# Particle Characterisation

## Particle shape and shape descriptors

The primary measuring information of image analysis methods is the contour of a particle. This makes up a difference to other methods, for example, to laser diffraction, ultrasonic extinction or dynamic light scattering, because the images contain information about the shape of the particles (though two-dimensional only) in addition to the particle size.

Many algorithms have been published to condense irregular contour data, into a single value, the particle size, expressed by a diameter. The most important of these have been implemented in the Sympatec QICPIC evaluation software. Other parameters allow for the compact characterisation of the particle shape in terms of a shape descriptor.

The description of fibres is a relatively young and strongly developing field of image analysis. As a consequence, some definitions of fibre size and fibre shape descriptors are not yet standardised, especially not the algorithms for their calculation. Some methods to calculate the length and the diameter of a fibre as well as its respective shape descriptors are explained below.

#### Diameter of a circle of equal projection area / EQPC

This is the diameter of a circle that has the same area as the projection area of the particle. It is widely used for the evaluation of particles sizes from the projection area A of a non-spherical particle.

#### Diameter of a circle of equal perimeter / PED

This is the diameter of a circle that has the same perimeter as the particle image.

#### Feret Diameter

This is not a diameter in its actual sense but the common basis of a group of diameters derived from the distance of two tangents to the contour of the particle in a well-defined orientation. In simpler words, the method corresponds to the measurement by a slide gauge (slide gauge principle). In general it is defined as the distance between two parallel tangents of the particle at an arbitrary angle. In practice the Minimum xF_{min} and Maximum Feret Diameter xF_{max}>, the Mean Feret Diameter and the Feret Diameters obtained at 90° to direction of the Minimum and Maximum Feret Diameters xF_{max}90 are used. The minimum Feret diameter is often used as the diameter equivalent to a sieve analysis.

Maximum Feret diameter | FERET_{max}

The Feret diameters for a sufficient number of angles are calculated, and their maximum is selected. If a particle has an irregular shape, the Feret diameter varies more than with regularly shaped particles. The maximum Feret diameter is always larger, than the diameter of the equivalent circle (EQPC).

Minimum Feret diameter | FERET_{min}

The Feret diameters for a sufficient number of angles are calculated, and their minimum is selected. If a particle has an irregular shape, the Feret diameter varies more than with regularly shaped particles. The minimum Feret diameter is always smaller, than the diameter of the equivalent circle (EQPC).

Mean value of the Feret diameters over all orientations according to the principle before.

First, the maximum Feret diameter, FERET_{max}, is calculated. The result is the Feret diameter measured at an angle of 90 degrees to that of the maximum Feret diameter.

First, the minimum Feret diameter, FERET_{min}, is calculated. The result is the Feret diameter measured at an angle of 90 degrees to that of the minimum Feret diameter.

The equivalent diameter, FERET_{vol}, represents the diameter of a sphere having the same volume as the cylinder constructed by FERET_{min} as the cylinder diameter and FERET_{max} as its length.

The calculation of the smallest encasing rectangle is based on the Feret diameter. The value is calculated as the minimum of the product of every possible pair of (x_{Feret}, x_{Feret90}).**Minimum Area Bounding Rectangle, Length / BRmax**

The larger dimension of the smallest encasing rectangle.**Minimum Area Bounding Rectangle, Width / BRmin**

The smaller dimension of the smallest encasing rectangle.

#### Sphericity

The sphericity S is the ratio of the perimeter of the equivalent circle, P_{EQPC}, to the real perimeter, P_{real}. The result is a value between 0 and 1. The smaller the value, the more irregular is the shape of the particle. This results from the fact that an irregular shape causes an increase of the perimeter. The ratio is always based on the perimeter of the equivalent circle because this is the smallest possible perimeter with a given projection area.

#### Aspect ratio

The ratio of the minimum to the maximum Feret diameter is another measure for the particle shape. The Aspect Ratio ψ_{A} (0 < ψ_{A}≤ 1) is defined by the ratio of the Minimum to the Maximum Feret Diameter ψA = x_{Feret min} / x_{Feret max}. It gives an indication for the elongation of the particle. Some literature also used 1/ψ_{A} as the definition of sphericity.

A fibre shaped particle is characterised by a length that is typically much larger than its diameter and an irregular shape. Consequently both, length and diameter, are necessary to properly describe the size of a fibre. The above definition of a fibre is imprecise but there is no better one, nor is there a standardised criterion which shape of a particle projection is considered as "fibre-shaped" and which is not. The evaluation methods described below can therefore be applied to particles of any shape if desired. It is then up to the user to judge the usefulness of the evaluated results.

#### Length of fibre / LEFI

The length of a fibre is defined as the direct connection between its opposite ends, this is the longest direct path from one end to another within the particle contour. ("Direct" means without loops or deviations.) A technique used to calculate this value is called "skeletonizing", it means to reduce the dimensions of the fibre from all directions until one or more lines of one pixel width remain. The black line in the fibre images below represents the longest direct path along their skeleton. Its length is the result of the LEFI calculation.

A very simple contour of a fibre is shown in image 1 below. It is simple because it has no branches or nooses, and its length-to-diameter ratio is large. Its opposite ends can clearly be defined, and there is not much discussion about what should be their connecting path.

Matters get a bit more complicated for image 2. The algorithm for the identification of the opposite ends has to try two branches and select the longer one.

Image 3 shows a complex fibre with branches and nooses. The effect of the skeletonizing algorithm can clearly be seen in this picture. The clue of the path finding algorithm is to avoid loops.

#### Diameter of fibre / DIFI and DIFIX

One could imagine a number of ways to describe the diameter of a fibre by one mean value. The method implemented in the evaluation software for the QICPIC image analyser is to divide the projection area by the sum of all lengths of the branches of the fibre.

The calculation of DIFI is applied to those fibres only that are completely within the image frame, whereas the calculation of DIFIX also includes fibres touching the edge of the image.

#### Volume based fibre diameter / VBFD

This diameter is defined as the diameter of a sphere which has the same volume as the respective fibre. It is calculated with x_{D}, the fibre diameter (DIFI) and x_{L}, the fibre length (LEFI). The volume based fibre diameter is very useful if sample material consists of a mixture of granulate and fibres, and a distribution diagram of volume over particle size is desired. Neither LEFI nor DIFI can be used appropriately for the x-axis of a volume distribution diagram but VBFD serves to an informative representation.

#### Elongation

This is the ratio of diameter and length of a fibre as defined by the formula, DIFI / LEFI. This parameter is also called eccentricity.

#### Convexitiy

The convexity is an important shape parameter describing the compactness of a particle. The figure below shows a particle with projection area A (grey/light) leaving open a concave region of area B (red/dark) on its right hand side. The convexity is the ratio of the projection area itself (A) and the area of the convex hull (A+B). The maximum theoretical convexity is 1, if there are no concave regions. Due to the detector design of a digital camera (square pixels), however, all particles seem to have small concave regions, corresponding to the tiny steps with every pixel in the perimeter line. Therefore, the maximum convexity calculated in reality is mostly limited to 0.99.

#### Straightness of fibre shaped particles

Most fibres, especially longer ones, tend to curl, and there have been several efforts to describe this phenomenon in terms of a single parameter. One of the possible definitions is the straightness (according to ISO standard 9276-6):

STRAIGHT = FERETmax / LEFI

The definitions of LEFI, the fibre length, and FERETmax, the "slide gauge compliant" outer dimension of the fibre, can be found above in this text. A value of 1 of the straightness represents a perfectly straight particle while values close to zero represent a greater deformation (curled fibres). An older definition of straightness is the Curl Index:

CURLindex = LEFI / FERETmax - 1

which reflected the tradition in some industries, mainly the wood processing industry.