Joviša unić

A Hu moment invariant as a shape circularity measure

In this paper we propose a new circularity measure which defines the degree to which a shape differs from a perfect circle. The new measure is easy to compute and, being area based, is robust-e.g., with respect to noise or narrow intrusions. Also, it satisfies the following desirable properties:*it ranges over (0,1] and gives the measured circularity equal to 1 if and only if the measured shape is a circle; *it is invariant with respect to translations, rotations and scaling. Compared with the most standard circularity measure, which considers the relation between the shape area and the shape perimeter, the new measure performs better in the case of shapes with boundary defects (which lead to a large increase in perimeter) and in the case of compound shapes. In contrast to the standard circularity measure, the new measure depends on the mutual position of the components inside a compound shape. Also, the new measure performs consistently in the case of shapes with very small (i.e., close to zero) measured circularity. It turns out that such a property enables the new measure to measure the linearity of shapes. In addition, we propose a generalisation of the new measure so that shape circularity can be computed while controlling the impact of the relative position of points inside the shape. An additional advantage of the generalised measure is that it can be used for detecting small irregularities in nearly circular shapes damaged by noise or during an extraction process in a particular image processing task.

Notes on shape orientation where the standard method does not work

In this paper we consider some questions related to the orientation of shapes with particular attention to the situation where the standard method does not work. There are irregular and non-symmetric shapes whose orientation cannot be computed in a standard way, but in the literature the most studied situations are those where the shape under consideration has more than two axes of symmetry or where it is an n-fold rotationally symmetric shape with n>2. The basic reference for our work is [W.H. Tsai, S.L. Chou, Detection of generalized principal in rotationally symmetric shapes, Pattern Recognition 24 (1991) 95-104]. We give a very simple proof of the main result from [W.H. Tsai, S.L. Chou, Detection of generalized principal in rotationally symmetric shapes, Pattern Recognition 24 (1991) 95-104] and suggest a modification of the proposal on how the principal axes of rotationally symmetric shapes should be computed. We show some desirable property in defining the orientation of such shapes if the modified approach is applied. Also, we give some comments on the problems that arise when computing shape elongation.

Measuring linearity of open planar curve segments

In this paper we define a new linearity measure for open planar curve segments. We start with the integral of the squared distances between all the pairs of points belonging to the measured curve segment, and show that, for curves of a fixed length, such an integral reaches its maximum for straight line segments. We exploit this nice property to define a new linearity measure for open curve segments. The new measure ranges over the interval (0,1], and produces the value 1 if and only if the measured open line is a straight line segment. The new linearity measure is invariant with respect to translations, rotations and scaling transformations. Furthermore, it can be efficiently and simply computed using line moments. Several experimental results are provided in order to illustrate the behaviour of the new measure.

Linearity measure for curve segments

In this paper, we consider linearity measure for a bounded length curves. First, we define a new linearity measure for open curve segments, and then extend method to closed curves (contours). The derived measures (for both, open curve segments and closed curves) are invariant with respect to similarity transformations. The linearity measure for open curve segments picks the value 1 if and only if the measured open line segment is a perfect straight line segment while the established linearity measures for closed curves never reach 1, as preferred.

Shape ellipticity based on the first Hu moment invariant

Article history: Received 29 March 2013 Accepted 23 July 2013 Available online 30 July 2013 Communicated by Jinhui Xu

Orientation and anisotropy of multi-component shapes from boundary information

We define a method for computing the orientation of compound shapes based on boundary information. The orientation of a given compound shape S is taken as the direction @a that maximises the integral of the squared length of projections, of all the straight line segments whose end points belong to particular boundaries of components of S to a line that has the slope @a. Just as the concept of orientation can be extended from single component shapes to multiple components, elongation can also be applied to multiple components, and we will see that it effectively produces a measure of anisotropy since it is maximised when all components are aligned in the same direction. The presented method enables a closed formula for an easy computation of both orientation and anisotropy.

The distance between shape centroids is less than a quarter of the shape perimeter

In this paper we consider the distance between the shape centroid computed from the shape interior points and the shape centroid computed from the shape boundary points. We show that the distance between those centroids is upper bounded by the quarter of the perimeter of the shape considered. The obtained upper bound is sharp and cannot be improved. Next, we introduce the shape centredness as a new shape descriptor which, informally speaking, should indicate to which degree a shape has a uniquely defined centre. By exploiting the result mentioned above, we give a formula for the computation of the shape centredness. Such a computed centredness is invariant with respect to translation, rotation and scaling transformations.

Curvature weighted gradient based shape orientation

Determining the orientation of a shape is a common task in many image processing applications. It is usually part of the image preprocessing stages and further processing may rely on an adequate method to determine the orientation. There are several methods for computing the orientation of a shape, each of them with its own strengths and weaknesses; a method which performs outstandingly for one application may have a poor performance for a different application. In this paper we present a new method for computing shape orientation based on the projection of the tangent vectors of a shape onto a line and weighting them using a function of the curvature. Some of the results from Zunic (2008) [14] are particular cases of the results presented here.

Shape elongation from optimal encasing rectangles

Let S be a shape with a polygonal boundary. We show that the boundary of the maximally elongated rectangle R(S) which encases the shape S contains at least one edge of the convex hull of S. Such a nice property enables a computationally efficient construction of R(S). In addition, we define the elongation of a given shape S as the ratio of the length of R(S) (determined by the longer edge of R(S)) and the width of R(S) (determined by the shorter edge of R(S)) and show that a so defined shape elongation measure has several desirable properties. Several examples are given in order to illustrate the behavior of the new elongation measure. As a by-product, of the method developed here, we obtain a new method for the computation of the shape orientation, where the orientation of a given shape S is defined by the direction of the longer edge of R(S).