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##### image.moments - Function

1.1.1 Compute moments from an image

Description

Summary

The primary goal of this function is to enable you to analyze a multi-dimensional image by generating moments of a specified axis. This is a time-honoured spectral-line analysis technique used for extracting information about spectral lines.

You can generate one or more output moment images. The return value of this function is an on-the-fly Image tool holding the first of the output moment images.

The word ‘moment’ is used loosely here. It refers to collapsing an axis (the moment axis) to one pixel and setting the value of that pixel (for all of the other non-collapsed axes) to something computed from the data values along the moment axis. For example, take an RA-DEC-Velocity cube, collapse the velocity axis by computing the mean intensity at each RA-DEC pixel. This function offers many different moments and a variety of automatic methods to compute them.

We try to make a distinction between a ‘moment’ and a ‘method’. This boundary is a little blurred, but it claims to refer to the distinction between what you are computing, and how the pixels that were included in that computation were selected. For example, a ‘moment’ would be the average value of some pixel values in a spectrum. A ‘method’ for selecting those pixels would be a simple pixel value range specifying which pixels should be included.

There are many available moments, and you specify each one with an integer code as it would get rather cumbersome to refer to them via strings. In the list below, the value of the ith pixel of the spectrum is Ii, the coordinate of this pixel is vi (of course it may not be velocity), and there are n pixels in the spectrum. The available moments are:

• -1 – the mean value of the spectrum

• 0 – the integrated value of the spectrum

where Δv is the width (in world coordinate units) of a pixel along the moment axis

• 1 – the intensity weighted coordinate (this is traditionally used to get ’velocity fields’)

• 2 – the intensity weighted dispersion of the coordinate (this is traditionally used to get ’velocity dispersion fields’)

• 3 – the median of I
• 4 – the median coordinate. Here we treat the spectrum as a probability distribution, generate the cumulative distribution, and then find the coordinate corresponding to the 50% value. This moment is not very robust, but it is useful for quickly generating a velocity field in a way that is not sensitive to noise. However, it will only give sensible results under certain conditions. The generation of the cumulative distribution and the finding of the 50% level really only makes sense if the cumulative distribution is monotonic. This essentially means only selecting pixels which are positive or negative. For this reason, this moment type is only supported with the basic method (see below – i.e. no smoothing, no windowing, no fitting) with a pixel selection range that is either all positive, or all negative
• 5 – the standard deviation about the mean of the spectrum

• 6 – the root mean square of the spectrum

• 7 – the absolute mean deviation of the spectrum

• 8 – the maximum value of the spectrum
• 9 – the coordinate of the maximum value of the spectrum
• 10 – the minimum value of the spectrum
• 11 – the coordinate of the minimum value of the spectrum

Smoothing

The purpose of the smoothing functionality is purely to provide a mask. Thus, you can smooth the input image, apply a pixel include or exclude range, and generate a smoothed mask which is then applied before the moments are generated. The smoothed data are not used to compute the actual moments; that is always done from the original data.

Basic Method

The basic method is to just compute moments directly from the pixel values. This can be modified by applying pixel value inclusion or exclusion ranges (arguments includepix and excludepix).

You can then also convolve the image (arguments smoothaxes, smoothtypes, and smoothwidths) and find a mask based on the inclusion or exclusion ranges applied to the convolved image. This mask is then applied to the unsmoothed data for moment computation.

Window Method

The window method (invoked with argument method=’window’) does no pixel-value-based selection. Instead a window is found (hopefully surrounding the spectral line feature) and only the pixels in that window are used for computation. This window can be found from the convolved or unconvolved image (arguments smoothaxes, smoothtypes, and smoothwidths).

The moments are always computed from the unconvolved data. The window can be found (for each spectrum) automatically. The automatic methods are via Bosma’s converging mean algorithm (method=’window’) or by fitting Gaussians and taking ±3σ as the window (method=’window,fit’). In Bosma’s algorithm, an initial guess for a range of pixels surrounding a spectral feature is refined by widening until the mean of the pixels outside of the range converges (to the noise).

Fit Method

The fit method (method=’fit’) fits Gaussians to spectral features automatically. The moments are then computed from the Gaussian fits (not the data themselves).

Other Arguments

• outfile - If you are creating just one moment image, and you specify outfile, then the image is created on disk with this name. If you leave outfile empty then a temporary image is created. In both cases, you can access this image with the returned Image tool. If you are making more than one moment image, then theses images are always created on disk. If you specify outfile then this is the root for the output file names. If you don’t specify it, then the input image name is used as the root.
• smoothing - If you smooth the image to generate a mask, you specify the kernel widths via the smoothwidths argument in the same way as in the sepconvolve function. See it for details.
• stddev - Some of the automatic methods also require an estimate of the noise level in the image. This is used to assess whether a spectrum is purely noise or not, and whether there is any signal worth digging out. If you don’t give it via the stddev argument, it will be worked out automatically from a Gaussian fit to the bins above 25% from a histogram of the entire image.
• includepix, excludepix - The vectors given by arguments includepix and excludepix specify a range of pixel values for which pixels are either included or excluded. They are mutually exclusive; you can specify one or the other, but not both. If you only give one value for either of these, say includepix=b, then this is interpreted as includepix=[-abs(b),abs(b)].

The convolving point-spread function is normalized to have a volume of unity. This means that point sources are depressed in value, but extended sources that are large with respect to the PSF remain essentially on the same intensity scale; these are the structures you are trying to find with the convolution so this is what you want. If you convolve the image, then arguments like includepix select based upon the convolved image pixel values. If you are having trouble getting these right, you can output the convolved image (smoothout) and assess the validity of your pixel ranges. Note also that if you are Hanning convolving (usually used on a velocity axis), then the width for this kernel must be 3 pixels (triangular smoothing kernels of other widths have no valid theoretical basis).

• doppler - If you compute the moments along a spectral axis, it is conventional to compute the world coordinate (needed for moments 0, 1 and 2) along that axis in ”km/s”. The argument doppler lets you specify what doppler convention the velocity will be calculated in. You can choose from doppler=radio, optical, true. See function summary for the definitions of these codes. For other moment-axis types, the world coordinate is computed in the native units.
• mask - The total input mask is the combination of the default pixel mask (if any) and the OTF mask. Once this mask has been established, then the moment method may make additional pixel selections.
• drop - If this is true (the default) then the moment axis is dropped from the output image. Otherwise, the output images have a moment axis of unit length and coordinate information that is the same as for the input image. This coordinate information may be totally meaningless for the moment images.

Finally, if you ask for a moment which requires the coordinate to be computed for each profile pixel (these are the intensity weighted mean coordinate [moment 1] and the intensity weighted dispersion of the coordinate [moment 2]), and the profile axis is not separable then there will be a performance loss. Examples of non-separable axes are RA and Dec. If the axis is separable (e.g. a spectral axis) there is no penalty. In the latter case, the vector of coordinates for one profile is the same as the vector for another profile, and it can be precomputed (once).

Note that this function has no “virtual” output file capability. All output files are written to disk. The output mask for these images is good (T) unless the moment method fails to generate a value (e.g. the total input pixel mask was all bad for the profile) in which case it will be bad (F).

If an image has multiple (per-channel beams) and the moment axis is equal to the spectral axis, each channel will be convolved with a beam that is equal to the beam having the largest area in the beamset prior to moment determination.

Arguments

 Inputs moments List of moments that you would like to compute. Default is integrated spectrum. allowed: intArray Default: 0 axis The moment axis. Default is the spectral axis if there is one. allowed: int Default: -10 region Region selection. See ”help par.region” for details. Default is to use the full image. allowed: any Default: variant mask Mask to use. See help par.mask. Default is none. allowed: any Default: variant method List of windowing and/or fitting functions you would like to invoke. Vector of strings from ’window’ and ’fit’. The default is to not invoke the window or fit functions. allowed: stringArray Default: smoothaxes List of axes to smooth. Default is no smoothing. allowed: intArray Default: -1 smoothtypes List of smoothing kernel types, one for each axis to smooth. Vector of strings from ’gauss’, ’boxcar’, ’hanning’. Default is no smoothing. allowed: any Default: variant smoothwidths List of widths (full width for boxcar, full width at half maximum for gaussian, 3 for Hanning) in pixels for the smoothing kernels. Vector of numeric. Default is no smoothing. allowed: doubleArray Default: 0.0 includepix Range of pixel values to include. Vector of 1 or 2 doubles. Default is include all pixel. allowed: doubleArray Default: -1 excludepix Range of pixel values to exclude. Default is exclude no pixels. allowed: doubleArray Default: -1 peaksnr The SNR ratio below which the spectrum will be rejected as noise (used by the window and fit functions only) allowed: double Default: 3.0 stddev Standard deviation of the noise signal in the image (used by the window and fit functions only) allowed: double Default: 0.0 doppler Velocity doppler definition for velocity computations along spectral axes allowed: string Default: RADIO outfile Output image file name (or root for multiple moments). Default is input + an auto-determined suffix. allowed: string Default: smoothout Output file name for convolved image. Default is don’t save the convolved image. allowed: string Default: overwrite Overwrite (unprompted) pre-existing output file? allowed: bool Default: false drop Drop moments axis from output images? allowed: bool Default: true stretch Stretch the mask if necessary and possible? allowed: bool Default: false async Run asynchronously? allowed: bool Default: false
Returns
image

Example

"""
#
print "\t----\t moments Ex 1 \t----"
ia.fromshape(shape=[32,32,32,32]) # replace with your own cube
im2 = ia.moments(moments=[-1,1,2], axis=2, smoothaxes=[0,1,2],
smoothtypes=["gauss","gauss","hann"],
smoothwidths=[5.0,5.0,3], excludepix=[1e-3],
smoothout=’smooth’, overwrite=true)
im2.done()
ia.close()
#
"""

In this example, standard moments (average intensity, weighted velocity
and weighted velocity dispersion) are computed via the convolve (spatially
convolved by gaussians and spectrally by a Hanning kernel) and clip
method (we exclude any pixels with absolute value less than \$0.001\$).
The output file names are automatically created for us and
the convolved image is saved.   The returned image tool holds the first
moment image.

Example

"""
#
print "\t----\t moments Ex 2 \t----"
ia.fromshape(shape=[32,32,32,32])
im2 = ia.moments(moments=[3], method=["window"])
im2.done()
ia.close()
#
"""

In this example, the median of each spectrum is computed, after pixel
selection by the automatic window method. The output
image is temporary and accessed via the returned Image tool.

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