Efficient two-dimensional Haar ^+ synopsis construction for the maximum absolute error measure

Several wavelet synopsis construction algorithms were previously proposed for optimal Haar ^+ synopses. Recently, we proposed the OptExtHP-EB algorithm to find an optimal one-dimensional Haar^+ synopsis. By utilizing the novel properties of optimal synopses, OptExtHP-EB represents the set of optimal synopses in a node of a Haar^+ tree by a set of extended synopses. While it is much faster than the previous Haar^+ synopsis construction algorithms, it can handle only one-dimensional data. In this paper, we propose the OptExtHP-EB2D algorithm for two-dimensional Haar^+ synopses by extending OptExtHP-EB. While a one-dimensional Haar^+ tree has only two child nodes and three coefficients in a node, a two-dimensional Haar^+ tree is much more complex in that it has four child nodes and seven coefficients per node. Thus, for each possible subset of the coefficients selected in a node, we develop the efficient methods to compute a set of optimal synopses denoted by extended synopses. Our experiments confirm the effectiveness of our proposed OptExtHP-EB2D algorithm.