  In this paper we analyze the joint rate distortion function (RDF), for a
tuple of correlated sources taking values in abstract alphabet spaces (i.e.,
continuous) subject to two individual distortion criteria. First, we derive
structural properties of the realizations of the reproduction Random Variables
(RVs), which induce the corresponding optimal test channel distributions of the
joint RDF. Second, we consider a tuple of correlated multivariate jointly
Gaussian RVs, $X_1 : \Omega \rightarrow {\mathbb R}^{p_1}, X_2 : \Omega
\rightarrow {\mathbb R}^{p_2}$ with two square-error fidelity criteria, and we
derive additional structural properties of the optimal realizations, and use
these to characterize the RDF as a convex optimization problem with respect to
the parameters of the realizations. We show that the computation of the joint
RDF can be performed by semidefinite programming. Further, we derive
closed-form expressions of the joint RDF, such that Gray's [1] lower bounds
hold with equality, and verify their consistency with the semidefinite
programming computations. We also verify our expressions reproduce the
closed-form formula of the joint RDF of scalar-valued RVs (i.e., $p_1=p_2=1$)
derived by Xiao and Luo [2].
