  For quaternionic signal processing algorithms, the gradients of a
quaternion-valued function are required for gradient-based methods. Given the
non-commutativity of quaternion algebra, the definition of the gradients is
non-trivial. The HR gradient operator provides a viable framework and has found
a number of applications. However, the applications so far have been mainly
limited to real-valued quaternion functions and linear quaternion-valued
functions. To generalize the operator to nonlinear quaternion functions, we
define a restricted version of the HR operator. The restricted HR gradient
operator comes in two versions, the left and the right ones. We then present a
detailed analysis of the properties of the operators, including several
different product rules and chain rules. Using the new rules, we derive
explicit expressions for the derivatives of a class of regular nonlinear
quaternion-valued functions, and prove that the restricted HR gradients are
consistent with the gradients in real domain.
