© ESO, 2013

1. Introduction

In a previous paper (Men’shchikov et al. 2012, hereafter referred to as Paper I), we described the multi-scale, multi-wavelength source extraction method getsources. Developed primarily for large far-infrared and submillimeter surveys of star-forming regions with Herschel, getsources can also be applied to other types of astronomical images.

Instead of following the traditional approach of extracting sources directly in the observed images, getsources analyzes filtered single-scale decompositions of detection images over a wide range of spatial scales. The algorithm separates the peaks of real sources from those produced by the noise and background fluctuations and constructs wavelength-independent sets of combined single-scale detection images preserving spatial information from all wavebands. Sources are detected in the waveband-combined images by tracking the evolution of their segmentation masks across all scales. Source properties are measured in the observed (background-subtracted and deblended) images at each wavelength. Based on the results of an initial extraction, detection images are flattened to produce more uniform noise and background fluctuations in preparation for the second, final extraction. The method has been thoroughly tested on many simulated benchmarks and real-life images obtained in the Herschel Gould Belt (André et al. 2010) and HOBYS (Motte et al. 2010) surveys. The overall benchmarking results (Men’shchikov et al., in prep.) have shown that getsources consistently performs very well in both the completeness and reliability of source detection and the accuracy of measurements.

Fig. 1 Main processing blocks of getsources and getfilaments. For a complete extraction, both methods require two runs (left): the initial and final extractions (red blocks; the preparation and flattening steps are shown in blue; cf. Paper I). A more detailed presentation of the processing steps (right) shows that both algorithms would have had lots of identical actions, if conceived and coded separately. The getfilaments algorithm is essentially localized in only the cleaning and combining steps of getsources (highlighted in yellow). In practice, getfilaments is an integral part of the source extraction code, activated by a single configuration parameter of getsources.

The wealth of high-sensitivity far-infrared images obtained with Herschel over the past three years have demonstrated that a substantial part of interstellar medium exists in the form of a fascinating web of omnipresent filamentary structures (see, e.g., Men’shchikov et al. 2010, for illustrations). This anisotropic component further complicates the very difficult problem of the systematic detection and measurement of dense cores in the strongly variable backgrounds of molecular clouds. The observational evidence that stars form in dense, cold filaments (e.g., André et al. 2010; Men’shchikov et al. 2010) creates severe problems for automated source extraction methods that must find as many real sources as possible from the images in several photometric bands, reliably distinguishing them not only from fluctuating backgrounds and noise, but also from the filamentary structures. The latter tend to “amplify” insignificant background or noise fluctuations that fall on top of them, confusing source extraction algorithms. The benchmarking results (Men’shchikov et al., in prep.) suggest that source extraction methods that do not take the presence of filaments into account always tend to create significant numbers of spurious sources along the filaments.

Although getsources showed very good results in the benchmarks, it still created some spurious sources in simulated images with strong unresolved filamentary structures. In order to improve the performance of getsources in the observed filamentary backgrounds, a multi-scale, multi-wavelength filament extraction method getfilaments has been developed that solves this problem and substantially improves reliability of source extraction. The main idea behind the new approach is to carefully extract filaments (i.e., separate their intensity distribution from sources and largely isotropic backgrounds) and subtract them from the original images before detecting and measuring sources. Depending on the accuracy of the reconstructed intensity distribution, this procedure removes filamentary structures from observed images or (at least) greatly reduces their contribution. The absence of filaments in detection images makes source extraction results much more reliable, practically eliminating spurious sources.

The getfilaments algorithm was developed within the framework of the multi-scale and multi-wavelength approach of getsources (Paper I) as an integral part of the source extraction method. Both getfilaments and getsources can be described by the processing blocks shown in Fig. 1. The filament extraction method is essentially localized in only the cleaning and combining steps of getsources. As the getfilaments algorithm adds only a relatively small number of image manipulations to the original version of the source extraction method, there is no need in creating a separate code for the extraction of filaments. Moreover, the intimate physical relationship between forming stars and filaments seen in Herschel observations demands that accurate filament extraction methods must remove the contribution of sources and, conversely, accurate source extraction methods must be able to remove underlying filamentary structures.

This paper follows conventions and definitions introduced in Paper I. The term noise is used to refer to the statistical instrumental noise including possible contributions from any other signals that are not astrophysical in nature, i.e. which are not related to the emission of the areas in space one is observing. The term background refers to the largely isotropic astrophysical backgrounds, whereas the term filaments describes significantly elongated structures¹. Filaments are anisotropic in the sense that their profiles and widths are very dissimilar in different directions.

Explicit distinction is made between the morphologically-simple (convex, not very elongated) sources of emission defined by source extraction methods and objects of specific astrophysical nature. In its present state, getsources does not know anything about the nature or true physical properties of the objects that produced the emission of significant peaks detected as sources. Like most of the other existing methods, it can only detect sources (that are possibly harboring our objects of interest) and determine their apparent two-dimensional intensity distributions above the variable background, filaments, and noise, measuring their apparent properties at each wavelength as accurately as possible.

2. Extracting filamentary structures: getfilaments

The fundamental problem in extracting filaments (or sources) is that all spatial scales in the images are mixed together and the intensity of any pixel contains unknown contributions from different components². Following the approach formulated in Paper I, getfilaments analyzes decompositions of original images (in each waveband) across a wide range of spatial scales separated by a small amount (typically ~5%). Each of the “single scales” contains non-negligible signals from only a relatively narrow range of spatial scales, mostly only from those structures that have widths (sizes) similar to the scale considered. In effect, this automatically filters out their contributions on irrelevant (much smaller and larger) spatial scales. An immediate benefit of such filtering is that one can manipulate entire single-scale images and use thresholding to separate filaments from other structures (sources, background, and noise).

Fig. 2 Profiles of the simulated filament (Sect. 2.1) used to illustrate the getfilaments method. The filament has the inner half-width R0 = 37″ (at half-maximum) and the outer power-law profile Iλ(r) ∝ r-1 (blue). Gaussian-shaped sources (cyan) and background (green) have FWHM sizes of 30′′ and 4000′′, respectively. Random pixel noise (black) was convolved to a resolution of 18′′ (FWHM) and scaled to have a standard deviation σnoise = 1.25 MJy/sr (dashed). Three profiles sample the full simulated image (Fig. 3) across the top of the filament (brown), just below its midpoint (magenta), and through the position of the uppermost source (red).

Fig. 3 Simulated image and its components (Sect. 2.1). (a) Straight filament with the profile displayed in Fig. 2, normalized to $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{I}}_{\mathrm{P}}\end{array}\mathrm{=}\mathrm{100}$. (b) Identical Gaussian sources with an intrinsic FWHM size of 24′′, convolved to 18′′ resolution and normalized to IP = 100. (c) Isotropic Gaussian background with a size of 4000′′ (FWHM), normalized to IP,b = 10. (d) Random instrumental noise with σnoise = 1.25. (e) Full simulated image ℐλO (≡ℐλD) with S/N = 80 and resolution of 18′′. Three horizontal lines indicate the locations and direction of the profiles shown in Fig. 2. Each panel’s dimensions are 600 × 2820 pixels (), pixel size Δ = 2″.

2.1. Simulated filament

The getfilaments algorithm is illustrated below using simulated images of a straight filament, a string of sources, a simple background, and a moderate-level noise³ (Figs. 2, 3), resembling the filaments observed with Herschel (e.g., Arzoumanian et al. 2011; Palmeirim et al. 2013). The filament profile, shown in Fig. 2, adopts the functional form (Moffat 1969) used in getsources to define deblending shapes (Paper I): ${\mathit{I}}_{\mathit{\lambda }}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}{\mathit{I}}_{\mathrm{P}}\hspace{0.17em}{\mathrm{(}\mathrm{1}\mathrm{+}\mathit{f}\mathrm{(}\mathit{\zeta }\mathrm{)}\hspace{0.17em}\mathrm{(}\mathit{r}\mathit{/}{\mathit{R}}_{\mathrm{0}}{\mathrm{)}}^{\mathrm{2}}\mathrm{)}}^{\mathrm{-}\mathit{\zeta }}\mathit{,}$(1)where I_P is the peak intensity, r the radial distance from the filament’s crest (in the orthogonal direction), R₀ the filament’s half-width at half-maximum (HWHM), ζ a power-law exponent, and f(ζ) = (2^1/ζ−1) normalizes the profile width to R₀ for all values of ζ. The function defined by Eq. (1) has Gaussian shape in its core, smoothly transforming into a power-law profile I_λ(r) ∝ r^{− 2   ζ} for large distances r ≫ R₀.

The parameters of Eq. (1) were fixed at I_P = 100 MJy/sr, ${\mathit{R}}_{\mathrm{0}}\mathrm{=}\mathrm{18}\stackrel{\mathrm{″}}{.}\mathrm{75}$, and ζ = 0.5. To simulate a 250 μm Herschel image, the model filament (Fig. 2) was convolved to an angular resolution of 18′′ (full width at half maximum, FWHM), preserving its peak intensity and yielding an image of the filament with a width of D₀ = 75″ (FWHM) and a power-law profile I_λ(r) ∝ r^-1 at large distances (Fig. 3a). A string of identical Gaussian-shaped sources, with an intrinsic FWHM size of 24′′, were convolved to the same angular resolution of 18′′, scaled to have the same peak intensity of 100 MJy/sr, and placed along the lower half of the filament (Fig. 3b). An isotropic background was modeled as a large Gaussian (4000′′ FWHM), normalized to 10 MJy/sr (Fig. 3c). A noise image was created by assigning random values to each pixel, convolving the resulting image to the resolution of 18′′, and scaling it to have the standard deviation σ_noise = 1.25 MJy/sr (Fig. 3d). The simulated components were added together to produce the “observed” 250 μm image of the filament with a signal-to-noise ratio S/N = 80 (Fig. 3e). Dimensions of all images are 4800 × 4800 pixels (, pixel size Δ = 2″), although only the central area of 600 × 2820 pixels, centered on the filament, is shown in this paper.

As in Paper I, images will be denoted by capital calligraphic characters (e.g., ) to make a clear distinction between the images and various other parameters; all symbols and definitions are listed in Appendix A. Below, the filament extraction method (illustrated in Fig. 1) is described in full detail⁴.

2.2. Preparing observed and detection images

The first step in the filament (source) extraction (Fig. 1) is to convert the original images ℐ_λ at all wavelengths λ to the same grid and align them across wavebands, producing the observed images. This is done by resampling all images to the same (finest) pixel size (using SWarp, Bertin et al. 2002).

Both getfilaments and getsources distinguish between the actual observed images and detection images (denoted as ℐ_λO and ℐ_λD, respectively). Most of the processing is done on detection images and, as the name suggests, they are used when detecting sources or filaments; observed images are used mostly for measuring and visualizing of sources. In simple cases, both ℐ_λO and ℐ_λD can be the same. Like getsources (Paper I), getfilaments uses convolution , where is a smoothing Gaussian with its FWHM size chosen to slightly degrade (by ~5%) the image resolution O_λ. This suppresses pixel-to-pixel noise in real-life images ℐ_λO (on spatial scales smaller than the observational beam size O_λ) and small-scale artifacts that would otherwise become enhanced in decomposed images.

The last part of the preparation is to create the observational masks ℳ_λ with pixel values of either 1 or 0 that define the areas in the original images that one is interested in. They exclude from processing all pixels of ℐ_λO and ℐ_λD in which the mask has zero values. In the simplest case of a perfect (simulated) image, ℳ_λ has values of 1 in all pixels. Very noisy areas (usually closer to edges) can affect the cleaning and detection algorithms and one needs to exclude them using carefully-prepared observational masks. The mask images ℳ_λ should not have isolated holes: all zero pixels must be connected to each other and to the image edges.

Fig. 4 Spatial decomposition (Sect. 2.3). Single scales ℐλDj of the simulated image ℐλD (Fig. 3) are shown for j = 18,32,46,60,74, NS = 99, fS = 1.079, ${\mathit{S}}_{\mathrm{1}}\mathrm{=}\mathrm{5}\stackrel{\mathrm{″}}{.}\mathrm{56}$, SNS = 9609″. The scales Sj are separated by a factor of 3 to illustrate the spatial decomposition; negative areas surrounding the filament are the direct consequence of the subtraction in Eq. (2). Scale sizes Sj are visualized and annotated here and in all subsequent similar figures.

2.3. Decomposing detection images in spatial scales

The spatial decomposition is done by convolving the original images with circular Gaussians of progressively larger sizes and subtracting them from one another (Fig. 4): ${\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}}\mathrm{=}{\mathrm{𝒢}}_{\mathit{j}\mathrm{-}\mathrm{1}}\mathrm{\ast }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}}\mathrm{-}{\mathrm{𝒢}}_{\mathit{j}}\mathrm{\ast }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}}\hspace{0.17em}\hspace{0.17em}\mathrm{\left(}\mathit{j}\mathrm{=}\mathrm{1}\mathit{,}\mathrm{2}\mathit{,}\mathit{...}\mathit{,}{\mathit{N}}_{\mathrm{S}}\mathrm{\right)}\mathit{,}$(2)where ℐ_λD is the detection image (Sect. 2.2), ℐ_λDj are its “single-scale” decompositions, and are the smoothing Gaussian beams ( is a two-dimensional delta-function). The beams have FWHM sizes S_j = f_S   S_j − 1 in the range 2   Δ ≲ S_j ≲ S_max, where Δ is the pixel size, f_S > 1 is the scale factor, and S_max is the maximum spatial scale considered. The number of scales N_S depends on the values of f_S (typically ≈    1.05) and S_max. The value of S_max is determined by the maximum sizes of filaments (sources) in the extraction and its upper limit is the size of the image along its smallest dimension. For large values of f_S, the single scales actually contain mixtures of wide ranges of scales, where faint small-scale structures become completely diluted by the contribution of irrelevant (much larger) scales. Smaller values of f_S ensure better spatial resolution of the set of single scales, just like fine mesh sizes better resolve structures in numerical methods. For values f_S that are too close to unity, images on scales j and j + 1 become almost identical⁵.

Equation (2) implicitly assumes that the convolved images are properly rescaled to conserve their total flux; therefore, the original image can be recovered by summing up all scales: ${\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}}\mathrm{=}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{S}}}{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}}\mathrm{+}{\mathrm{𝒢}}_{{\mathit{N}}_{\mathrm{S}}}\mathrm{\ast }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}}\mathit{.}$(3)Before convolution, the images ℐ_λD are expanded from the edges of the areas covered by the observational masks ℳ_λ towards the image edges and the entire images are expanded on all sides by a large enough number of pixels (2   S_j/Δ) to avoid undesirable border effects. Both expansions are performed using the pixel values at the edges of the masks and images, respectively, and extrapolating them outwards in four main directions (horizontal, vertical, and two diagonals). After convolution, the images are reduced back to their original size.

The single-scale decomposition of Eq. (2) filters out emission on all irrelevant scales and thus ℐ_λDj reveal structures with a much higher contrast than ℐ_λD does. The decomposition naturally selects filaments (sources) of specific widths, which become most visible in the images containing similar scales. The negative areas surrounding bright filaments in Fig. 4 are the direct consequence of the successive unsharp masking in Eq. (2), i.e., the subtraction of an image convolved with a larger smoothing beam from the one convolved with a smaller beam.

2.4. Cleaning single scales of noise, background, and sources

Before one can use the single-scale detection images ℐ_λDj for filament extraction, they must be cleaned of the contributions of noise, background, and sources to make sure that most (if not all) non-zero pixels belong to real filamentary structures.

2.4.1. Iterative cleaning algorithm

As in Paper I, single-scale cleaning is done by the global intensity thresholding of ℐ_λDj. Unlike the original images ℐ_λO or ℐ_λD that often have a very strong and highly variable background, the entire single-scale images are “flat” in the sense that signals on considerably larger scales have been removed or greatly suppressed (see Fig. 4). Another advantage of this single-scale cleaning is that the noise contribution depends very significantly on the scale. For example, the small-scale noise gets heavily diluted on large scales, where extended sources become most visible. In effect, in the reconstructed clean images ℐ_λD   C =  ∑ _jℐ_λDj   C, one can see large structures better (deeper) than in ℐ_λD.

Fig. 5 Single-scale removal of noise and background (Sect. 2.4). The same set of spatial scales (Fig. 4) is displayed in the single-scale images ℐλDj   C, cleaned of noise and background with an iterative procedure described in full detail in Paper I. Noise fluctuations visible in Fig. 4 have been removed by zeroing pixels with Iλj < ϖλj.

Paper I described an iterative cleaning algorithm that automatically finds (on each scale) a cut-off level separating significant signals from those of the noise and background. On the first scale (j    =    1), it computes the cut-off (threshold) ϖ_λj = n_λj   σ_λj, where σ_λj is the standard deviation over the entire image ℐ_λDj   ℳ_λ, and n_λj is a variable factor having an initial value of n_λ1 = 6. Then the procedure masks out all pixels with the values | I_λj| ≥ ϖ_λj and repeats the calculation of σ_λj over the remaining pixels, estimating a new threshold, which is generally lower than the one in the previous iteration. The procedure masks out bright pixels again and iterates further, always computing σ_λj at | I_λj| < ϖ_λj, outside the peaks and hollows, until ϖ_λj converges (δϖ_λj < 1%) to a stable threshold (see Paper I).

Having obtained the single-scale thresholds ϖ_λj distinguishing between the significant and insignificant signals, one can create clean single-scale images ℐ_λDj   C, where all faint pixels with I_λj < ϖ_λj are zeroed. This (ideally) leaves non-zero only those pixels that belong to significantly bright structures (sources, filaments). The resulting clean single-scale images of the simulated filament are illustrated in Fig. 5.

When sources are being extracted, ϖ_λj is the deepest level that getsources can descend to (2.5   σ_λj ≤ ϖ_λj ≤ 6   σ_λj, cf. Paper I). At fainter levels, there is no reliable way of distinguishing between sources and peaks produced by noise and background fluctuations, while there is a real danger of creating spurious sources. In the case of filaments considered in this paper, it is possible to use the fact that filaments are substantially elongated structures (as opposed to sources that are not very elongated, Sect. 1) and analyze much fainter signals. Numerous tests have shown that $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$ is a good choice for detecting filaments. Very faint filaments are only found in the intermediate range of intensities ($\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ to ϖ_λj), where they are strongly eroded by noise and background fluctuations, as well as altered by sources. Considerably brighter filaments that rise well above the source detection threshold ϖ_λj (Fig. 5) are much less affected by fluctuations at the σ_λj level.

Fig. 6 Single-scale detection of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here below the intensity threshold $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$ for detection of filaments. The filament’s base is clearly visible in the single scales, although the latter are strongly contaminated by noise and background fluctuations.

2.4.2. Cleaning algorithm for detecting filaments

Paper I has emphasized great benefits of detecting sources in single-scale images: the spatial decomposition of Eq. (2) is based on convolution and the latter acts as a natural selector of scales in decomposed images (cf. Sect. 2.3). As a consequence, resolved isolated circular sources with a FWHM size A would have their maximum peak intensity in single-scale images with smoothing beams S_j ≈ A. Indeed, convolving with small beams (S_j ≪ A) would have almost no effect on the source, whereas using extended beams (S_j ≫ A) would greatly dilute the source. At both these extremes, spatial decomposition produces decreasing peak intensities, while creating the strongest signal for the sources with sizes A ≈ S_j. Completely unresolved sources are the brightest on spatial scales S_j ≲ O_λ. In effect, sizes of all significant structures seen in single-scale images are very similar to the size S_j of the smoothing beam.

Lengths L of filaments are significantly greater (at least several times) than their widths W, which makes their single-scale properties quite different from those of sources, allowing one to distinguish them from the contributions of all other components (noise, background, and sources). The spatial decomposition of Eq. (2) selects the filaments with widths similar to the smoothing beam (W ≈ S_j), whereas their greater lengths (L ≫ S_j) are mostly unaffected by the convolution. This means that filaments occupy much larger areas in single-scale images than any contribution from sources and fluctuations of noise or largely isotropic backgrounds.

Fig. 7 Single-scale masking of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here exactly on the intensity threshold $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$ for detection of filaments, and all lower level signals were set to zero. The simulated filament is clearly visible on all scales, as is the abundant contamination of the images by noise. Noise-free black zones that appear to surround the filament are the consequence of the negative areas seen in Figs. 4 and 6.

When clipped at the filament detection threshold $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$, the single-scale images with intensities ${\mathit{I}}_{\mathit{\lambda j}}\mathrm{\le }\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ clearly display the base of filamentary structures (Fig. 6). The structures can also be seen in a much simpler way (as mask images) when one zeroes all pixels with intensities ${\mathit{I}}_{\mathit{\lambda j}}\mathit{<}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ (Fig. 7). The low level of thresholding leads to strong contamination by noise peaks (in general, also by background fluctuations and sources) that have to be removed before the images could be used for filament extraction. Such cleaning is a simple procedure based on the comparison of the area of connected non-zero pixels with the area of the smoothing beam.

As illustrated in Fig. 7, non-filamentary (insignificantly elongated) structures always occupy relatively small areas, when decomposed into single-scale images and considered above the threshold level $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$. The decomposition of Eq. (2) naturally selects structures with characteristic scales similar to S_j, filtering out both much smaller and much larger scales. Above the level $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$, sufficiently bright filaments connect relatively large areas of pixels, because it is the filament width that becomes similar to the decomposition scale S_j. The longer dimension of filaments is practically unaffected by the convolution at S_j and thus stays almost the same over a much wider range of scales (cf. Fig. 4).

Cleaning of the single-scale images of noise, background, and sources is done with the TintFill algorithm (Smith 1979)⁶ used in getsources for detecting sources (Paper I). The algorithm finds clusters of pixels connected to each other by their sides and fills all the pixels with a new value⁷. To remove connected clusters from images, the new value is set to zero.

Fig. 8 Single-scale masks of filaments (Sect. 2.4). The images of spatial scales from Fig. 7 are shown here at their base level as the normalized filament masks ℳλj after the removal of relatively small clusters of pixels of non-filamentary nature (including noise, background, and sources) with ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}\mathit{<}{\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$. In the five images displayed, structures with less than 1.31 × 103, 8.15 × 104, 6.85 × 105, 5.76 × 106, and 5.65 × 107 pixels have been removed (cf. Fig. 7).

Distinguishing between the real filamentary structures and smaller peaks of non-filamentary nature, in order to remove the latter from single-scale images, getfilaments employs a lower limit ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$ on the number of connected pixels N_Πλj that are allowed to remain in the clean images of filaments: ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }\mathrm{=}\begin{array}{c}\mathrm{˜}\\ \mathit{f}\end{array}\hspace{0.17em}{\mathit{N}}_{\mathrm{B}}\hspace{0.17em}\mathit{\pi }\hspace{0.17em}{\mathrm{(}\mathrm{3}{\mathit{O}}_{\mathit{\lambda }}^{\mathrm{6}}\mathrm{+}{{\mathit{S}}_{\mathit{j}}^{\mathrm{6}}}^{\mathrm{)}}}^{\mathrm{2}\mathit{/}\mathrm{6}}{\mathrm{\Delta }}^{-2}\mathit{,}$(4)where O_λ is the observational beam size, S_j the smoothing (decomposition) beam, Δ the pixel size, N_B the number of cleaning beam areas (N_B = 30), and $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{f}\end{array}$ a shape factor defined in Eq. (5) below (assume $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{f}\end{array}\mathrm{=}\mathrm{1}$ for a moment). Clusters of pixels with ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}\mathit{<}{\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$ are removed from the decomposed images on each spatial scale (see Fig. 8).

The combination of two beams in Eq. (4) defines the effective cleaning beam designed to change smoothly and rapidly between the regimes of small and large spatial scales. On small scales, the cleaning beam becomes almost constant (approaching 1.2O_λ), which is necessary to offset enhanced noisiness of small-scale images and minimize the chances of false detections. This raises the effective beam substantially above S_j on small scales (S_j ≲ O_λ), which may lead to removal of small real filaments. To recognize filaments in small-scale structures better, one can examine shapes of the latter in addition to their areas.

The shape factor $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{f}\end{array}$ in Eq. (4) is designed to fine-tune ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$ depending on various shapes of structures. To quantify them, getfilaments employs images of masks (Fig. 7), defining an ellipse for each cluster of connected pixels by computing their major and minor sizes (a,b) from intensity moments (e.g., Appendix F in Paper I). Simple, relatively straight filaments can be quantified by their elongation $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{E}\end{array}$, which is defined as the ratio a/b. However, most of the actual filaments observed with Herschel are curved, warped, twisted, or shaped irregularly otherwise, reflecting complex dynamical (possibly violent) processes that created them. Elongation $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{E}\end{array}$ alone cannot be used to quantify strongly curved, not very “dense” clusters of connected pixels that meander around (e.g., a spiral structure). To describe such a filament, one can define sparsity $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{S}\end{array}$ as the ratio of the elliptical area πab to the total area occupied by all non-zero pixels belonging to the filament. Although $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{E}\end{array}$ may well be very close to unity for sparse clusters of connected pixels, a high value of $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{S}\end{array}$ for such structures would indicate that they are filaments.

The above considerations, along with some experimentation, led to the following empirical definition of the shape factor: $\begin{array}{c}\mathrm{˜}\\ \mathit{f}\end{array}\mathrm{=}{\mathrm{(}\begin{array}{c}\mathrm{˜}\\ \mathit{E}\end{array}\begin{array}{c}\max \end{array}{\mathrm{\{}\begin{array}{c}\mathrm{˜}\\ \mathit{S}\end{array}\mathit{,}{\mathrm{1}}^{\mathrm{\}}}}^{\mathrm{)}}}^{-1}\exp \mathrm{(}\mathrm{20}\hspace{0.17em}\mathrm{(}\mathrm{1.2}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ \mathit{E}\end{array}\mathrm{)}\mathrm{+}{\mathrm{1}}^{\mathrm{)}}\mathit{.}$(5)The elongation $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{E}\end{array}$ and sparsity $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{S}\end{array}$ lower the value of $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{f}\end{array}$, hence the required number of connected pixels in Eq. (4) for structures that are increasingly elongated and sparse. Besides, the exponential factor in Eq. (5) raises a steep barrier for structures with $\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{E}\end{array}\lesssim \mathrm{1.3}$.

With the shape factor defined in Eq. (5), the simple area condition of Eq. (4) works very well for all simulated and Herschel images tested. The cleaning procedure of getfilaments removes non-filamentary structures (noise and background fluctuations, sources), revealing clean filaments, such as the ones shown as (normalized) mask images ℳ_λj in Fig. 8. The masks define the maximum area of single-scale filaments at their base level, allowing reconstruction of their intensity distribution (Sect. 2.4.3).

The above method of detecting real filamentary structures in single-scale images produces very few (if any) spurious filaments and only for strongly-variable backgrounds. Even if a few spurious filaments are found, they are practically always quite faint and should not present any real problem since they can be easily removed, if necessary. Experience shows that the filament threshold $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$ is a good choice: decreasing it to even lower levels would result in more spurious filaments. Indeed, at progressively lower levels, small-scale noise or background fluctuations merge into longer, randomly-oriented elongated chains, similar to the white structures within the noise on the smallest spatial scales in Fig. 6. Some of them would have ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}\mathit{>}{\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$ and thus contaminate clean images of filaments with faint spurious structures.

Fig. 9 Single-scale intensities within filament masks (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here in pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathit{>}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$. Roughly representing the intensity distribution of clean filaments, such images do not take negative areas into account, and they are contaminated by the emission of bright sources.

After applying the cleaning procedure to ℐ_λDj (Fig. 4), one can derive intensity distributions within the filament masks on each spatial scale (Fig. 9) and determine their full intensities by summing up all scales. However, one has to overcome several complications to properly reconstruct the intrinsic intensity distribution of filaments. On one hand, it is necessary to include negative areas of single-scale filaments (Fig. 4) resulting from the spatial decomposition of Eq. (2), as accumulating only positive intensities would incorrectly give substantially wider filaments. On the other hand, observed intensity distributions are very often altered by the bright sources spatially associated with filaments (Figs. 3–5). To include negative areas and remove practically entire contribution of sources, getfilaments reconstructs clean filament intensities $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$ in a more elaborate way.

2.4.3. Reconstructing intrinsic intensities of filaments

The cleaning algorithm described in Sect. 2.4.2 detects all filamentary structures that exist in decomposed single-scale images ℐ_λDj, by separating the filamentary component of the images from all other structures of non-filamentary nature that are outside the filament masks. The problem is that the filaments in $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$ are still contaminated by the noise and background fluctuations and by the sources that are inside their masks (Fig. 9). Substantial work is required to reconstruct full intrinsic intensity distribution of filaments, removing all non-filamentary peaks. In real-life observations, individual filaments and different segments of a filament usually have significantly varying intensities. This further complicates the problem, since getsources and getfilaments process entire images and neither individual filaments nor their parts.

Removing sources from filaments in each single scale, getfilaments splits images ℐ_λDj between their maximum and $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ by a number of intensity levels l, spaced by a factor of 1.05. At each level, the filament reconstruction procedure works on a sequence of differential images $\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}}^{\mathrm{+}}\mathrm{=}{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}\hspace{0.17em}\mathrm{+}\hspace{0.17em}\mathrm{1}}\mathrm{-}{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}}$(6)from the bottom to the top, starting with $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ at l = 1. Clusters of connected pixels with ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}\mathit{<}{\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$ are removed from the images using the TintFill algorithm, producing clean images $\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}\hspace{0.17em}\mathrm{C}}^{\mathrm{+}}$. This cleaning in the process of reconstruction of the intrinsic intensities of filaments is essentially the same as the procedure used above for obtaining clean masks of filaments, with the only difference being that the decomposition beam S_j in Eq. (4) is replaced by min    { S_j,1.8   O_λ }. The latter softens the degree of cleaning of the differential images on spatial scales larger than 3   O_λ, as the noise and background fluctuations and sources usually make smaller contribution to the filaments on large spatial scales. This softening of the cleaning allows an accurate reconstruction of the intrinsic filament profiles including their faintest outskirts on the largest scales.

Fig. 10 Reconstructed positive areas in single-scale images of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here in pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathit{>}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ after the removal of small structures (sources) from the filament (cf. Figs. 4, 9). This gives a relatively good approximation to its intrinsic intensity distribution of filaments; however, the images do not take the negative areas into account.

The above approach worked very well in all benchmarks and real-life images where it has been tested. It was found to become less accurate only in a model image that simulated the limiting case of extremely large sources on top of comparably wide filaments, in which case the filament intensities under the source were overestimated and the source intensities were underestimated by up to ~30%. This is related to the fundamental difficulty of distinguishing between components on very large scales (approaching the image size), where signals from the components blend together so much that it is impossible to separate them without any additional assumptions. One may consider this case unrealistic, since all observed images tested were very far from displaying such combinations of sources and filaments.

Clean images δℐ_{λDj   l   C} are summed up further with those from all lower levels. When all levels have been processed and accumulated, the resulting images (Fig. 10) contain reconstructed positive intensity distributions of all filamentary structures present in ℐ_λDj, with most of the peaks from noise, background, and sources removed: $\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{+}}\end{array}\mathrm{=}\sum _{\mathit{l}\hspace{0.17em}\mathrm{=}\hspace{0.17em}\mathrm{0}}^{{\mathit{N}}_{\mathrm{L}}^{\mathrm{+}}}\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}\hspace{0.17em}\mathrm{C}}^{\mathrm{+}}\mathit{,}$(7)where ${\mathit{N}}_{\mathrm{L}}^{\mathrm{+}}$ is the number of levels and the l = 0 base-level clean differential image is obtained from ℐ_λDj by only taking pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathrm{\le }\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$.

Fig. 11 Reconstructed negative areas in single-scale images of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here in pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathit{<}\mathrm{-}\hspace{0.17em}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ after the removal of small clusters with large negative values around bright non-filamentary peaks (sources) within the filament (cf. Figs. 4, 6). The images are shown here with positive values (multiplied by − 1) for better visibility.

The negative areas around decomposed filaments also require a careful treatment. They cannot be taken directly from the decomposed images (Fig. 4), because the latter are non-locally affected by the negative areas produced by sources and other peaks of non-filamentary nature. To obtain the clean negative areas of the filaments, the algorithm multiplies the single-scale images by −1 and applies the same cleaning procedure described above for the positive areas. It splits the images − ℐ_λDj between their maximum and $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ by a number of intensity levels $\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}}^{\mathrm{-}}\mathrm{=}\mathrm{-}\mathrm{(}{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}\mathrm{+}\mathrm{1}}\mathrm{-}{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}}\mathrm{)}\mathit{,}$(8)starting with $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ at l = 1, and produces clean images $\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}\hspace{0.17em}\mathrm{C}}^{\mathrm{-}}$ by removing small clusters of connected pixels. All processed levels are accumulated, which gives reconstructed negative areas (Fig. 11) produced by all filamentary structures present in ℐ_λDj, with most of the negatives from noise, background, and sources removed: $\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{-}}\end{array}\mathrm{=}\sum _{\mathit{l}\hspace{0.17em}\mathrm{=}\hspace{0.17em}\mathrm{0}}^{{\mathit{N}}_{\mathrm{L}}^{\mathrm{-}}}\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}\hspace{0.17em}\mathrm{C}}^{\mathrm{-}}\mathit{,}$(9)where ${\mathit{N}}_{\mathrm{L}}^{\mathrm{-}}$ is the number of levels and the l = 0 base-level clean differential image is obtained from ${\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}}^{\mathrm{-}}$ by taking only pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathrm{\le }\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$.

Fig. 12 Reconstructed clean single-scale filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here as $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$ after the removal of noise and background fluctuations. Reconstructed positive and negative areas from Figs. 10 and 11 have been added together to produce the images.

The reconstructed positive and negative components are slightly convolved using a small Gaussian beam with a size of 0.1max    { S_j,Δ }. The convolution is beneficial for avoiding abrupt intensity jumps to zero below $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ and it does not alter the filament intensity distribution above $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ because of the small beam. After computing the clean positives and negatives, one can easily obtain their intensity distributions on each scale (Fig. 12): $\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}\mathrm{\equiv }\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{+}}\end{array}\mathrm{-}\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{-}}\end{array}\mathit{,}$(10)where all significant contributions of non-filamentary components have been removed. Reconstruction of the full intrinsic intensity distribution of all significant filaments detected in ℐ_λD on all spatial scales reduces to $\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}\mathrm{=}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{S}}}\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}\mathrm{+}{\mathrm{𝒢}}_{{\mathit{N}}_{\mathrm{S}}}\mathrm{\ast }\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{S}}}\begin{array}{c}\mathrm{˜}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}\mathit{,}$(11)where the second term is an estimate of the contribution of the clean filaments to the largest scales (S > S_max) that are always left out of the single-scale decomposition (cf. Eqs. (2), (3)).

To improve the quality of measurements, image flattening, and final extraction (Sects. 2.6, 2.7), the full reconstructed filaments $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$ are subtracted from the original detection images ℐ_λD and measurement images ℐ_λO, producing their filament-subtracted counterparts ℐ_λD   FS and ℐ_λO   FS.

Fig. 13 Single-scale skeletons (Sect. 2.4). Red and yellow pixels belong to a clean reconstructed filament, whereas the blue pixels have been set to zero during the cleaning. The filament is measured at each pixel in the four main directions to determine the direction where the filament has the smallest half-maximum width. Along those directions (indicated by short straight lines), the brightest pixels (marked by yellow color) belong to the filament’s crest and define the skeleton.

2.4.4. Deriving skeletons of filaments

For studying properties of filaments, it is useful to determine their skeletons, i.e., the lines of connected pixels tracing filaments’ crests. While at the cleaning step, getfilaments creates single-scale skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$ using a simple algorithm illustrated in Fig. 13. At each pixel of the reconstructed filament $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$, half-maximum widths of the latter in four main directions (two image axes and two diagonals) are analyzed to find the direction of the narrowest intensity profile. The brightest pixel of the profile in that direction defines the skeleton pixel, the value of which is set to 1. Since the skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$ are obtained independently on each scale, their location may somewhat fluctuate between scales. To reduce the influence of that uncertainty on accumulated skeletons, the skeleton width is increased to three pixels. This means that each pixel of the skeleton shown in Fig. 13 will eventually spread its value over all eight surrounding pixels.

Full skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$ accumulated over all scales are obtained by summation, similarly to Eq. (11). Such skeletons contain detailed information on their significance, as the value of any non-zero pixel is proportional to the number of spatial scales, where the pixel belongs to the skeleton. Using the same algorithm, getfilaments creates another version of skeletons, tracing the crests of the skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$, instead of those of the filaments $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$. The skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}^{\mathrm{\prime }}\end{array}$ are one-pixel wide and their pixel values are equal to 1. They are further used to produce the segmentation images of filamentary structures, where all pixels belonging to a filament have the value of the filament’s number (see, e.g., Figs. A.1f, A.1f in Appendices A, A).

2.5. Combining clean single scales over wavelengths

The cleaning algorithm outlined in Sect. 2.4 is applied to the single-scale detection images ℐ_λDj independently for each wavelength λ. Combining information across several wavebands in the process of source extraction with getsources significantly improves the source detection and measurement qualities (Paper I). There are similar benefits of combining filaments obtained independently at each wavelength, because the robustness of the detection of filamentary structures (their significance) increases with the number of wavebands where the structures appear. Since only monochromatic images of the simulated filament are used in this paper, the combination of filaments over wavebands is not illustrated here.

The positive component $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{+}}\end{array}$ of filamentary structures, produced by getfilaments (Sect. 2.4), is subtracted from the clean decomposed images ℐ_λDj   C created by getsources for each spatial scale, which improves the reliability of source detection in filamentary backgrounds. In practice, the subtraction is done just before getsources produces combined detection images ℐ_Dj   C and ${\mathrm{ℐ}}_{\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{\prime }}$ (Sect. 2.4 in Paper I).

For an overview of filamentary structures in all wavebands in multi-wavelength extractions, getfilaments accumulates clean filaments $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$ over all detection wavelengths, creating a combined image $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$. Similarly, the images of skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$ are accumulated over all bands in a combined image $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$. Although no such combined images are directly involved in either source or filament extraction, they provide cumulative views of the filaments’ appearance and their significance across wavelengths that are useful when studying filamentary structures.

2.6. Detecting, measuring, and visualizing sources

Source detection, measurements, and visualization in getsources (Fig. 1; Paper I) are practically unaffected by getfilaments, except that the filament-subtracted versions of the respective images are used to improve the source detection and measurement qualities. An additional benefit of the filament extraction is that linear scanning artifacts or radial spikes of the diffraction pattern that may be contaminating observed images are also detected as (spurious) filaments and removed.

Sources are detected in the filament-subtracted single-scale detection images ℐ_{Dj   C   FS}, which greatly reduces the chances of creating spurious sources in strongly filamentary backgrounds. Measurements of the sources’ properties are also performed in the filament-subtracted images ℐ_λO   FS, which considerably improves the interpolation and subtraction of backgrounds, because the latter become largely isotropic. The visualization step produces a number of additional images, where the sources are overlaid on the images of filaments and skeletons, useful when studying various objects and processes associated with filamentary structures (not only forming stars, but also galaxies or their clusters; cf. Appendices A, A).

2.7. Flattening background and noise fluctuations

The Herschel images of Galactic regions display highly variable backgrounds; standard deviations of the combined background and noise fluctuations (outside of sources) sometimes differ by orders of magnitude between various areas of a large image ℐ_λD. Any global thresholding method would have difficulty handling such images, because the thresholds would not be equally good for all areas. This is why getsources employs a special approach (Fig. 1; Paper I): completing an initial source extraction, then flattening detection images based on the local intensity fluctuations outside sources, and then performing the final source extraction using flattened images.

With getfilaments, the flattening procedure for detection images remains essentially the same as described in Paper I, except that the scaling (flattening) image ℐ_λF is computed from the filament-subtracted detection images ℐ_λD   FS. The images ℱ_λ now include the footprints of both sources and filaments, to avoid any possibility that imperfect extraction of filaments in the initial extraction would affect the accuracy of flattening. The flattened filament-subtracted images are used in getsources throughout the final extraction, replacing the original detection images ℐ_λD that were used in the initial extraction.

3. Extraction results

Results of both initial and final extractions of filaments and sources in the simulated image (Fig. 3e), used to illustrate getfilaments in this paper, are shown in Figs. 14, 15. The power-law filament is extracted quite well already after the initial run. Away from the ends of the filament, the reconstructed filament is slightly overestimated owing to imperfect cleaning and separation of the isotropic background (Fig. 14b). Closer to the ends of the filament, where the background intensity becomes lower, the filament is somewhat underestimated in the initial extraction. After the final run, the filament profile is very accurate, as is demonstrated by the profiles f₁, f₇ in Fig. 15. Although a small fraction of the background ended up in the filament, maximum deviations from the true model filament intensity distribution are still within 5% of its peak.

Fig. 14 Reconstruction of the simulated filament (Figs. 2, 3) obtained in the final extraction. (a) Intensity distribution of the clean reconstructed filament. (b) Residuals after the subtraction of the true model filament from the reconstructed filament. (c) Residuals after the subtraction of the reconstructed filament from the original simulated image. (d) Extracted sources, filament- and background-subtracted. (e) Skeleton, integrated over single scales, shown in those pixels that belong to it in more than 5 spatial scales.

Only at the locations of sources in the original image, the deviations increase to about 20% levels, due to imperfect separation of the sources in the process of the filament reconstruction, as indicated by the profiles f₁–f₃ in Fig. 15. The accuracy level depends on the relative properties of sources, filaments, background, and noise. Maximum deviations are expected in the most difficult cases when the sources have size and brightness similar to those of the filament in which they are embedded, as in the present simulation. The more sources and filaments are dissimilar in their widths, the easier it is to separate them; the brighter a component is, the better it can be extracted and the less accurate the extraction of the fainter components.

Without any background (see below), the final extraction brings substantial improvements to the faint outskirts of the filament intensity profile. However, in the presence of the relatively bright background, it also leads to a slight increase in the background emission incorporated into the filament (Fig. 14c). There is a fundamental difficulty in separating contributions of structural components on the largest scales, since their contributions blend together and become very similar. The power-law intensity profile I_λ(r) ∝ r^-1 of the simulated filament makes the problem especially difficult. For steeper (e.g., Gaussian) filaments, the reconstruction and separation of all components are much more accurate.

Fig. 15 Profiles of the components of the simulated image (Figs. 2, 3) obtained in the initial and final extractions (left and right, respectively). Orthogonal profiles of a model source (the second one from the bottom in Fig. 14d) are labeled s1 and s2; the narrower one (red) cuts through the source in a direction perpendicular to the filament, whereas the wider one (blue) is the source profile along the filament. Profiles f1 to f7 show cuts through the reconstructed filament (Fig. 14a) at 7 equidistant locations from its bottom to the top. The first three profiles (f1–f3) pass through the locations of the 2nd, 8th, and 14th sources; all other profiles (f4–f7) display the upper half of the filament, unaffected by sources.

Intensities of the extracted sources (Fig. 14d) are also somewhat affected by the process of separation: their profiles along the filament are slightly overestimated due to the emission from the filament, whereas in the orthogonal direction, they are somewhat underestimated (cf. profiles s₁, s₂ in Fig. 15). Despite the differences, the fluxes measured by getsources are sufficiently accurate, considering large total uncertainties involved in source extraction and measurements in highly structured and variable backgrounds⁸. The peak intensities of 19 sources were overestimated by ~1–7%, whereas their total fluxes were underestimated by ~10–20%. The variations in the accuracies of the fluxes are mainly caused by noise fluctuations, while their average levels are the consequence of the imperfection of the separation of the sources from the filament. Separating structural components from each other will always be a source of additional uncertainties, because the components are completely blended together, and their intrinsic intensity distributions are fundamentally unknown.

Extracting sources in filamentary images without first removing the filaments gives much less accurate results. To demonstrate the difference, another getsources extraction was performed on the simulated image (Fig. 3e), with getfilaments turned off. On average, the sources were found to be substantially (by 60%) elongated along the filament. The peak intensities and total fluxes were overestimated by ~30% and ~100%, respectively. Local uncertainties of the fluxes were overestimated by almost an order of magnitude. The reason for the erroneous results is very simple: the sources were not separated from the filament and the fluxes and local intensity fluctuations include the signal from the filament. As is clear from Fig. 2, footprints of the model sources reach radial distances of 40′′, where the filament intensity drops to 45 MJy/sr, by more than a factor of two. Assuming that the true source footprints are determined correctly, the background subtraction is to be done at that level of intensities, effectively incorporating the upper half of the filament into the sources. Along the filament, however, the baseline for background subtraction lies at 100 MJy/sr, the peak intensity of the filament. For such anisotropic “background” as the filament is, any approach to background subtraction that does not closely approximate the filament profile is bound to give inaccurate results.

The simulated filamentary image (Fig. 3e) was made relatively simple to demonstrate all features of the getfilaments method as clearly as possible; however, with four structural components it is not very simple (see much more complex cases in Appendices A, A). To illustrate the filament extraction with getfilaments in the most straightforward case, a simplified version of the filament image was created that combines just two of the components, the filament and noise (Fig. 3a, d). The intensity distribution and residuals of the extracted filament are displayed in Figs. 16 and 17. Only the results of the final extraction are presented because those of the initial run would appear almost indistinguishable. The small deficit visible in the reconstructed image and profile can be approximated by just a constant value of ~1 MJy/sr. This minor discrepancy is caused by the difficulties in recovering the entire signal on the largest scales close to image size.

Compared to the simple simulation created for this paper, images obtained from complex three-dimensional simulations or Herschel observations are much more challenging for extracting and studying filamentary structures. Filaments in the interstellar medium appear to be very complex structures with greatly varying shapes, intensities, and profiles at different locations along their crests. Often they blend together with crowds of sources and with highly structured, bright, and variable backgrounds, as well as with other nearby filaments⁹. In multi-wavelength observations, the same filamentary area may appear quite differently owing to the contribution of structures with different temperatures and to optical depth effects. Unknown orientations of observed filaments in three-dimensional space greatly complicate their detailed studies and increase total uncertainties of results.

Fig. 16 Results of the final extraction for a simplified version of the simulated image (Figs. 2, 3) with only noise but without background and sources. (a) Intensity distribution of the clean reconstructed filament. (b) Residuals after the subtraction of the true model filament from the reconstructed filament. (c) Residuals after the subtraction of the reconstructed filament from the original simulated image.

4. Conclusions

Herschel observations have demonstrated that the interstellar medium is highly structured on all spatial scales and that its significant fraction emerges in omnipresent filamentary structures. Filamentary backgrounds present serious complications for source extraction methods since the filaments tend to amplify insignificant background or noise fluctuations that fall on top of the structures and thus create spurious sources. This paper describes the filament extraction method getfilaments, which shares the general multi-scale and multi-wavelength philosophy and approach with the source extraction method getsources (Paper I). Although both methods were designed primarily for use in large far-infrared and submillimeter surveys of star-forming regions with Herschel, they are applicable to other types of images.

Fig. 17 Profiles of the reconstructed filament for a simplified version of the simulated image (Figs. 2, 3) with only noise but without background and sources, obtained in the final extraction. Profiles f1 to f7 show cuts through the extracted filament (Fig. 16a) at 7 equidistant locations from its bottom to the top.

Instead of tracing filaments directly in the observed images, getfilaments analyzes highly filtered decompositions of original images over a wide range of spatial scales (Sect. 2.3). The algorithm identifies filaments on each spatial scale as significantly elongated structures and reconstructs their full intrinsic intensity distributions, which are practically unaffected by sources and largely isotropic backgrounds (Sect. 2.4). Additionally, it determines single-scale and accumulated skeletons of the filaments, tracing the crests of their intensity distributions. Furthermore, it produces segmentation images of the filamentary structures, where each filament is identified by its sequential number. For an overview of all filaments, getfilaments creates combined images of clean filaments and their skeletons over all wavebands (Sect. 2.5). Based on the results of the initial extraction, detection images are flattened to produce much more uniform fluctuations of noise and non-filamentary background in preparation for the second, final extraction (Sect. 2.7).

Because it is incorporated into getsources, the getfilaments method brings substantial improvements to source extraction

in filamentary backgrounds. Extraction of sources is also essential for an accurate reconstruction of the intrinsic intensity distribution of filaments. The intimate physical relationship between forming stars and filaments seen in Herschel observations demands that accurate filament extraction methods remove the contribution of sources, and conversely, accurate source extraction methods must be able to remove underlying filamentary structures. The images of clean filaments are now subtracted by getsources from the original images during source extraction, significantly improving the robustness of the method and reducing the chances of spurious sources contaminating extraction catalogs. An important benefit of the improved source extraction method is that, in addition to the catalogs and images of sources, it provides researchers with clean images of the filamentary structures that are the birthplace of stars.

Both getsources and getfilaments methods have been thoroughly tested using many simulated benchmark images and real-life observations. The source and filament extraction code is automated, very flexible, and easy-to-use. The latest version of the code with an installation guide and a quick start guide will soon be freely available upon request and downloadable from a web page¹⁰.

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
${\mathrm{ℐ}}_{\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{\prime }}$	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
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$	filaments cleaned of sources, noise, and background
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{+}}\end{array}$	positive component of reconstructed filaments $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}^{\mathrm{-}}\end{array}$	negative component of reconstructed filaments $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$
ℐλD   C	full images of sources reconstructed from ℐλDj   C
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$	image of filaments combined over wavelengths
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$	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
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$	image of skeletons combined over wavelengths
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$	skeletons of clean single-scale filaments
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$	full accumulated skeletons of clean filaments
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathrm{D}\hspace{0.17em}\mathrm{C}}^{\mathrm{\prime }}\end{array}$	wavelength-combined skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}^{\mathrm{\prime }}\end{array}$
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}^{\mathrm{\prime }}\end{array}$	skeletons tracing crests of the full skeletons $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{𝒮}}_{\mathit{\lambda }\mathrm{D}\hspace{0.17em}\mathrm{C}}\end{array}$
a	major size of a filament mask
A	major FWHM size of a source
${\mathit{A}}_{\mathit{\lambda }}^{\max }$	maximum FWHM sizes of sources to be extracted
b	minor size of a filament mask
D0	filament width: FWHM of the inner Gaussian core
$\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{E}\end{array}$	elongation of the clusters of connected pixels
$\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{f}\end{array}$	empirical shape factor of filamentary structures
fS	scale factor defining relative spacing between scales
f(ζ)	width normalization factor of a simulated filament
${\mathit{I}}_{\mathit{\lambda j}}^{\mathrm{\ast }}$	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
${\mathit{N}}_{\mathrm{L}}^{\mathrm{\pm }}$	number of intensity levels in filament reconstruction
NS	number of spatial scales in the image decomposition
${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$	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
$\begin{array}{c}{\mathrm{˜}}_{}\\ \mathit{S}\end{array}$	sparsity of the clusters of connected pixels
W	width of a filament
$\mathit{\delta }{\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathit{l}}^{\mathrm{\pm }}$	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)
$\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$	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.

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.

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.

Acknowledgments

The author employed SAOImage DS9 (by William Joye) and WCSTools (by Jessica Mink) developed at the Smithsonian Astrophysical Observatory (USA), the CFITSIO library (by William D. Pence) developed at HEASARC NASA (USA), the STILTS library (by Mark Taylor) developed at Bristol University (UK), the PSPLOT library (by Kevin E. Kohler) developed at Nova Southeastern University Oceanographic Center (USA), SWarp (by Emmanuel Bertin) developed at Institut d’Astrophysique de Paris (France), and ImageJ (by Wayne Rasband) developed at the National Institutes of Health (USA). Web sites of the computational projects Starformat and Horizon provided images used for illustrations in this paper. Multi-wavelength observations of star-forming regions obtained in the frame of the Herschel Gould Belt and HOBYS Key Projects were useful in developing and testing both getsources and getfilaments; the work of all those who contributed to the production of the images (directly or indirectly) is highly appreciated. Useful tests were done on small ground-based (sub-) millimeter images of star-forming regions observed by Frédérique Motte and Philippe André. Two of a series of simulated filamentary images used in the development of getfilaments were created by Doris Arzoumanian, and some comparisons with observed filaments were made by Pedro Palmeirim. Comments on a draft made by Alana Rivera-Ingraham, Sarah Sadavoy, Nick Cox, Martin Hennemann, Nicola Schneider, Pierre Didelon, Nicolas Peretto, and Arabindo Roy are acknowledged.

References

All Figures

Fig. 1 Main processing blocks of getsources and getfilaments. For a complete extraction, both methods require two runs (left): the initial and final extractions (red blocks; the preparation and flattening steps are shown in blue; cf. Paper I). A more detailed presentation of the processing steps (right) shows that both algorithms would have had lots of identical actions, if conceived and coded separately. The getfilaments algorithm is essentially localized in only the cleaning and combining steps of getsources (highlighted in yellow). In practice, getfilaments is an integral part of the source extraction code, activated by a single configuration parameter of getsources.

In the text

Fig. 2 Profiles of the simulated filament (Sect. 2.1) used to illustrate the getfilaments method. The filament has the inner half-width R0 = 37″ (at half-maximum) and the outer power-law profile Iλ(r) ∝ r-1 (blue). Gaussian-shaped sources (cyan) and background (green) have FWHM sizes of 30′′ and 4000′′, respectively. Random pixel noise (black) was convolved to a resolution of 18′′ (FWHM) and scaled to have a standard deviation σnoise = 1.25 MJy/sr (dashed). Three profiles sample the full simulated image (Fig. 3) across the top of the filament (brown), just below its midpoint (magenta), and through the position of the uppermost source (red).

In the text

Fig. 3 Simulated image and its components (Sect. 2.1). (a) Straight filament with the profile displayed in Fig. 2, normalized to $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{I}}_{\mathrm{P}}\end{array}\mathrm{=}\mathrm{100}$. (b) Identical Gaussian sources with an intrinsic FWHM size of 24′′, convolved to 18′′ resolution and normalized to IP = 100. (c) Isotropic Gaussian background with a size of 4000′′ (FWHM), normalized to IP,b = 10. (d) Random instrumental noise with σnoise = 1.25. (e) Full simulated image ℐλO (≡ℐλD) with S/N = 80 and resolution of 18′′. Three horizontal lines indicate the locations and direction of the profiles shown in Fig. 2. Each panel’s dimensions are 600 × 2820 pixels (), pixel size Δ = 2″.

In the text

Fig. 4 Spatial decomposition (Sect. 2.3). Single scales ℐλDj of the simulated image ℐλD (Fig. 3) are shown for j = 18,32,46,60,74, NS = 99, fS = 1.079, ${\mathit{S}}_{\mathrm{1}}\mathrm{=}\mathrm{5}\stackrel{\mathrm{″}}{.}\mathrm{56}$, SNS = 9609″. The scales Sj are separated by a factor of 3 to illustrate the spatial decomposition; negative areas surrounding the filament are the direct consequence of the subtraction in Eq. (2). Scale sizes Sj are visualized and annotated here and in all subsequent similar figures.

In the text

	Fig. 5 Single-scale removal of noise and background (Sect. 2.4). The same set of spatial scales (Fig. 4) is displayed in the single-scale images ℐλDj   C, cleaned of noise and background with an iterative procedure described in full detail in Paper I. Noise fluctuations visible in Fig. 4 have been removed by zeroing pixels with Iλj < ϖλj.
In the text

Fig. 6 Single-scale detection of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here below the intensity threshold $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$ for detection of filaments. The filament’s base is clearly visible in the single scales, although the latter are strongly contaminated by noise and background fluctuations.

In the text

Fig. 7 Single-scale masking of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here exactly on the intensity threshold $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}\mathrm{=}{\mathit{\sigma }}_{\mathit{\lambda j}}$ for detection of filaments, and all lower level signals were set to zero. The simulated filament is clearly visible on all scales, as is the abundant contamination of the images by noise. Noise-free black zones that appear to surround the filament are the consequence of the negative areas seen in Figs. 4 and 6.

In the text

Fig. 8 Single-scale masks of filaments (Sect. 2.4). The images of spatial scales from Fig. 7 are shown here at their base level as the normalized filament masks ℳλj after the removal of relatively small clusters of pixels of non-filamentary nature (including noise, background, and sources) with ${\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}\mathit{<}{\mathit{N}}_{\mathrm{\Pi }\mathit{\lambda j}}^{\hspace{0.17em}\min }$. In the five images displayed, structures with less than 1.31 × 103, 8.15 × 104, 6.85 × 105, 5.76 × 106, and 5.65 × 107 pixels have been removed (cf. Fig. 7).

In the text

Fig. 9 Single-scale intensities within filament masks (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here in pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathit{>}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$. Roughly representing the intensity distribution of clean filaments, such images do not take negative areas into account, and they are contaminated by the emission of bright sources.

In the text

Fig. 10 Reconstructed positive areas in single-scale images of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here in pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathit{>}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ after the removal of small structures (sources) from the filament (cf. Figs. 4, 9). This gives a relatively good approximation to its intrinsic intensity distribution of filaments; however, the images do not take the negative areas into account.

In the text

Fig. 11 Reconstructed negative areas in single-scale images of filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here in pixels with ${\mathit{I}}_{\mathit{\lambda j}}\mathit{<}\mathrm{-}\hspace{0.17em}\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathit{\varpi }}_{\mathit{\lambda j}}\end{array}$ after the removal of small clusters with large negative values around bright non-filamentary peaks (sources) within the filament (cf. Figs. 4, 6). The images are shown here with positive values (multiplied by − 1) for better visibility.

In the text

Fig. 12 Reconstructed clean single-scale filaments (Sect. 2.4). The images of spatial scales from Fig. 4 are shown here as $\begin{array}{c}{\mathrm{˜}}_{}\\ {\mathrm{ℐ}}_{\mathit{\lambda }\mathrm{D}\mathit{j}\hspace{0.17em}\mathrm{C}}\end{array}$ after the removal of noise and background fluctuations. Reconstructed positive and negative areas from Figs. 10 and 11 have been added together to produce the images.

In the text

Fig. 13 Single-scale skeletons (Sect. 2.4). Red and yellow pixels belong to a clean reconstructed filament, whereas the blue pixels have been set to zero during the cleaning. The filament is measured at each pixel in the four main directions to determine the direction where the filament has the smallest half-maximum width. Along those directions (indicated by short straight lines), the brightest pixels (marked by yellow color) belong to the filament’s crest and define the skeleton.

In the text

Fig. 14 Reconstruction of the simulated filament (Figs. 2, 3) obtained in the final extraction. (a) Intensity distribution of the clean reconstructed filament. (b) Residuals after the subtraction of the true model filament from the reconstructed filament. (c) Residuals after the subtraction of the reconstructed filament from the original simulated image. (d) Extracted sources, filament- and background-subtracted. (e) Skeleton, integrated over single scales, shown in those pixels that belong to it in more than 5 spatial scales.

In the text

Fig. 15 Profiles of the components of the simulated image (Figs. 2, 3) obtained in the initial and final extractions (left and right, respectively). Orthogonal profiles of a model source (the second one from the bottom in Fig. 14d) are labeled s1 and s2; the narrower one (red) cuts through the source in a direction perpendicular to the filament, whereas the wider one (blue) is the source profile along the filament. Profiles f1 to f7 show cuts through the reconstructed filament (Fig. 14a) at 7 equidistant locations from its bottom to the top. The first three profiles (f1–f3) pass through the locations of the 2nd, 8th, and 14th sources; all other profiles (f4–f7) display the upper half of the filament, unaffected by sources.

In the text

Fig. 16 Results of the final extraction for a simplified version of the simulated image (Figs. 2, 3) with only noise but without background and sources. (a) Intensity distribution of the clean reconstructed filament. (b) Residuals after the subtraction of the true model filament from the reconstructed filament. (c) Residuals after the subtraction of the reconstructed filament from the original simulated image.

In the text

	Fig. 17 Profiles of the reconstructed filament for a simplified version of the simulated image (Figs. 2, 3) with only noise but without background and sources, obtained in the final extraction. Profiles f1 to f7 show cuts through the extracted filament (Fig. 16a) at 7 equidistant locations from its bottom to the top.
In the text

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.

In the text

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.

In the text