Optimal representation of piecewise Hölder smooth bivariate functions by the easy path wavelet transform. The easy path wavelet transform (EPWT) has recently been proposed by the first author [Multiscale Model. Simul. 7, No. 3, 1474--1496 (2009; Zbl 1175.65158)] as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we aim to provide a theoretical understanding of the performance of the EPWT. In particular, we derive conditions for the path vectors of the EPWT that need to be met in order to achieve optimal (N)-term approximations for piecewise Hölder smooth functions with singularities along curves.