\;.\end{aligned}$$ Note that the weight function $w$ needs to be adapted using the inverse map $T^{-1}$ which obviously exists.

Specifically for $R=\mathbb{R}$ we notice that the symmetric definitions , are also equivariant under strictly monotonically decreasing maps $T$.

For the median of continuous densities or over $\mathbb{R}$ the equivariance w.r.t. strictly monotone transformations $T$ is given by $$\begin{aligned}
T\left(\operatorname{med}(\gamma)\right) &=
\operatorname{med}\bigl(\gamma^T)\bigr) \;.
\label{medd-equv}\end{aligned}$$ where $\gamma^T$ is given as $$\begin{aligned}
\gamma^T(x) &:= \frac{\mathrm{d}}{\mathrm{d}y}
\left(\int\nolimits_{-\infty}^{T^{-1}(y)}\gamma(z)\,\mathrm{d}z\right)(x)\;,\end{aligned}$$ i.e., the uniquely determined density for which $$\begin{aligned}
\int\nolimits_{-\infty}^{T(x)}\gamma^T(y)\,\mathrm{d}y
&= \int\nolimits_{-\infty}^x\gamma(y)\,\mathrm{d}y\quad
\text{for all $x\in\mathbb{R}$.}\end{aligned}$$ In the case of , the equivariance property again holds for strictly monotonically decreasing maps $T$, too.

Algorithmic Aspects
-------------------

In the processing of images by median filtering, see Section \[sec-medf\], median computations will have to be carried out multiple times. Efficient algorithms are therefore relevant for the practicability of such filters.

**Unweighted median.**

Finding the median of a finite multiset $\mathcal{X}$ of size $N$ is what is known in algorithmics as a *selection problem.* Whereas in principle one could sort the data in $\mathcal{X}$ based on their total order (for which several algorithms with $\mathcal{O}(N\log N)$ computational complexity exist), the selection problem does actually not require the complete order of the data, 