NL2023430B1

NL2023430B1 - Target Tracking Method Based on Improved Particle Swarm Optimization Algorithm

Info

Publication number: NL2023430B1
Application number: NL2023430A
Authority: NL
Inventors: Sun Hui; Li Jiabin; Deng Rui; Li Meng; Xia Longlong; Zou Shigui; Liao Xiaolong; Wang Xuyu
Original assignee: Chengdu Qitai Zhilian Information Tech Co Ltd
Priority date: 2019-06-05
Filing date: 2019-07-03
Publication date: 2020-10-06
Also published as: LU101298B1; BE1027208B1; CN110288634A; BE1027208A1

Abstract

The present invention provides a target tracking method based on an improved particle swarm optimization algorithm, and relates to the technical field of digital image processing. The method comprises the steps of firstly, performing box selecting on a target to be tracked in an image sequence, and through solving the one-dimensional features of a target region in an HSV color space, describing the target region, then adopting a linearly decreasing inertia weight adjustment strategy, adjusting inertia weight in a particle swarm optimization algorithm, and balancing the development and research capacity for particles in the particle swarm optimization algorithm, and finally, tracking the target in the image sequence by a two-swarm particle swarm optimization algorithm. According to the target tracking method based on an improved particle swarm optimization algorithm, the two-swarm particle swarm optimization algorithm is used for target tracking in an image sequence, so that the learning ability of particles is balanced between individual optimal position and global optimal position, which is favorable for position updating of the particles, and the tracking efficiency and tracking accuracy can be further improved.

Description

Target Tracking Method Based on Improved Particle Swarm Optimization Algorithm

TECHNICAL FIELD

[0001] The present invention relates to the technical field of digital image processing, in particular to a target tracking method based on an improved particle swarm optimization algorithm.

BACKGROUND ART

[0002] A target tracking method is a research focus of special interest in the field of computer vision. Main target tracking methods at present comprise a centroid tracking method, a correlation tracking method, an optical flow method, a mean shift tracking method, a Kalman filter tracking method, a particle filter tracking method, and the like. With constant researches by researchers, a variety of new tracking methods have emerged to be used for target tracking in image sequences or videos. The target tracking methods can be used in all aspects of vehicle tracking, pedestrian tracking, medical image processing, and the like. Solving the problem that the target tracking accuracy is not high at present appears to be particularly important.

[0003] The particle swarm optimization (PSO) algorithm is a new swarm intelligent optimization algorithm. The PSO algorithm is simulated and abstracted from the predatory behavior of bird swarms or fish swarms. The particle swarm optimization algorithm has the characteristic of being simple and easy to use, does not depend on specific problems and has certain applicability to various problems. However, the particle swarm optimization algorithm has the phenomenon that diversity of the particles loses, and the particles are easy to be in the state of local optimization so as to cause premature convergence. The current target tracking methods based on the particle swarm optimization algorithm comprise: 1. Yin Hongpeng, Liu Zhaodong, Luo Xianke, et al., A target tracking feature selection algorithm based on the particle swarm optimization. Computer Engineering and Applications, 2013, 49(17): 164-168; 2. Nouiri M, Bekrar A, Jemai A, et al. An effective and distributed particle swarm optimization algorithm for flexible job-shop scheduling problem. Journal of Intelligent Manufacturing, 2018:1-13; 3. Bae C, Teung HW F, Chung Y Y. Effective object tracking framework using weight adjustment of particle swarm optimization. International Conference on Information NETWORKING.

IEEE Computer Society, 2018:831-833. In order to solve the problem of particle degeneracy; 4.

Guo Siqiu, Xu Tingfa, Wang Hongqing, et al. An improved particle swarm optimization target tracking method. China optics, 2014, 7(5): 759-767 puts forward an improved PSO algorithm used for target tracking. This algorithm timely adjusts inertia weight mainly according to different states of particles in iteration. However, the above methods still cannot solve the problem that the target tracking accuracy is not high.

Summary of the Invention

[0004] In order to solve the techncial problem, in accordance with defects existing in the prior art, the present invention provides a target tracking method based on an improved particle swarm optimization algorithm so as to realize accurate tracking of targets in image sequence.

[0005] In order to solve the technical problem, the present invention adopts the technical scheme of a target tracking method based on an improved particle swarm optimization algorithm, wherein the target tracking method comprises the following steps of

[0006] Step 1: reading an image sequence to be processed, and performing box-selecting on a target to be tracked in the first frame of image so as to obtain the position of the target at the first frame, namely determining the target to be tracked,

[0007] Step 2: converting the image in a target region from an RGB image into an HSV image according to the box-selected target, and calculating one-dimensional features of the target region in the HSV color space so as to describe the target region;

[0008] converting the image in the target region from an RGB image into an HSV image, namely converting an RGB color space into anHSV color space, as shown in the following formula: V =max(R,G,B) 5 max (R,G,B)-min(R,G,B) | max (R,G, B)

60x(G—B)/(SxV),S #0,max(R,G,B)=R H =<60x(2+(B-R)/(SxV)),S #0,max(R,G,B)=G 60x(2+(B-R)/(S xV)), S #0,max(R,G,B) = B

[0009] wherein in the above formula, R represents red, G represents green, B represents blue, the value range of R, G and B is [0, 255], H represents hue, S represents saturation, V represents visibility, the value range of H is [0,360], the value range of S and F is [0,255]:

[0010] and based on the converted HSV color space and according to the visual resolving power of human eyes, dividing the hue color space H into 8 parts, dividing the saturation space S' into 3 parts, dividing the visibility value 7° space into 3 parts, and constructing the one-dimensional features of the target region on this basis to realize the feature extraction of the selected target region, wherein

[0011] The construction formula of the one-dimensional features M of the target region is as follows: M =9H +3S+V

[0012] Step 3: adjusting inertia weight w in the particle swarm optimization algorithm by using a linearly decreasing inertia weight adjustment method according to the one-dimensional features of the target region in HSV color space, and balancing the development and exploration abilities of particles in the particle swarm optimization algorithm, wherein

[0013] the linearly decreasing inertia weight adjustment method is shown as follows: Woax Win ; WZW oo 2 Biter max [ter

[0014] wherein % ís the maximum inertia weight value in the particle swarm optimization algorithm obtained through calculation based on the one-dimensional features of the target region, w_. is the minimum inertia weight value in the particle swarm optimization algorithm obtained through calculation based on the one-dimensional features of the target region, max_ [fer is the maximum number of iterations of the particle swarm optimization algorithm, and iter is the current number of iterations.

[0015] Step 4: tracking the target in the image sequence by using a two-swarm particle swarm optimization algorithm, and outputting tracking results,

[0016] wherein the particle velocity updating formula in the two-swarm particle swarm optimization algorithm is shown as follows: vil =wvl ton ( pbest; -x) +e, ( gbest —x)

[0017] wherein, ;=1,2,L ‚n and n are the swarm size in the particle swarm algorithm, w is the linearly decreasing inertia weight, c, and c¢, are accelerating factors, # and * are both random numbers with value range of [0,1], v is the velocity of particle / at the 7th iteration, v/"' is the velocity of particle / at the 741th iteration, pbest! is the optimal position of particle 7at thes th iteration, gbest is the global optimal position of the particle in the particle swarm at the {th iteration, and x! is the position of the target point at the 7th iteration.

[0018] The position updating formula of the particle is shown as follows: vil — Vv +

[0019] wherein x'"' represents the position of the target point at the £+1 th iteration.

[0020] The target tracking method based on the improved particle swarm optimization algorithm, adopting the technical scheme, has the following beneficial effects that (1) in a traditional target tracking method based on a PSO algorithm, the inertia weight w in the PSO algorithm is a constant which is unchanged in the entire tracking process; a linearly decreasing inertia weight adjustment strategy is adopted to constantly change the magnitude of inertia weight during iteration, i.e. in a frame of image, particles initially start to search the position of a target in a global space and later determine the approximate position of the target, so that the algorithm can accurately determine the position of the target. The method can reduce the number of iterations and improve the operation efficiency of the algorithm. (2) The two-swarm particle swarm optimization algorithm is used for target tracking in an image sequence, so that the learning ability of particles is balanced between individual optimal position and global optimal position, which 1s favorable for position updating of the particles, and the tracking efficiency and tracking accuracy can be further improved.

DESCRIPTION OF DRAWINGS

[0021] Fig. 11s a flow chart of a target tracking method based on an improved particle swarm 5 optimization algorithm, provided by the embodiment of the present invention;

[0022] Fig.2 shows target tracking results obtained by using method provided by the embodiment of the present invention and the particle swarm optimization algorithm to track the 20th frame of an image sequence, wherein (a) is the tracking result of the method, and (b) is the tracking result of the particle swarm optimization algorithm;

[0023] Fig3 shows target tracking results obtained by using method provided by the embodiment of the present invention and the particle swarm optimization algorithm to track the 40th frame of an image sequence, wherein (a) is the tracking result of the method, and (b) is the tracking result of the particle swarm optimization algorithm;

[0024] Fig.4 shows the tracking time comparison diagram of two algorithms provided by the embodiment of the present invention;

[0025] Fig.5 shows the tracking error comparison diagram of two algorithms provided by the embodiment of the present invention.

DETAILED DESCRIPTION

[0026] The specific implementation mode of the present invention is further described in details through combination with drawings and an embodiment. The following embodiment is used for explaining the present invention, rather than limiting the scope of the present invention.

[0027] A 70-frame image sequence is taken as an example in the embodiment, and the target tracking method based on the improved particle swarm optimization algorithm of the present invention is used for tracking a target in the image sequence.

[0028] As shown in Fig.1, the target tracking method based on the improved particle swarm optimization algorithm comprises the following steps:

[0029] Step 1: reading an image sequence to be processed, and performing box-selecting on a target to be tracked in the first frame of image through the box selection mode by a mouse so as to obtain the position of the target at the first frame, namely determining the target to be tracked;

[0030] Step 2: converting the image in a target region from an RGB image into an HSV image according to the box-selected target, and calculating one-dimensional features of the target region in the HSV color space so as to describe the target region;

[0031] converting the image in the target region from an RGB image into an HSV image, namely converting an RGB color space into an HSV color space, as shown in the following formula: V =max(R, G,B) S= max (R, G,B) —min (R, G,B) u max (R,G, B) 60x(G-B)/(SxV),S #0, max(R,G, B) =R H =+60x(2+(B-R)/(SxV)),S #0, max(R,G,B)=G 60x(2+(B-R)/(S xV)),S #0,max (R,G,B)=B

[0032] wherein in the above formula, R represents red, G represents green, B represents blue, the value range of R, G and B is [0, 255], H represents hue, S represents saturation, V represents visibility, the value range of H is [0,360], the value range of S and Fis [0,255];

[0033] and based on the converted HSV color space and according to the visual resolving power of human eyes, dividing the hue color space H into 8 parts, dividing the saturation space § into 3 parts, dividing the visibility value J space into 3 parts, and constructing the one-dimensional features of the target region on this basis to realize the feature extraction of the selected target region, wherein

[0034] The construction formula of the one-dimensional features M of the target region is as follows: M=9H +35 +}

[0035] In the embodiment, hue space H is divided into 8 parts, the saturation S space is divided into 3 parts and the visibility value P is divided into 3 parts, specifically represented as:

0 He[316,20] 1H e[21,40] 2 He[41,75] y_3 e[76,155] |4 He[156,190] 5 He[191,270] 6 He[271,295] 7 He[296,315] |° S e[0,0.2] S=+1 Se[0.2,0.7] |; S e[0.7,1] K V e[0,0.2] V=41 Ve[0.2,0.7] |; V e[0.7,1]

[0036] Step 3: adjusting inertia weight w in the particle swarm optimization algorithm by 5 using a linearly decreasing inertia weight adjustment method according to the one-dimensional features of the target region in HSV color space, and balancing the development and exploration abilities of particles in the particle swarm optimization algorithm, wherein

[0037] In the particle swarm optimization algorithm, the inertia weight is adjusted according to the number of iterations. The inertia weight is great at the initial stage of iterations and is used for searching a target in the global region. The inertia weight is small at the later stage of iterations and is used for searching around the target region so as to accurately find the position with globally optimal solution. The inertia weight adjustment strategy adopted in the present invention is the linearly decreasing inertia weight adjustment strategy;

[0038] the linearly decreasing inertia weight adjustment method is shown as follows: wo Won WZW on ier max_ Jer

[0039] wherein %‚ is the maximum inertia weight value in the particle swarm optimization algorithm obtained through calculation based on the one-dimensional features of the target region, w_. 1s the minimum inertia weight value in the particle swarm optimization algorithm obtained through calculation based on the one-dimensional features of the target region, max Jer is the maximum number of iterations of the particle swarm optimization algorithm,

and ier is the current number of iterations.

[0040] Step 4: tracking the target in the image sequence by using a two-swarm particle swarm optimization algorithm, and outputting tracking results,

[0041] wherein the particle velocity updating formula in the two-swarm particle swarm optimization algorithm is shown as follows: vw ten ( pbest; —x!) 4e, ( gbest x!)

[0042] wherein, i=1,2,L ‚n and # are the swarm size in the particle swarm algorithm, w is the linearly decreasing inertia weight, c, and c¢, are accelerating factors, # and +, are both random numbers with value range of [0.1]. v is the velocity of particle 7 at the rth iteration, v'"' is the velocity of particle i at the ¢+1th iteration, pbest is the optimal position of particle iat thes th iteration, ghest’ is the global optimal position of the particle in the particle swarm at the ¢th iteration, and x] is the position of the target point at the ¢th iteration.

[0043] The position updating formula of the particles is shown as follows: x =x!

[0044] wherein x/”! represents the position of the target point at the #+1 th iteration.

[0045] In the embodiment, in the two-swarm particle swarm optimization algorithm, the accelerating factors of one swarm are ¢, =0.5 and ¢, =2.3 and the accelerating factors of the other swarm are ¢, =23 and ¢, =0.5 to make sure the two swarms to have different motion trails, so that a larger searching solution space exists to improve the calculation efficiency of the entire algorithm.

[0046] In the embodiment, the target in the 70-frame image sequence is tracked by using the method and the particle swarm optimization algorithm respectively, and the results are shown in Fig.2 and Fig.3. Fig.2 shows the target tracking results of the 20th frame of image by the two methods; Fig.2(a) shows the tracking result of the target in the image sequence by using the particle swarm optimization algorithm; Fig.2(b) shows the tracking result of the target in the image sequence by using the improved particle swarm optimization algorithm of the present invention; Fig.3 shows the target tracking results of the 40th frame of image by the two methods; Fig.3(a) shows the tracking result of the target in the image sequence by using the particle swarm optimization algorithm; and Fig.3(b) shows the tracking result of the target in the image sequence by using the improved particle swarm optimization algorithm provided by the present invention. From the above figures, it can be seen that the method of the present invention has better tracking accuracy.

[0047] In order to objectively evaluate the target tracking effect of the improved particle swarm optimization algorithm provided by the present invention, the errors of the tracking results and the tracking time between the two tracking methods are compared. As shown in Fig.4 and Fig.5, Fig.4 shows the comparison of target tracking time between the two methods. As shown in Fig.4, it can be seen that the improved particle swarm optimization algorithm provided by the present invention has less tracking time. Fig.5 shows the comparison of target tracking errors between the two methods. As shown in Fig.5, it can be seen that the method of the present invention has better tracking accuracy.

[0048] In the embodiment, experimental comparison of target tracking in the image sequence verifies that the improved particle swarm optimization algorithm of the present invention has better tracking accuracy on the target in the image sequence and has shorter tracking time.

[0049] Finally, it should be noted that the above embodiment is only used for illustrating the technical scheme of the present invention, rather than limiting the present invention; those ordinary skills in the field should understand that the technical scheme recorded in the embodiment can still be modified, or the part of the technical features or all the technical features can be equivalently replaced with the present invention is illustrated in details with reference to the above embodiment; and the essence of the corresponding technical scheme does not depart from the scope limited by the claims of the present invention due to the modification or replacement.

Claims

CONCLUSIESCONCLUSIONS

1.- Doelzoekingswerkwijze op basis van een verbeterd deeltjeszwerm-optimalisatiealgoritme, gekenmerkt doordat zij de volgende stappen omvat: Stap 1: het aflezen van een beeldsequentie die moet worden verwerkt, en het uitvoeren van vak-selectie op een doel dat moet worden gevolgd in het eerste beeldkader om de positie van het doel aan het eerste kader te verkrijgen, namelijk het bepalen van het doel dat moet worden gevolgd; Stap 2: het omzetten van het beeld in een doelgebied van een RGB-beeld in een HSV-beeld volgens het vak-geselecteerde doel, en het berekenen van eendimensionale kenmerken van het doelgebied in de HSV-kleurenruimte om het doelgebied te beschrijven; Stap 3: het bijstellen van inertiegewicht W in het deeltjeszwerm-optimalisatiealgoritme aan de hand van een lineair afnemende inertiegewicht-regelwerkwijze volgens het eendimensionale kenmerk van het doelgebied in HSV-kleurenruimte, en het uitbalanceren van de ontwikkelings- en exploratiemogelijkheden van deeltjes in het deeltjeszwerm-optimalisatiealgoritme; en Stap 4: het zoeken van het doel in de beeldsequentie aan de hand van een deeltjeszwerm-optimalisatiealgoritme met twee zwermen, en het uitvoeren van de zoekingsresultaten.1.- Target finding method based on an improved particle swarm optimization algorithm, characterized in that it comprises the following steps: Step 1: reading an image sequence to be processed, and performing box selection on a target to be tracked in the first image frame to obtain the position of the target at the first frame, namely determining the target to be tracked; Step 2: converting the image in a target area from an RGB image into an HSV image according to the box-selected target, and calculating one-dimensional features of the target area in the HSV color space to describe the target area; Step 3: Adjusting inertial weight W in the particle swarm optimization algorithm using a linearly decreasing inertial weight control method according to the one-dimensional characteristic of the target area in HSV color space, and balancing the development and exploration capabilities of particles in the particle swarm optimization algorithm; and Step 4: searching for the target in the image sequence using a two-swarm particle swarm optimization algorithm and executing the search results.

2.- Doelzoekingswerkwijze op basis van een verbeterd deeltjeszwerm-optimalisatiealgoritme volgens conclusie 1, gekenmerkt doordat in de stap 1 vak-selectie wordt uitgevoerd op een doel dat moet worden gevolgd in het eerste beeldkader door de vak-selectiemodus met behulp van een muis.A target search method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that in the step 1 box selection is performed on a target to be tracked in the first frame by the box selection mode using a mouse.

3.- Doelzoekingswerkwijze op basis van een verbeterd deeltjeszwerm-optimalisatiealgoritme volgens conclusie 1, gekenmerkt doordat in de stap 2 de werkwijze specifiek omvat: het omzetten van het beeld in het doelgebied van een RGB-beeld in een HSV-beeld, namelijk het omzetten van een RGB-kleurenruimte in een HSV-kleurenruimte, zoals weergegeven in de volgende formule: V = max (R, G,B ) go max (R, G,B)-min (R, G,B) max (R,G, B)3. A target search method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that in the step 2, the method specifically comprises: converting the image in the target area from an RGB image into an HSV image, namely converting an RGB color space in an HSV color space, as shown in the following formula: V = max (R, G, B) go max (R, G, B) -min (R, G, B) max (R, G , B)

it 60x(G-B)/(SxV),S #0,max(R,G,B) =R H =<60x(2+(B-R)/(SxV)),S #0,max(R,G,B)=G 60x(2+(B-R)/(S xV))S #0,max(R,G,B)=B waarbij in de bovenstaande formule, R staat voor rood, G staat voor groen, B staat voor blauw, RG B [0,255] .it 60x (GB) / (SxV), S # 0, max (R, G, B) = RH = <60x (2+ (BR) / (SxV)), S # 0, max (R, G, B ) = G 60x (2+ (BR) / (S xV)) S # 0, max (R, G, B) = B where in the above formula, R stands for red, G stands for green, B stands for blue , RG B [0.255].

het waardebereik van ££ © en 1s, H staat voor kleurtoon, S staat voor saturatie, , : : yg [0.360] . § V staat voor zichtbaarheid, het waardebereik van is, het waardebereik van © en sv [0.255] is en op basis van de omgezette HSV-kleurenruimte en in overeenstemming met het visuele resolutievermogen van het menselijke oog, het verdelen van de kleurtoon-kleurenruimte H in 8 delen, het verdelen van de saturatieruimte © in 3delen, het verdelen van de zichtbaarheidswaarde ! -ruimte in 3 delen, en het construeren van de eendimensionale kenmerken van het doelgebied op deze basis voor het realiseren van de eigenschapextractie van het geselecteerde doelgebied, waarbij de constructieformule van het eendimensionale kenmerk B van het doelgebied als volgt is: B=9H +35+Fthe value range of ££ © and 1s, H stands for hue, S stands for saturation,,:: yg [0.360]. § V stands for visibility, the value range of is, the value range of © and sv [0.255] is and based on the converted HSV color space and in accordance with the visual resolution capability of the human eye, dividing the hue-color space H into 8 parts, dividing the saturation space © into 3 parts, dividing the visibility value! space in 3 parts, and constructing the one-dimensional features of the target area on this basis to realize the property extraction of the selected target area, where the construction formula of the one-dimensional feature B of the target area is as follows: B = 9H +35 + F

4.- Doelzoekingswerkwijze op basis van een verbeterd deeltjeszwerm-optimalisatiealgoritme volgens conclusie 1, gekenmerkt doordat in de stap 3 de lineair afnemende inertiegewicht-regelwerkwijze is weergegeven als volgt: Wo Won WZW on iter max [ter waarbij Wane de maximale inertiegewichtwaarde in het deeltjeszwerm optimalisatiealgoritme dat werd verkregen door berekening op basis van de eendimensionale kenmerken van het doelgebied is, Wain de minimale inertiegewichtwaarde in het deeltjeszwerm-optimalisatiealgoritme dat werd verkregen door berekening op basis van de . . . . max [ter . . .Target finding method based on an improved particle swarm optimization algorithm according to claim 1, characterized in that in the step 3 the linearly decreasing inertial weight control method is represented as follows: Wo Won WZW on iter max [ter where Wane is the maximum inertial weight value in the particle swarm optimization algorithm which was obtained by calculation based on the one-dimensional characteristics of the target area, Wain is the minimum inertial weight value in the particle swarm optimization algorithm obtained by calculation based on the. . . . max [ter. . .

eendimensionale kenmerken van het doelgebied is, — het maximumaantal iteraties van het deeltjeszwerm-optimalisatiealgoritme is, en #€7 het huidige aantal iteraties is.is one-dimensional characteristics of the target area, - is the maximum number of iterations of the particle swarm optimization algorithm, and # $ 7 is the current number of iterations.

5.- Doelzoekingswerkwijze op basis van een verbeterd deeltjeszwerm-optimalisatiealgoritme volgens conclusie 4, gekenmerkt doordat in de stap 4 de deeltjessnelheid-updatingformule in het deeltjeszwerm-optimalisatiealgoritme met twee zwermen is weergegeven als volgt: vil wvl ten ( pbest; —x) +, ( gbest' x) .. i=12 . . . .. , waarbij, / L2L.n on fl de zwermgrootte in het deeltjeszwermalgoritme zijn, " het lineair afnemende inertiegewicht is, “en © versnellingsfactoren zijn, i en 2 beide [BI] bl Lj: P willekeurige getallen met een waardebereik zijn, : de deeltjessnelheid ? is aan de . Vit i i +1 . . phest’ . -de iteratie, : de deeltjessnelheid 1s aan de Tl -de iteratie, ‚ de optimale deeltjespositie Î is aan def -de iteratie, ghest de globale optimale positie van het deeltje in £ de deeltjeszwerm is aan de !-de iteratie, en Yi de positie van het doelpunt aan de 7 -de iteratie is.5. Targeting method based on an improved particle swarm optimization algorithm according to claim 4, characterized in that in the step 4 the particle velocity updating formula in the particle swarm optimization algorithm with two swarms is represented as follows: felt wvlten (pbest; -x) +, (gbest 'x) .. i = 12. . . .., where, / L2L.n on fl are the swarm size in the particle swarm algorithm, "is the linearly decreasing inertial weight," and © are acceleration factors, i and 2 are both [B1] bl Lj: P are arbitrary numbers with a value range,: the particle velocity? is at the. Vit ii +1. phest '. -th iteration,: the particle velocity 1s at the T1 -th iteration, ‚the optimal particle position Î is at def-th iteration, ghest the global optimal position of the particle in the particle swarm is at the! -th iteration, and Yi is the position of the target at the 7th iteration.

De positie-updateformule van het deeltje is weergegeven als volgt: =d tn it == waarbij * de positie van het doelpunt aan de £+1 -de iteratie voorstelt.The position update formula of the particle is presented as follows: = d tn it == where * represents the position of the target point at the £ + 1 th iteration.