As suggested in the answer above, in general, decoherence increases the entropy associated with a quantum system and as such has the same type of time-reversal asymmetry that appears in thermodynamics. The question, however, is also concerned with how an "uncollapse" would look like. Here I want to illustrate how this can be done in principle.
The net effect of a projective measurement on a pure quantum system is a nonlinear mapping from an initial state $|\psi\rangle$ to a final state $|\psi'\rangle$. The nonlinearity arises from the fact that the final state must be normalized.
Nevertheless, what is important is that the final state is also a pure state of unit norm, and there always exists a reversible unitary mapping connecting the two. Hence, it is possible to simply apply the reverse unitary on the state after the measurement to get back the original state.
Here is an example. Suppose we start with the state $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and measure in the computational basis $\{|0\rangle,|1\rangle\}$. After the measurement, the state will be described either by $|0\rangle$ or $|1\rangle$ with equal probability. For the sake of argument let's assume it is $|0\rangle$.
Then all we need to retrieve our original state is to apply a Hadamard transform
$H=\left(\begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\\
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}
\end{array}\right)$
to retrieve the initial state. This follows from the relation
$H |0\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$. Note that something like this can be routinely implemented in the lab.
In the case of mixed states and general measurements the situation is a bit more complicated, but by introducing an auxiliary system one could also perform a mapping between the state after a measurement and before it.
