# Two-Dimensional Shape Analysis of Complex Geometry Based on Photogrammetric Models of Iconostases
## 4 Method for 2D Shape Analysis Based on Photogrammetric Models
### Two-Dimensional Shape Analysis
#### 4.3.2 An Application of 2D Shape Descriptors  
Considering the given measurements, the shape descriptors were further employed to quantitatively describe the shape of icons andthe shapes of the front and side cross-sections of the iconostasis' main constructive elements (base, column, and molding). They present shape features characterized by a numerical value. In this paper, we used some of the existing global shape descriptors: eccentricity, elongation, compactness, convexity, solidity, rectangularity, and ellipticity. We defined the first two basic shape descriptors as the measurements of the shape aspect ratio:  
The eccentricity \(Exc(S)\) is defined as the ratio of the length of the shortminor axis (\(b\)) to the length of the longmajor axis (\(a\)) of the shape [68; 69]:  
\[Ecc(S)=\frac{b}{a}. \tag{6}\]  
The result is a measure of the shape eccentricity, given as a value between 0 and 1. The low eccentricity tends to the value of 1, while as the ratio increases from 1, the shape eccentricity is higher.  
The elongation \(Elo(S)\) is defined as the ratio between the length of the shorter edge (\(W\)) and the length of the longer edge (\(L\)) of the minimum bounding rectangle \(R(S)\):  
\[Elo(S)=1-\frac{W}{L}. \tag{7}\]  
The result is a measure of the shape elongation, given as a value between 0 and 1. The lowest possible measure is equal to 0 and implies the square, and as the ratio decreases from 0, the shape becomes more elongated.  
The compactness \(Cst(S)\) is a numerical quantity representing the degree to which a shape is compact [70]. It also defines a measure of the sharpness of the shape [71]. It ranges over 0-1 and is defined as follows [68; 70; 71; 72]:  
\[Cst(S)=\frac{4\pi\times A(S)}{P(S)^{2}}, \tag{8}\]  
The value of 1 is used for a circle as it is the object with the most compact shape.  
The convexity \(C(S)\) measures the irregularity and roughness of the shape boundary. It can be obtained as the ratio of the perimeter of the convex hull of the shape \(P(CH(S))\) to the perimeter of the shape itself \(P(S)\)[68; 72]:  
Figure 24: Shape measurements applied to the cross sections (front and side) of the base, column, and molding of ICONC.  
\[C(S)=\frac{P(CH(S))}{P(S)}, \tag{9}\]  
where \(P(CH(S))\) and \(P(S)\) are the Euclidean perimeters of the boundaries of \(CH(S)\) and \(S\), respectively. It satisfies the following properties [68; 72]:  
* the convexity measure is a number from 0-1 and
* the convexity measure of a given shape equals 1 if the shape is convex and will be less than 1 if the shape is not convex, such as by having an irregular boundary.  
The solidity \(S(S)\) shape descriptor is defined as the ratio of the area of the shape \(A(S)\) to the area of the convex hull of the same shape, \(A(CH(S))\)[68]:  
\[S(S)=\frac{A(S)}{A(CH(S))}. \tag{10}\]  
It measures the density of the shape and satisfies the following properties [68]:  
* the solidity measure is a number from 0-1 and
* the value of 1 signifies a solid shape and a value of less than 1 will represent a shape with an irregular boundary or containing holes.  
The rectangularity the \(R(S)\) shape descriptor is defined as follows:  
\[R(S)=\frac{A(S)}{A(R)}. \tag{11}\]  
It is represented as the ratio of the shape area \(A(S)\) to the area of the minimum rectangle that bounds the shape \(A(R)\). The rectangularity has a value of 1 for a perfectly rectangular object.  
The ellipticity \(Ell(S)\) shape descriptor [73] is defined as the ratio of the area of the maximum inscribed ellipse \(A(Ell(S))\) and the observed shape \(A(S)\):  
\[Ell(S)=\frac{A(Ell(S))}{A(S)}. \tag{12}\]  
It ranges over 0-1, where a value of 1 signifies an ellipse.