Online material

Appendix A: List of symbols

For the convenience of readers, this section lists and defines all symbols introduced in Sect. 2 of this paper (images are denoted by capital calligraphic characters):

ℱλ	images of source footprints in measurement iterations
	smoothing Gaussians in successive unsharp masking
	smoothing Gaussians used to create detection images
ℐDj   C	clean detection images combined over wavelengths
ℐDj   C   FS	filament-subtracted combined detection images
	clean detection images combined over wavelengths
ℐλ	original observed images produced by a map-maker
ℐλDF	flattened detection images for the final extraction
ℐλD	detection images: either ℐλO or transformed ℐλO
ℐλD   FS	filament-subtracted detection images
ℐλDj	single-scale decompositions of the images ℐλD
ℐλDj   C	single-scale images cleaned of noise and background
	filaments cleaned of sources, noise, and background
	positive component of reconstructed filaments
	negative component of reconstructed filaments
ℐλD   C	full images of sources reconstructed from ℐλDj   C
	image of filaments combined over wavelengths
	full images of filaments reconstructed from ℐλDj
ℐλF	scaling image smoothed by convolution
ℐλO	measurement images: ℐλ resampled to pixel Δ
ℐλO   FS	filament-subtracted measurement images
ℳλ	observational mask images defining areas of interest
ℳλj	mask of a single-scale filament
	image of skeletons combined over wavelengths
	skeletons of clean single-scale filaments
	full accumulated skeletons of clean filaments
	wavelength-combined skeletons
	skeletons tracing crests of the full skeletons
a	major size of a filament mask
A	major FWHM size of a source
	maximum FWHM sizes of sources to be extracted
b	minor size of a filament mask
D0	filament width: FWHM of the inner Gaussian core
	elongation of the clusters of connected pixels
	empirical shape factor of filamentary structures
fS	scale factor defining relative spacing between scales
f(ζ)	width normalization factor of a simulated filament
	minimum peak intensity of detected filaments
Iλj	pixel intensity in a single-scale detection image
Iλ(r)	intensity profile of a simulated filament
IP	peak intensity of a simulated filament
j	running number of a decomposed spatial scale
l	running number of an intensity sub-level
L	length of a filament
nλj	variable number of standard deviations σλj in ϖλj
NB	number of cleaning beam areas
	number of intensity levels in filament reconstruction
NS	number of spatial scales in the image decomposition
	minimum value of NΠλj for cleaning filaments
NΠλ	number of pixels in a cluster of connected pixels
Oλ	observational angular resolution: FWHM beam size
r	radial distance from the peak of a filament
R0	filament radius: HWHM of the inner Gaussian core
Sj	spatial scale: FWHM of a smoothing Gaussian beam
Smax	largest spatial scale in a single-scale decomposition
	sparsity of the clusters of connected pixels
W	width of a filament
	differential images in filament reconstruction
Δ	pixel size (same for all images in an extraction)
λ	wavelength (central wavelength of a waveband)
ϖλj	iterated cleaning thresholds (cut-off levels)
	filament detection thresholds (=σλj)
σλj	standard deviation in a single-scale image
σnoise	standard deviation of simulated random noise
ζ	power-law exponent of a simulated filament

Appendix A: Filaments in MHD simulations

Fig. A.1 Filaments in MHD simulations of colliding flows (Hennebelle et al. 2008). The upper panels display the original image of column densities (a), extracted filaments on all spatial scales (b), and filament-subtracted image (c). The lower panels show the filaments partially reconstructed up to 20′′ scale (d), 3-color (red, green, blue) composite image of the filaments partially reconstructed up to 2000′′, 160′′, and 10′′ scales (e), as well as the segmentation image of skeletons that appear on more than 5 spatial scales (f). Pixel values in panel (f) represent the skeleton number.

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This section illustrates application of getfilaments to images obtained from three-dimensional magnetohydrodynamic (MHD) simulations of the formation of molecular clouds in colliding flows of warm diffuse gas (Hennebelle et al. 2008). Gravity, atomic cooling, photoelectric heating on dust grains, and initially uniform magnetic field were included in the simulations. Two opposite flows of diffuse neutral gas with the initial density of 1 cm^-3 and velocity of 13.35 km s^-1 were set up to collide in the YZ plane of the computational box. On a time scale of a few million years, a dense gas phase (10²–10⁴ cm^-3) developed under the influence of cooling, ram pressure, and gravity. All details of the simulation (labeled as Slower Flow) and corresponding images can be found on their web site¹¹.

A snapshot of the column density in the YZ plane corresponding to a time of 9.737 Myr from the start of the simulation was cut to a size of 1000 × 1000 pixels. The image was arbitrarily assigned a 2′′ pixel size; the image values were scaled to a maximum of 100 (in arbitrary units) and some noise at a level of 0.5 has been added. The resulting image was convolved to a 5′′ resolution.

The filamentary structures clearly visible in the original column density image (Fig. A.1a) are cleanly and almost fully extracted (Fig. A.1b), leaving only low-level filamentary residuals in the filament-subtracted image (Fig. A.1c). The latter shows mostly compact density enhancements (sources, intersections of the filaments) but no significant filamentary structures. An image of filaments reconstructed only up to a spatial scale of 20′′ (Fig. A.1d) reveals the web of thin filaments that are largely diluted in panel b by the contribution of much larger scales. Although large filaments may appear as regular and smooth entities, many of them become heavily substructured on smaller scales. The composite image of the filaments (Fig. A.1e) uses the red, green, and blue colors to make the large, medium, and small-scale structures more visible. The segmented image of skeletons (Fig. A.1f) traces and numbers the crests of the filaments. All these images, as well as many other images and multi-wavelength catalogs of sources automatically produced by getsources and getfilaments, can be very useful for detailed studies of the properties of the filaments in the interstellar medium and their relationship with star formation.

Appendix A: Filaments in cosmological simulations

Fig. A.1 Filaments in the MareNostrum simulation of the formation of galaxies. The upper panels display the original image (a), extracted filaments on all spatial scales (b), and filament-subtracted image (c). The lower panels show the filaments partially reconstructed up to 20′′ scale (d), 3-color (red, green, blue) composite image of the filaments partially reconstructed up to 2000′′, 160′′, and 10′′ scales (e), as well as the segmentation image of skeletons that appear on more than 5 spatial scales (f). Pixel values in panel (f) represent the skeleton number.

Open with DEXTER

This section illustrates application of getfilaments to images obtained from the Horizon MareNostrum simulation of the formation of galaxies at high redshifts (Ocvirk et al. 2008; Devriendt et al. 2010) performed on the MareNostrum supercomputer at the Barcelona Supercomputer Center. Galactic winds, chemical enrichment, ultraviolet background heating, radiative cooling, star formation, and supernovae feedback were included in this large-scale and high-resolution simulation with up to five levels of adaptive mesh refinement. Impressive networks (cosmic web) of filamentary structures linking clusters of galaxies have been created and visualized in the simulation.

One of the images of a piece of the Universe corresponding to a redshift of 2.5 was downloaded from the project’s web site¹², converted from JPG to FITS format using the ImageMagick utility, and reduced in size to 1000 × 1000 pixels. As in Appendix A, the image was arbitrarily assigned a 2′′ pixel size, scaled to a maximum of 100 (in arbitrary units), and added with pixel noise at a level of 0.5. The resulting image was also convolved to a 5′′ resolution.

The filament extraction results on cosmological scales are similar to those presented in Appendix A. The fascinating cosmic web visible in the original image (Fig. A.1a) is quite well

extracted on all spatial scales (Fig. A.1b), with low filamentary residuals in the filament-subtracted image (Fig. A.1c) that shows mostly compact peaks (galaxies, clusters of galaxies). An image of filaments reconstructed up to a spatial scale of 20′′ (Fig. A.1d) reveals thin filaments that are substantially diluted in panel b by the contribution of all larger scales; many large filaments are also substructured on smaller scales. The composite image of the filaments (Fig. A.1e) makes the large, medium, and small-scale structures more visible by combining the red, green, and blue colors on the same image. The segmented image of skeletons (Fig. A.1f) traces and numbers the crests of the filaments. Such images, as well as other images and multi-wavelength source catalogs produced by getfilaments and getsources, can readily be used for further studies of the cosmic web and the properties and formation processes of galaxies and their clusters.

© ESO, 2013