Iterative Closest Point Ronen Gvili. The Problem Align two partially- overlapping meshes given...

transcript

Iterative Closest Point

Ronen Gvili

The Problem

Align two partially-overlapping meshesgiven initial guessfor relative transform

Data Types Point sets Line segment sets (polylines) Implicit curves : f(x,y,z) = 0 Parametric curves : (x(u),y(u),z(u)) Triangle sets (meshes) Implicit surfaces : s(x,y,z) = 0 Parametric surfaces

(x(u,v),y(u,v),z(u,v)))

Motivation Shape inspection Motion estimation Appearance analysis Texture Mapping Tracking

Motivation Range images registration

Range Scanners

Aligning 3D Data

Corresponding Point Set Alignment

Let M be a model point set. Let S be a scene point set.

We assume :1. NM = NS.

2. Each point Si correspond to Mi .

Corresponding Point Set Alignment

The MSE objective function :

The alignment is :

qsqRmN

TranssRotmN

),(),,( SMdtransrot mse

Aligning 3D Data If correct correspondences are known,

can find correct relative rotation/translation

Aligning 3D Data How to find correspondences: User

input? Feature detection? Signatures? Alternative: assume closest points

correspond

Aligning 3D Data How to find correspondences: User

input? Feature detection? Signatures? Alternative: assume closest points

correspond

Aligning 3D Data Converges if starting position “close

enough“

Closest Point Given 2 points r1 and r2 , the

Euclidean distance is:

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

2212121 )()()(),( zzyyxxrrrrd

),(min),( 1..1

ardArd

Finding Matches The 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 :

smdMsdMm

min),(

Finding Matches

ysdsmdMsdMm

),(min),(

C – the closest point operator

Y – 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 :

),(),,( YSdtransrot

transSrotSnew )(

The AlgorithmInit the error to ∞

Calculate correspondence

Calculate alignment

Apply alignment

Update error

If error > threshold

Y = CP(M,S),e

(rot,trans,d)

S`= rot(S)+trans

d` = 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 Theorem The ICP algorithm always

converges monotonically to a local minimum with respect to the MSE distance objective function.

Convergence Theorem Correspondence error :

Alignment error:

ikiokik

Sk TranssRoty

Convergence Theorem

Calculate alignment

Apply alignment

S`= rot(S)+trans

Calculate alignment

Convergence Theorem Proof :

ikiokik

TranssRotyN

TransSRotS

Convergence Theorem Proof : 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 Theorem Proof : After the last alignment, the closest

point operator is applied :It is clear that:

Thus :

),( 11 kk SMCY

kiikkiki

1,1,1,

kkkk eded 110

Time analysis

Each 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 Algorithm

The 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 Algorithm K-D Tree : Construction time: O(knlogn)

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

Optimizing 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 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 Algorithm The Optimized K-D Tree :

Construction time :

Tn = 2Tn/2+kN = O(KNlogN)

Search time: O(logN) Expected.

Optimizing the Algorithm The Optimized K-D Tree :

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

Optimizing the Algorithm As the ICP algorithm proceeds a

sequence of vectors is generated : q1, q2, q3, q4 …

Optimizing the Algorithm Let be a small angular tolerance. Suppose :

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

1121 ,,0 kkkkkk vqvqvv

Time analysis

Each 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:

1. Selecting sample points (from one or both meshes)2. Matching to points in the other mesh3. Weighting the correspondences4. Rejecting certain (outlier) point pairs5. Assigning an error metric to the current transform6. Minimizing the error metric w.r.t. transformation

Performance of Variants Can analyze various aspects of

performance: Speed Stability Tolerance of noise and/or outliers Maximum initial misalignment

ICP Variants

1. Selecting sample points (from one or both meshes).

2. Matching to points in the other mesh.3. Weighting the correspondences.4. Rejecting certain (outlier) point pairs.5. Assigning an error metric to the

current transform.6. Minimizing the error metric w.r.t.

transformation.

Selection of points Use 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 points

Uniform SamplingUniform Sampling Normal-Space SamplingNormal-Space Sampling

ICP Variants

transformation.

Points matching Closest 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 matching Closest point :

Points matching Normal Shooting

Points matching Projection (reverse calibration)

Project the sample point onto the destination mesh , from the point of view of the destination mesh’s camera.

Points matching

ICP Variants

transformation.

Weighting of pairs Constant weight. Assigning lower weights to pairs

with greater point-to-point distance :

Weighting based on compatibility of normals :

Scanner uncertainty

21 ),(1

ppDistWeight

21 nnWeight

Weighting of pairs

The rectangles and circles

indicate the scanner

reflectance value.

ICP Variants

transformation.

Rejecting Pairs Corresponding 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*σ. Rejection of pairs that are not consistent with

their neighboring pairs [Dorai 98] : (p1,q1) , (p2,q2) are inconsistent iffthresholdqqDistppDist ),(),( 2121

Rejecting Pairs

Distance thresholding

Rejecting Pairs

Points on end vertices

Rejecting Pairs

Inconsistent Pairs

ICP Variants

transformation.

Error metric and minimization Sum of squared distances between

corresponding points .There exist closed form solutions for rigid body transformation :

1. SVD2. Quaternions3. Orthonoraml matrices4. Dual quaternions.

Error metric and minimization Sum 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 minimization Using 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

Point to plane

Search 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

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

Registering multiple Images Sequential :1. Less memory is needed.2. Cheap computation cost.3. Each alignment step is not

affected by number of images.4. Less accurate.

Registering multiple Images Sequential :

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

Registering multiple Images Simultaneous1. Diffusively distribute the alignment

errorover all overlaps of each range images.

2. Large Computational cost.

Registering multiple Images Simultaneous

The total alignment error is diffusively distributed among all pairs.

The AlgorithmArray KDTrees, Scenes, PointMates, Transforms

foreach r in AllRangeImages foreach s in AllRangeImage-r

Scenes[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 Up During 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 Rejection Outlier thresholding:

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

Outliers Rejection Median/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 Rejection M-Estimators

Instead of minimizing the sum of square residuals , the square residuals are replaced by ρ(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

Iterative Closest Point Ronen Gvili. The Problem Align two partially- overlapping meshes given...

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