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, ultrasonics extinction, or PCCS, 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 serve to the compact description of the particle shape in terms of a shape factor.

The description of fibres is a relatively young and strongly developing field of image analysis. Consequently, the definitions of fibre size and fibre shape parameters are not yet standardized, nor are the algorithms for their calculation. Some methods to calculate the length and the diameter of a fibre and the straightness for its shape are explained below.

Diameter Definitions

Diameters Derived from the Equivalent Circle

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.

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).

here: orientation horizontal (0°)

Feret Diameter, Maximum (FERET_MAX) or Minimum (FERET_MIN)

Maximal or minimal Feret diameter after consideration of all possible orientations (0°...180°). The Feret diameters for a sufficient number of angles are calculated, and their maximum or minimum is selected. If a particle has an irregular shape, the Feret diameter usually varies much more than with regularly shaped particles. The maximum can therefore be significantly larger, the minimum significantly smaller than the diameter of the equivalent circle.

Feret Diameter, Mean Value (FERET_MEAN)

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

Feret Diameter, 90° to the Maximal Feret Diameter (FERET_MAX90)

First, the maximal Feret diameter, FERET_MAX, is calculated. The result is the Feret diameter measured at an angle of 90 degrees to that of the maximal Feret diameter.

Feret Diameter, 90° to the Minimal Feret Diameter (FERET_MIN90)

First, the minimal Feret diameter, FERET_MIN, is calculated. The result is the Feret diameter measured at an angle of 90 degrees to that of the minimal Feret diameter.

Minimum Area Bounding Rectangle

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 (BR_MAX)

The larger dimension of the smallest encasing rectangle.

Minimum Area Bounding Rectangle, Width (BR_MIN)

The smaller dimension of the smallest encasing rectangle. This dimension corresponds quite well to the results of a sieve analysis.

Chord Length

This is not an diameter in its actual sense but the common basis of a group of diameters.

A chord length is defined by the distance of two points of the contour, measured exactly across the centre of gravity of the projection area. This is why all methods of evaluating the chord length imply an evaluation of the centre of gravity of the projection area.

Warning:

Chord length methods are problematic with strongly concave contours of a particle.

Example:

x_cv = Chord length vertical

x_ch = Chord length horizontal

Chord Length, Vertical (CHORD_VERTICAL)

The result is the chord length measured vertically across the centre of the projection area.

Chord Length, Horizontal (CHORD_HORIZONTAL)

The result is the chord length measured horizontally across the centre of the projection area.

Chord Length, Maximal (CHORD_MAX)

The result is the largest chord length measured across the centre of the projection area.

Chord Length, Minimal (CHORD_MIN)

The result is the smallest chord length measured across the centre of the projection area.

Chord Length, 90° to the Maximalen Chord Length (CHORD_MAX90)

First, the maximal chord length, CHORD_MAX, is calculated. The result is the chord length at an angle of 90 degrees to that of the maximal chord length.

Chord Length, 90° to the Minimalen Chord Length (CHORD_MIN90)

First, the minimal chord length, CHORD_MIN, is calculated. The result is the chord length at an angle of 90 degrees to that of the minimal chord length.

Chord Length, Mean Value (CHORD_MEAN)

First, chord length values for a sufficient number of orientations are calculated. Their mean value is output.

Martin Diameter

This is not an diameter in its actual sense but the common basis of a group of diameters.

The Martin diameter, xM, is that chord dividing the projection area of the particle into two equal halves.

Warning:

The Martin diameter is problematic if a particle has many concave parts of the contour and should be avoided in such cases.

orientation of measurement

Martin Diameter, Maximal (MARTIN_MAX), or Minimal (MARTIN_MIN)

This is the maximal or minimal Martin diameter after consideration of all possible orientations (0°...180°). The Martin diameters for a sufficient number of orientations are calculated, and their maximum or minimum, respectively, is selected.

Martin Diameter, Mean Value (MARTIN_MEAN)

This is the mean value of the Martin diameters of all possible orientations according to the priciple described above.

Size Definitions for Fibre-Shaped Particles

A fibre shaped particle is characterized 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 standardized 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 Sympatec WINDOX 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 by

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.

Limitations of the Evaluation Method for Fibres

The algorithm implemented in the WINDOX evaluation is one of the fastest available while maintaining a good coverage of even difficult fibre geometries. It has certain limits, however, as the following examples show:

4.		A perfect ring has no ends, so the LEFI calculation will yield no result.
5.		This fibre has one end only, but the LEFI calculation needs a minimum of two ends.
6.a)		This fibre forms a hook with a gap of exactly one pixel. "Holes" of one pixel are regarded as imaging errors and filled.
6.b)		This is the result of the closing algorithm. "Closing" is a commonly applied method to correct imaging errors. In most cases it is very advantageous but not in this example.
7.		The DIFI calculation of very small particles (example to the left: 4 x 3 pixels with a "gap") often yields a value greater than LEFI. Very small particles, in general, cannot be fibre-shaped, so it is useless to treat them as such.

Shape Factor Definitions

Shape Factor Derived from the Equivalent Circle

Sphericity (SPHERICITY)

The sphericity, S, is the ratio of the perimeter of the equivalent circle, P_EQPC, to the real perimeter, P_real.

P = perimeter

A = area

The sphericity is defined by the formula below:

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.

Shape Factor Derived from the Feret Diameter

Aspect Ratio

The ratio of the minimal to the maximal Feret diameter is another measure for the particle shape.

Other Shape Parameters

Convexity

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 defined as follows

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 (proposed for the coming ISO standard 9276-6):

STRAIGHT = FERET_MAX / 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:

CURL_INDEX = LEFI / FERET_MAX - 1

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

Elongation

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