# br SPF showed overall greater accuracy and

SPF showed overall greater accuracy and lower computational time with respect to classical variational level set methods, essen-tially thanks to its ability to escape local minima and converge faster (Abdelsamea & Tsaftaris, 2013). However, as a global level set method which does not exploit prior knowledge SPF cannot suc-cessfully segment regions with overlapping intensity distributions. Moreover, SPF is not able to deal with the changes in the inten-sity distribution of the region to be segmented. Here, we propose to incorporate prior knowledge in the SPF level set formulation to cope with the inherent complexity of histology images.

3.1.2. Fuzzy Signed Pressure Force method

The main idea of the FSPF is to incorporate a probabilistic model of the epithelium into the SPF level sets formulation. Given a contour C, x the pixel location in the image I(x), our proposed

Table 1

The Precision, Recall, and Dice metrics for the FSPF and SPF approaches applied to TMA images of Fig. 6.

Image FSPF Classical SPF method

where

EPI

EPI

inside and outside the contour, and PEPI(x) is the local Epithelium
Fig. 6
(h)
100

probabilistic model that guides the movement of the contour. The

latter is the optimal membership function results from the mini-

mization of the following objective function:

Table 2

The Precision, Recall, and Dice metrics for our approach applied to WSIs of Fig. 8.

Image

Single training sample

Multiple training samples

where c is the number of classes, Ilocal(x) is the local intensity ap-

proximation function calculated at each pixel location x, P2 (x) are

j

the intensity centroids of the Moniliformin and are defined as follows:
Fig. 8
(d)
100

3.2. Histological image feature representation

(8)
We propose a novel set of invariant shape and appearance fea-

c

tures, specifically designed for histology ROIs.

(9)
imizes the integral of the square of the distances between each

point in the region and the line. The ALI passes through the cen-

with gσ

a Gaussian smoothing kernel of standard deviation σ

.
troid of the region and is orientation independent.

For each connected component of the extracted epithelium we

(10)
first estimate the ALI and then project each point, I, onto it. The Eu-

where

clidean distance between the projected points furthest from each

other is the length of the ALI. The ALI splits the ROI, I(x) x, into

P

I

left and right domains, left and right respectively (see Fig. 3).

We propose the following ALI-based descriptors:

PEPImin
− PEPImax

sign I

n

α (PI(x), EPI) is responsible for creating the external pressure
forces for the evolution of the contour and spf(I(x), PEPI) modulates the signs of the pressure forces.

Compare to SPF approach, FSPF is able to cope with (i) inten-sity inhomogeneities; and (ii) high visual variability of different tis-sues in the image. This is by integrating the probabilistic models of different tissues/classes (i.e., Pj(x)), which are calculated based on local information (i.e., Ilocal(x)), into the segmentation framework (Eq. (12)). For epithelium segmentation, the probabilistic model of epithelium (i.e., PEPI(x)) is used to control the contour evolution for an effective segmentation. Moreover, one can benefit from this approach, as a multi-class object segmentation, by considering the relative probabilistic model to target a specific tissue (e.g., stroma only).
and n is the number of pixels in the region.

n

n

All three descriptors quantify the similarity of the left and right-hand domains: ALI by considering the distances to the ALI, ALI by considering the distances to the centroid of the region, and ALIζ by considering the distributions of the projected points.