A new form of image decomposition is derived that uses compound systems to target critical TCC components while at the same time providing the usual least-squares optimal match to the TCC as a whole. Significant improvements in accuracy under a given runtime budget are obtained by intensively correcting those portions of the TCC which are most recalcitrant to the standard coherent decomposition used in OPC today (e.g. SOCS or OCS). In particular, the non-coherent structure of our new decomposition systems is well-suited to extract any near-Toeplitz components present in the spatial-domain TCC. Such components are difficult to capture with coherent decomposition, and we show that TCCs for lithographic systems in fact contain strong Toeplitz-like components that arise from slope discontinuities associated with the sharp aperture of the projection lens. 1D tests show that for a given kernel-count budget in the typical e.g. 10-100 range, image calculation error can routinely be reduced by at least 5X if our new systems are included in the decomposition.