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We show that functions in $GSBV^p$ in three-dimensional space with smallvariation in $2$ of $3$ directions are close to a function of one variableoutside an exceptional set. Bounds on the volume and the perimeter in these twodirections of the exceptional sets are provided. As a key tool we prove anapproximation result for such functions by functions in $W^{1,p}$. For this wepresent a two-dimensional countable ball construction that allows to carefullyremove the jumps of the function. As a direct application, we show$\Gamma$-convergence of an anisotropic three-dimensional Mumford-Shah model toa one-dimensional model.

An anisotropic Poincar\'e inequality in $GSBV^p$ and the limit of strongly anisotropic Mumford-Shah functionals