We will present a new paradigm for wavelet representations. One highlight of the novel approach is a new inversion algorithm that bypasses the need to find dual filters. Another highlight is that the approach bypasses the need to construct highpass filters altogether. A third highlight is that no construction is needed at all! On top of all, no lowpass convolution is needed in the decomposition step. Furthermore, the decomposition and reconstruction algorithms are very fast: linear complexity with tiny constants that decay, rather than grow, with the spatial dimension. Moreover, the representation is very local, and with the size of the underlying filters being very small (4-6 taps is the rule) and being independent of the spatial dimension. And all that is topped with performance analysis that shows that the coefficients of those new representations characterize isotropic smoothness in much the same way that mainstream (slow and non-local) wavelet representations do.

Prior familiarity with the wavelet representation and the notion of refinability is assumed; however a brief overview of these notions may be provided. The talk is based in joint work with Y. Hur (MIT), S. Nam (Wisconsin) and V. Temlyakov (S. Carolina).