English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the RMS distance (sum of squared distances between pairs ofpoints) under translation between two point sets in the plane, in two differentsetups. In the partial-matching setup, each point in the smaller set is matchedto a distinct point in the bigger set. Although the problem is not known to bepolynomial, we establish several structural properties of the underlyingsubdivision of the plane and derive improved bounds on its complexity. Theseresults lead to the best known algorithm for finding a translation for whichthe partial-matching RMS distance between the point sets is minimized. Inaddition, we show how to compute a local minimum of the partial-matching RMSdistance under translation, in polynomial time. In the Hausdorff setup, eachpoint is paired to its nearest neighbor in the other set. We develop algorithmsfor finding a local minimum of the Hausdorff RMS distance in nearly linear timeon the line, and in nearly quadratic time in the plane. These improvesubstantially the worst-case behavior of the popular ICP heuristics for solvingthis problem.

Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms