A morphological transformation T of an image (point set) A in 2-dimensional Euclidean space \mathbb{E}^2 by a smaller image (point set of smaller cardinality) B can be formulated in terms of a relation of the point set A with so called structuring element B. The application of morphological transformation T(A) means that the structuring element moves systematically across the entire set A with respect to a local origin.
Define translation X_h of point set X by a vector h as
X_h = \{ p \in \mathbb{E}^2, p=x+h \text{ for some } x \in X \}
Translation of X by h
The erosion operation combines two point sets by Minkowski set subtraction [1].
X \ominus B = \{ p \in \mathbb{E}^2: p+b \in X \| b \in B \}
that is equivalent to taking the minkowski sum of a shape and the mirror of another shape.
every point p from the image is tested, and the result of the erosion is given by those pixels p for which all possible p + b are in X. Erosion can be expressed as an intersection of all translations of the image X by the vectors -b \in B :
X \ominus B = \bigcap_{b \in B} X_{-b}.
Example: given point sets
X = \{ (1, 0),(1, 1),(1, 2),(0, 3),(1, 3),(2, 3),(3, 3),(1, 4) \}
and
B = {(0, 0),(1, 0)}
the result of erosion will be
X \ominus B = {(0,3),(1,3),(2,3)}
Erosion of X with structuring element B
By default, Nyxus uses the 3 \times 3 structuring element in the erosion operations.
Default Nyxus structuring element
Feature EROSIONS_2_VANISH calculates the number of erosion operations on a ROI to vanish in its axis aligned bounding box in a sense that each pixel intensity becomes =0.
Example: 5 erosion operations applied to a star-shaped ROI
Unlike EROSIONS_2_VANISH which is measured with respect to a ROI's axis aligned bounding box, this feature calculates the number of erosion operations on a ROI to vanish in its convex hull.
- Hadwiger, H. (1957) Vorlesungen uber Inhalt, Oberflache und isoperimetrie. Springer Verlag, Berlin. http://dx.doi.org/10.1007/978-3-642-94702-5