In this paper we have developed a family of shape measures. All the measures from the family evaluate the degree to which a shape looks like a predefined convex polygon. A quite new approach in designing object shape based measures has been applied. In most cases such measures were defined by exploiting some shape properties. Such properties are optimized (e.g., maximized or minimized) by certain shapes and based on this, the new shape measures were defined. An illustrative example might be the shape circularity measure derived by exploiting the well-known result that the circle has the largest area among all the shapes with the same perimeter. Of course, there are many more such examples (e.g., ellipticity, linearity, elongation, and squareness measures are some of them). There are different approaches as well. In the approach applied here, no desired property is needed and no optimizing shape has to be found. We start from a desired convex polygon, and develop the related shape measure. The method also allows a tuning parameter. Thus, there is a new 2-fold family of shape measures, dependent on a predefined convex polygon, and a tuning parameter, that controls the measure's behavior. The measures obtained range over the interval (0,1] and pick the maximal possible value, equal to 1, if and only if the measured shape coincides with the selected convex polygon that was used to develop the particular measure. All the measures are invariant with respect to translations, rotations, and scaling transformations. An extension of the method leads to a family of new shape convexity measures.

中文翻译：

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使用所需的凸多边形测量形状


在本文中，我们开发了一系列形状测量方法。该族中的所有度量都会评估形状与预定义凸多边形的相似程度。已经应用了一种基于对象形状的测量设计的全新方法。在大多数情况下，此类度量是通过利用某些形状属性来定义的。这些属性通过某些形状来优化（例如，最大化或最小化），并且基于此，定义了新的形状度量。一个说明性的例子可能是通过利用众所周知的结果得出的形状圆度度量，即在具有相同周长的所有形状中，圆具有最大的面积。当然，这样的例子还有很多（例如，椭圆度、线性度、伸长率和方形度测量就是其中的一些）。也有不同的方法。在此应用的方法中，不需要所需的属性，也不必找到优化形状。我们从所需的凸多边形开始，并开发相关的形状度量。该方法还允许调整参数。因此，有一个新的二重形状测量系列，依赖于预定义的凸多边形和控制测量行为的调整参数。当且仅当测量的形状与用于开发特定测量的所选凸多边形一致时，获得的测量范围在区间 (0,1] 内，并选择最大可能值，等于 1。所有测量都是不变的关于平移、旋转和缩放变换，该方法的扩展导致了一系列新的形状凸性度量。