In this paper we study a variational problem in the space of functions of
bounded Hessian. Our model constitutes a straightforward higher-order extension
of the well known ROF functional (total variation minimisation) to which we add
a non-smooth second order regulariser. It combines convex functions of the
total variation and the total variation of the first derivatives. In what
follows, we prove existence and uniqueness of minimisers of the combined model
and present the numerical solution of the corresponding discretised problem by
employing the split Bregman method. The paper is furnished with applications of
our model to image denoising, deblurring as well as image inpainting. The
obtained numerical results are compared with results obtained from total
generalised variation (TGV), infimal convolution and Euler's elastica, three
other state of the art higher-order models. The numerical discussion confirms
that the proposed higher-order model competes with models of its kind in
avoiding the creation of undesirable artifacts and blocky-like structures in
the reconstructed images -- a known disadvantage of the ROF model -- while
being simple and efficiently numerically solvable.

本文研究了一个在 Hessian 有界函数空间中的变分问题，从 ROF 总变差（total variation）最小化模型扩展到更高阶模型，结合了总变差和一阶导数的总变差的凸函数。我们证明了该组合模型的最小值的存在唯一性，并通过采用分裂 Bregman 方法展示了相应离散问题的数值解。该论文通过图像去噪、去模糊以及图像修复的应用展示了该模型，并将得到的数值结果与其他最先进的高阶模型进行了比较。数值结果表明该模型避免了在重建的图像中产生不良的伪影和方块状结构，而且其简单易解和高效。