tokens: above even a high percentile point necessarily belong to a real filament. The histograms in Fig. 4 g are plotted for the intersection of two sets of pixels, those that are above the 95% percentile of significance (given by the and values) and those that coincide with a real filament (see Fig. 4 d). The RHT parameters are this time optimised manually, only keeping = 10 fixed. This leads to values = 5 and = 0.65. By visual inspection of the results, TM is slightly more successful in highlighting the location of filaments (Fig. 4 c) and estimating their position angles (Fig. 4 g). Fig. 4 Comparison of RHT and TM results in the case of non-linear filamentary structures on a fluctuating background. For the description of individual frames see Fig. 2 . Figure 5 shows how the results change as a function of the parameter values. For TM, the values of and are varied in 30% steps. For RHT, the parameter is kept at a value of 0.65, which was found to give the best results in Fig. 4 , but and are also change in 30% steps around the best parameter combination. The results of TM are less sensitive to these changes in the input parameters. Fig. 5 Change in RHT and TM results as a function of the method parameters. RHT parameters and are varied and parameter is kept at a value of 0.65. 3.2. Reliability and completeness We address questions of TM reliability estimation in the case of the simulation shown in Fig. 4 . equals 2.8 pixels and any two estimates that are separated by approximately five pixels can be considered independent. We first analyse maps similar to those in Fig. 4 but without filaments and with different realisations of noise and of large-scale fluctuations. According to this Monte Carlo exercise, 84%, 97.7%, and 99.9% confidence limits correspond to values of 0.0457, 0.0638, and 0.0850, respectively. This can be compared to the values of the same percentiles that are obtained directly from the input map of Fig. 4 . Those values are 0.057, 0.23, and 0.43 and they would thus result in lower estimates of the statistical significance of a given value. For example, , the apparent value corresponding to the 84% confidence limit, is in reality closer to the 97.7% confidence limit. The values are higher because of the presence of real elongated structures. The more real structures the map has, the more conservative are the limits derived from the statistics of the image itself. For a 360 360 pixel image, one would expect 2981 spurious pixels above the given threshold of . We count as spurious all pixel whose distance to real filaments exceeds three pixels. The number of such pixels is 3174, by 6.5% higher than expected. However, there are still some pixels with that are associated to filaments but are at distances exceeding three pixels. This suggests that the threshold obtained from the Monte Carlo simulations gives a rather accurate estimate of the reliability. For the 97.7% confidence limit, we expect the map to contain 68 instances where two
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