Figure 9.

Illustrations of the decomposition’s limitations assuming a two function approximation in 2D. In (a) the MGDM decomposotion can perfectly represent the distances to 3 different objects (including the current object) at a point, when 2 distance functions are used. The white object is the current label at the centerpoint. When one or two other object (blue and pink) are nearby, the true boundary locations (black lines) to those objects can be represented perfectly with two distance functions. However, when a third (green) or fourth (orange) object is nearby but further than the first two objects, MGDM approximates their distances (gray dotted lines) with the distance to the second neighbor and incurs errors, the magnitudes of which are indicated by the red arrows. The errors can be eliminated by storing additional levels of the decomposition. In (b), the 1D plot of the reconstructed level set function for object O₁ (along the dotted line), shows how approximations appear when we are further from the object’s boundaries than from other object’s boundaries.