Consider selecting points on a contour in the x-y plane. In shape analysis, this is frequently referred to as contour sampling. It is important to select the points such that they effectively represent the shape of the contour. Generally, the stroke order and number of strokes are informative for that purpose. Several effective methods exist for sampling contours drawn with a certain stroke order and number of strokes, such as the English alphabet or Arabic figures. However, many contours entail an uncertain stroke order and number of strokes, such as pictures of symbols, and little research has focused on methods for sampling such contours. This is because selecting the points in this case typically requires a large computational cost to check all the possible choices. In this paper, we present a sampling method that is useful regardless of whether the contours are drawn with a certain stroke order and number of strokes or not. Our sampling method thereby expands the application possibilities of contour processing. We formulate contour sampling as a discrete optimization problem that can be solved using a type of direct search. Based on a geometric graph whose vertices are the points and whose edges form rectangles, we construct an effective objective function for the problem. Using different shape datasets, we demonstrate that our sampling method is effective with respect to shape representation and retrieval.

A two-dimensional shape is generally represented with line drawings or object contours in a digital image. Shapes can be divided into two types, namely ordered and unordered shapes. An ordered shape is an ordered set of points, while an unordered shape is an unordered set. As a result, each type typically uses different attributes to define the local descriptors involved in representing the local distributions of points sampled from the shape. Throughout this paper, we focus on unordered shapes. Since most local descriptors of unordered shapes are not scale-invariant, we usually make the shapes in an image data set the same size through scale normalization, before applying shape matching procedures. Shapes obtained through scale normalization are suitable for such descriptors if the original whole shapes are similar. However, they are not suitable if parts of each original shape are drawn using different scales. Thus, in this paper, we present a scale-invariant descriptor constructed by von Mises distributions to deal with such shapes. Since this descriptor has the merits of being both scale-invariant and a probability distribution, it does not require scale normalization and can employ an arbitrary measure of probability distributions in matching shape points. In experiments on shape matching and retrieval, we show the effectiveness of our descriptor, compared to several conventional descriptors.