Compared to other spanning trees, the significant superiorities of the cross-trees are that the trees׳ constructions are efficient and the trees are exactly unique since the constructions are independent on any local or global property of the image itself.
It follows immediately that the mapping admits a unique fixed point which is exactly the unique solution of (7) for (rho=rho_{0}+delta).
By virtue of Theorem 2.1, there exists a unique such that and where is a functional defined implicity in (2.24) and. is just a unique solution of (2.12) and is exactly a unique solution of (2.11).
Note that if we have some smoothness conditions of to guarantee the existence and uniqueness of the solutions for (2.4), then is exactly the unique solution of (2.4).
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