Particle detection - image processing

author: Tereza Štanglová

Table of contents

• Particles' generator
• Area
• Perimeter
• Automatic particle counting
  • Simple method
  • Histogram method
  • Method of circular tracks
• Results
• User documentation
• Easter egg

Particles' generator

This is the first step of whole process. You can specify the number of particles. The particles' sizes are random, between 10 and 25 pixels. The particles' positions are also random. The particles can overlap each other. When particles overlap they seems to look like one. The final image is an input for particle detection.

Area

Algorithm:

1. Analyze the image line by line and search for red points (pixels).
2. Check the area around the red point.
3. If there is another red point in the area around, add its coordinates to the stack. Change the color of the current point to the color of background (black in this case).
4. Is the stack empty?

• No - Pop the first element and go to 2.
• Yes - End.

Perimeter

At this stage we know the coordinates of all points of each particle. This knowledge can be used for detection of internal boundaries of particles (perimeter). Internal boundary can be detect by erosion - morphological transformation. Internal boundary is defined by:

H_r = A \ (A ⊖ B),

where A is a matrix of points for each particle, B is a kernel (structuring element) and ⊖ is a vector difference.

Kernel for erosion.

Points of input image are called foreground pixels. Each foreground pixel is superimposed by the structuring element on top of the input image. If for every pixel in the structuring element, the corresponding pixel in the image underneath is a foreground pixel, then the input pixel is left as it is. Pixels that were removed are a part of the internal boundary.

1. Original image.	2. Kernel application on point (1;1).
3. The result of kernel application.	4. Final result of erosion.

Automatic particle counting

The following methods are used for counting particles.

Simple method

For this method we have to know the number of particles N and theirs area A_i. The area of each particle is compared to reference area A_ref. A_ref is the average area of all A_i.

Estimation of particles' multiplicity n_i is defined by:

Then the estimated number of particles Particles_single can be computed as:

If A_i area is smaller than A_ref area, then n_i=1. That could be a problem.

Methods for A_ref selection:

• Median.
• The upper quartile.
• The lower quartile.

Histogram method

For this method we have to know the number of particles N and theirs area A_i. The next step is to create a histogram of A_i distribution.

Algorithm:

1. Divide the histogram into intervals , which will have a size q.

2. Select an interval with most particles. 3. Select *A_ref* as a lower limit or a middle value of interval. 4. For each particle, if *A_i > A_ref*, calculate its multiplicity *n_j*. For smaller particles *n_j=1*.

5. Then the estimated number of particles *Particles_single* can be computed as:

Method of circular tracks

For this method we need to define circularity of objects. Circularity is the measure of how closely the shape of an object approaches that of a circle. The higher it goes, the less circular it is. Circularity in descrete space can be defined as:

Algorithm:

1. For each particle calculate A_i, P_i and VOL_i (parameter of circularity) defined by:

,

where *C* is a constant. *C* is approximately *4π*.

2. Calculate a reference area A_ref of particles where Round(VOL_i)=1

3. For each A_i calculate a multiplicity w_i defined by:

4. The estimated number of particles Particles_i in A_i can be computed as:

,

where *k* and *l* are constants that can be changed. Default value is 1.

5. Then the estimated number of particles Particles_single can be computed as:

Results

In the table below, one can see results for different number of particles. In the first section are compared the number of particles generated and the number of particles found. With the increasing number of particles generated is reduced the number of particles found. It is caused by overlapping of particles.

Simple method had the best result with median. Lower and upper quartile proved to be not very useful.

Histogram method proved to have better result with increasing number of intervals. But there was also a limit for that number. After that limit, the effectiveness of this method rapidly declined.

Method of circular tracks had the best result with k and l = 1.

Particles
Generated	10	50	75	100	125	150	175	200
Found	10	47	60	82	89	102	114	125
Simple method
Average	10	47	60	82	88,99	102	113,9	124,9
Median	10,14	48,33	67,65	83,33	103,46	122,03	143	158,8
Lower quartile	15,65	80,11	132	157,54	195,58	271,62	318,27	255,8
Upper quartile	8,63	35,23	39,74	69,28	66,56	70,31	90,1	94,61
Histogram method (param: number of intervals)
n=3	2,15	33,75	38,92	36,48	59,29	78,56	84	100
n=5	6	42,93	59,28	59,93	91,93	128,44	129,3	147,42
n=8	6	6,1	70,82	74,22	132	163,43	163,42	197,85
Method of circular tracks (params: k and l)
k=1, l=1	10	53	79	103	126	157	183	190
k=1, l=10	11	50	70	97	126	147	174	188
k=1, l=50	11	46	67	94	120	150	176	180
k=10, l=1	10	50	72	93	116	134	165	172
k=50, l=1	10	50	71	93	114	131	162	169

User documentation

User interface is devided to two sections. The first section includes a preview image. The second section includes all control buttons.

1. In the first box you can enter a number of particles that will be generated on canvas. The box contains button for generating particles, saving and loading image.
2. The number of particles, perimeters and areas are computed after generating.
3. In the Simple method box you can choose a method for A_ref selection.
4. In the Histogram method box you can enter a number of intervals.
5. In the Circularity box you can enter values for k and l constants.

Easter egg

If you start the program with command line argument deadpool, random particles will look like a little Deadpool.