Fuzzy Connections - Algorithms

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 μ_α for the adjacency relation, and μ_κ for the intensity relation.

Fuzzy Connector Tool

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

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

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.