We study a general class of infimal convolution type regularisation functionals suitable for applications in image processing. These functionals incorporate a combination of the total variation seminorm and \(\mathrm {L}^{p}\) norms. A unified well-posedness analysis is presented and a detailed study of the one-dimensional model is performed, by computing exact solutions for the corresponding denoising problem and the case \(p=2\). Furthermore, the dependency of the regularisation properties of this infimal convolution approach to the choice of p is studied. It turns out that in the case \(p=2\) this regulariser is equivalent to the Huber-type variant of total variation regularisation. We provide numerical examples for image decomposition as well as for image denoising. We show that our model is capable of eliminating the staircasing effect, a well-known disadvantage of total variation regularisation. Moreover as p increases we obtain almost piecewise affine reconstructions, leading also to a better preservation of hat-like structures.

中文翻译：

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BV 和 [公式：见文本] 空间的 Infimal Convolution 正则化泛函：第一部分：有限 [公式：见文本] 案例。

我们研究了适用于图像处理应用的一类通用的小卷积类型正则化函数。这些泛函结合了总变分半范数和\(\mathrm {L}^{p}\)范数。通过计算相应去噪问题和情况\(p=2\)的精确解，提出了统一的适定性分析，并对一维模型进行了详细研究。此外，研究了这种小卷积方法的正则化属性对p选择的依赖性。事实证明，在这种情况下\(p=2\)这个正则化器相当于总变差正则化的 Huber 型变体。我们提供了图像分解和图像去噪的数值示例。我们证明了我们的模型能够消除阶梯效应，这是总变化正则化的一个众所周知的缺点。此外，随着p的增加，我们获得了几乎分段的仿射重建，这也导致了对帽状结构的更好保存。