The present article puts forth a novel method for extraction of a quaternion from a given rotation matrix. The proposed method improves over other time performing methods through a more accurate orthonormalization of noisy rotation matrices. To arrive at said method, it is here first shown that a duality between a quaternion extraction and the extraction of a reflection (or projection) direction can be established, by virtue of the fact that the reflection (or projection) along a given rotation quaternion can be uniquely identified with the corresponding rotation matrix. Under this duality, a geometric interpretation and equivalence of known quaternion extraction methods is put forth, wherein the methods are seen to correspond to a reflection (or projection) extraction in a four dimensional space.