Due to the success in image denoising

the nonlocal strategies have recently been introduced to extend morphology from local to nonlocal.And the operators with adjunction property (adjunction operators for short) are the basic operator form in traditional morphology

which are fundamental in morphological analyzing.However

the adjunction property of operators has usually been missed in the extended works.To overcome the problem

some extensions for nonlocal morphology with adjunctions operators have been provided.Unfortunately

as declared in this paper

another important property

i.e.

order-preservation

has been lost in these extensions.To make up the defect of losing order-preservation

by adapting the nonlocal weights acquisition

a novel nonlocal morphology is proposed in this study.Meanwhile

it has been proved that adjunction and order-preservation properties are both kept in the proposed extension.Finally

the experimental results on synthetic and natural images validate its feasibility and effectiveness.