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# br x Lx br Multiscale and multidimensional percolation br

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4.2.3. Multiscale and multidimensional percolation

The percolation was calculated from the colon images as well the respective curvelet sub-images, similarly to the other multi-scale and multidimensional approaches. The definition of percola-tion consists of the existence of a path of connected pixels (cluster)

The average coverage ratio of the largest cluster function was calculated by the coverage ratio of the largest cluster in each eval-uated box of size L. The calculation of the average coverage ratio was performed by dividing the sum of the coverage in each box by the quantity of boxes T Eq. (5). The division of its largest cluster (γ i) by the number of pixels in the box L2 consists of the coverage

The percolating boxes ratio Q was calculated by counting the number of boxes percolated in a scale L. A box qi was considered as a percolating box if the ratio between the number of pixels la-belled as pores ( i) and the total number of pixels inside the box (L2) exceeded the percolation threshold p Eq. (7). Thus, the func-tion Q(L) was obtained by dividing the total number of percolating boxes qi by the total number of boxes T in a specified scale L, as in Eq. (8).

from one extremity to the other of an image, by considering it Lovastatin as the squared lattice. The applied method was described in detail by Roberto et al. (2017).

The first step consisted of a multiscale analysis with the appli-cation of the gliding box algorithm. A square box with a side of L was used to cover all the pixels of an image given as input. The size of L was increased after completely covering the image. The box total T was obtained considering width W and height H of the image, in function of a box size of L, as shown in Eq. (5).

In the next step, the multidimensional analysis was performed by positioning the central pixel Fc from a box with a side L over each pixel of the analysed image, as in the approach described in (Ivanovici & Richard, 2011; Ivanovici et al., 2009). The most rel-evant channel was selected for performing comparisons between the central and the remaining pixels in the box, by applying the Minkowski distance Eq. (2). This process was similar to that ap-plied for calculating the FDp attribute. Next, when the Minkowski distance δ was less or equal to L, the value of P was defined as −1 in order to indicate that the pixel represents a pore in the analysed box.

Following this, the third step consisted of associating the per-colation theory from the results obtained in the multiscale and multidimensional analyses. Thus, the height and the width of the matrix and the probability p defines a pore where a fluid may flow through the path. The presence of a percolating cluster is guaranteed when p is greater than the percolating threshold (p = 0.59275) (Bird & Perrier, 2010). The cluster labeling algorithm of Hoshen–Kopelman was applied to obtain the percolating clusters based on the value of P = −1 (Hoshen & Kopelman, 1976). After labelling each pore, the clusters were defined in case the same la-bel was associated to a set of pixels. When the first cluster is fin-ished, the algorithm advances to the next pore pixel that has still not been labelled.

Three different functions were considered for analysing the H&E images: cluster average C, percolating box ratio Q and average cov-erage ratio of the largest cluster . The number of clusters in a single box is given by ci and the average number of clusters per box C(L) is presented in Eq. (6), where the sum of the number of clusters in a scale L was divided by the number of boxes T.

T

The behaviour for each function C(L), Q(L) and (L) were ob-tained considering the metrics area under curve (ARC), skewness (SKW), area ratio (AR), maximum point (MP) and the scale of the maximum point (SMP). The ARC was obtained through the numer-ical integration using the trapezoidal method. The SKW consisted of an asymmetry indication compared to the average value. The negative values of skewness indicated that the sample was con-centrated to the left of the average value. The positive values of SKW indicated that the sample was concentrated to the right of the average value. Considering a case of perfectly symmetrical sample the skewness obtained is 0. The AR considered the ratio between the right side and the left side areas under the function curve. The MP provided the value of the maximum point of each function and SMP the scale of the maximum point. The sizes of the boxes cho-sen for the obtainment of the percolation features were L = 3 to L = 45.