( ( ( The ( optimum ( displacement function ) ) ) ( for ( ( any pair ) ( of signatures ) ) ) ) ( ( is ( extracted ( using ( minimization ( of ( ( a functional ) ( ( defined ( as ( ( the ( weighted sum ) ) ( of ( ( the ( squared ( Euclidean distance ) ) ) ( between ( ( ( two signatures ) and ) ( a ( penalty term ) ) ) ) ) ) ) ) ) ( requiring ( smoothness ( of ( the ( displacement function ) ) ) ) ) ) ) ) ) ) ) ) . ) )
(ROOT (S (NP (NP (DT The) (JJ optimum) (NN displacement) (NN function)) (PP (IN for) (NP (NP (DT any) (NN pair)) (PP (IN of) (NP (NNS signatures)))))) (VP (VBZ is) (VP (VBN extracted) (S (VP (VBG using) (NP (NP (NN minimization)) (PP (IN of) (NP (NP (DT a) (NN functional)) (VP (VBN defined) (PP (IN as) (NP (NP (DT the) (JJ weighted) (NN sum)) (PP (IN of) (NP (NP (DT the) (NNP squared) (NNP Euclidean) (NN distance)) (PP (IN between) (NP (NP (CD two) (NNS signatures)) (CC and) (NP (DT a) (NN penalty) (NN term)))))))) (S (VP (VBG requiring) (NP (NP (NNS smoothness)) (PP (IN of) (NP (DT the) (JJ displacement) (NN function)))))))))))))) (. .)))
The optimum displacement function for any pair of signatures is extracted using minimization of a functional defined as the weighted sum of the squared Euclidean distance between two signatures and a penalty term requiring smoothness of the displacement function.
(ROOT (S (NP (NNP Distance)) (VP (VBZ is) (VP (VBN defined) (PP (IN as) (NP (NP (DT the) (NN magnitude) (CC or) (NN size)) (PP (IN of) (NP (NP (NN displacement)) (PP (IN between) (NP (CD two) (NNS positions))))))))) (. .)))