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A Novel High Breakdown M-estimator for Visual Data Segmentation
Reza Hoseinnezhad
Swinburne University of Technology
Victoria, Australia
Alireza Bab-Hadiashar
Swinburne University of Technology
Victoria, Australia
[email protected]
[email protected]
Abstract
Most robust estimators, designed to solve computer vision problems, use random sampling to optimize their
objective functions. Since the random sampling process
is patently blind and computationally cumbersome, other
searches of parameter space using techniques such as
Nelder Meade Simplex or gradient search techniques have
been also proposed (particularly in combination with PbMestimators). In this paper, we introduce a novel high breakdown M-estimator having a differentiable objective function
for which a closed form updating formula is mathematically
derived (similar to redescending M-estimators) and used to
search the parameter space. The resulting M-estimator has
a high breakdown point and is called High Breakdown Mestimator (HBM). We show that this objective function can
be optimized using an iterative reweighted least squares regression similar to redescending M-estimators. The closed
mathematical form of HBM and its guaranteed stability
combined with its high breakdown point and fast convergence speed make this estimator an outstanding choice for
segmentation of multi-structural data. A number of experiments, using both synthetic and real data have been conducted to show and benchmark the performance of the proposed estimator both in terms of accurate segmentation of
numerous structures in the data and also the convergence
speed. Moreover, the computational time of HBM, ASSC,
MSSE and PbM are compared using the same computing
platform and the results show that HBM significantly outperforms aforementioned techniques.
1. Introduction
Since the introduction of RANSAC [3], a quarter of century ago, several high breakdown robust estimators have
been specially designed to solve computer vision problems
(e.g. RESC[14], ALKS[6], MSSE[1], ASSC[13] and Projection based M-estimators [2, 8, 9, 7] also called PbM).
All such estimators include three main steps:
• Optimization:
Searching the parameter space to
find the parameter values which optimize the objective
function of the estimator.
• Segmentation:
Extracting an inlier-outlier dichotomy using the parameters given by the searching
process.
• Refinement: Updating the parameter estimates with
a least-squares fit to the extracted inliers.
The robust estimators reported in computer vision so far,
mainly differ in their objective functions and the way they
extract an inlier-outlier dichotomy.
For the objective function optimization, almost all robust estimators (except PbM) use random sampling. The
main reason is that the objective functions used in those
high breakdown robust estimators are non-differentiable
and optimization methods based on gradient and iterative
reweighted least-squares regressions (as in redescending Mestimators1 ), cannot be employed.
Random sampling is a random search scheme in the sample space for the best elemental subset (p-tuple) that gives
rise to the parameter values which optimize the objective
function. An elemental subset is a subset of p data samples (p is the dimension of parameter space) that defines a
full rank system of equations from which a model candidate
can be computed. If N elemental subsets are randomly selected, then with a probability of:
N
Psuccess = 1 − [1 − p ]
(1)
at least one of them is a good elemental subset (i.e. all its
samples belong to the inlier structure), where is the ratio
of inliers samples. Thus, for a given success probability
Psuccess , at least:
N=
log(1 − Psuccess )
log(1 − p )
(2)
1 It is important to note that redescending M-estimators do not have high
breakdown points and cannot be efficiently employed to solve visual data
segmentation problems particularly with several data structures.
elemental subsets should be randomly examined.
2. High Breakdown M-estimator
Two important observations are highlighted here: Firstly,
the value of N given by equation (2) is a lower bound as it
implies that any elemental subset which contains only inliers provides a suitable model candidate. This assumption
is not always true, specially if the measurement noise is significant [7]. Secondly, for cases involving multi-structural
data, the above minimum number of random p-tuples can
be substantial and the computational load of segmentation
would be too high for real-time (or near real-time) applications. It is important to note that the inlier ratio is not priorly
known and in equation (2), should be taken as the smallest
possible ratio of inliers in the application.
Consider a vision problem that involves segmentation of
several data structures. From each structure, ni measurement samples denoted by {yi ; i = 1, . . . , ni } are available
and each sample yi ∈ Rp is corrupted with independent and
identically distributed (i.i.d.) noise:
The number of required elemental subsets can be significantly reduced when information regarding the reliability of the data points is available (either provided by user
or derived from the data through an auxiliary estimation
scheme). Guided sampling techniques, choose the elemental subsets by directing the samples toward the points having higher probabilities of being inliers [11, 12]. However,
in most visual data segmentation problems, sufficiently reliable information to guide the sampling is not available [7].
An alternative approach to random sampling proposed as
an optimization strategy for PbM is to use techniques like
Nelder-Mead Simplex search [7, 2]. Simplex is a heuristic search technique and it is highly sensitive to its initialized search point in parameter space. Therefore, substantial
number of initializations are commonly required to guarantee that the global minimum (or maximum) of the objective
function would be found by the Simplex search. Subbarao
and Meer [9, 10] have proposed using a local search (based
on the first order conjugate gradient method of a Grassman manifold of the parameter vector θ ∈ Rp satisfying
θ > θ = 1) in the neighborhood of each elemental subset.
However, since the objective function of PbM estimator is
not differentiable, the dependence of the α parameter (in the
common errors-in-variables regression model as explained
later in this paper) on the parameter vector θ has to be ignored. Therefore, the procedure of local optimization needs
to be repeated for several elemental subsets.
In this paper, we introduce a new high breakdown estimator with a differentiable objective function that can be
optimized through an iterative reweighted least square regression scheme. Since the redescending M-estimators employ similar continuous updating formulas for their search
scheme we call the new technique: High Breakdown Mestimator or HBM estimator for short. Our studies show
that the proposed technique can segment structures with
population ratios of less than 20% significantly faster than
other modern high breakdown techniques.
yi = yio + δyi ; δyi ∼ GI(0, σ 2 Ip )
(3)
where yio is the true value of yi , GI(.) stands for a general
symmetric distribution of independent measurement noise
samples and σ is the unknown scale of noise. Usually,
noise distribution is assumed to be normal however the measurement noise does not necessarily have to be normally
distributed. Indeed, characterizing the distribution by its
first two central moments in equation (3) implies normality
assumption as only a normal distribution can be uniquely
characterized this way.
Each data structure can be modeled by the following linear errors-in-variables (EIV) regression model:
>
yio
θ − α = 0 ; i = 1, . . . , ni
(4)
where θ ∈ Rp and α are the model parameters yet to be estimated for each structure and the following constraints are
imposed to eliminate the ambiguity of the model parameters
being defined up to a multiplicative constant:
||θ|| = 1 ; α ≥ 0.
(5)
Since the proposed HBM estimator does not calculate the
θ̂ and α̂ estimates separately, we augment those parameters
and rewrite the model as below:
x>
io Θ = 0 ; i = 1, . . . , ni
(6)
> >
where xio = [1 yio
] and Θ = [α θ > ]> . Thus, the measurements are denoted by xi = [1 yi> ]> and we slightly
modify the constraints (5) as shown below:
Θ(1) ≥ 0 ; ||Θ|| = 1.
(7)
For a given parameter estimate Θ̂, each data sample xi
corresponds to an algebraic distance ri = x>
i Θ̂. With traditional regression models, these distances are called residuals and we also use this popular term in this paper. In the
least k-th order statistics (LkOS) estimator, the objective
function is the k-th order statistics of the squared residuals:
2
JLkOS (Θ̂) = rk:n
(8)
where n is the total number of available data samples. The
order k is given by k = dne where is the minimum possible ratio of inliers in the application. The breakdown point
of LkOS estimator can be higher than 50%. More precisely,
provided there are moderate number of samples in the target
structure, the breakdown point is (1 − ) × 100%. The objective function (8) is usually optimized using random sampling.
In the proposed HBM estimator, the functional form of
the k-th order statistics of the squared residuals is chosen
as objective function. For a given parameter estimate Θ̂, the
squared residuals {zi = ri2 ; i = 1, . . . , n} have a statistical
distribution that can be estimated by the following kernel
density estimator:
n
fΘ̂ (z) =
1 X
K
nh i=1
z − zi
h
(9)
K(u)du = 1
(10)
K(u) = K(−u) ≥ 0
K(u1 ) ≥ K(u2 ) for |u1 | ≤ |u2 |
(11)
(12)
and h is the kernel bandwidth. The value of the bandwidth
has a weak influence on the result of the M-estimation [7]
and we use the following formula to calculate it based on a
median of absolute differences (MAD) estimate [7, 2, 8, 9,
10]:
1
h = n− 5 medi |zi − medj zj |.
(13)
The objective function of the HBM estimator is given by:
JHBM (Θ̂) = z = FΘ̂−1 ()
(14)
where FΘ̂−1 (.) is the inverse cumulative distribution function (inverse CDF) of the squared residuals. The CDF of
the squared residuals is the following differentiable function of Θ̂:
n Z
1 X z
α − zi
K
dα.
FΘ̂ (z) =
nh i=1 −∞
h
(15)
Therefore, the inverse CDF is also differentiable and can be
optimized by solving the following equation:
Θ̂=Θ∗
=
=
∂FΘ̂ (z )
∂ Θ̂
Pn
1
i=1
nh
Pn
1
i=1
nh
R z
α−zi
∂
K
dα
h
−∞
∂ Θ̂
∂z
i
K( z −z
h )+
∂ Θ̂
R z ∂
i
K α−z
dα
h
−∞ ∂ Θ̂
(18)
To optimize the objective function, the condition (16)
should be satisfied. Thus, in the above equation we replace
with zero:
the term ∂z
∂ Θ̂
=
=
Pn R z ∂
α−zi
1
dα
i=1 −∞ ∂ Θ̂ K
nh
h
Pn R z ∂zi 0 α−zi −1
K
dα
i=1 −∞
nh2
h
i
Pn ∂zi h ∂ Θ̂z −zi 1
K
−
K(−∞)
.
2
i=1 ∂ Θ̂
h
h
(19)
+∞
−∞
∂FΘ̂−1 () ∂ Θ̂ 0=
0=
where K(.) is a kernel function with the following properties:
Z
By differentiating both sides of (17), the following equations are derived:
∂z =
∂ Θ̂ = 0.
(16)
Θ̂=Θ∗
For any parameter estimate we have:
= FΘ̂ (JLkOS )
= FΘ̂ (z )
(17)
The dependence of the bandwidth on the parameter estimates has been ignored in the above derivations as the bandwidth given by equation (13) does not substantially vary
with Θ̂ (and ∂∂h
is small) and the size of bandwidth (and
Θ̂
therefore its variations) do not substantially affect the performance of the estimator [7, 10, 2, 8, 9]. From the kernel
properties (10)-(12) we have K(−∞) = 0.
i
i
with 2ri ∂r
the following
By replacing the term ∂z
∂ Θ̂
∂ Θ̂
equation is derived:
n
X
1
z − zi
∂ri
= 0.
K
ri
2
h
h
∂ Θ̂
i=1
(20)
As it is the case for redescending M-estimators, the above
equation can be iteratively solved by updating the parameters through iterative reweighted least squares regression on
the data with the following weights:
1
z − ri2
wi = 2 K
.
(21)
h
h
Provided there are moderate number of data samples, the
functional form of the k-th order statistics, z can be approximated with its sample value:
2
1
rk:n − ri2
wi = 2 K
.
(22)
h
h
This is equivalent to an M-estimator with the objective funcP
r 2 −r 2
tion i ρ( k:nh i ) where ρ(.) is proportional to the integral
of the chosen kernel function. For example, for a Gaussian
2
1
kernel K(u) = √12π exp( −u
2 ) we have ρ(u) = h2 Φ(u)
where Φ(.) is the CDF of standard normal variables. Figure 1 shows the ρ(.) function plotted versus sample residuals. It is important to note that in contrast to redescending
M-estimators, ρ(.) does not merely depend on r but also on
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4- Calculate
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the
from
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residuals (r)
Figure
1. The
curve)
modified (lower
Figure
1. original
The ρ(.)(upper
function
plot and
for Gaussian
kernels.curve)
ρ(r) functions for Gaussian kernels.
491
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the k-th order statistics of all squared residuals. Therefore,
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502
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−2
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ing close data structures [5].
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3. Simulation Results
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533
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Figure 5. (a) A pic
3 for(with
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443
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− rk:n 0
2
5rk:n
2 and
5- Sort
Sortthe
theresiduals,
residuals,find
find
rk:n
and
calculate the weights wi using
calculate the weights wi using
equation (23). Reorder the data in an
equation (22). Reorder the data in an
ascending order of their corresponding
ascending order of their corresponding
residuals and calculate
√
√ calculate
residuals
and
w 1 x1 · · · w
X=[ √
√k xk ]⊤ ;
X = [ w1 x1 · · · wk xk ]> ;
6- Calculate the singular value
6- decomposition
Calculate theofsingular
value
X and find
the
decomposition
of
X and find the
eigenvector
v = [v
1 . . . vp+1 ] corresponding
eigenvector
v =eigenvalue;
[v1 . . . vp+1 ] corresponding
to
the smallest
to the smallest v
eigenvalue;
7- Calculate Θ̂new = ||v||
sign(v1 ) where
v
7- sign(v
Calculate
Θ̂
=
new
1 ) where
≥ 0 sign(v
and −1
1 ) is +1 for v1||v||
sign(v1 ) is +1 for v1 ≥ 0 and −1
otherwise;
otherwise;
8- If
||Θ̂ − Θ̂
|| is larger than a given
487
488
568
569
570
571
572
573
574
tributed in [−1, +1] × [−4, +4]. Then for each dataset, the
linear structure was segmented using HBM and PbM (with
conjugate gradient local search scheme) and the success
rate (the ratio of successful segmentations) was recorded.
A snapshot of the synthetic data and the segmented line is
M
ASSC (ra
PBM (ran
MSSE (ra
PBM (con
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pa-paFigure
A snapshot
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simulationsinintwo
two
dimensional
rameter
space
to
examine
the
required
number
of
random
initialFigure
4.
The
fit
returned
by
HBM
in
the
two
dimensional
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553
rameter space to examine the required number of random initial553
izations.
izations.
554
554
555
555
557
557
558
558
559
559
560
560
561
561
562
562
563
563
564
Success
Rate
(%)
Success
Rate
(%)
556
556
0.8
597
598
597
598
599
599
600
601
600
601
602
602
603
604
603
604
605
606
605
606
608
609
610
1
0.8
595
596
607
1
HBM
HBM
PbM
611
612
PbM
613
614
0.6
0.6
615
0.25
0.251
1
15 30 45 60 75 90
Number
Initialisations
15
30 of45
60 75 90
616
617
(b)
594
595
596
618
607
608
609
610
611
612
613
614
615
616
617
Number of Initialisations
619
618
(b)
(b)
4. Success
ratesofofsegmentation
segmentation ininthethe
two-dimensional
566
620
565
619
FigureFigure
5. Success
rates
two-dimensionalFigure 5. (a) A picture of the scanned pentagonal pyramid (b) Re3 for HBMinand
PbM estimators.
simulation
study shown
inof
Figure
Figure
4. Success
rates
segmentation
the
567
566
620of
Figure
6.
(a) A by
picture
of thepentagonal
scanned pyramid
polyhedra
(b) 621
Results
of range
segmentation
HBM
estimator.
3 for HBM
andtwo-dimensional
PbM estimators.sultsFigure
simulation
study
shown
in Figure
5. (a)
A picture
of
the
scanned
(b) Re565
564
568
567
569
568
570
569
simulation study shown in Figure 3 for HBM and PbM estimators.
tributed in [−1, +1] × [−4, +4]. Then for each dataset, the
linear
structure
HBM
PbM
local tributed
search
scheme)
shown
inusing
Figure
4 and
-each
and
the(with
success
571
in
[−1,was
+1]-segmented
×
[−4, +4].
Then
for
dataset,
the
570
conjugate
gradient
search
scheme) andwas
the recorded.
success
572
rate (the
ratio
of
successful
segmentations)
linear
structure
waslocal
segmented
using
HBM and
PbM
(with
571
rate (the ratio of successful segmentations) was recorded.
573
574
segmentation
estimator.
sultsrange
of range
segmentationby
by HBM
HBM estimator.
622
623
Method
Processing Time
624
ASSC (random
sampling)
308Processing
s Time
Method
Processing
Method
Time
625
PBM
(random
sampling)
261
s
ASSC
(random
sampling)
308 s
626s
ASSC
(random
sampling)
308
MSSE (random sampling)
234 s
627
PBMPBM
(random
sampling)
261 s
(random
261
s
PBM (conjugate
gradient) sampling) 83 s
628
MSSE (random sampling)
234 s
HBM (random sampling) 22 s
MSSE
234 s
621
622
623
624
625
conjugate
gradient
local search
scheme)
and the successthe
For
different
numbers
random
initializations,
626
A snapshot
of the
syntheticofdata
and the segmented
line is
rate
(the
ratio
of
successful
segmentations)
was recorded.
573aboveshown
627
procedure
was
repeated
and
the
success
rates
were
in
Figure
3.
575
629
PBM (conjugate gradient)
83 s
A snapshot of the synthetic data and the segmented line is
574recorded.
628
PBMHBM
(conjugate gradient)
83
s
5 shows
theofrecorded
success rates the
plottedTable 1. Comparative
ForFigure
different
numbers
random initializations,
576
630
22 s
processing times elapsed for segmentation
shown in Figure 3.
575
629
procedureof
was
repeatedinitialization
and the success
rates were
HBM
22
577
631s
versusabove
the number
random
applied
in eachof the five patches scanned from
the pentagonal pyramid in FigFor different
random
initializations,
576
630
shows theof
recorded
success
rates plottedthe ure Table
recorded.
Figure 4numbers
578
632
1.
Comparative
processing
times
elapsed
for
segmentation
5.
method.
It is observed
that
in this
the PbM
estimator
above
repeated
andcase,
the success
were
577
631
versus procedure
the numberwas
of random
initialization
appliedrates
in each
579
633
of the
five patches
scanned from
the pentagonal
pyramid
infor
FigTable
1.
Comparative
processing
times
elapsed
segmentation
needsrecorded.
at leastItFigure
60
trials
ofthat
random
initializations
toplotted
achieve ure 5.
4 shows
the
recorded
success
578
method.
is observed
in this
case,
the
PbMrates
estimator
580
634
of the five patches scanned from the pentagonal pyramid in 632
Figa success
of 60
98%
thisinitialization
can
be achieved
by
HBMpling) and
versus
number
ofwhile
random
applied
in each
needsrate
atthe
least
trials
of random
initializations
to achieve
579
633
581
635
6. (once using random sampling and once with
urePbM
method.
Itrate
isonly
observed
that this
inThe
this
the PbM
a success
of 98%
while
cancase,
be achieved
byestimator
HBM
estimator
with
20 trials.
gradient
search
appliedconjugate gradient search). For each estimator, the pro580
634
582
636
estimator
with 60
only
20 trials.
The
gradient
applied
needs
at least
trials
of random
initializations
to
achieve
times
elapsed
for using
the required
pling)
and PbM
(once
randomcomputations
sampling andwere
once with 637 635
583
581
in PbM
estimator
needs
more
search
trialssearch
perhaps
due tocessing
PbM estimator
many this
search
as becausebyofHBM
its
1. ForFor
comparison
purposes,
as listed
in Table
ainsuccess
of needs
98% while
cantrials
be and
achieved
conjugate
gradient
search).
each estimator,
pro- 638 636
584
582
its locality
andrate
inaccurate
assumptions,
many trials
arerecorded
4600 as
calculated
from
equation
(2)
withthe =
0.10, p = 3
inaccuracywith
and only
locality,
is highly
likely
to either
performed
using
the same
computing were 639
estimator
20 each
trials.trial
The
gradient
search
applied all computations
585
cessing timeswere
elapsed
for the
required
computations
583
637
likely
to
either
diverge
or
be
trapped
in
a
local
minimum.
and
P
=
0.99.
For
HBM
estimator
and
PbM
success
diverge
be trapped
in amany
local minimum.
and as
were
programmed
MATLAB
586
in
PbM or
estimator
needs
search trials as because of its platform
For comparison
purposes, 640 (using
recorded
listed
in Table in1.MathWorks’
584
638
conjugate
gradient
search),
theofminimum
number of
programming
environment.
The
number
random
time
ofofHBM
waslikely
also compared
587 Computation
641random
Computation
time
HBM
estimator
was
compared
to
inaccuracy
and
locality,
eachestimator
trial
is highly
to either
all computations
were performed
using
the
same samcomputing
585
639
forinitializations
MSSE,
(using
random
sampling)
588
otherestimators
estimators
inina number
of real
range
for PbM
the 99%
rate
have
been 642
applied.
to other
a innumber
of 3D
real
3Dsegmentation
range segmen-plesplatform
diverge
or be trapped
a local
minimum.
andASSC
were and
programmed
insuccess
MathWorks’
MATLAB
586
640
were
4600 as calculated
from equation
(2) with of
ǫ =
0.10, sam- 643
Onetime
of
them
shownestimator
in Figure
5 involves
589
programming
environment.
The number
random
The results
show that
HBM
estimator
is substantially
587
641
tationexperiments.
experiments.
The
of
one of
those
issegshown
Computation
ofresults
HBM
was
compared
to
p =ples
3 and
0.99.and
For PbM
HBM(using
estimator
and PbM
590
644
mentation of five patches scanned from a pentagonal pyrasuccess =
forPMSSE,
ASSC
random
sampling)
588in Figure
642
other
estimators
ininvolves
a numbersegmentation
of real 3D rangeofsegmentation
faster
than
other
high
breakdown
robust
estimators
that
use
6
which
five
patches
(using conjugate gradient search), the minimum number of
591
645
mid as depicted in Figure 5(a). The result of segmentation
wererandom
4600 assampling.
calculated from
equation (2)
with
ǫ = 0.10, robustness
experiments.
One of
of them
shown polyhedra
in Figure 5 as
involves
seg- inrandom
589scanned
643
In
addition,
the
theoretical
from
a
side
a
uniform
depicted
initializations have been applied.
by HBM is shown in Figure 5(b) and similar results were
592
p = 3 and Psuccess = 0.99. For HBM estimator and PbM 646 644
590
mentation of five patches scanned from a pentagonal pyraand
stability
of HBM
HBMestimator
estimator
(as discussed in647
previous
Figure
6(a). The
of segmentation
by HBM
is samshown in The
593
obtained
usingresult
MSSE[1]
and ASSC [13] (with
random
results
show that
is substantially
(using conjugate gradient search), the minimum number of
591
645
mid as depicted in Figure 5(a). The result of segmentation
section)
means
that
it
requires
far
less
random
initializaFigure
6(b)
and
similar
results
were
obtained
using
MSSE
random initializations have been applied.
by HBM is shown in Figure 5(b) and similar results were
592
646
tionsresults
compared
to PbM
conjugate gradient
ASSCusing
[13]MSSE[1]
(with random
sampling)
PbMsam(once
6
593[1] and
647
obtained
and ASSC
[13] (withand
random
The
show that
HBMestimator
estimator using
is substantially
search, and hence is considerably faster.
using random sampling and once with conjugate gradient
We have also compared the performance of HBM estisearch). For each estimator, the processing time is recorded
6
mator with pbM and MSSE for solving the fundamental
and listed in Table 1. For comparison purposes, all computations were performed using the same computing platmatrix estimation problem using both synthetic and real image pairs [4]. This is a much higher dimensional problem
form and were programmed in MathWorks’ MATLAB pro(compared to range segmentation) and therefore, the cost
gramming environment. The number of random samples
of computation by a RANSAC-based technique is also subfor MSSE, ASSC and PbM (using random sampling) were
572
stantially higher as a large number of random samples is
required to solve this problem. The results of our study
again show that HBM is substantially faster than the other
two techniques while the segmentation performances of the
three estimators (in terms of small estimation error and correct number of inliers) are comparable.
4. Conclusions
A computationally efficient high breakdown robust estimator, called HBM estimator, was introduced for solving
multi-structural data segmentation problems encountered in
various computer vision applications. HBM estimator has
a novel differentiable objective function for which a closed
form updating formula can be mathematically derived (similar to redescending M-estimators) and used to optimize its
objective function. The resulting M-estimator has a high
breakdown point as it minimizes the functional form of
the k-th smallest squared residual. We have mathematically proved that optimization of this objective function can
be achieved by solving a weighted least squares problem.
Thus, instead of minimizing the single k-th order statistics of the squared residuals (as in LkOS estimators), this
estimator minimizes a smoothed window of the residuals
around the k-th order statistics of the squared residuals.
The closed mathematical form of HBM and its guaranteed stability (theoretically supported by stability properties of redescending M-estimators) combined with its high
breakdown point (evidenced by LkOS and ALKS estimators) and its fast convergence speed make this estimator an
excellent choice for solving the problem of segmenting of
multi-structural data.
A number of experiments, using both synthetic and real
data have been conducted to benchmark the performance of
the proposed estimator. The computational time of HBM,
MSSE, ASSC and PbM are compared using the same computing platforms (CPU, memory, software, etc.) and the
results show that HBM outperforms aforementioned techniques.
Acknowledgement
This research was supported by the Australian Research
Council and Pacifica Group Technologies (PGT) through
the ARC Linkage Project grant LP0561923.
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