In this paper we study a class of fast geometric image inpainting methods based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels are filled by assigning them a color equal to a weighted average of their already filled neighbors. However, there is flexibility in terms of the order in which pixels are filled, the weights used for averaging, and the neighborhood that is averaged over. Varying these degrees of freedom leads to different algorithms, and indeed the literature contains several methods falling into this general class. All of them are very fast, but at the same time all of them leave undesirable artifacts such as “kinking” (bending) or blurring of extrapolated isophotes. Our objective in this paper is to build a theoretical model in order to understand why these artifacts occur and what, if anything, can be done about them. Our model is based on two distinct limits: a continuum limit in which the pixel width \(h \rightarrow 0\) and an asymptotic limit in which \(h > 0\) but \(h \ll 1\). The former will allow us to explain “kinking” artifacts (and what to do about them) while the latter will allow us to understand blur. Both limits are derived based on a connection between the class of algorithms under consideration and stopped random walks. At the same time, we consider a semi-implicit extension in which pixels in a given shell are solved for simultaneously by solving a linear system. We prove (within the continuum limit) that this extension is able to completely eliminate kinking artifacts, which we also prove must always be present in the direct method. Finally, we show that although our results are derived in the context of inpainting, they are in fact abstract results that apply more generally. As an example, we show how our theory can also be applied to a problem in numerical linear algebra.

本文基于从边界向内填充连续壳中的修复区域这一思想，研究了一类快速几何图像修复方法。通过为图像像素分配颜色等于其已填充邻居的加权平均值来填充图像像素。但是，在填充像素的顺序，用于平均的权重以及被平均的邻域方面，存在灵活性。改变这些自由度会导致使用不同的算法，实际上，文献中包含了几种属于此类的方法。它们都非常快，但同时它们都留下了不良的伪影，例如“扭结”（弯曲）或外推的等渗线模糊。本文的目的是建立一个理论模型，以了解为什么会出现这些假象以及可以对它们进行任何处理。我们的模型基于两个不同的限制：连续限制，其中像素宽度\（h \ rightarrow 0 \）和一个渐近极限，其中\（h> 0 \）但\（h \ ll 1 \）。前者将使我们能够解释“扭曲”伪像（以及如何处理它们），而后者将使我们能够理解模糊。这两个限制都是基于所考虑的算法类别与停止的随机游走之间的联系而得出的。同时，我们考虑一个半隐式扩展，其中通过求解线性系统同时求解给定外壳中的像素。我们证明（在连续范围内）该扩展能够完全消除弯折伪影，而且我们还证明了直接方法中必须始终存在这种情况。最后，我们表明，尽管我们的结果是在修复的背景下得出的，但实际上它们是抽象的结果，适用于更广泛的应用。例如，我们说明了我们的理论如何也可以应用于数值线性代数中的问题。