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Find boundaries using parabolic model



boundaries = findParabolicLaneBoundaries(xyBoundaryPoints,approxBoundaryWidth) uses the random sample consensus (RANSAC) algorithm to find parabolic lane boundary models that fit a set of boundary points and an approximate width. Each model in the returned array of parabolicLaneBoundary objects contains the [A B C] coefficients of its second-degree polynomial equation and the strength of the boundary estimate.

[boundaries,boundaryPoints] = findParabolicLaneBoundaries(xyBoundaryPoints,approxBoundaryWidth) also returns a cell array of inlier boundary points for each boundary model found.

[___] = findParabolicLaneBoundaries(___,Name,Value) uses options specified by one or more Name,Value pair arguments, with any of the preceding syntaxes.


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Find lanes in an image by using parabolic lane boundary models. Overlay the identified lanes on the original image and on a bird's-eye-view transformation of the image.

Load an image of a road with lanes. The image was obtained from a camera sensor mounted on the front of a vehicle.

I = imread('road.png');

Transform the image into a bird's-eye-view image by using a preconfigured sensor object. This object models the sensor that captured the original image.

bevSensor = load('birdsEyeConfig');
birdsEyeImage = transformImage(bevSensor.birdsEyeConfig,I);

Set the approximate lane marker width in world units (meters).

approxBoundaryWidth = 0.25;

Detect lane features and display them as a black-and-white image.

birdsEyeBW = segmentLaneMarkerRidge(im2gray(birdsEyeImage), ...

Obtain the image coordinates corresponding to the lane candidate positions. The find function returns pixel indices that correspond to the candidate lane positions. By convention, the order of the image coordinates is always reversed relative to the pixel indices. For more information about image coordinates, see Coordinate Systems.

Obtain the corresponding lane boundary points in vehicle coordinates by using the imageToVehicle function.

[imageY,imageX] = find(birdsEyeBW);
xyBoundaryPoints = imageToVehicle(bevSensor.birdsEyeConfig,[imageX,imageY]);

Find lane boundaries in the image by using the findParabolicLaneBoundaries function. By default, the function returns a maximum of two lane boundaries. The boundaries are stored in an array of parabolicLaneBoundary objects.

boundaries = findParabolicLaneBoundaries(xyBoundaryPoints,approxBoundaryWidth);

Use insertLaneBoundary to overlay the lanes on the original image. The XPoints vector represents the lane points, in meters, that are within range of the ego vehicle's sensor. Specify the lanes in different colors. By default, lanes are yellow.

XPoints = 3:30;

sensor = bevSensor.birdsEyeConfig.Sensor;
lanesI = insertLaneBoundary(I,boundaries(1),sensor,XPoints);
lanesI = insertLaneBoundary(lanesI,boundaries(2),sensor,XPoints,'Color','green');

View the lanes in the bird's-eye-view image.

BEconfig = bevSensor.birdsEyeConfig;
lanesBEI = insertLaneBoundary(birdsEyeImage,boundaries(1),BEconfig,XPoints);
lanesBEI = insertLaneBoundary(lanesBEI,boundaries(2),BEconfig,XPoints,'Color','green');

Input Arguments

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Candidate boundary points, specified as an [x y] vector in vehicle coordinates. To obtain the vehicle coordinates for points in a birdsEyeView image, use the imageToVehicle function to convert the bird's-eye-view image coordinates to vehicle coordinates.

Approximate boundary width, specified as a real scalar in world units. The width is a horizontal y-axis measurement.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'MaxSamplingAttempts',200

Maximum number of lane boundaries that the function attempts to find, specified as the comma-separated pair consisting of 'MaxNumBoundaries' and a positive integer.

Function to validate the boundary model, specified as the comma-separated pair consisting of 'ValidateBoundaryFcn' and a function handle. The specified function returns logical 1 (true) if the boundary model is accepted and logical 0 (false) otherwise. Use this function to reject invalid boundaries. The function must be of the form:

isValid = validateBoundaryFcn(parameters)

parameters is a vector corresponding to the three parabolic parameters.

The default validation function always returns 1 (true).

Maximum number of attempts to find a sample of points that yields a valid parabolic boundary, specified as the comma-separated pair consisting of 'MaxSamplingAttempts' and a function handle. findParabolicLaneBoundaries uses the fitPolynomialRANSAC function to sample from the set of boundary points and fit a parabolic boundary line.

Output Arguments

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Lane boundary models, returned as an array of parabolicLaneBoundary objects. This table shows the properties of the each output boundary object.


Coefficients for a parabolic model of the form y = Ax2 + Bx + C, specified as a real-valued vector of the form [A B C].


Type of lane boundary, specified as a LaneBoundaryType enumeration. Supported lane boundary types are:

  • Unmarked

  • Solid

  • Dashed

  • BottsDots

  • DoubleSolid

Lane boundary objects always return BoundaryType as type Solid. Update these types to match the types of the lanes that are being fitted. To update a lane boundary type, use the LaneBoundaryType.BoundaryType syntax. For example, this code sample shows how to update the first output lane boundary to type BottsDots:

boundaries(1) = LaneBoundaryType.BottsDots;


Strength of the boundary model, specified as a real scalar. Strength is the ratio of the number of unique x-axis locations on the boundary to the length of the boundary specified by the XExtent property. A solid line without any breaks has a higher strength than a dotted line that has breaks along the full length of the boundary.


Length of the boundary along the x-axis, specified as a real-valued vector of the form [minX maxX] that describes the minimum and maximum x-axis locations.

Inlier boundary points, returned as a cell array of [x y] values. Each element of the cell array corresponds to the same element in the array of parabolicLaneBoundary objects.


  • To fit a single boundary model to a double lane marker, set the approxBoundaryWidth argument to be large enough to include the width spanning both lane markers.


  • This function uses fitPolynomialRANSAC to find parabolic models. Because this algorithm uses random sampling, the output can vary between runs.

  • The maxDistance parameter of fitPolynomialRANSAC is set to half the width specified in the approxBoundaryWidth argument. Points are considered inliers if they are within the boundary width. The function obtains the final boundary model using a least-squares fit on the inlier points.

Extended Capabilities

Version History

Introduced in R2017a