  We revisit the representation of generalized quantum observables by
establishing a geometric picture in terms of their positive operator-valued
measures (POVMs). This leads to a clear geometric interpretation of Born's rule
by introducing the concept of contravariant operator-valued measures. Our
approach is applied to the theory of array detectors, which is a challenging
task as the finite dimensionality of the POVM substantially restricts the
available information about quantum states. Our geometric technique allows for
a direct estimation of expectation values of different observables, which are
typically not accessible with such detection schemes. In addition, we also
demonstrate the applicability of our method to quantum-state reconstruction
with unbalanced homodyne detection.
