Perception for Autonomous Systems 31392:
Lecturer: —PhD
10 Feb. 2020 DTU Electrical Engineering 2
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Fitting Data to a Model (Handling Outliers)
• Let’s work with the line example
– Fitting a model without (or with minimum) outliers data points – Fitting data through voting ( )
– RANdom SAmple Consensus (RANSAC)
• What about data not in a line?
10 Feb. 2020 DTU Electrical Engineering
Least Squares
• Let the line depicted here be described by: ax+by+c =0
• Then the distance of a point xi,yj is defined as:
|axi + byi + c|
• Therefore, we can find the line that best matches our data
by minimizing the following function:
xi,yi xi,yi
ax+by+c=0 xi,yi
E=n (ax+by+c)2
• However, in the presence of a lot of outlier data this problem is not directly solvable (in a closed form solution)
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• Assuming we want to fit a line in our data we can use the Hough transform as follows:
– Formulate the problem as a bounded one – Create a grid of parameter values
– Each data point votes on the grid
– Find local-maxima in the grid and track back to lines in image
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• Lets consider the line equation: Y=aX+b
• The problem with the above equation is that a,b are unbounded, Therefore we consider the following formulation (polar transformation):
xcos − ysin = d d : perpendicular distance from
line to origin
θ : angle the perpendicular makes with the x-axis
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• Next step:
– Initialize the grid using (d,θ)
H: accumulator array (votes)
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• Populate the grid by passing through the whole image and adding votes.
• See the following video:
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• Identify maxima and track lines back to image:
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RANdom SAmple Consensus (RANSAC)
Algorithm:
1. Sample (randomly) the number of points required to fit the model
2. Solveformodelparametersusingsamples
3. Score by the fraction of inliers within a preset threshold of the model
• Repeat 1-3 until the best model is found with high confidence
• δ- is the threshold upon whitch a sample is considered to not fit to the selected model
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RANdom SAmple Consensus (RANSAC)
• Select the models with most inliers to create lines
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What about data not in a line?
• Same approach is followed for more complex models eg: circle model is:
which requires 3 parameters in the Hough grid
• However the complexity grows exponentially (usually up to 4 parameters is advised)
• Ransac can handle higher order models.
10 Feb. 2020 DTU Electrical Engineering 12
Perception for Autonomous Systems 31392:
Lecturer: —PhD
10 Feb. 2020 DTU Electrical Engineering 13
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