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On the ICP Algorithm. Esther Ezra, Micha Sharir Alon Efrat. The problem: Pattern Matching. Input: A = {a 1 , …, a m }, B = {b 1 , …, b n } A,B  R d Goal: Translate A by a vector t  R d s.t.  (A+t,B) is minimized. The (1-directional) cost function:.

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on the icp algorithm

On the ICP Algorithm

Esther Ezra, Micha Sharir

Alon Efrat

the problem pattern matching
The problem: Pattern Matching

Input:

A = {a1, …, am}, B = {b1, …, bn} A,B  Rd

Goal:

TranslateAby a vectort  Rds.t.(A+t,B)is minimized.

The (1-directional) cost function:

The cost function (measures resemblance).

The nearest neighbor of ainB.

A , B

the algorithm besl mckay 92
The algorithm [Besl & Mckay 92]

Use local improvements:

Repeat: At each iteration i, with cumulative translation ti-1

  • Assign each point a+ti-1 A+ti-1to its NN b = NB(a+ti-1)
  • Compute the relative translationti that minimizes with respect to this fixed (frozen) NN-assignment.Translate the points of A by ti(to be aligned to B). ti ti-1 +ti .
  • Stop when the value of does not decrease.

The overall translation vector previously computed. t0 = 0.

Some of the points of A may acquire new NN in B.

the algorithm convergence
The algorithm: Convergence

[Besl & Mckay 92].

  • The value of=2decreases at each iteration.
  • The algorithm monotonically converges to a local minimum.

This applies for =as well.

A local minimum of the ICP algorithm.

a1

a2

a3

b1

b2

b3

b4

The global minimum is attained when a1, a2, a3 are aligned on top of b2, b3, b4 .

nn and voronoi diagrams
NN and Voronoi diagrams

Each NN-assignment can be interpreted by the Voronoi diagram V(B) of B.

b is the NN of a

a is contained in the Voronoi cell V(b) of V(B).

the algorithm an example in r 1
The algorithm: An example in R1

(A+t,B) = RMS(t) =

Iteration 1:

Solving the equation: RMS’(t)=0

(A,B) 3

m=4, n=2.

a1 = -3-

a2 = -1

a3 = 1

a4 = 3

b1 = 0

b2 = 4

t1 1

(b1+b2)/2 = 2

an example in r 1 cont
An example in R1:Cont.

Iteration 2:

(A+t1,B) 2

a1+t1

a4+ t1

a2+ t1

a3+ t1

b1 = 0

b2 = 4

t2 = 1

(b1+b2)/2 = 2

Iteration 3:

(A+t1+t2,B) 1

a1+ t1 + t2

a2+ t1 + t2

a3+ t1 + t2

a4+ t1 + t2

b1 = 0

b2 = 4

NN-assignment does not change.t3 = 0

(b1+b2)/2 = 2

number of iterations an upper bound
Number of iterations: An upper bound

The value of is reduced at each iteration, since

  • The relative translation t minimizes  with respect to the frozen NN-assignment.
  • The value of  can only decrease with respect to the new NN-assignment (after translating by t).

No NN-assignment arises more than once!

#iterations = #NN-assignments (that the algorithm reaches)

number of iterations a quadratic upper bound in r 1
Number of iterations: A quadratic upper bound in R1

am

a1

a2

ai

Critical event:The NN of some point ai is changed.

Two “consecutive” NN-assignments differ by one pair (ai,bj).

#NN-assignments  |{(ai,bj) | ai  A, bj  B}| = nm.

b1

b2

bj

bj+1

bn

t

aicrosses into a new Voronoi cell.

the bound in r 1
The bound in R1

Is the quadratic upper bound tight???

A linear lower bound construction (n=2): (m)

m=2k+2, n=2

a2 = -2k+1

a1 = -2k-1

am-1 = 2k-1

am = 2k+1

b1 = 0

b2 = 4k

Using induction on i=1,…, k/2 .

(b1+b2)/2 = 2k

Tight! (when n=2) .

structural properties in r 1
Structural properties in R1

Lemma:

At each iteration i2 of the algorithm

Corollary (Monotonicity):

The ICP algorithm always moves A in the same (left or right) direction.Either ti 0for eachi0, orti 0for eachi0.

Use induction on i.

a super linear lower bound construction in r 1
A super-linear lower bound construction in R1.

The construction (m=n):

a1 = -n - (n-1), ai = (i-1)/n -1/2 + , bi = i-1, for i=1, …, n.

m=n

b1 = 0

b2 = 1

1 = 1/2

Theorem:

The number of iterations of the ICP algorithm under the above construction is (n log n).

slide14

Iteration 2:

b2 = 1

b3= 2

Iteration 3:

b3 = 2

b2 = 1

b4= 3

At iteration 4, the NN of a2 remains unchanged (b3):

Only n-2 points of A cross into V(b4).

the lower bound construction in r 1
The lower bound construction in R1

Round j:n-1 right pts of A assigned to bn-j+1 and bn-j+2

Consists of steps where pts of A cross the bisector

n-j+1 = (bn-j+1 + bn-j+2) / 2

Then at each such step i,

# pts of A that cross n-j+1is exactly j

(except for the last step).

At the last such step, # pts of A that cross n-j+1or n-j+2is exactly j-1.

# steps in the round n/j .

At the next round, # steps in the round n/(j-1) .

the lower bound construction in r 11
The lower bound construction in R1

The overall number of steps:

the lower bound construction in r 1 proof of the claim
The lower bound construction in R1:Proof of the claim

Induction on j:

Consider the last step i of round j :

l< j pts have not yet crossed n-j+1

Black pts still remain in V(bn-j+2).

bn-j+2

bn-j+1

bn-j+3

The points a2, …, al+1 cross n-j+1, and an-(j-l-2), …,an cross n-j+2 .

Overall number of crossing points =l + j - l -1 = j - 1.

QED

general structural properties in r d
General structural properties in Rd

Theorem:

Let A = {a1, …, am}, B = {b1, …, bn} be two point sets in Rd.

Then the overall number of NN-assignments, over all translated copies of A, is O(mdnd). Worst-case tight.

Proof:

Critical event: The NN of ai+t changes from b to b’

t lies on the common boundary of V(b–ai) and

V(b’–ai)

t

b - ai

b’ - ai

Voronoi cells of the shifted diagram V(B-ai) .

b’’ - ai

nn assignments in r d
# NN Assignments in Rd

Critical translations t for a fixedai

= Cell boundaries in V(B-ai) .

All critical translations =

Union of all Voronoi edges =

Cell boundaries in the overlayM(A,B)of V(B-a1),…, V(B-am).

# NN-assignments = # cells in M(A,B)

Each cell of M(A,B) consists of translations with a common NN-assignment

nn assignments in r d1
# NN Assignments in Rd

Claim:

The overall number of cells of M(A,B) is O(mdnd).

Proof:

  • Charge each cell ofM(A,B) to some vertex v of it(v is charged at most 2d times).
  • v arises as the vertex in the overlay Md of only d diagrams.
  • The complexity of Mdis O(nd) [Koltun & Sharir 05].
  • There are d-tuples of diagrams V(B-a1),…, V(B-ad).

QED

lower bound construction for m a b
Lower bound construction for M(A,B)

1/m

d=2

A

Vertical cells: Minkowski sums of vertical cells of V(B) and (reflected) horizontal points of A .

1

B

M(A,B) contains (nm) horizontal cells that intersect (nm) vertical cells:

(n2m2) cells.

Horizontal cells: Minkowski sums of horizontal cells of V(B) and (reflected) vertical points of A .

nn assignments in r d2
# NN Assignments in Rd

Lower bound construction can be extended to any dimension d

Open problem:

Does the ICP algorithm can step through all (many) NN-assignments in a single execution?

d=1:

Upper bound: O(n2).

Lower bound: (n log n).

Probably no.

monotonicity
Monotonicity

d=1:

The ICP algorithm always moves A in the same (left or right) direction.

Generalization to d2:

 = connected path obtained by concatenating the ICP relative translations t. ( starts at the origin)

Monotonicity: does not intersect itself.

monotonicity1
Monotonicity

Theorem:

Let t be a move of the ICP algorithm from translation t0 to t0+ t . Then RMS(t0+ t)is a strictly decreasing function of .

 does not intersect itself.

Cost function monotone decreasing

along t.

t

proof of the theorem
Proof of the theorem

The Voronoi surface, whose minimization diagram is V(B-a).

SB-a(t) = minbB||a+t-b||2 = minbB (||t||2 + 2t·(a-b) + ||a-b||2)

RMS(t) =

SB-a(t) - ||t||2is the lower envelope of n hyperplanes

SB-a(t) - ||t||2is the boundary of a concave polyhedron.

Q(t) = RMS(t) - ||t||2is the average of {SB-a(t) - ||t||2}aA

Q(t) is the boundary of a concave polyhedron.

The average of m Voronoi surfaces S(B-a).

proof of the theorem1
Proof of the theorem

Replace f(t) by the hyperplane h(t) containing it.

h(t) corresponds to the frozen NN-assignment.

NN-assignment at t0 defines a facet f(t) of Q(t), which contains the point (t0,Q(t0)).

t0

t

The value of h(t)+||t2|| along t is monotone decreasing

Since Q(t) is concave, and h(t) is tangent to Q(t) at t0, the value of Q(t)+||t2||along tdecreases faster than the value of h(t)+||t2|| .

QED

more structural properties of
More structural properties of 

Corollary:

The angle between any two consecutive edges of  is obtuse.

Proof:

Let tk, tk+1 be two consecutive edges of  .

Claim:

tk+1  tk  0

b’ = NB(a+tk)

a

tk

a must cross the bisector of b, b’. b’ lies further ahead

b = NB(a+tk-1)

QED

more structural properties potential
More structural properties: Potential

Lemma:

At each iteration i1 of the algorithm,

(i)

(ii)

Corollary:

Let t1, …,tk be the relative translations computed by the algorithm. Then

Due to (i)

Due to (ii)

Cauchy-Schwarz inequality

more structural properties potential1
More structural properties: Potential

Corollary:

d=1:

Trivial

d2:

Given that ||ti||   , for each i1, then

tk

Since

 does not get too close to itself.

the relative translation
The relative translation

Lemma:

Let Di-1 be the smallest enclosing ball of {a+ti-1 – NB(a+ti-1) | a  A}. Then ti moves the center of Di-1 to the origin.

Proof:

Any infinitesimal move of D increases the (frozen) cost.

The radius of D determines the (frozen) costafter the translation.

a3

a2

a3-b3

a3-b3+t

a2-b2

a2-b2+t

b2

b3

o

o

D

b1

t

b4

D

a4-b4+t

a4-b4

a4

QED

a1-b1

a1

a1-b1+t

no monotonicity d 2
No monotonicity (d  2)

Lemma:

In dimension d2 the cost function does not necessarily decrease along t .

Proof:

Final distances:

||a2+t-b|| = ||a3+t-b|| = r ,

|| a1+t-b’ || < r .

Initially, ||a1-b|| = max ||ai-b|| > r , i=1,2,3 .

The distances of a2, a3 from b start increasing

t

a2

a2+t

r

b

a3

t

c

a3+t

b

a1

a1+t

b’

Distance of a1 from itsNN always decreases.

b’

QED

the one dimensional problem
The one-dimensional problem

Lemma (Monotonicity):

The algorithm always moves A in the same (left or right) direction.

Proof:

Let |a*-b*|=maxaA |NB(a)-a| .

Suppose a* < b*.

a*-b* is the left endpoint

of D0; its center c < 0.

t1 > 0, |t1| < |a*-b*| .

a*+t1 < b*, so b* is still the

NN of a*+t1 .

|a*+t1-b*|=maxaA |a+t1-NB(a)|> |a+t1-NB(a+t1)|

a*+t1-b* is the left endpoint of D1.

The lemma follows by induction.

Initial minimum enclosing “ball”.

t

D0

c

0

a–NB(a)

a*-b*

a*, b* still determine the next relative translation.

0

a*+t1-b*

a+t1–NB(a)

QED

number of iterations an upper bound1
Number of iterations: An upper bound

Ratio between the diameter of B and the distance between its closest pair.

Theorem:

Let B be the spread of B.

Then the number of iterations that the ICP algorithm executes is O((m+n) log B / log n).

Corollary:

The number of iterations is O(m+n) when B is

polynomial in n

the upper bound proof sketch
The upper bound: Proof sketch

Use: the pair a*, b* that satisfies |a*-b*|=maxaA |NB(a)-a| always determines the left endpoint of Di.

Put Ik = |b*-(a*+tk)| .

Classify each relative translation tk

  • Short if
  • Long – otherwise.Overall: O(n log B / log n).

Easy

the upper bound proof sketch1
The upper bound: Proof sketch

Short relative translations:

a A involved in a short relative translation at the k-th iteration,

|a+tk-1 – NB(a+tk-1)|is large.

a A involved in an additional short relative translation

a has to be moved further by |a+tk-1 – NB(a+tk-1)| .

Ik = b*-(a*+tk) must decrease significantly.

b1

bj

bj+1

Ik-1

a1

a

tk

number of iterations a lower bound construction
Number of iterations: A lower bound construction

Theorem:

The number of iterations of the ICP algorithm under the following construction (m=n) is (n).

  • At the i-th iteration:
  • ti = 1/2i, fori=1,…,n-2 .
  • Only ai+1crosses into the next cell V(bi+2).
  • The overall translation length is< 1.

b2

b1

bn

bi

n-1

n

an

ai

a2

a1

many open problems
Many open problems:
  • Bounds on # iterations
  • How many local minima?
  • Other cost functions
  • More structural properties
  • Analyze / improve running time,

vs. running time of directly computing

minimizing translation

And so on…