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We propose a generic scattering power factorization framework (SPFF) for polarimetric synthetic aperture radar (PolSAR) data to directly obtain N scattering power components along with a residue power component for each pixel. Each scattering power component is factorized into similarity (or dissimilarity) using elementary targets and a generalized volume model. The similarity measure is derived using a geodesic distance between pairs of 4 × 4 real Kennaugh matrices. In standard model-based decomposition schemes, the 3 × 3 Hermitian-positive semi-definite covariance (or coherency) matrix is expressed as a weighted linear combination of scattering targets following a fixed hierarchical process. In contrast, under the proposed framework, a convex splitting of unity is performed to obtain the weights while preserving the dominance of the scattering components. The product of the total power (Span) with these weights provides the nonnegative scattering power components. Furthermore, the framework, along with the geodesic distance (GD) is effectively used to obtain specific roll-invariant parameters such as scattering-type parameter (αGD), helicity parameter (τGD), and purity parameter (PGD). A PGD/αGD unsupervised classification scheme is also proposed for PolSAR images. The SPFF, the roll invariant parameters, and the classification results are assessed using C-band RADARSAT-2 and L-band ALOS-2 images of San Francisco. [ABSTRACT FROM AUTHOR]