Measuring linearity of curves in 2D and 3D

In this paper we define a new linearity measure for open curve segments in 2D and 3D . The measure considers the distance of the curve end points to the curve centroid. It is simple to compute and has the basic properties that should be satisfied by any linearity measure. The new measure ranges over the interval (0,1], and produces the value 1 if and only if the measured curve is a perfect straight line segment. Also, the new linearity measure is invariant with respect to translations, rotations and scaling transformations. The new measure is theoretically well founded and, because of this, its behaviour can be well understood and predicted to some extent. This is always beneficial because it indicates the suitability of the new measure to the desired application. Several experiments are provided to illustrate the behaviour and to demonstrate the efficiency and applicability of the new linearity measure

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 α 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 α. 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

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

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

Interactive histogenesis of axonal strata and proliferative zones in the human fetal cerebral wall

Development of the cerebral wall is characterized by partially overlapping histogenetic events. However, little is known with regards to when, where, and how growing axonal pathways interact with progenitor cell lineages in the proliferative zones of the human fetal cerebrum. We analyzed the developmental continuity and spatial distribution of the axonal sagittal strata (SS) and their relationship with proliferative zones in a series of human brains (8-40 post-conceptional weeks; PCW) by comparing histological, histochemical, and immunocytochemical data with magnetic resonance imaging (MRI). Between 8.5 and 11 PCW, thalamocortical fibers from the intermediate zone (IZ) were initially dispersed throughout the subventricular zone (SVZ), while sizeable axonal "invasion" occurred between 12.5 and 15 PCW followed by callosal fibers which "delaminated" the ventricular zone-inner SVZ from the outer SVZ (OSVZ). During midgestation, the SS extensively invaded the OSVZ, separating cell bands, and a new multilaminar axonal-cellular compartment (MACC) was formed. Preterm period reveals increased complexity of the MACC in terms of glial architecture and the thinning of proliferative bands. The addition of associative fibers and the formation of the centrum semiovale separated the SS from the subplate. In vivo MRI of the occipital SS indicates a "triplet" structure of alternating hypointense and hyperintense bands. Our results highlighted the developmental continuity of sagittally oriented "corridors" of projection, commissural and associative fibers, and histogenetic interaction with progenitors, neurons, and glia. Histogenetical changes in the MACC, and consequently, delineation of the SS on MRI, may serve as a relevant indicator of white matter microstructural integrity in the developing brain

An alternative approach to computing shape orientation with an application to compound shapes

We consider the method that computes the shape orientation as the direction α that maximises the integral of the length of projections, taken to the power of 2N, of all the straight line segments whose end points belong to the shape, to a line that has the slope α. We show that for N=1 such a definition of shape orientation is consistent with the shape orientation defined by the axis of the least second moment of inertia. For N>1 this is not the case, and consequently our new method can produce different results. As an additional benefit our approach leads to a new method for computation of the orientation of compound objects

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

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