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Iterative Closest Point

Ronen Gvili

The ProblemAlign two partially-overlapping meshesgiven initial guessfor relative transform

Data TypesPoint setsLine segment sets (polylines)Implicit curves : f(x,y,z) = 0Parametric curves : (x(u),y(u),z(u))Triangle sets (meshes)Implicit surfaces : s(x,y,z) = 0Parametric surfaces (x(u,v),y(u,v),z(u,v)))

MotivationShape inspectionMotion estimationAppearance analysisTexture MappingTracking

MotivationRange images

registration

MotivationRange images registration

Range Scanners

Aligning 3D Data

Corresponding Point Set AlignmentLet M be a model point set. Let S be a scene point set.

We assume :NM = NS.Each point Si correspond to Mi .

Corresponding Point Set AlignmentThe MSE objective function :

The alignment is :

Aligning 3D DataIf correct correspondences are known, can find correct relative rotation/translation

Aligning 3D DataHow to find correspondences: User input? Feature detection? Signatures?Alternative: assume closest points correspond

Aligning 3D DataHow to find correspondences: User input? Feature detection? Signatures?Alternative: assume closest points correspond

Aligning 3D DataConverges if starting position close enough

Closest PointGiven 2 points r1 and r2 , the Euclidean distance is:

Given a point r1 and set of points A , the Euclidean distance is:

Finding MatchesThe scene shape S is aligned to be in the best alignment with the model shape M.The distance of each point s of the scene from the model is :

Finding Matches

C the closest point operatorY the set of closest points to S

Finding Matches Finding each match is performed in O(NM) worst case.Given Y we can calculate alignment

S is updated to be :

The Algorithm

Init the error to Calculate correspondenceCalculate alignmentApply alignmentUpdate errorIf error > thresholdY = CP(M,S),e(rot,trans,d)S`= rot(S)+transd` = d

The Algorithmfunction ICP(Scene,Model)beginE` + ;(Rot,Trans) In Initialize-Alignment(Scene,Model);repeat E E`; Aligned-Scene Apply-Alignment(Scene,Rot,Trans); Pairs Return-Closest-Pairs(Aligned-Scene,Model); (Rot,Trans,E`) Update-Alignment(Scene,Model,Pairs,Rot,Trans);Until |E`- E| < Thresholdreturn (Rot,Trans);end

Convergence TheoremThe ICP algorithm always converges monotonically to a local minimum with respect to the MSE distance objective function.

Convergence TheoremCorrespondence error :

Alignment error:

Convergence Theorem

Calculate correspondenceCalculate alignmentEk+1Dk+1

Convergence TheoremProof :

Convergence TheoremProof :

If not - the identity transform would yield a smaller MSE than the least square alignment. Apply the alignmentqk on S0 Sk+1 .Assuming the correspondences are maintained : the MSE is still dk.

Convergence TheoremProof :

After the last alignment, the closest point operator is applied :It is clear that:

Thus :

Time analysisEach iteration includes 3 main steps A. Finding the closest points : O(NM) per each point O(NM*NS) total.B. Calculating the alignment: O(NS)C. Updating the scene: O(NS)

Optimizing the AlgorithmThe best match/nearest neighbor problem : Given N records each described by K real values (attributes) , and a dissimilarity measure D , find the m records closest to a query record.

Optimizing the AlgorithmK-D Tree :

Construction time: O(knlogn) Space: O(n) Region Query : O(n1-1/k+k )

Optimizing the AlgorithmThe Bounds Overlap Ball query :Function BOP Local d,sumbeginFor d = 1 to Kif Xq[d] < B-[d]sum += dist(d,Xq[d],B-[d])else if Xq[d] > B+[d]sum += dist(d,Xq[d],B+[d])if sum > PQ[0] return falsereturn true;

- Optimizing the AlgorithmFunction SEARCH(node)Local p,d,tempBeginif node == leaf if ball within bounds update PQend d Disc[node] , p Part[node]If Xq[d]
Optimizing the AlgorithmOptimizing the K-D Tree :

Motivation: In each internal node we can exclude the sub K-D tree if the distance to the partition is greater than the ball radius .Adjusting the discriminating number, the partition value , and the number of records in each bucket.

Optimizing the Algorithm

Optimizing the Algorithm

Optimizing the Algorithm

Optimizing the AlgorithmOptimizing the K-D Tree :

We choose in each internal node the key with the largest spread values as the discriminator and the median as the partition value.

Optimizing the AlgorithmThe Optimized K-D Tree :

Construction time : Tn = 2Tn/2+kN = O(KNlogN) Search time: O(logN) Expected.

Optimizing the AlgorithmThe Optimized K-D Tree :

The algorithm can use the m-closest points to cache potentially closest points.

Optimizing the AlgorithmAs the ICP algorithm proceeds a sequence of vectors is generated :

q1, q2, q3, q4

Optimizing the AlgorithmLet be a small angular tolerance.Suppose :

Instead of 50 iterations in the ICP , this accelerated variant converges in less than 20 iterations.

Time analysisEach iteration includes 3 main steps A. Finding the closest points : O(NM) per each point O(NMlogNS) total.B. Calculating the alignment: O(NS)C. Updating the scene: O(NS)

ICP Variants

Variants on the following stages of ICPhave been proposed:

Selecting sample points (from one or both meshes)Matching to points in the other meshWeighting the correspondencesRejecting certain (outlier) point pairsAssigning an error metric to the current transformMinimizing the error metric w.r.t. transformation

Performance of VariantsCan analyze various aspects of performance:SpeedStabilityTolerance of noise and/or outliersMaximum initial misalignment

ICP VariantsSelecting sample points (from one or both meshes).Matching to points in the other mesh.Weighting the correspondences.Rejecting certain (outlier) point pairs.Assigning an error metric to the current transform.Minimizing the error metric w.r.t. transformation.

Selection of pointsUse all available points [Besl 92].Uniform subsampling [Turk 94].Random sampling in each iteration

[Masuda 96].Ensure that samples have normals distributed as uniformly as possible [Rusinkiewicz 01].

Selection of pointsUniform SamplingNormal-Space Sampling

ICP VariantsSelecting sample points (from one or both meshes).Matching to points in the other mesh.Weighting the correspondences.Rejecting certain (outlier) point pairs.Assigning an error metric to the current transform.Minimizing the error metric w.r.t. transformation.

Points matchingClosest point in the other mesh [Besl 92].Normal shooting [Chen 91].Reverse calibration [Blais 95].Restricting matches to compatible points (color, intensity , normals , curvature ..) [Pulli 99].

Points matchingClosest point :

Points matchingNormal Shooting

Points matchingProjection (reverse calibration)Project the sample point onto the destination mesh , from the point of view of the destination meshs camera.

Points matching

ICP VariantsSelecting sample points (from one or both meshes).Matching to points in the other mesh.Weighting the correspondences.Rejecting certain (outlier) point pairs.Assigning an error metric to the current transform.Minimizing the error metric w.r.t. transformation.

Weighting of pairsConstant weight.Assigning lower weights to pairs with greater point-to-point distance :

Weighting based on compatibility of normals : Scanner uncertainty

Weighting of pairs

The rectangles and circles indicate the scanner reflectance value.

Rejecting PairsCorresponding points with point to point distance higher than a given threshold.Rejection of worst n% pairs based on some metric. Pairs containing points on end vertices.Rejection of pairs whose point to point distance is higher than n*.Rejection of pairs that are not consistent with their neighboring pairs [Dorai 98] :

(p1,q1) , (p2,q2) are inconsistent iff

Rejecting PairsDistance thresholding

Rejecting PairsPoints on end vertices

Rejecting PairsInconsistent Pairs

Error metric and minimizationSum of squared distances between corresponding points .

There exist closed form solutions for rigid body transformation :SVDQuaternionsOrthonoraml matricesDual quaternions.

Error metric and minimizationSum of squared distances from each sample point to the plane containing the destination point (Point to Plane) [Chen 91].

No closed form solution available.

Error metric and minimizationUsing point-to-plane distance instead of point-to-point lets flat regions slide along each other [Chen & Medioni 91]

Error metric and minimization

Closest Point

Error metric and minimization

Point to plane

Error metric and minimizationSearch for alignment :Repeatedly generating set of corresponding points using the current transformations and finding new transformations that minimizes the error metric [Chen 91].The above method combined with extrapolation in transform space [Besl 92].

Real Time ICP

Robust Simultaneous Alignment of Multiple Range Images

Motivation

Motivation

Motivation Graph of the twenty-seven registered scans of the Cathedral data set. The nodes correspond to the individual range scans. The edges show pair wise alignments. The directed edges show the paths from each scan to the pivot scan that is used as an anchor.

Motivation

Motivation

Registering multiple ImagesSequential :

Less memory is needed.Cheap computation cost.Each alignment step is not affected by number of images. Less accurate.

Registering multiple ImagesSequential :

as we progress in the alignment the accumulated error is noticeable.

Registering multiple ImagesSimultaneous

Diffusively distribute the alignment errorover all overlaps of each range images. Large Computational cost.

Registering multiple ImagesSimultaneous

The total alignment error is diffusively distributed among all pairs.

The AlgorithmArray KDTrees, Scenes, PointMates, Transforms

foreach r in AllRangeImages foreach s in AllRangeImage-rScenes[s] = s foreach i in Pointsof(r)foreach s in Scenes PointMates[i] += CorrespondenceSearch(i,KDTree[s]) Transforms[r] = TransformationStep(PointMates)

TransformAll(AllRangeImages, Transforms)

Speeding UpDuring the first iterations it is more important to bring the sets of points closer to each other than to accurately calculate the transform .Random sub sampling of the points.

Outliers RejectionOutlier thresholding:

standard deviation of the error.Eliminate matches with error > |k|Some valid points might be classified as outliers, and some outliers might be classified as valid points.

Outliers RejectionMedian/Rank estimation:

Calculate the median, which is (almost guaranteed) valid point and use its error as estimation, i.e. Least Median of Squares. Requires exhaustive search.

Outliers RejectionM-Estimators

Instead of minimizing the sum of square residuals , the square residuals are replaced by (ri).Each point is assigned with a likelihood probability (weight) and after each iteration the probability is updated with respect to the residual.

The End

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