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13D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
The image depicts how our robot Irma3D sees itself in a mirror. The laser looking into itself
creates distortions as well as changes in intensity that give the robot a single eye,
complete with iris and pupil. Thus, the image is called
"Self Portrait with Duckling".
3D Point CloudProcessing
Prof. Dr. Andreas Nüchter
Planes
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23D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
SIFT Example (1)
• Typical 30000 features in animage of 3.6 Megapixel
• Example:
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33D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
SIFT Example (2)
• Typical 30000 features in animage of 3.6 Megapixel
• Example:
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43D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Remember: Feature-Based Registration
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53D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Minimization using Least Squares
• This minimization does not tolerate “outliers”
0 2 4 6 8 10 12-2
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Outlier
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Correct line
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63D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Robust Estimator for Data with Outlier
The M-estimator and Least-Median-of-Squares (LMedS) estimator cannot cope with 50% outliers
• Solution: The RANSAC (RANdom SAmple Consensus) Algorithm
Developed by Fischler and Bolles
One of the most importtant techniques in computer viso
Can cope with 50% or more outlier
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73D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
The RANSAC Algorithm – In General
• Generate M (a predefined number) of Model hypotheses, from which all a calculated by a minimal set of points
• Evaluate all hypotheses
• Calculate the remaining error using all data.
• Points with errors smaller than a threshold are classified as “Inlier”
• The hypothesis with a maximal number of “Inlier” is chosen. Aterwards the hypotheses is re-estimated, using only the inlier.
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83D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
The RANSAC Algorithm – Formal
k := 0
Repeat until P(better solutions exists) < η
(cost function C and step counter k)
k := k + 1
I. Hypothesis estimation
(1) Select random set
(2) Calculate parameter
II. Verification
(3) calculate costs
(4) if C* < Ck then C*:=Ck, p := pk
end
mSUS kk ||,)( kk Sfp
Ux kk xpc ),(
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93D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Example: Line Detection with RANSAC (1)
• Given the following set of points
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103D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Example: Line Detection with RANSAC (2)
• Select points @ random
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113D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Example: Line Detection with RANSAC (3)
• Estimate a line based of the chosen points
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123D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Example: Line Detection with RANSAC (4)
• Calculate the error that is made in this estimation
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133D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Example: Line Detection with RANSAC (5)
• Apply a threshold; estimate a new line based on the red and green points
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143D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Example: Line Detection with RANSAC (7)
• Repeat everything for different random points
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153D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
RANSAC for Plane Detection
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163D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
RANSAC for Plane Detection
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173D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
RANSAC for Plane Detection
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183D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Detecting Shapes with RANSAC
• Improve selection of sample points– Choose points with higher likelihood if in close proximity– Lower number of draws required
• Speed up validation of hypothesis
R. Schnabel, R. Wahl, and R. Klein. Efficient RANSAC for Point-Cloud Shape Detection. Computer Graphics Forum, 2007.
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193D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Detection of Lines (1)
• Example: Accumulator with point pairs
Representation using the Hessian normal form:
We need a discrete accumulator H[ρ][l] : Both ρ and l are represented with finite
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203D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Detection of Lines (2)
Hough Transformation of a line(based on edge point pairs)
• Input: edge points (x1, y1), (x2, y2), . . . , (xn, yn)
• Output: lines that go through these edge points
• One can expect that there will be peaks in the accumulator array H[ρ][l]. All peaks correspond to a line in the image.
• Huge number of edge point pairs: O(n^2)
Set all elements of the accumulator H[ρ][l] to zero;for (all pairs of edge points in k(r, c)) {
calculate the corresponding ρ and l;H[ρ][l] ++; / Consider discretization of ρ and l /∗ ∗
}Detect peaks in H[ρ][l];
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213D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Detection of Lines (3)
• Accumulator with single edge points:Reduction of the work load: Construct the Accumulator H[ρ][l] with “hints” to possible lines based on single edge points. Unfortunately, every edge point (xi, yi) implies not one edge, but a set of edges (ρ, l):
i.e., a sinusoidal curve in the ρl space.
• Solution: All possible parameters (ρ, l), that fulfill the above constraint (I.e. they represent a line with the current edge point (xi, yi)) are considered and the corresponding counter in the array is incremented.
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223D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Detection of Lines (4)
• Input: Edge points (x1, y1), (x2, y2), . . . , (xn, yn)
• Output: Lines, that go though these edge points1. Transform all adge points (xi, yi) according to
l = xi cos ρ + yi sin ρ from the xy-space to the ρl-space.for (ρ = 0; ρ < 2π; ρ+=dρ)for (l = 0; l < lmax; l+=dl)
H[ρ][l] = 0;for (i = 1; i ≤ n; i++)for (ρ = 0; ρ < 2π; ρ+=dρ) {
l = xi cos ρ + yi sin ρ;H[ρ][l] ++; / Regard the discretisation of l /∗ ∗
/ Increment according to edge width s(xi, yi) poss. /∗ ∗
}2. Search for cluster points in ρl-space, i.e. in H[ρ][l]. 3. All cluster points (ρ0, l0) define a line
l0 = x cos ρ0+y sin ρ0 in xy-space.
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233D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Detection of Lines (5)
• Example:
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243D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Iterations (1)
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253D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Iterations (2)
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263D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Results
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273D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transformation: Example
• Detection of Lines:
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283D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
From 2D to 3D – From Lines to Planes
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293D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Polar Coordinates
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303D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transform (HT)
• Hough Space: - Space–
–
–
• Cartesian Space: (x,y,z) – Space
– -∞ < x,y,z < ∞• Hough Transform
– Cartesian Space → Hough Space
– For a point (x,y,z) HT yields all planes that go through (x,y,z)
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313D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Transform – Example
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323D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Discretization – Accumulator Array
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333D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Discretization – Related Work
[Censi, 2004] [Zaharia, 2002]
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343D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Accumulator Ball
• discretize with stepwidth
with and
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353D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparing Accumulators (1)
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363D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Variants - SHT
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373D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Hough Variants• Probablistic Hough Transform (PHT)
– Use p% of the input points only
• Progressive Probabilistic Hough Transform (PPHT)– Pick points randomly and perform HT– Quit when one cell has been voted by p% of the points
• Adaptive Probabilistic Hough Transform (APHT) – Pick points randomly and perform HT– Obtain a list of maxima– Quit when list of maxima remains stable
• Randomized Hough Transform (RHT)– Pick three points randomly– Accumulate the cell corresponding to the plane spanned by
these points– Delete points of plane when threshold is reached
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383D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparing Accumulators (2)
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393D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparing Hough Variants (1)
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403D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparing Hough Variants (2)
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413D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparison with Related Work (1)
• Randomized Hough Transform (RHT)
• Region Growing (RG) [Poppinga, 2008]
• Hierarchical Fitting Primitives (HFP)
[Attene, 2006]
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423D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparision with Related Work (2)
RHT
RG
HFP
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433D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparision with Related Work (3)
RHT
RG
HFP
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443D Point Cloud ProcessingDr. Andreas NüchterJuly 23, 2014
Robotics and Telematics
Comparision with Related Work (4)