Applicability of Ant Colony Optimization in particle ...

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 07-10 July, 2008 ＃

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Applicability of Ant Colony Optimization in particle tracking velocimetry

Kazuo Ohmi1, Achyut Sapkota

2, Sanjeeb Prasad Panday

2

1: Dept. of Information Systems Engineering, Osaka Sangyo University, Japan, [email protected]

2: Dept. of Information Systems Engineering, Graduate School of Osaka Sangyo University, Japan, [email protected]

Abstract A new concept algorithm based on the Ant Colony Optimization is developed for the use in 2-D and 3-D particle tracking velocimetry (PTV). In the particle matching process of PTV, the Ant Colony Optimization is usually aimed at minimization of the sum of the distances between the first-frame and second-frame particles. But this type of minimization often goes unsuccessfully in the regions where the particles are located very close to each other. In order to avoid this flaw, a new type of minimization is attempted using a physical property corresponding to the flow consistency or the quasi-rigidity of particle distribution patterns. Specifically, the Ant Colony Optimization is now aimed at minimization of the sum of the relaxation of neighbor particles. The new algorithm is applied to sets of 2-D and 3D synthetic particle images and the results are compared with the theoretical values.

1. Introduction

Nowadays, the particle tracking velocimetry (PTV), together with its counterpart particle image

velocimetry (PIV) has been widely accepted as a reliable whole-field velocity measurement

technique in every branch of the fluid engineering. Their algorithms have been improved for more

than three decades and their known drawbacks have been resolved one after another. But if

compared to the PIV, the PTV algorithms are still subject to new ideas and concepts because in

PTV there is always more room for discussing various physical constraint conditions about particle

motions. The PIV algorithms, in this respect, have become more expertise only from the viewpoint

of signal processing and seem to lack in breakthrough ideas.

The most classical algorithms for particle tracking velocimetry would be the multi-frame

geometrical tracking method (Kobayashi et al. 1989), typically using four consecutive particle

image frames, and the binary-image cross correlation method (Uemura et al. 1989) using two

frames instead of four. As might be expected, the two frame method is usually preferred but the

cross correlation pattern matching is sensitive to the temporal deformation of the interrogation

window. In order to solve this problem new algorithms using a concept of particle cluster matching

have been proposed. In their spring model particle tracking, Okamoto et al. (1995) uses the spring

residual force of particle clusters to match individual particles. By contrast, Ishikawa et al. (2000)

makes use of the velocity gradient tensor with respect to the central particle of the cluster.

Another new idea of particle tracking is the use of various kinds of cost functions, most of which

are related to the algorithms for optimal solution problems. A typical example is the genetic

algorithm PTV implemented by Ohyama et al. (1993) and by Ohmi et al. (2001). Another example

is the Hopfield neural network PTV implemented by Knaak et al. (1997). Together with the use of

these cost functions, there is a current trend of using Fuzzy logics and neural networks in the

particle tracking velocimetry. The Fuzzy logic particle tracking has been attempted by Wernet

(1993) since long time. One more classical approach is the multi-layer neural network application

by Grant and Pan (1995). The Hopfield neural network PTV by Knaak et al. (1997) is already

mentioned above as a cost function method. The cellular neural network PTV by Ohmi et al. (2006)

is considered as an improved version of this Hopfield neural network PTV. The self-organizing

maps (SOM) neural network PTV proposed by Labonté (1999) is simple in algorithm but works

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well with a relatively small number of particles.

In the present work, the authors attempt to use the Ant Colony Optimization (ACO) method for

the particle tracking. This is a kind of algorithm for optimal solution problems and seems attractive

for the particle tracking in the sense that the method uses a concept of group intelligence. This

means that the algorithm incorporates the idea of particle cluster matching in its mechanism of

individual particle matching. This method has already been used by Takagi (2007) in his attempt of

2D and 3D particle tracking but the results seem largely dependent on the number of particles to be

tracked. So the objective of the present work is the improvement of the ACO particle tracking with

a new idea of its implementation. The performance of the new algorithm is demonstrated by using

the 2D and 3D synthetic particle image data from the PIV Standard Image project (Okamoto et al.

2000a and Okamoto et al. 2000b) of the Visualization Society of Japan.

2. Ant colony optimization

2.1 Basic principle

The ant colony optimization (ACO) is an algorithm that imitates the behavior of a group of ants

searching for foods and bringing it back to their nest (Dorigo et al. 1996). Their food collection is a

cooperative work of the ants going on a scouting mission and those collecting foods. First, at

beginning of a day, the scout ants go out of the nest for their daily missions in the vicinity. When

they find out foods, they come back to the nest while diffusing the itinerary pheromone on the route

to be traced by other ants following them.

The pheromone is a kind of volatile chemical substance recognized as an odor. The food-

collecting ants make their way to the food location by sniffing out this itinerary pheromone. On the

way to the foods, the food-collecting ants also diffuse the itinerary pheromone on the route. As

might be expected, the food-collecting ants coming later make their way pursuing a route with

stronger odor or with more pheromone. The routes on which the ants do not go often lose their

pheromone by evaporation and are gradually abandoned. Only those routes which more ants follow

are to survive. If there are two routes on the same destination but with different distances, the

pheromone evaporates more on the longer route because the ants need more time to pass through it.

Naturally, in this way any longer routes are discarded and only the shortest-distance routes are

selected.

Individual ants conduct themselves according to two simple rules: (1) they travel at constant

speed while marking their route with pheromone; (2) they make their way on the routes with

stronger pheromone. However if the ants are viewed as a group, they also behave as if there were

intelligence in the mass of ants. And this sort of group intelligence can be described as the essence

of the ant colony optimization. In order to reproduce such behaviors of ants as individuals and as a

group, the scenario of the ant colony optimization is evolved in the following way.

First of all, a number of agents imitating individual ants are prepared for work. These ant agents

act independently in the space of the problem to be solved. Then they go on a travel from different

points in the space searching for the solution which they try to find out by combining two kinds of

information. The first one is the short-sighted information obtained from direct views of the

problem and the second one is the global information drawn from the group activities of ant agents.

Figure 1 shows a typical problem to which the ant colony optimization is applied. The ant agents

start from different numbered points and travel independently with a mission of visiting all the

numbered points only once. The total number of the routes for the agents to visit all 9 points (in

Figure1) is equal to factorial of 9. Of this large number of possible routes, only one with the best

estimate will be selected as an optimal solution. And if the best estimate is based on the total

distance traveled by the agents, the mission of the agents is to find out a traveling route with the

shortest total distance.

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Fig.1 Search of travel routes in the ant colony optimization

However, if all the agents go simply at random, this comes down to the “try all possibilities”

strategy. Therefore, the agents should convey their own acquired information to each other. This

mission can be realized by the agents leaving pheromone on the route. This pheromone information

is volatile like the itinerary pheromone left by real ants. Therefore, more pheromone is stored on the

routes on which more ants travel while pheromone is gradually lost on the unfrequented routes.

In addition, the agents’ action is determined stochastically not only by the amount of pheromone

but also by their own scout missions. In this way, even if two agents get the same information, their

action could be different. Since their action is controlled stochastically, they try to make their way

according to one of the two information sources if it can be considered as definitely better or

otherwise, according to more global information. In short, the ant colony optimization is designed

for more efficient search by combining all theses features of ant agents like stochastic action

selection, pheromone diffusion and evaporation and group intelligence produced by a mass of

agents.

2.2 Application of ACO in traveling salesman problem

The traveling salesman problem (TSP) is a problem which requires that one should find the shortest

route visiting each of a given set of cities and returning to the starting point without visiting any

cities twice. The algorithm in which the ant colony optimization (ACO) is applied to solve this

traveling salesman problem is often called “ant system or in an abbreviated form AS” (Dorigo et al.

1996). And in the present work, the particle tracking velocimetry (PTV) algorithm will be

developed on the basis of this ant system. The ant system can be outlined as follows.

First, let n be the number of cities of TSP and m be the number of agents. On the route

connecting every pair of cities the pheromone is set, the amount of which is indicated by non-

negative real number values. This amount of pheromone on the route connecting cities i and j is

designated by τ(i, j). The pheromone amount on every route at an initial time is initialized by a

constant τ0. When the agent k selects the next city to visit, he follows the selection probability

defined by the following equation.

( ) ( )[ ] ( )[ ]( )[ ] ( )[ ]βα

βα

ητ

ητ

lili

jijijip

kNl

k

,,

,,,

∈∑

= (1)

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where η(i, j) stands for the information specific to the problem space and in the case of TSP, is

usually given by the reciprocal of the distance d(i, j) between cities i and j. α and β are both non-

negative real number parameters, which determine the weight balance of the local information

prescribed by the physical constraints and of the global information provided by the pheromone

amount. If α is set larger then the global information by pheromone is given more importance while

if β is larger, the local information becomes more dominant.

After all the agents complete their circuit route, the pheromone amount τ(i, j) on every portion of

route is updated according to the following equations.

( ) ( ) ( ) ( )

( )( )

∈

=∆

∆+−← ∑=

otherwise

TjiifLji

jijiji

kk

k

m

k

k

　0

,1,

,,1,1

τ

ττρτ

(2)

where Tk stands for the sets of portions of route included in the whole itinerary traveled by the agent

k and Lk designates the total distance of the circuit route. Since the pheromone amount is

determined by the reciprocal of the total distance of the circuit route, more pheromone is left on the

routes with smaller total distances and less amount of pheromone is left on the other routes. Since

the amount of pheromone left by each agent is accumulated, the total amount of pheromone works

as a source of global information for the agents to come. In addition, ρ in equation (2) stands for the

evaporation rate of pheromone and is usually defined in the range 0<ρ<1. This ρ intends to

represent the loss rate of the pheromone left earlier on every portion of travel route.

In practice, each time when every individual agent completes his circuit route, the pheromone

amount is updated on every portion of route, all the circuit routes traveled by the agents are cleared

off and the agents restart their route search from their respective start cities. And this 5-step cycle,

composed of next city selection, completion of circuit route, update of pheromone information,

clear off of circuit route and restart of travel, is iterated until the optimal solution is obtained.

2.3 Application of AS to two-dimensional PTV

The ant system (AS) is designed for the traveling salesman problem where there is only one

problem area. But in the particle tracking velocimetry (PTV), there are usually two problem areas

so that the ant system cannot be applied to this problem directly. So in the present study, when

applying the ant system into the two dimensional PTV, the agents’ travel rules in the traveling

salesman problem have been modified to be used in the particle matching process of PTV.

(a) First exposure image (b) Second exposure image

Fig.2 Agent's travel rule in the particle tracking velocimetry

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In the first place, similarly to the ant agent in the TSP traveling to the next city according to the

values of selection probability, the agent in the PTV also travels stochastically from the first image

particle to the second image one. This travel distance is counted as a part of the total distance of the

circuit route traveled by the agent. Next, the agent goes to another first image particle (according to

some simple rule like the particle id number) without addition of the travel distance. Then, from this

first image particle, the agent goes to another second image particle determined stochastically with

addition of the travel distance. This process is iterated until all the agents visit all the first image

particles and their respective second image partners.

This travel rule is schematically illustrated in Figure 2. In this figure, the heavy arrow lines

indicate the travel of agents according to the selection probability and the thin arrow lines denote

the simple uncounted movement of agents. In reality, however, the travel distance of the agents is

not measured in this parallel arrangement of the two time-differential images but in a single

superposed image.

2.4 Improvement of the AS implementation

According to the preliminary tests by the present authors, this type of the AS implementation in the

2D particle tracking velocimetry works well as long as the number of particles to be tracked does

not exceed the order of several hundreds. But the results of this classical (or conventional) strategy

are rapidly deteriorated for larger numbers of particles. And the deterioration comes not only with

the number of particles but also with the particle distances to be tracked. In many PTV experiments,

the particle velocity has a relatively wide velocity range and the large displacement of particles in

high velocity regions is more likely to deteriorate the PTV results with this conventional strategy.

So a new strategy is proposed here for the ant system used in the particle tracking velocimetry. In

the new algorithm, the travel of ant agent does not aim at the smallest sum of simple time-

differential particle distances but at the smallest sum of the relaxation lengths for a group of

particles. Before the start of the ant system, every individual particle in the first image forms a

cluster of particles with its neighbours and the sum of the relaxation lengths of this cluster is

calculated with respect to every candidate partner particle in the second image. A similar idea is

used in the fitness function of the genetic algorithm PTV implemented by Ohmi et al. (2001) with

nicely successful results. Hereafter this new strategy is referred to as “relaxation minimization”

while the other one is called as “conventional minimization”.

3. Test results

3.1 2D particle tracking

The tests of 2D particle tracking are all conducted by using the synthetic particle images of the PIV

Standard Image project (Okamoto et al. 2000a). From their library, a set of time-series particle

images, numbered as #301, are used with different time intervals. The first two exposures of this set

of particle images are shown in Figure 3. By using this set of particle images, two pairs of time-

differential sets of particle coordinates data are established for the particle tracking test. The first

pair is composed of the first and second exposure images (Frame0-1) and the second pair of the first

and third exposure images (Frame0-2). As a matter of fact, the time interval of the second pair of

images is as twice long as the first pair and thereby, the particle tracking of this second one is a

more difficult task for any algorithms.

Both sets of particle coordinates data are analyzed by the two types (conventional or relaxation

minimization) of AS particle tracking algorithm. The particle match results are compared with the

known exact data of particle displacement and thereby the performance of the particle tracking is

evaluated. The calculation parameters of the present ant system are as follows:

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(a) First exposure image (b) Second exposure image

Fig.3 PIV standard image # 301 (with 1/30 s time interval)

Number of ant agents m : 20

Number of particles per image n : 100, 250, 500 or 1000

Number of iterations k : 5

Weight coefficient α : 1.0

Weight coefficient β : 5.0

Pheromone evaporation rate ρ : 0.5

Number of cluster particles nc : 4 or 6 (applicable only in the relaxation minimization)

The results of the AS particle tracking are not completely reproducible because the ant agents are

initially distributed at random particle positions. So the final results are obtained from the best value

in 10 independent trials. Typical particle tracking results at n=250, 500 and 1000, together with

their original particle positions, are shown in Figures 4 to 9. The relevant statistical data of these

particle tracking results are given in Tables 1 and 2.

It is recognized from all these figures and tables that under any experimental conditions, the new

AS implementation with a relaxation minimization strategy give rise to definitely better particle

tracking results than the conventional minimization strategy. The difference of the two strategies is

even more pronounced with the dataset Frame0-2, where the tracking results with the conventional

minimization strategy are rapidly deteriorated with respect to those with the dataset Frame0-1. The

longer time interval of two exposure images (i.e. larger displacement of individual particles) makes

a more difficult task for such a conventional ant system tracking algorithm. The general decrease of

the number of correct particle tracking with time interval is also discernible from the statistical data

in Tables 1 and 2. Even the new strategy results are not completely free from errors with this longer

time interval. Another point to be remarked in Tables 1 and 2 is that the AS particle tracking makes

a rather time-consuming process if the number of particles per image is increased. For typical

values, the computation is finished in only 0.7 to 0.8 second (PC with a Dual Core E2180 CPU)

with 100 particles, whereas it takes nearly 8.5 to 10.5 minutes with 1000 particles. This is evidently

due to the power function in equation (1), of which the computational load increases exponentially

with the number of particles.

As far as the correct match rates of particle tracking are concerned, excellent results are obtained

at n=100 or 250, where 100% of the first image particles of Frame0-1 are correctly matched with

the second image particles with the relaxation minimization strategy and this percentage goes down

only to 98 or 88% with the conventional minimization strategy. Excellent results are still obtained

also with the dataset Frame0-2. The particle tracking with the relaxation strategy still produces no

errors while that with the conventional gives rise to 36.8% error at n=250. The correct tracking rates

are generally decreased at n=500 or 1000 with both strategies but the performance of the relaxation

strategy still keeps a very high level. The correct tracking rates with the conventional strategy are

rapidly deteriorated and in the worst case (with dataset Frame0-2) fall down to only 29.4%.

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(a) Original particle positions (b) Conventional minimization (c) Relaxation minimization

Fig.4 Particle tracking results of 2D PIV Standard Image #301

(250 particles between Frames 0 and 1)







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This degraded performance of the conventional strategy is probably due to that fact that in a pair

of densely distributed particle images, the stochastically smallest sum of particle distances can be

smaller than the sum of the real particle distances. One evidential example is seen in the particle

tracking results of Figure 7 (b). In top center of this figure, one sees two displacement vectors (short

and long) crossing with each other. The sum of the particle distances in such a crossed displacement

pattern may be smaller than that in another parallel displacement pattern. And in order to avoid

erroneous particle matching in such a parallel displacement area, the new relaxation minimization

strategy must be a highly powerful alternative.

Table 1 Performance of particle tracking for dataset Frame0-1

Minimization Number of

particles Correct pairs Correct rate CPU time (s)

100 98 98% 0.6

250 220 88% 7.5

500 366 73.2% 59 Conventional

1000 568 56.8% 503

100 100 100% 0.7

250 250 100% 8.0

500 500 100% 66 Relaxation

1000 998 99.8% 531

Table 2 Performance of particle tracking for dataset Frame0-2


particles Correct pairs Correct rate CPU time (s)

100 86 86% 0.6

250 158 63.2% 8.3


1000 294 29.4% 548

100 100 100% 0.7

250 250 100% 9.1

500 498 99.6% 76 Relaxation

1000 988 98.8% 630

3.2 3D particle tracking

In the next step, the test of the 3D particle tracking is conducted by using the library of the 3D

PIV Standard Image implemented by Okamoto et al. (2000b). This library usually offers various

type sets of stereoscopic particle images together with their respective calibration images. So the 3D

particle coordinates should be computed after establishing the stereo pair matching between a pair

of coinstantaneous stereoscopic images. But since this stereo vision process is not the objective of

the present work, the 3D coordinates of the distributed particles are obtained directly from their

numerical library as exact values.

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The 3D particle match results are compared to the known displacement data of the particles and

thereby the performance of the particle tracking is evaluated. The calculation parameters of the

present ant system are as follows:

Number of ant agents m : 20

Number of particles per image n : 500 or 1000

Number of iterations k : 5

Weight coefficient α : 1.0

Weight coefficient β : 5.0

Pheromone evaporation rate ρ : 0.5

Number of cluster particles nc : 4 (applicable only in the relaxation minimization)

Typical particle tracking results at n=500 and 1000, together with the respective original particle

positions, are shown in Figures 10 and 11. The relevant statistical data of these tracking results are

given in Table 3. It is recognized from these figures and tables that the particle tracking results with

a 3D particle dataset are less sensitive to the minimization strategy used in the computation. The

reason for this is that the 3D particle coordinates provide naturally more exploitable information for

the particle matching process than the 2D particle coordinates. In other words, 3D particle tracking

is an easier task for any particle tracking algorithm, as long as the 3D particle coordinates are given

with precision.

4. Conclusions

A new particle matching algorithm using the ant colony optimization (ACO) has been developed

for the use in the particle tracking velocimetry (PTV). The test results in the present work are

obtained only from 2D and 3D synthetic particle images. The algorithm works quite well with the

minimization strategy of the particle cluster relaxation. By contrast the performance of the

conventional minimization strategy is rather poor, especially in the case of the 2D particle tracking

where the information of particle displacement is decreased by one dimension. The only problem to

be solved in the future is the computation time. In order to reduce the computational load, the

selection probability in equation (1) must be calculated with less time-consuming functions. At the

same time the update formulae of the pheromone amount (2) should be more simplified.

Another perspective of the present work is that the ant colony optimization may be applicable to

the PIV-PTV hybrid system as a basic algorithm. Since the algorithm attempts to find the final

results by combining the locally viewed optimal solutions and the globally obtained group

intelligence, the two parts of this combination may be shared by the relaxation PTV of this work

and the cross correlation PIV. This type of PIV-PTV hybrid would be a highly effective system for

validating both the PIV and PTV results thereby solving the problem of which one of the two

should be done first.

Table 3 Performance of the 3D particle tracking with conventional and new minimization strategy


particles Correct pairs Correct rate CPU time (S)


1000 994 99.4% 340

500 500 100% 61 Relaxation

1000 1000 100% 476

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(1000 pairs of particles between Frames 0 and 1)

References Dorigo M, Maniezzo V, Colorni A (1996) The ant system: optimization by a colony of cooperating agents,

IEEE Trans. on Systems, Man, and Cybernetics Part B, 26(2): 29-41

Dorigo M, Gambardella LM (1997) Ant colony system: a cooperative learning approach to the traveling

salesman problem, IEEE Trans. on Evolutionary Computation, 1(1): 53-66

Grant I, Pan X (1995) An investigation of the performance of multi-layer neural networks applied to the

analysis of PIV images, Experiments in Fluids, 19(3): 159-166

Ishikawa M, Murai Y, Wada A, Iguchi M, Okamoto K, Yamamoto F (2000) A novel algorithm for particle

tracking velocimetry using the velocity gradient tensor, Experiments in Fluids, 29: 519–531

Knaak M, Rothlübbers C, Orglmeister R (1997) A Hopfield neural network for flow field computation based

onparticle image velocimetry/particle tracking velocimetry image sequences, Proc. IEEE International

Conference on Neural Networks, Vol.1: 48-52

Kobayashi T, Saga T, Segawa S (1989) Multipoint velocity measurement for unsteady flow field by digital

image processing, Flow Visualization V, Hemisphere Pub, 197-202

Labonté G (1999) A new neural network for particle tracking velocimetry, Experiments in Fluids, 26(4):

340-346

Ohmi K, Yoshida N, Huynh Thai Hoang (2001) Genetic algorithm PIV, Proc. 4th Int. Symp. on Particle

Image Velocimetry, P-1051

Ohmi K (2003) 3-D particle tracking velocimetry using a SOM neural network, Proc. 5th Int. Symp. on

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Particle Image Velocimetry, #3112

Ohmi K, Sapkota A (2006) Cellular neural network based PTV, Proc. 13th Int. Symp. on Applications of

Laser Techniques to Fluid Mechanics, #16.3

Ohmi K, Sapkota A (2007) Fuzzy logic based particle tracking velocimetry, Proc. 2nd Int. Symp. on

Advanced Fluid/ Solid Science and Technology in Experimental Mechanics, #62

Ohyama R, Takagi T, Tsukiji T, Nakanishi S, Kaneko K (1993) Particle tracking technique and velocity

measurement of visualized flow fields by means of genetic algorithm (in Japanese), Journal of the

Visualization Soc. Japan, 13(S1): 35-38

Okamoto K, Schmidl WD, Hassan YA (1995) Spring model tracking algorithm for three-dimensional

particle image velocimetry, ASME FED, Vol.209: 91-97

Okamoto K, Nishio S, Saga T, Kobayashi T (2000a) Standard images for particle image velocimetry, Meas.

Sci. Technology, Vol.11: 685-691

Okamoto K, Nishio S, Kobayashi T, Saga T, Takehara K (2000b) Evaluation of the 3D-PIV Standard Images

(PIV-STD Project), J. Visualization, 3(2): 115-124

Takagi T (2007) Study on particle tracking velocimetry using ant colony optimization (in Japanese), J.

Visualization Soc. Japan, 27(S2): 89-90

Uemura T, Yamamoto F, Ohmi K (1989) High speed algorithm of image analysis for real time measurement

of two-dimensional velocity distribution, Flow Visualization: ASME FED-85, 129-134

Wernet MP (1993) Fuzzy logic particle tracking velocimetry, NASA Technical Memorandum, #106194

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