Consider now the process of constructing a new partition tree for the columns. The distance between two columns, when viewed as functions on the rows, in the dual Besov space, refines the affinity structure on the columns, leading to a new, smoother column tree organization and a new corresponding Haar-like basis. This updated basis can then be utilized to identify the new dual Besov space on the rows. The construction of a new row tree organization follows a similar procedure with this newly defined Besov space. Each step of this iteration refines and smoothens the geometry of the columns as well as that of the rows.