Nizhny Tagil Technological Institute of Ural Federal University carries out investigations of shape detection of irregular plate objects with image analysis.

Problem of research includes three subproblems:

1. Find contour of object on a picture. It is considered that the source image is a binary-colored one. Binarization of multicolored images is the subject of another research. The object perimeter and area are formed by using contour. 2. Detect standard shape, in which it looks like. In this work, the following four shapes are examined: circle, rectangle, rhombus, and ellipse. Nevertheless, real objects can have very complex shape. 3. Find characteristic sizes of chosen standard shape, using equation (1). Characteristic sizes are shown in Table 1.

The opencv library is used to find contour of the object and its characteristics. The following three criteria of contour are used to detect the standard shape:

rectness that characterizes closeness of contour to rectangle; it is calculated by Eq. 2

where rec is the rectness, S contour is the area of a contour, S rect is the area of minimum rectangle that contains this contour. It can be found by the method GetMinAreaRect [2] in opencv. rec ≤ 1 (rec = 1 for the rectangle). circless that characterizes closeness of the contour to a circle; it is calculated by Eq. ( 3)

where cir is the circless, S contour is the area of a contour, S cirle is the area of enclosing circle that contains this contour; it can be found by method MinEnclosingCircle [3] in opencv. cir ≤ 1 (cir = 1 for the circle); compactness [1] is an universal shape criterion; if the contour is close to the standard shape, their compactnesses are approximataly equal; it is calculated by Eq. ( 4)

where C is the compactness, P is the perimeter of the contour, S is the area of the contour.

All standard shapes are convex ones, but real irregular objects can have a lot of local concavities. So before calculations, a source contour must be smoothed and perimeter must be decreased to make it closer to perimeter of the standard form. Real and smoothed contours are shown in Fig. (1). It can be done by the GetConvexHull [4] method in opencv. It is interesting to note that perimeter of the source contour is 6804px, but perimeter of the smoothed contour is only 3702px. The coefficient k = S contour S hull is used to return to real contour in the next calculations. So, characteristic size a of a real contour can be calculated from size of the smoothed contour with this coefficient:

Combination of three criteria (rectness, circless, and compactness) does not allow to detect one of the standard shape uniquely. The next strategy is used: if rec > 0.8 or cir > 0.8, than contour is counted as a rectangle or a circle respetively. In other cases, it is neccesary to choose a shape, which compactness is closest to the smoothed contour compactness.

Compactness of standard shapes can be calculated by the following equations:

circle (see Eq. 5)

the circle compactness is constant; it is minimum possible compactness; in all cases C ≥ 4π. The characteristic parameter (diameter) can be calculated using Eq. ( 1) as d = S contour 4π ;

rhombus (see Eq. 6)

rhombus compactness C ≥ 16 (C = 16 in the case of α = π/2, i.e. quadrate);

rectangle (see Eq. 7):

rectangle compactness C ≥ 16 (C = 16 in the case of a = b, i.e. quadrate) ellipse perimeter is calculated with elliptic integral [5] (see Eq. 8)

This intergal cannot be expressed in terms of elementary functions, so the following approximation (Eq. 9) of ellipse perimeter with the maximun error of 0.63% is used. The strcucture of this formula can help to simplify the following mathematical transformations,

In this case, the ellipse compactness can be calculated by Eq. ( 10)

The ellipse compactness C ≥ 4π (C = 4π in the case of a = b, i.e. circle).

In the cases of rectangle and ellipse, compactness depends on two parameters, so, characteristic sizes cannot be determined uniquely. But if relation e = b/a is used, compactness depends on one parameter, and all similar rectangles and ellipses have the same compactness. Compactness equations are shown on Table 2. In the case of circle, characteristic size (d) is calculated uniquely using Eq. ( 1) with Eq. ( 11)

In the case of rhombus, characteristic sizes (a, h) are calculated with Eq. (6) by Eq. (12 and Eq. 13)

In the cases of rectangle and ellipse, characteristic sizes can be calculated with well-known area equations S = ab and S = πab respectively. Using the compactness parameter e = b/a, these equations are transformed to S = a In the case of rectangle, e can be expressed with equation in Table 2 by Eq. ( 14)

In the case of rhombus, dependence C(e) is very complex, but this equation can be solved numerically.

This algorithm was applied to some real irregular objects. In this section some examples are introduced. All results have gathered in tables. The first column contains the original image, image with contour, and detected shape. The second column represents the calculated data. Bold font marks the base for shape detection.

Notations in tables are -S is the area of contour (smoothed) -P is the perimeter of contour (smoothed) -C is the compacntess of contour (smoothed) rec is the rectness cir is the circless -C r is the compacntess of rectangle -C c is the compacntess of circle -C rh is the compacntess of rhombus -C e is the compacntess of ellipse a, b, h, α, d are the characteristic parameters of detected figure (depending on shape).