Fuzzy Connection Algorithm

The algorithms used by the Fuzzy Connections tool are based on those described in:
J.K. Udupa and S. Samarasekera. "Fuzzy Connectedness and Object Definition: Theory, Algorithms, and Application in Image Segmentation". Graphical Models and Image Processing 58:246-261 (1996).

In this work, term affinity is used to express a relationship between two pixels. The affinity between two pixels c and d depends on the degree of adjacency between the pixels and the similarity of their intensity values. These two properties are given the symbols μα for the adjacency relation, and μκ for the intensity relation.

Fuzzy Connector Tool

The Fuzzy Connector tool uses the following, as in the reference above.

  • μα=1 if c and d are edge connected (4-adjacency, or 6-adjacency for propagation in 3-D).
    μα=0 otherwise.

  • μκ = exp(  −1 T2 [(I(c) + I(d))/2 − m]t S−1 [(I(c) + I(d))/2 − m] )

where I(c) and I(d) are r-component intensity column vectors; r is the number of input images; m is the r-component mean intensity vector over all seed pixels; and S is the r×r covariance matrix of intensities.

The affinity is the product of μα and μκ.

In the above, T is a number that is set to 2 by default. However, setting the intensity hints modifies T as follows:

  • If the intensity hint is set to "Brighter" for an image, and the intensities of pixels c and d are both brighter than the mean intensity of the seeds in that image, then T is incremented by 2.

  • If the intensity hint is set to "Darker" for an image and the intensities of pixels c and d are both darker than the mean intensity of the seeds in that image, then T is incremented by 2.

In this way, setting an intensity hint gives a greater affinity between pixels for a given deviation from the mean intensity of the seeds, if that deviation is in the right direction according to that hint.