In this paper, we introduce a novel class of systems called channeled partial Mueller matrix polarimeters (c-pMMPs). Their analysis benefits greatly by drawing from the concepts of generalized construction of channeled polarimeters as described by the modulation matrix. The modulation matrix resembles that of the data reduction method of a conventional polarimeter, but instead of using Mueller vectors as the bases, attention is focused on the Fourier properties of the measurement conditions. By leveraging the understanding of the measurement's structure, its decomposition can be manipulated to reveal noise resilience and information about the polarimeter's ability to measure the aspect of polarization that are important for any given task. We demonstrate the theory with a numerical optimization that designs c-pMMPs for the task of monitoring the damage state of a material as presented earlier by Hoover and Tyo [Appl. Opt.46, 8364 (2007)APOPAI0003-693510.1364/AO.46.008364]. We select several example systems that produce a fewer-than-full-system number of channels yet retain the ability to discriminate objects of interest. Their respective trade-offs are discussed.