Maximizing a Sum Related to Image Segmentation Evaluation

Abstract

Consider the set S of points in the plane consisting of the ordered pairs (i, j ), where $1 \leqslant i \leqslant m$ and $1 \leqslant j \leqslant n$. A problem related to the study of segmentation evaluation of visual images concerns finding a permutation σ of the points of S for which the sum

$$ \sum\limits_{s \in S}d(s, \sigma(s)) $$

is maximal among all possible permutations of S, where d denotes the Euclidean metric. In this note, we show that this maximum is achieved by exactly those permutations of S for which the line joining the points s and σ(s) passes through the point $(\frac{m+1}{2},\frac{n+1}{2})$ for all s ∈ S. In fact, the result applies to any point-symmetric set in any dimension for all L ^p metrics, 1 < p < ∞. In addition, we provide asymptotic estimates as m and n grow large for the actual maximum value achieved by the above sum.