Detection of Symmetry Points in Images

Abstract

Abstract:This article proposes a new method for detecting symmetry points in images. Like other symmetry detection algorithms, it assigns a "symmetry score" to each image point. Our symmetry measure is only based on scalar products between gradients and is therefore both easy to implement and of low runtime complexity. Moreover, our approach also yields the size of the symmetry region without additional computational effort. As both axial symmetries as well as some rotational symmetries can result in a point symmetry, we propose and evaluate different methods for identifying the rotational symmetries. We evaluate our method on two different test sets of real world images and compare it to several other rotational symmetry detection methods.

The Gradient Product Transform: An Image Filter for Symmetry Detection

“…To understand this phenomenon, we have computed the symmetry score as a function of the radius r with square shaped regions (r x = r y = r) for all ground truth symmetry points in the data set [3]. To make the measurements comparable, we have normalized all radii with the ground truth radius r 0 , and all scores with the score s(r 0 ).…”

“…Whilst Table 1 demonstrates the discriminating suitability of the four features for a particular example, the scatterplots in Figure 6 confirm these tendencies for a greater number of symmetry center points. These points were taken from the data set published with [3], which includes a list of ground truth points that are labeled as either axial or rotational symmetry centers. In each individual scatterplot of Figure 6, there is, however, considerable overlap between the classes.…”

“…where the index i ∈ {a, r} denotes the class (axial or rotational), and ξ is the row vector of the four features in the same order as in the list above. The parameters µ i , w i and W i have been determined with the labeled ground truth data from [3] used as training data via the function qda as implemented in the MASS package or the R language for statistical computing 5 . The values reported in [4] are: The second aforementioned question, i.e., how to define a threshold for the symmetry score representing an actual symmetry, has not yet been discussed in the literature.…”

“…The algorithm was originally defined in [3] for square shaped symmetry regions and a simple criterion for discriminating between rotational and axial symmetries was suggested. In [4], this was extended to rectangular regions, a more robust method for determining the size of the symmetry region was introduced, a quadratic discriminant analysis with more features was suggested to discriminate between axial and rotational symmetries, and an optional modification was proposed to make the GPT applicable to blood vessel extraction.…”

“…For object recognition, we propose a new threshold on a normalized score for identifying actual symmetry points. This problem was circumvented in [3,4] by simply looking for the highest symmetry score in an image, which always yielded exactly one symmetry point, irrespective of the actual number of symmetric objects. We now give a threshold recommendation for detecting more than one (or, which might also happen: none) symmetry.…”