In this paper, we first propose a new family of geometrical image transforms that decompose images both radially and angularly. Our construction comprises two stages of filter banks that are non-redundant and perfect reconstruction and therefore lead to an overall non-redundant and perfect reconstruction transform. Using the wavelet transform as the first stage, we apply directional filter banks to the wavelet coefficients in such a way to maintain the anisotropy scaling law. Furthermore, we propose a new image coding scheme based on the proposed transform, the wavelet-based contourlet transform (WBCT), using a new contourlet-based set partitioning in hierarchical trees (CSPIHT) algorithm that provides an embedded code. Due to differences in parent-child relationships between the WBCT coefficients and wavelet coefficients, under CSPIHT, we developed an elaborated repositioning algorithm for the WBCT coefficients in such a way that we could scan spatial orientation trees that are simil...