# About the computation of the Tree of Shapes on n-dimensional images in quasi-linear time.

## Abstract

The Tree of Shapes is an useful image transform used to process digital images in a self-dual way. We recently introduced a new algorithm to compute this tree in ${\displaystyle n}$-dimensional gray scale images with a quasi-linear time complexity. However, no proof nor benchmark were given. In addition, this algorithm requires to increase dramatically the size of the input image. Such increase comes with an important cost in term of memory usage and computation times. We discuss the initialization of the algorithm and provide a proof. We also address its memory and computation time issues when used on two dimensional images.