Particle Measurements

This section contains tables that list and describe the NI Vision particle measurements. The tables include definitions, symbols, and equations for particle measurements.

Note  Some equation symbols may be defined inside tables later in the section.

Particle Concepts

The following table contains concepts relating to particle measurements.

Concept	Definition
Bounding Rect	Smallest rectangle with sides parallel to the x-axis and y-axis that completely encloses the particle.
Perimeter	Length of a boundary of a region. Because the boundary of a binary image is comprised of discrete pixels, NI Vision subsamples the boundary points to approximate a smoother, more accurate perimeter. Boundary points are the pixel corners that form the boundary of the particle. Refer to the introduction for an illustration of pixel corners.
Particle hole	Contiguous region of zero-valued pixels completely surrounded by pixels with nonzero values. Refer to the particle holes section for more information.
Angle	Degrees of rotation measured counter-clockwise from the x-axis, such that 0 ≤ θ < 180.
Equivalent Rect	Rectangle with the same perimeter and area as the particle.
Equivalent Ellipse	Ellipse with the same perimeter and area as the particle.
Max Feret Diameter	Line segment connecting the two perimeter points that are the furthest apart. Max Feret Diameter Start—Highest, leftmost of the two points defining the Max Feret Diameter Max Feret Diameter End—Lowest, rightmost of the two points defining the Max Feret Diameter Max Feret Diameter Orientation Particle Perimeter Max Feret Diameter
Convex Hull	Smallest convex polygon containing all points in the particle. The following figure illustrates two particles, shown in gray, and their respective convex hulls, the areas enclosed by black lines.
Max Horiz. Segment Length	Longest row of contiguous pixels in the particle. This measurement is always given as a pixel measurement.
Sum	Moments of various orders relative to the x-axis and y-axis.
Moment of Inertia	Moments about the particle center of mass. Provides a representation of the pixel distribution in a particle with respect to the particle center of mass. Moments of inertia are shift invariant.
Norm. Moment of Inertia	Moment of Inertia normalized with regard to the particle area. Normalized moments of inertia are shift and scale invariant.
Hu Moment	Moments derived from the Norm. Moment of Inertia measurements. Hu Moments are shift, scale, and rotation invariant.

Particle Holes

A particle hole is a contiguous region of zero-valued pixels completely surrounded by pixels with nonzero values. A particle located inside a hole of a bigger particle is identified as a separate particle. The area of a hole that contains a particle includes the area covered by that particle.

Particle #	Area	Area of Hole	Area of Particle & Holes
1	A	B + C	A + B + C
2	D	0	D
3	E	F + G	E + F + G
4	G	0	G

Hole measurements are valuable when analyzing particles similar to the one in figure A. For example, if you threshold a cell with a dark nucleus (figure A) so that the nucleus appears as a hole in the cell (figure B), you can make the following cell measurements:

• Area of Hole—Returns the size of the nucleus.
• Area of Particle and Hole—Returns the size of the entire cell.
• Area of Hole/Area of Particle and Hole—Returns the percentage of the cell that the nucleus occupies.

Coordinates

The following table lists the NI Vision particle measurements relating to coordinates.

Measurement	Definition	Symbol	Equation
Center of Mass	Point representing the average position of the total particle mass, assuming every point in the particle has a constant density. The center of mass can be located outside the particle if the particle is not convex.	—	—
First Pixel	Highest, leftmost particle pixel. The first pixel is always given as a pixel measurement. The black squares in the following figure represent the first pixel of each particle.	—	—
Center of Mass x	X-coordinate of the particle Center of Mass.	x
Center of Mass y	Y-coordinate of the particle Center of Mass.	y
First Pixel x	X-coordinate of the first particle pixel.	—	—
First Pixel y	Y-coordinate of the first particle pixel.	—	—
Bounding Rect Left	X-coordinate of the leftmost particle point.	BL	—
Bounding Rect Top	Y-coordinate of highest particle point.	BT	—
Bounding Rect Right	X-coordinate of the rightmost particle point.	BR	—
Bounding Rect Bottom	Y-coordinate of the lowest particle point.	BB	—
Max Feret Diameter Start x	X-coordinate of the Max Feret Diameter Start.	Fx1	—
Max Feret Diameter Start y	Y-coordinate of the Max Feret Diameter Start.	Fy1	—
Max Feret Diameter End x	X-coordinate of the Max Feret Diameter End.	Fx2	—
Max Feret Diameter End y	Y-coordinate of the Max Feret Diameter End.	Fy2	—
Max Horiz. Segment Length Left	X-coordinate of the leftmost pixel in the Max Horiz. Segment. Max Horiz. Segment Length Left is always given as a pixel measurement.	—	—
Max Horiz. Segment Length Right	X-coordinate of the rightmost pixel in the Max Horiz. Segment. Max Horiz. Segment Length Right is always given as a pixel measurement.	—	—
Max Horiz. Segment Length Row	Y-coordinate for all of the pixels in the Max Horiz. Segment. Max Horiz. Segment Length Row is always given as a pixel measurement.	—	—

Lengths

the following table lists the NI Vision particle relating to length.

Measurement	Definition	Symbol	Equation
Bounding Rect Width	Distance between Bounding Rect Left and Bounding Rect Right.	W	BR – BL
Bounding Rect Height	Distance between Bounding Rect Top and Bounding Rect Bottom.	H	BB – BT
Bounding Rect Diagonal	Distance between opposite corners of the Bounding Rect.	—
Perimeter	Length of the outer boundary of the particle. Because the boundary is comprised of discrete pixels, NI Vision subsamples the boundary points to approximate a smoother, more accurate perimeter.	P	—
Convex Hull Perimeter	Perimeter of the Convex Hull.	PCH	—
Hole Perimeter	Sum of the perimeters of each hole in the particle.	—	—
Max Feret Diameter	Distance between the Max Feret Diameter Start and the Max Feret Diameter End.	F
Equivalent Ellipse Major Axis	Length of the major axis of the Equivalent Ellipse.	E2a
Equivalent Ellipse Minor Axis	Length of the minor axis of the Equivalent Ellipse.	E2b
Equivalent Ellipse Minor Axis (Feret)	Length of the minor axis of the ellipse with the same area as the particle, and Major Axis equal in length to the Max Feret Diameter.	EF2b
Equivalent Rect Long Side	Longest side of the Equivalent Rect.	Ra
Equivalent Rect Short Side	Shortest side of the Equivalent Rect.	Rb
Equivalent Rect Diagonal	Distance between opposite corners of the Equivalent Rect.	Rd
Equivalent Rect Short Side (Feret)	Shortest side of the rectangle with the same area as the particle, and longest side equal in length to the Max Feret Diameter.	RFb
Average Horiz. Segment Length	Average length of a horizontal segment in the particle. Sum of the horizontal segments that do not superimpose any other horizontal segment. Average Horiz. Segment Length is always given as a pixel measurement.	—	A SH
Average Vert. Segment Length	Average length of a vertical segment in the particle. Sum of the vertical segments that do not superimpose any other vertical segment. Average Vert. Segment Length is always given as a pixel measurement.	—	A SV
Hydraulic Radius	Particle area divided by the particle perimeter.	—	A P
Waddel Disk Diameter	Diameter of a disk with the same area as the particle.	—	2 A π

Ellipses

• Equivalent Ellipse Major Axis—Total length of the major axis of the ellipse that has the same area and same perimeter as a particle. This length is equal to 2a.

This definition gives the following set of equations:

Area = πab

For a given area and perimeter, only one solution (a, b) exists.

• Equivalent Ellipse Minor Axis—Total length of the minor axis of the ellipse that has the same area and same perimeter as a particle. This length is equal to 2b.
• Ellipse Ratio—Ratio of the major axis of the equivalent ellipse to its minor axis, which is defined as

The more elongated the equivalent ellipse, the higher the ellipse ratio. The closer the equivalent ellipse is to a circle, the closer the ellipse ratio is to 1.

Rectangles

• Equivalent Rect Long Side—Length of the long side (R_a) of the rectangle that has the same area and same perimeter as a particle

This definition gives the following set of equations:

A = Area = R_aR_b

P = Perimeter = 2(R_aR_b)

This set of equations can be expressed so that the sum R_a + R_b and the product R_aR_b become functions of the parameters Particle Area and Particle Perimeter. R_a and R_b then become the two solutions of the following polynomial equation:

2x² – Px + 2A = 0

Notice that for a given area and perimeter, only one solution (R_a, R_b) exists.

• Equivalent Rect Short Side—Length of the short side of the rectangle that has the same area and same perimeter as a particle. This length is equal to R_b.
• Equivalent Rect Diagonal—Distance between opposite corners of the Equivalent Rect.

• Rectangle Ratio—Ratio of the long side of the equivalent rectangle to its short side, which is defined as

The more elongated the equivalent rectangle, the higher the rectangle ratio.

The closer the equivalent rectangle is to a square, the closer to 1 the rectangle ratio.

Hydraulic Radius

A disk with radius R has a hydraulic radius equal to

Areas

The following table lists the NI Vision particle area measurements.

Image Area

Figure A shows an image of a calibration grid. The image exhibits nonlinear distortion. Figure B shows an image of coins taken with the same camera setup used in figure A. The dashed line around figure B defines the image area in pixels. Figure C illustrates the image of coins after image correction. The dashed line around figure C defines the image area in calibrated units.

Quantities

The following table lists the NI Vision particle measurements relating to quantity.

Angles

The following table lists the NI Vision particle angle measurements. The equations are given in radians. The results are given in degrees that are mapped into the range 0 to 180, such that 0 ≤ θ < 180.

The Orientation angle is measured counterclockwise from the horizontal axis, as shown in the following figure. The value can range from 0° to 180°. Angles outside this range are mapped into the range. For example, a 190° angle is considered to be a 10° angle.

1  Line with Lowest Moment of Inertia 2  Orientation in Degrees	3  Horizontal Axis

Note  Refer to the max feret diameter entry in the for an illustration of Max Feret Diameter Orientation.

Ratios

The following table lists the NI Vision particle ratio measurements.

Factors

The following table lists the NI Vision particle factor measurements.

Sums

The following table lists the NI Vision particle sum measurements.

Moments

The following table lists the NI Vision particle moment measurements.