## 4 Discussion
### Technical Features of the Method  
Our approach is based on a model of sample-light interaction, which reproduces the sources of origin of the birefringence signal of collagen, when it is imaged in a linear polarization setting. The regular spacing of collagen fibrils into fibers provides them with positive intrinsic birefringence [8]. This intrinsic birefringence can be enhanced by a specific staining process to point out even thin collagen fibers. For instance, picrosirius red staining is usually the preferred staining, since its peculiar chemical properties results in higher specificity and selectivity for collagen and less fading problems with respect to other techniques, such as van Gieson and trichrome techniques [11; 12; 13]. The intrinsic and/or staining-induced birefringence results in collagen fibers appearing bright versus a dark background when illuminated in polarized light. Figure 8: Sensitivity analysis on the effects of collagen threshold \(t_{col}\) on fibrosis detection. Fibrosis fraction for the sinus rhythm (SR, green) and atrial fibrillation (AF, red) groups are displayed as a function of the threshold \(t_{col}\). Both parameters displayed a progressive decrease at the increase of the threshold, but differences between the two groups remained statistically significant at all threshold values. Data are expressed as mean (dot) \(\pm\) standard deviation (whiskers) over the two subgroups. \({}^{*}\), \(p<0.05\). N = 14. Figure 7: Proof of concept application of the algorithm to cardiac sample image database. Algorithm-estimated values for the amount (**left** panel) and heterogeneity (**right** panel) of mid-wall fibrosis are compared in patients with sinus rhythm (SR) and in patients with atrial fibrillation (AF). AF patients displayed larger and more heterogeneous fibrosis content. In boxplots, red lines are median values, blue boxes indicate interquartile ranges (IQR) and black whiskers 1.5*IQR. Individual values are superimposed as black dots. \({}^{*}\), \(p<0.05\). N = 14. Based on a physical model of polarized light-sample interaction, we aimed to improve the estimation of collagen birefringence as an index of collagen amount by combining the information in both bright-field and dark-field images, and by reducing the effects of artifacts and confounding factors deriving from variability sources in sample preparation as well as in microscopy set-up. The combination of bright-field and dark-field image information occurred at different levels of our algorithmic flow. Specifically, both bright-field and dark-field tissue images were used to construct a signal proportional to the transmittance, which was used for the automatic segmentation of the tissue from the background and for the estimation of sample thickness. Bright- and dark-field images of a white image were also entered in the signal construction to correct the field illumination nonuniformity and to improve tissue segmentation from the background. Both bright-field and dark-field images were combined in the construction of the birefringence signal, which was used for the semi-automatic segmentation of collagen. In particular, the birefringent signal was also corrected for variations in sample thickness according to the thickness estimation from the transmittance signal. The combination of bright- and dark-field images for the segmentation of background and collagen is an original aspect of our algorithm with respect to the literature [9; 10; 15; 22; 23; 24]. The majority of studies using picrosirius red staining in combination with polarized light employed bright-field illumination to detect collagen [15; 22; 23; 24]. In the study by [PERSON] et al. [15], a semi-automatic threshold-based approach was applied to detect collagen from bright-field images. A color transformation from the RGB to the CIE LAB color space was performed and specific thresholds, associated with the staining, were applied to detect each specific tissue compartment (i.e., lumen, cardiomyocytes, or collagen fibers).