In the paper, we present a new interval-based set inversion algorithm which takes into account the continuity of the problem. In the case where the set $\mathbb{Y}$ to be inverted has some volume, we show that inverting the boundary $\partial \mathbb{Y}$ of $\mathbb{Y}$ is sufficient to reconstruct the preimage $\mathbb{X}={\mathbf{f}}^{-1}\left(\mathbb{Y}\right)$. The inversion of $\partial \mathbb{Y}$ separates the domain of $\mathbf{f}$ into two regions : one inside $\mathbb{X}$ and one outside. To detect which part is inside or outside, we show that we can retro-propagate the information coming from $\mathbb{Y}$ at negligible cost. The efficiency of the approach is illustrated on a localization problem.

中文翻译：

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集反转的边界方法

在本文中，我们提出了一种新的基于区间的集反转算法，该算法考虑了问题的连续性。在设置的情况下$\mathbb{ÿ}$ 被反转有一定的体积，我们证明了反转边界 $\partial \mathbb{ÿ}$ 的 $\mathbb{ÿ}$ 足以重建原像 $\mathbb{X}={\mathbf{F}}^{-\mathrm{1个}}（\mathbb{ÿ}）$。倒置$\partial \mathbb{ÿ}$ 分隔域 $\mathbf{F}$ 分为两个区域：一个内部 $\mathbb{X}$还有一个在外面 为了检测哪个部分在内部还是外部，我们证明了我们可以反向传播来自$\mathbb{ÿ}$费用微不足道。在定位问题上说明了该方法的效率。