determined. This corrects for non-coplanar baselines which is a problem for large arrays and wide-field imaging (see Chapter 3 for discussions). This is the algorithm used in theAIPS taskIMAGR5.
Maximum Entropy Algorithm
Another algorithm is the MEM, a form of information theory, which selects a probability distribution that best fits the data within the noise level from all the possible distributions and also has maximum entropy. Maximum entropy is defined as a positive image with a compressed range of pixel values forcing the image to be “smooth” (Cornwell et al. 1999). Entropy takes the general form:
H = −X k Ik ln Ik Mk (1.20)
where Ik is the reconstructed image and Mk is the expected or a priori image.
Each visibility cannot be exactly fitted by the probability distribution to produce a positive value and therefore the data is constrained by a χ2fit of the probability distibution
to the observed image being equal to the expected value
χ2 = X k |V (uk, vk) − V (u\k, vk)|2 σV (u2 k, vk) , (1.21)
where V (uk, vk) is the probability distribution, V (u\k, vk) is the observed image and
σV (u2
k, vk) is the variance of the image, i.e. Gaussian noise (Cornwell et al. 1999).
Images with around one million pixels take a similar amount of time for MEM and CLEAN, with MEM being faster for larger images and CLEAN faster for smaller images. MEM can also be faster than CLEAN when the image is filled with emission, but CLEAN is faster for sources which are well represented by a small number of point sources and for images with high dynamic ranges (Cornwell and Evans 1985).
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