Quasiconformal mappings and the rank of $\frac{dDf}{d\mathrm{\mid }Df\mathrm{\mid }}$ for $f\in BV({\mathbb{R}}^n;{\mathbb{R}}^n)$

We define a relaxed version $H_f^{\text{fine}}$ of the distortion number $H_f$ that is used to define quasiconformal mappings. Then we show that for a BV function $f\in BV({\mathbb{R}}^n;{\mathbb{R}}^n)$, for $\mathrm{\mid }Df\mathrm{\mid }$-a.e. $x\in {\mathbb{R}}^n$ it holds that $H_{f^{\ast }}^{\text{fine}}(x)<\mathrm{\infty }$ if and only if $\frac{dDf}{d\mathrm{\mid }Df\mathrm{\mid }}(x)$ has full rank.