Measuring squareness and orientation of shapes

In this paper we propose a measure which defines the degree to which a shape differs from a square. 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 squareness equal to 1 if and only if the measured shape is a square; •it is invariant with respect to translations, rotations and scaling. In addition, we propose a generalisation of the new measure so that shape squareness can be computed while controlling the impact of the relative position of points inside the shape. Such a generalisation enables a tuning of the behaviour of the squareness measure and makes it applicable to a range of applications. A second generalisation produces a measure, parameterised by δ, that ranges in the interval (0,1] and equals 1 if and only if the measured shape is a rhombus whose diagonals are in the proportion 1:δ. The new measures (the initial measure and the generalised ones) are naturally defined and theoretically well founded—consequently, their behaviour can be well understood. As a by-product of the approach we obtain a new method for the orienting of shapes, which is demonstrated to be superior with respect to the standard method in several situations. The usefulness of the methods described in the manuscript is illustrated on three large shape databases: diatoms (ADIAC), MPEG-7 CE-1, and trademarks

Nanoparticle shapes: Quantification by elongation, convexity and circularity measures

The goal of the nanoparticle synthesis is, first of all, the production of nanoparticles that will be more similar in size and shape. This is very important for the possibility of studying and applying nanomaterials because of their characteristics that are very sensitive to size and shape such as, for example, magnetic properties. In this paper, we propose the shape analysis of the nanoparticles using three shape descriptors – elongation, convexity and circularity. Experimental results were obtained by using TEM images of hematite nanoparticles that were, first of all, subjected to segmentation in order to obtain isolated nanoparticles, and then the values of elongation, convexity and circularity were measured. Convexity C x ( S ) is regarded as the ratio between shape’s area and area of the its convex hull. The convexity measure defines the degree to which a shape differs from a convex shape while the circularity measure defines the degree to which a shape differs from an ideal circle. The range of convexity and circularity values is (0, 1], while the range of elongation values is [1, ∞). The circle has lowest elongation (ε = 1), while it has biggest convexity and circularity values ( C x = 1; C = 1). The measures ε( S ), C x ( S ), C ( S ) proposed and used in the experiment have the few desirable properties and give intuitively expected results. None of the measures is good enough to describe all the shapes, and therefore it is suggested to use a variety of measures so that the shapes can be described better and then classify and control during the synthesis process

Ellipticity and circularity measuring via Kullback-Leibler divergence

Using the Kullback-Leibler divergence we provide a simple statistical measure which uses only the covariance matrix of a given set to verify whether the set is an ellipsoid. Similar measure is provided for verification of circles and balls. The new measure is easily computable, intuitive, and can be applied to higher dimensional data. Experiments have been performed to illustrate that the new measure behaves in natural way

Orientation and anisotropy of multi-component shapes

There are many situations in which several single objects are better considered as components of a multi-component shape (e.g. a shoal of fish), but there are also situations in which a single object is better segmented into natural components and considered as a multi-component shape (e.g. decomposition of cellular materials onto the corresponding cells). Interestingly, not much research has been done on multi-component shapes. Very recently, the orientation and anisotropy problems were considered and some solutions have been offered. Both problems have straightforward applications in different areas of research which are based on a use of image based technologies, from medicine to astrophysics.The object orientation problem is a recurrent problem in image processing and computer vision. It is usually an initial step or a part of data pre-processing, implying that an unsuitable solution could lead to a large cumulative error at the end of the vision system’s pipeline. An enormous amount of work has been done to develop different methods for a spectrum of applications. We review the new idea for the orientation of multi-component shapes, and also its relation to some of the methods for determining the orientation of single-component shapes. We also show how the anisotropy measure of multi-component shapes, as a quantity which indicates how consistently the shape components are oriented, can be obtained as a by-product of the approach used

Measuring shapes with desired convex polygons

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 the most cases such measures were defined by exploiting some of shape properties. An illustrative example might be the shape circularity measure derived by exploiting the well-know result that the circle has the largest area among all the shapes with the same perimeter. In the approach applied here, no desired property is needed and no optimizing shape has to be found. We start from a desired/selected convex polygon, and develop the related shape measure. 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, used to develop the a particular measure. All the measures are invariant with respect to translations, rotations, and scaling transformations. The method used has an straightforward extension to a wider family of shape measures, dependent on a tuning parameter involved. Another extension leads to a family of the new shape convexity measures

Shape Descriptors

Every day we recognize a numerous objects and human brain can recognize objects under many conditions. The way in which humans are able to identify an object is remarkably fast even in different size, colours or other factors. Computers or robots need computational tools to identify objects. Shape descriptors are one of the tools commonly used in image processing applications. Shape descriptors are regarded as mathematical functions employed for investigating image shape information. Various shape descriptors have been studied in the literature. The aim of this thesis is to develop new shape descriptors which provides a reasonable alternative to the existing methods or modified to improve them. Generally speaking shape descriptors can be categorized into various taxonomies based on the information they use to compute their measures. However, some descriptors may use a combination of boundary and interior points to compute their measures. A new shape descriptor, which uses both region and contour information, called centeredness measure has been defined. A new alternative ellipticity measure and sensitive family ellipticity measures are introduced. Lastly familiy of ellipticity measures, which can distinguish between ellipses whose ratio between the length of the major and minor axis differs, have been presented. These measures can be combined and applied in different image processing applications such as image retrieval and classification. This simple basis is demonstrated through several examples