Distance transform, a central operation in image and video analysis, involves finding the shortest path between feature and non-feature entries of a binary image. The process may be implemented using chamfer-based sequential algorithms that apply small-neighborhood masks to estimate the Euclidean metric. Success of these algorithms depends on the cost function used to optimize chamfer weights. And, for years, mean absolute error and mean squared error have been used for optimization. However, studies have revealed weaknesses of these cost functions—sensitivity against outliers, lack of symmetry, and biasedness—which limit their application. In this work, we have proposed a robust and a more accurate cost function, symmetric mean absolute percentage error, which attempts to address some weaknesses. The proposed function averages the absolute percentage errors in a set of measurements and offers interesting mathematical properties (smoothness, differentiability, boundedness, and robustness) that allow easy interpretation and analysis of the results. Numerical results show that chamfer masks designed under our optimization criterion generate lower errors. The present work has also proposed an automatic algorithm that converts coefficients of the designed real-valued masks into integers, which are preferable in most practical computing devices. Lastly, we have modified the chamfer algorithm to improve its speed and then embedded the proposed weights into the algorithm to compute distance maps of real images. Results show that the proposed algorithm is faster and uses fewer number of operations compared with those consumed by the classical chamfer algorithm. Our results may be useful in robotics to address the matching problem.

中文翻译：

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利用对称平均绝对百分比误差的倒角掩模的优化设计

距离变换是图像和视频分析中的核心操作，它涉及查找二进制图像的特征和非特征条目之间的最短路径。可以使用基于倒角的顺序算法来实现该过程，该顺序算法应用小邻域掩码以估计欧几里得度量。这些算法的成功取决于用于优化倒角权重的成本函数。而且，多年来，平均绝对误差和均方误差已用于优化。但是，研究发现这些成本函数的缺点-对异常值的敏感性，缺乏对称性和偏见-限制了它们的应用。在这项工作中，我们提出了一个健壮且更准确的成本函数，即对称平均绝对百分比误差，它试图解决一些弱点。所提出的函数可对一组测量值中的绝对百分比误差取平均值，并提供有趣的数学属性（平滑度，微分性，有界性和鲁棒性），可轻松解释和分析结果。数值结果表明，根据我们的优化标准设计的倒角掩模产生的误差较小。本工作还提出了一种自动算法，该算法将设计的实值掩码的系数转换为整数，这在大多数实际的计算设备中都是优选的。最后，我们修改了倒角算法以提高其速度，然后将拟议的权重嵌入到算法中以计算真实图像的距离图。结果表明，与经典倒角算法所消耗的算法相比，该算法具有更快的速度和更少的运算量。我们的结果可能对解决匹配问题的机器人技术很有用。