Multigrid convergence of estimated properties in image analysis
by
Reinhard Klette
University of Auckland

Gauss, Jordan, Peano and others introduced digitizations of sets in the plane and in 3D space for the purpose of feature measurements. Features measured for digitized sets, such as perimeter, contents etc., should converge (for increasing grid resolution) towards the corresponding features of the given sets before digitization. This type of multigrid convergence is one option of evaluating approaches for feature measurement in image analysis with respect to correctness. Assume a family of sets S, a digitization model D_r(S) depending upon grid resolution r, and a feature F defined for this family of sets. Then an estimator M of this feature is convergent for this family of sets and this digitization model if there is a grid resolution r_S for any set S in this family such that an estimator value M(D_r(S)) is defined for any grid resolution r > r_S, and |M(D_r(S)) - F(S)| < f(r) for a function f converging towards 0 if r goes to infinity. The function f specifies the convergence speed, e.g. linear convergence for f(r)=1/r or quadratic convergence for f(r)=1/(r²).

The talk reviews work in multigrid convergence in the sketched context of digital image analysis with a main focus on new results in surface area estimation.

Date received: February 10, 2002