Soft Computinghttps://doi.org/10.1007/s00500-020-04704-2

FOCUS

Lane departure warning algorithm based on probability statisticsof driving habits

Jindong Zhang1,2,3 · Jiaxin Si1 · Xuelong Yin1 · Zhenhai Gao3 · Young Shik Moon4 · Jinfeng Gong5 ·Fengmin Tang6

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

AbstractFor the different degrees of danger caused by different driving habits, a lane departure warning algorithm based on probabilitystatistics of driving habits is proposed in this paper. According to the different driving habits of different drivers, the earlywarning mechanism can be adaptively adjusted through the method of probability statistics to make lane departure warningmore targeted and accurate. Firstly, each frame of image is preprocessed, including gray treatment, edge detection andbinarization. Then, Canny operator is used to detect the edge, and Hough transform is applied to detect the lines. And the lanemedian line equation for the detection and identification of lane also can be calculated. After that, the image coordinate systemis transformed into the world coordinate system by means of the formula and matrix of coordinate conversion. According tothe theory of Kalman filter, the statistics of lateral acceleration and lateral velocity are updated continuously, and the positionof the vehicle in the next moment is predicted by the state transition equation and the forecast equation. From the results ofexperiments and the comparison with exhaustive algorithms, the advantages of using Kalman filter to predict the location ofvehicles and the improved time-to-lane-crossing combined with probabilistic statistics to warning are illustrated clearly.

Keywords Image processing · Lane departure warning · Kalman filter · Probability statistics

1 Introduction

In recent years, with the development of road constructionand the improvement of vehicle popularization, the rate of

Communicated by Ching-Hsien Hsu.

B Jindong Zhangzhangjindong_100@163.com

1 College of Computer Science and Technology, JilinUniversity, Changchun 130012, China

2 Key Laboratory of Symbol Computation and KnowledgeEngineering of the Ministry of Education,Jilin University, Changchun 130012, China

3 State Key Laboratory of Automobile Simulation and Control,Jilin University, Changchun 130025, China

4 Division of Computer Science, College of Computing,Hanyang University, 1271 Sa-3 Dong, Ansan 426-791,Republic of Korea

5 China Automotive Technology and Research Center,Tianjin 300300, China

6 College of Mechanical Engineering, Hebei University ofTechnology, Tianjin 300132, China

traffic accidents grows, which brings a great threat to the eco-nomic security and people’s social life and property safetywith enormous casualties and economic losses. According tostatistics, 80–90% traffic accidents are directly or indirectlycaused by the driver’s mistakes, such as fatigue driving, lackof concentration, poor driving habits and so on. The conse-quence of the mistake is that the vehicle deviates from thecurrent driving lane, the vehicle contact or beyond the lane,and collides with other vehicles (Lei et al. 2016). In orderto solve this problem, the intelligent transportation has beendeveloped rapidly, and the lane departure warning (LDW)technology is an important part, which can effectively reducethe traffic accidents caused by the driver’s mistake.

With the development of intelligent transportation tech-nology, many countries have obtained fruitful achieve-ments in lane departure warning. Gianni et al. (2010)discussed data-fusion algorithms for computing the time-to-lane-crossing (TLC) of a vehicle traveling along a lane on thebasis of road images, collected by an on-board video cam-era, and kinematic data coming from sensors. Wang et al.(2010) applied fuzzy method to vision-based lane detectionand departure warning system. Pongtep et al. (2011) made

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use of the stochastic driver behavior model in lane depar-ture warning. Dahmani et al. (2011) employed the differencebetween the road curvature and the vehicle trajectory curva-ture with steering dynamics to detect vehicle lane departures.Chihli et al. (2011) designed a vehicle warning system forland departure and collision avoidance with fuzzy decisionmaking. The multi-camera approach for LDWwas proposedby Borkar et al. (2011). Jiang et al. (2011) introduced aweak lane model with the particle filter-based approach andan Euclidean distance transform. The road curvature esti-mation for vehicle lane departure detection using a robustTakagi–Sugeno fuzzy observer was designed by Dahmaniet al. (2013). Tapia Espinoza and Torres Torriti (2013) pro-posed an approach for lane segmentation and tracking thatwas robust to varying shadows and occlusions. Salari andOuyang (2013) presented camera-based forward collisionand LDW systems using SVM. The integration of scanningand image processing algorithms based on the fuzzy methodwas proposed by Zhang et al. (2015). Kyun Jeong and Jeong(2014) designed a hardware implementation method of theLDW system based on an around view monitoring image.Ozcan et al. (2014) introduced MonoSLAM-based methodfor LDW design, assuming that the camera was moving ina previously unknown scene. Sharma et al. (2014) used theLucas–Kanade optical flow and Hough transform to design aLDW system. Moon et al. (2014) proposed a lane departureassistance system based on DGPS-GIS to solve curvy lanedetection and speed up real-time performance. Chien et al.(2014) raised an Android-based solution for lane detectionand departure warning. Vijay and Shashikant (2015) appliedthree lane-related parameters based on theEuclideandistancetransform to estimate the departure measure. Madrid andHurtik (2016) designed a LDW system for mobile devicesbased on a fuzzy representation of images.

These methods have been improved in the study of lanedeparture warning systems, but they still need to be improvedin early warning decisionmaking. This paper proposes a lanedeparture warning algorithm based on probabilistic statisticsof driving habits. This warning method has better accu-racy. The image coordinate system is converted into theworld coordinate system by the coordinate transformationto acquire the accurate data. Kalman filter is used for theprediction of vehicle position to get the precise data in thenext moment. Combining the improved TLC algorithm withthe early warning decision-making algorithm based on prob-abilistic statistics, and adaptively adjusting the early warningmechanism based on the predicted data obtained by the TLCalgorithm and the probability statistics of different drivinghabits of different drivers, and making different early warn-ing decisions to reduce false warning.

The structures of the rest of the paper are as follows: Sect. 2introduces the algorithms applied in detail, including imagepreprocessing, the lane detection, coordinate transformation,

the lane trajectory prediction, combined with the improvedTLC algorithm and the deviation early warning decisionalgorithm based on the probabilistic statistical model of driv-ing habits. And the principles and steps of the improvedalgorithm are also comprehensively introduced. Section 3shows experimental results and analyses data of our methodand the existing method. Conclusions are given in Sect. 4.

2 Algorithm description

The algorithms of the LDW in the paper include the follow-ing parts: image preprocessing, the detection and recognitionof the lane based on Hough transform, coordinate transfor-mation, the lane trajectory prediction based on Kalman filterand the improved TLC with probability statistics of drivinghabits. Figure 1 shows the processes used in the proposedalgorithm.

2.1 Image preprocessing

The original image is abundant in color and noise, and theobjective features are not obvious, so it needs to be prepro-cessed in order to reduce computation load. The originalimage collected by the driving recorder is RGB color model,in which each pixel is composed of three channels (R, Gand B). We make R=G=B to convert the colorful imageto grayscale one, which can reduce storage space and raiseoperation efficiency. The Canny edge detection method isused to measure, detect and locate the gray level changes ofthe image for the reason that the determination of the imageedge is the basis of the recognition and segmentation. Afterthe image is processed by Gaussian filter, first-order partialderivative finite difference is used to calculate the magnitudeand direction of the gradient. Then, the gradient amplitude isprocessed by non-maxima suppression and the dual thresh-old algorithm is applied to detect and connect the edge. Thegray level of each pixel in the gray image is set to 0 (black)and 255 (white) to highlight the contour of region of inter-est (ROI), which separates the target lane from the imagebackground easily. Figure 2 shows the preprocessed image.

2.2 Lane detection and recognition

After image preprocessing, Hough transform is employedto detect lines and calculate the lane median equation to fitthe lane. Under the consideration of both straight lines andcurves, the ROI-containing lanes are divided into five parts,of which heights are highly increasing from top to bottomaccording to the principle of perspective. At this point, eachpart can be approximated as a straight line, and the lane isdetected and fitted one by one. By the duality of point line,Hough algorithmmapped the points of the image space to the

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Fig. 1 The flowchart of algorithm

parameter space. Through the simple accumulated statisticsin the parameter space, the approach for the peak accumu-lator is used to detect all lines. However, there are manyinterference lines and the first line met from the middle tothe side of the image is the lane by common sense. With theconstraint condition that each partition lane is connected, theinterference lines are ruled out. For the 5 partitions, the laneis judged from the bottom to the top, and the midline coor-dinates are obtained and fitted. With the extracted features,a straight line is found where most of the feature points aredistributed in the vicinity to realize curve fitting. Optimalfunction matching is calculated by minimizing the square ofthe error and the least square method, and the lane line isfitted.

2.3 Coordinate transformation

The image coordinate system is used in image processing,which is difficult to study the position relationship betweenthe vehicle and the lane. By the imaging principle and cameraimaging mode combined with the internal and external cam-era parameters, the image coordinate system is converted tothe camera coordinate system and then to the world coordi-nate system. The coordinate transformation algorithm is asfollows:Coordinate transformation algorithm:Input: Pi (u, v); Hc; dx, dy : (u0, v0); Rr

The direct coordinate systemU−V is established in pixelas the upper left corner of the image is the origin. The hori-zontal axisU is the number of columns in the image array, andthe vertical coordinate V is the row number. Pi (u, v) is thecoordinate of the system, and it is limited to the coordinateson the ground. The image coordinate system X − Y repre-sented by physical unit defines the intersection point betweenthe optical axis of the camera and the image plane (locatedat the center of the image plane) as the origin O . The X -axisand the Y -axis parallel withU -axis and V -axis, respectively,and dx and dy represent physical dimension of each pixel in

Fig. 2 The preprocessed image

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the horizontal X and Y , respectively. (u0,Kg(i)) representsthe coordinate in the coordinate systemU −V which is con-verted from the origin of the coordinate system X − Y thatis the intersection of the optical axis and the image. Hc is theheight of the camera, and Rr represents the rotation matrixwhich parallels to the optical axis and is perpendicular to theplane of the vehicle.Output: Ww(xw, yw, zw)

The origin of theworld coordinate system is the projectionpoint of the camera on the ground. X -axis is horizontal to theleft, Y -axis parallels to the optical axis, Z -axis is verticallyupward, and Ww(xw, yw, zw) represents the coordinate incoordinate system.

Step1: The coordinate system u − v which stands for therow and column of the pixel is converted into the coordinatesystem xp − yp with physical dimension, as follows:

xp = (u − u0) ∗ dx (1)

yp = (v − v0) ∗ dy (2)

Step 2: The coordinate system xp − yp is converted intothe camera coordinate system Bc(xb, yb, zb), as follows:

xb = Hc ∗ xp/yp (3)

yb = Hc (4)

zb = Hc ∗ xp/yb (5)

Step 3: The camera coordinate system is converted intothe world coordinate system Ww(xw, yw, zw), as follows:

Ww(xw, yw, zw, 1) = (xb, yb, zb) ∗ Rr ∗ Rt ∗ Tw (6)

The matrix Rt represents rotating 90◦ around the X -axisclockwise, and the matrix Tw is moving H downward.

2.4 The lane trajectory prediction based on Kalmanfilter

By usingKalman filter, the error in themeasurement and pre-diction of the distance between the vehicle and the lane canbe excluded. The relatively accurate distance can be obtainedwith the measured and predicted values. Meanwhile, the lat-eral velocity, the lateral acceleration and other constantlyupdated statistical data can be acquired as well. Then withthe help of the state transition equation and the predictionequation, the vehicle position in the next time can be esti-mated, which is the basis for subsequent warning operations.

Firstly, the velocity and acceleration of the moving targetare initialized according to the experience. According to thetheory of Kalman filter (Kobayashi et al. 1995), the equationsof velocity and acceleration of vehicle position are presented

to predict the possible location in the next frame. The regionalcenter with minimum error of predicted value is found fromthe detected region of the moving target in next frame. Thevelocity and acceleration of the target are updated. Throughthe prediction and amendment of the position in each frame,the tracking of moving objects is completed. Through thelinear state equation and the input and output observationdata (velocity, acceleration and position), prediction equa-tions and covariance equations are listed to predict the systemstate. Then, the predicted values are modified according tothe gain error and updated covariance. Finally, the optimalestimation is obtained. The Kalman filter algorithm is as fol-lows:

Step 1: If i > 3, calculate predicted value as follows:

Spre = S′i−1 + Vi ∗ Tt + 0.5 ∗ ai−1 ∗ T 2

t (7)

where Spre is the position prediction in i-th frame. vi is thevelocity in i-th frame. ai is the acceleration in i th frame.

Update the covariance of the error between the estimatedvalue and the true value as follows:

P(i |i − 1) = P(i − 1|i − 1) + Qk (8)

where Qk is the noise covariance in the prediction process.Update Kalman gain error Kg(i) as follows:

Kg(i) = P(i |i − 1)/(P(i |i − 1) + Rk) (9)

where Rk is the noise covariance in themeasurement process.The distance between the vehicle and the lane through the

filter is as follows:

S′i = Spre + Kg(i) ∗ (Si − Spre) (10)

where Si is the lateral distance between the vehicle and thelane in the i-th frame. S

′i is the lateral distance between the

vehicle and the lane after Kalman filter process.Update the covariance P(i |i) as follows:

P(i |i) = (1 − Kg(i)) ∗ P(i |i − 1) (11)

else, S′i = Si .

Step 2: If i > 2 then, update the velocity as follows:

Vi = (S′i − S

′i−1)/Tt (12)

Step 3: If i > 3 then, update the acceleration as follows:

ai = (vi − vi−1)/Tt (13)

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2.5 The improved TLC algorithm

According to the different driving habits of different drivers,we propose to improve the risk early warning decision algo-rithm according to different driving habits. We combine theprobabilistic statistical model based on driving habits withthe TLC algorithm to improve the traditional TLC algorithm.The improved TLC algorithm is composed of the earliestwarning line and the latest warning line. When the vehiclecrosses the latest warning line, the warning sounds directly.Andwhen it crosses the earliest line, the improved TLC algo-rithm is employed to estimate the probability of deviationaccording to recent driving habits’ statistical data of lateraldisplacement. If the probability exceeds a certain threshold,the warning is started. The improved TLC algorithm is asfollows:

Step 1: The precise distance from the lane and the lateralvelocity through Kalman filter.

(L p, Vy) = Kalman(L p) (14)

where L p is the distance between the vehicle and the lane inthe current frame.

Step 2: The predicted time when the vehicle beyond thelane.

Tp = L p/Vy (15)

where Le is the distance between the earliest warning lineand the lane.

Step 3: If La <= Lla then warning, where Lla is thedistance between the latest warning line and the lane. Else ifL p <= Le and Tp < TL where TL is the driver’s reactiontime, then Step 4.

Step 4: The times of moving to the predicted direction areinitialized, Times = 0.

Step 5:Update the times of moving to the predicted direc-tion, for each Hi , if Hi = Dire, Times = Times + 1, whereHi is the direction of the lateral displacement in i-th frame,and the value of it is assumed as 1 to the right, −1 to the left,Dire is the direction of prediction.

Step 6: Calculate the probability of driving habits as fol-lows:

Pw = Times/Hsize (16)

where Hsize is the size of the array H , which stores thedirection of the lateral displacements in recent Hsize frames.

Step 7: If Pw > WP, then Warning, else No warning,where WP is the threshold to determine whether the vehiclewill deviate from the lane.

3 Experiment

In order to test the effectiveness of the LDW algorithm,we have run the program on PC (AMD A8-4500M, 1.7GHz, 4 GB RAM). The ten videos are collected from theInternet, which are recorded by the tachographs on vehi-cles. The average length of the videos is 10 minutes. Thesevideos show the record of vehicles on the highway andinclude a lot of lane-change frames. The experimental resultsof Kalman&Improved TLC, ImprovedTLC, Kalman&TLC,and TLC are as shown in Figs. 3, 4 and 5.

In Fig. 3, the horizontal coordinate represents the time andthe vertical coordinate is the distance between the vehicle andthe lane. The green line stands for the image processed byKalman filter, and the red one represents the image using theTLC algorithm only. Compared with the actual movement

Fig. 3 The results of comparison with TLC and Kalman filter

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Fig. 4 The results of comparison with TLC and improved TLC algorithm

Fig. 5 The results of Kalman&TLC and Kalman&ImprovedTLC

Table 1 The warning results of different algorithm combinations

No. Times (s) Distance (m) Kalman&TLC Kalman&ImprovedTLC TLC ImprovedTLC

1 5.13 0.19 Y (false warning) N – –

2 5.20 0.12 Y (false warning) N – –

3 5.26 0.07 Y (false warning) Y (false warning) – –

4 23.80 0.15 Y Y – –

5 23.89 0.04 Y Y – –

6 3.93 0.18 – – Y (false warning) N

7 4.60 0.13 – – Y (false warning) N

8 5.13 0.19 – – Y (false warning) N

9 23.73 0.14 – – Y N (false warning)

10 24.86 0.03 – – Y N (false warning)

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Table 2 The algorithms processing time for per frame

Running time (s)

Fastest Slowest Average

Kalman&TLC 0.019 0.030 0.027

Kalman&ImprovedTLC 0.017 0.031 0.029

TLC 0.019 0.026 0.025

ImprovedTLC 0.018 0.029 0.026

of the vehicle, the vibration amplitude of the green line isgreater, which is more close to the actual trajectory. And it isindicated thatKalmanfilter can effectively predict themotionstate of the vehicle.

Figure 4 is a comparison of the experimental results usingthe TLC algorithm and the improved TLC algorithm, andFig. 5 is a comparison of the experimental results of predic-tion with Kalman filtering. In Figs. 4 and 5, 0.03 stands forthe latest warning line, and 0.2 is the earliest warning line.The red hollow circle represents the warning point with TLConly, and the green solid point represents the warning pointwith the probability statistics of driving habits. The deflec-tion probability is at least 0.6. In Figs. 4 and 5, the improvedTLC algorithm can eliminate a large number of false warn-ings, reduce the false-positive rate of the TLC algorithm,have a better accuracy rate and can improve the accuracyof LDW. Through the comparison of the above figures, it isconcluded that the data without Kalman filter are dependenton the experimental results, which are not reliable. And itsvelocity change is not obvious enough to provide effectivesupport for the probability analysis. And the data predictedby Kalman filter is closer to the actual data, which make theearly warning result more accurate. The comparison of warn-ing results is shown in Table 1. The algorithms processingtime per frame are shown in Table 2.

In Table 1,we can see thewarning results of different algo-rithm combinations. The warning accuracy of our improvedTLC algorithm is higher than that of the traditional TLCalgorithm. The early warning algorithm using Kalman filterfor prediction has higher accuracy than that without Kalmanfilter. In general, the algorithm combining Kalman filteringwith improved TLC has better accuracy.

We canfind thatKalman&ImprovedTLChas fewestwarn-ing errors, which is shown in Table 1. But there is little dif-ference in the running time betweenKalman&ImprovedTLCand other algorithms, which is shown in Table 2. By com-paring the warning results of different combinations ofthe above algorithms, we can find that the superiority ofalgorithms from large to small is Kalman&ImprovedTLC,ImprovedTLC, Kalman&TLC and TLC.

4 Conclusion

Although the existing lane departure warning algorithm canbe realized basically, it still has some problems in accuracy,including omission and false alarm. For the problem of warn-ing accuracy of existing lane departure warning algorithms,this paper proposes a lane departure warning algorithm basedon the probabilistic statistical model of driving habits, takingfull account of the different warning situations caused by dif-ferent driving habits. We used Kalman filter to make a moreaccurate prediction of the location of a moving vehicle at thenext moment. And probability statistics is used to estimatethe probability that the vehicle really deviates from the laneto avoid the false warning. The algorithm has high predictionaccuracy for the following reasons: The next location of thevehicle is predicted by Kalman filter with the state transferequation and the prediction equation which are establishedby a large number of the measured data to revise the dataconstantly. And the earliest warning line, the latest warningline and probability statistics are employed on the basis ofthe traditional TLC to improve the accuracy of the warning.Although our method has some improvement over the tradi-tional method, it can be further improved on the probabilisticstatistical model based on driving habits.

Funding This study was funded by the National Key Research andDevelopmentProgramofChina (2017YFB0102500, 2017YFB0102600),Natural Science Foundation of Jilin Province (20170101133JC), theKoreaFoundation forAdvancedStudies’ International ScholarExchangeFellowship for the academic year of 2017–2018 and Jilin University(5157050847, 2017XYB252).

Compliance with ethical standards

Conflict of interest Jindong Zhang declares that he has no conflict ofinterest. Jiaxin Si declares that he has no conflict of interest. XuelongYin declares that he has no conflict of interest. Zhenhai Gao declaresthat he has no conflict of interest. Young Shik Moon declares that hehas no conflict of interest. Jinfeng Gong declares that he has no conflictof interest. Fengmin Tang declares that he has no conflict of interest.

Ethical approval This article does not contain any studies with humanparticipants or animals performed by any of the authors.

References

Borkar A, Hayes M, Smith MT (2011) A new multi-camera approachfor lane departure warning. In: 13th international conference onadvanced concepts for intelligent vision, Aug 22–25, vol 6915, pp58–69

Chien TY, Chung SL (2014) Android-based driving assistant for lanedetection and departure warning. In: 33rd Chinese control anddecision conference, Nanjing, Peoples Republic of China, pp 174–179

Dahmani H, Chadli M, Rabhi A (2011) Vehicle dynamics and roadcurvature estimation for lane departure warning system using

123

J. Zhang et al.

robust fuzzy observers: experimental validation. Veh Syst Dyn53(8):1135–1149

Dahmani H, Chadli M, Rabhi A (2013) Road curvature estimation forvehicle lane departure detection using a robust Takagi–Sugenofuzzy observer. Veh Syst Dyn 51(5):581–599

Gianni C, Alessandro C, Giuseppe F (2010) Data fusion algorithms forlane departure warning systems. In: Proceedings of the Americancontrol conference, Baltimore, pp 5344–5349

Huo CL, Yu Y, Syu J (2011) Vehicle warning system for land depar-ture and collision avoidance: using fuzzy decision making. In:IEEE international conference on fuzzy systems, Taipei, Taiwan,pp 1554–1559

Jiang R, Klette R, Vaudrey T (2011) Lane detection and tracking using anew lanemodel and distance transform.MachVisAppl 22(4):721–737

Kobayashi K, Cheok KC, Watanabe K (1995) Estimation of absolutevehicle speed using fuzzy logic rule-based Kalman filter. Proc AmControl Conf 5:3086–3090

Kyun Jeong H, Jeong Y (2014) FPGA implementation of AVM-based lane departure warning system. J Korean Inst Inf Technol12(11):59–68

Lei J, Yang J, Zhao J et al (2016) Backstepping sliding mode lane keep-ing control of lateral position error with dynamic of tire steeringdevice. Optik Int J Light Electron Opt 127(5):2439–2443

Madrid N, Hurtik P (2016) Lane departure warning for mobile devicesbased on a fuzzy representation of images. In: 12th internationalconference on fuzzy set theory and applications, Liptovsky Jan,Slovakia, pp 144–159

Moon S, Lee S-G, KimM (2014) Assessment and reliability validationof lane departure assistance system based on DGPS-GIS usingcamera vision. Trans KSAE 22(6):49–58

Ozcan B, Boyraz P, Yigit CB (2014) A monoSLAM approach to lanedeparture warning system. In: IEEE ASME international confer-ence on advanced intelligent mechatronics, Besacon, France, pp640–645

Pongtep A, Ryuta T, Toshihiro W (2011) On the use of stochasticdriver behaviormodel in lane departure warning. IEEETrans IntellTransp Syst 12(1):174–183

Salari E, Ouyang D (2013) Camera-based forward collision and lanedeparturewarning systems usingSVM. In: 56th IEEE internationalMidwest symposium on circuits and systems conference proceed-ings, Ohio Union, Columbus, pp 1278–1281

Sharma R, Taubel G, Yang J-S (2014) An optical flow and Hough trans-form based approach to a lane departure warning system. In: 11thIEEE international conference on control and automation ICCA,Taichung, Taiwan, pp 688–693

Tapia Espinoza R, Torres Torriti M (2013) Robust lane sensingand departure warning under shadows and occlusions. Sensors13(3):3270–3298

VijayG, Shashikant L (2015)Lane departure identification for advanceddriver assistance. IEEE Trans Intell Transp Syst 16(2):910–918

Wang J, Lin C, Chen S (2010) Applying fuzzy method to vision-basedlane detection and departure warning system. Expert Syst Appl37(1):113–126

Zhang J, JiaX, Li J (2015) Integration of scanning and image processingalgorithms for lane detection based on fuzzymethod. J Intell FuzzySyst 2015(29):2779–2786

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