Gauss, Jordan, Peano and others introduced digitizations of sets in the plane and in the 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 now one way 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, eg. linear convergence for f(r)=1/r or quadratic convergence for f(r)=1/(r^2).

The talk reviews work in multigrid convergence in the sketched context of digital image analysis. In 2D, problems of area estimations and lower-order moment estimations do have "classical" solutions (Jordan, Peano et al.). The perimeter estimation problem (or length of a curve) and moment estimates of arbitrary order do have quite recent solutions in image analysis. The linearity of convergence is known for two techniques for curve length estimation based on regular grids. Estimates of moments of arbitrary order are converging with f(r)=r^(-15/11).

In 3D, for problems of volume estimations and lower-order moment estimations solutions are known for about one-hundred years (Jordan, Minkowski, Scherrer et al.). But the problem of multigrid surface contents measurement is still a challenge, and there is recent progress in this field. Local approximation techniques are still in use in 3D image analysis which are known to be non-convergent. Curves in 3D are another interesting subject, and a first algorithmic solution for convergent length estimation has been suggested quite recently.