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The orthogonal subspace projection (OSP) approach has received considerable interest in hyperspectral data exploitation recently. It has been shown to be a versatile technique for a wide range of applications. Unfortunately, insights into its design rationale have not been investigated and have yet to be explored. This work conducts a comprehensive study and analysis on the OSP from several signal processing perspectives and further discusses in depth how to effectively operate the OSP using different levels of a priori target knowledge for target detection and classification. Additionally, it looks into various assumptions made in the OSP and analyzes filters with different forms, some of which turn out to be well-known and popular target detectors and classifiers. It also shows how the OSP is related to the well-known least-squares-based linear spectral mixture analysis and how the OSP takes advantage of Gaussian noise to arrive at the Gaussian maximum-likelihood detector/estimator and likelihood ratio test. Extensive experiments are also included in this paper to simulate various scenarios to illustrate the utility of the OSP operating under various assumptions and different degrees of target knowledge.