Detection of generalized principal axes in rotationally symmetric shapes

doi:10.1016/0031-3203(91)90080-O

Pattern Recognition

Volume 24, Issue 2, 1991, Pages 95-104

https://doi.org/10.1016/0031-3203(91)90080-O Get rights and content

Abstract

Automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper. The inapplicability of a well-known equation, which is used to compute the direction of the principal axis of a given shape using moment functions, to a class of so-called degenerate shapes is first pointed out. Rotationally symmetric shapes often encountered in real applications are shown to belong to this class. This problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions. Analytic equations for computing the direction of high-order principal axes are derived. They include the well-known equation for computing the direction of the (second-order) principal axis as a special case. Some experimental results are included finally to show the effectiveness of the derived analytic equations.

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The classical approach (Jain et al., 1995) defined orientation as the axis of the least second moment of inertia. Tsai and Chou (1991) proposed to detect principal axes of shapes having symmetry properties. Lin (1993, 1996) defined shape orientation using universal principal axes, formed of half lines starting from the shape centroid.
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It is worth mentioning that the latter method is analogous to the standard method which defines the shape orientation by the line that minimises the integral of squared distances of the points in the shape to the line. A use of a higher exponent than 2 (in the standard method) is studied in [31,36]. Namely, it is well known that the standard method should be modified in order to be applicable to a wider class of shapes.
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There are already several definitions and methods developed to determine and compute shape orientation. Different techniques have been used, including those based on geometric moments, complex moments, principal component analysis, etc., but, there is still an ongoing demand for the new methods; as well as previously established methods [5–8,10], there are also very recent ones [2,3,9,13,14], as well. In this paper we define a new boundary based method for computing the shape orientation.
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Note also that, due to the diversity of shapes and the diversity of applications in different areas such as computer science, medicine, biology, robotics, etc., several methods are often developed for measuring the same shape property. As an illustration we list just a few known methods for measuring shape convexity: [1,11,19,21,26,37] and for computing the orientation of a shape: [3,8,13,27,28,33]. The latter is often used within a shape normalisation procedure that is sometimes required prior to further shape analysis.
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- •
  it ranges over $(0, 1]$ and gives the measured circularity equal to 1 if and only if the measured shape is a circle;
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  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.
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^§: Sheng-Lin Chou is also with Advanced Technology Center, Electronics Research and Service Organization, Industrial Technology Research Institute, Chutung, Hsinchu, Taiwan 31015, R. O. C.

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Detection of generalized principal axes in rotationally symmetric shapes

Abstract

Comput. Vision Graphics Image Process

Comput. Vision Graphics Image Process

Pattern Recognition

Recognitive aspects of moment invariants

IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6

Image normalization by complex moments

IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7