Statistical Modeling and Calibration of Triangulation Lidars

Anas Alhashimi, Damiano Varagnolo and Thomas Gustafsson

Control Engineering Group, Department of Computer Science, Electrical and Space Engineering,

Luleå University of Technology, Luleå 97187, Sweden

Keywords:

Maximum Likelihood, Least Squares, Statistical Inference, Distance Mapping Sensors, Lidar, Nonlinear

System, AIC.

Abstract:

We aim at developing statistical tools that improve the accuracy and precision of the measurements returned

by triangulation Light Detection and Rangings (Lidars). To this aim we: i) propose and validate a novel model

that describes the statistics of the measurements of these Lidars, and that is built starting from mechanical

considerations on the geometry and properties of their pinhole lens - CCD camera systems; ii) build, start-

ing from this novel statistical model, a Maximum Likelihood (ML) / Akaike Information Criterion (AIC) -

based sensor calibration algorithm that exploits training information collected in a controlled environment; iii)

develop ML and Least Squares (LS) strategies that use the calibration results to statistically process the raw

sensor measurements in non controlled environments. The overall technique allowed us to obtain empirical

improvements of the normalized Mean Squared Error (MSE) from 0.0789 to 0.0046.

1 INTRODUCTION

Lidars are ubiquitously used for mapping purposes.

Different types of Lidar technologies, such as Time

of Flight (ToF) and triangulation, have different sta-

tistical performance. For example, ToF Lidars have

generically lower bias and measurement noise vari-

ances than triangulation ones. At the same time, trian-

gulation Lidars are generally cheaper than ToF ones.

The market pull is then to increase the performance of

cheaper Lidars in a cost-effective way.

Improving the accuracy and precision of sensors

can then be done in different ways, e.g., by improv-

ing their mechanical properties. In this paper we have

a precise target: improve the performance indexes of

triangulation Lidars by removing their biases and ar-

tifacts through opportune statistical manipulations of

the raw information coming from the sensor.

The following literature review analyzes a set of

algorithms that are related to our aim.

Literature Review. It is convenient to categorize

the algorithms in the existing and relevant literature

as:

• procedures for the characterization or calibration

of the devices. Here characterization means a

thorough quantiﬁcation of the measurement nois-

iness of the device, while calibration means an

algorithm that aims at diminishing this noisiness

level;

• when dealing with calibration issues, procedures

for the intrinsic or extrinsic calibration. Here

intrinsic means that the focus is on estimating

the parameters of the Lidar itself, while extrinsic

means that the focus is on estimating the parame-

ters resulted from sensor positioning and installa-

tion.

Characterization issues: several papers discuss

Lidar characterization issues for both ToF (Kneip

et al., 2009; Reina and Gonzales, 1997; Lee and

Ehsani, 2008; Sanz-Cortiella et al., 2011; Tang et al.,

2009; Tuley et al., 2005; Ye and Borenstein, 2002;

Anderson et al., 2005; Alhashimi et al., 2015) and tri-

angulation Lidars (Lima et al., 2015; Campos et al.,

2016). Notice that, at the best of our knowledge, for

triangulation Lidars there exist only two manuscripts:

(Lima et al., 2015), that discusses the nonlinearity of

Neato Lidars, and (Campos et al., 2016), that analyzes

the effect of the color of the target on the measured

distance. Importantly, (Lima et al., 2015) models non-

linear effects on the measurements and the variance

of additive measurement noises as two independent

effects that can be modeled with a second order poly-

nomials on the actual distance. From a statistical per-

spectives the authors, therefore, decouple the learning

process into two separate parts.

308

Alhashimi, A., Varagnolo, D. and Gustafsson, T.

Statistical Modeling and Calibration of Triangulation Lidars.

DOI: 10.5220/0005965803080317

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 308-317

ISBN: 978-989-758-198-4

Calibration issues: as for the calibration issues

there is a relatively large number of papers describing

how to calibrate extrinsic parameters either using ad-

ditional sensors (such as cameras) (Zhang and Pless,

2004b; Mei and Rives, 2006; Jokinen, 1999; Tidde-

man et al., 1998), or just requiring knowledge on the

motion of the Lidar itself (Andreasson et al., 2005;

Wei and Hirzinger, 1998; McIvor, 1999; Zhang and

Pless, 2004a).

Still considering calibration issues, there has been

also a big effort on how to perform intrinsic cali-

bration for multi-beam Lidar systems, where the re-

sults from one beam is used to calibrate the intrinsic

parameters of other beams (Chen and Chien, 2012;

Muhammad and Lacroix, 2010; Atanacio-Jiménez

et al., 2011; Glennie and Lichti, 2010; Glennie and

Lichti, 2011; Glennie, 2012; Gordon and Meidow,

2013; Mirzaei et al., 2012; Gong et al., 2013; Park

et al., 2014). As for single-beam Lidar systems, in-

stead, (Mirzaei et al., 2012) proposes a method for

the intrinsic calibration of a revolving-head 3D Lidar

and the extrinsic calibration of the parameters with re-

spect to a camera. The technique involves an analyti-

cal method for computing an initial estimate for both

the Lidar’s intrinsic parameters and the Lidar-camera

transformation, that is then used to initialize an iter-

ative nonlinear least-squares reﬁnement of all of the

calibration parameters.

We also mention the topic of on-line calibration of

sensor parameters for mobile robots when doing Si-

multaneous localization and mapping (SLAM), very

useful in navigation tasks. In this category, (Küm-

merle et al., 2011) proposes an approach to simulta-

neously estimate a map of the environment, the po-

sition of the on-board sensors of the robot, and its

kinematic parameters. These parameters are subject

to variations due to wear of the devices or mechanical

effects like loading. An other similar methodology for

the intrinsic calibration of depth sensor during SLAM

is presented in (Teichman et al., 2013).

Statement of Contributions. We focus speciﬁcally

on triangulation Lidars for robotic applications, and

aim to increase their performance in a cost-effective

way through statistical processing techniques. Our

long term vision is to arrive at a on-line automatic

calibration procedure for triangulation Lidars like

in (Kümmerle et al., 2011; Teichman et al., 2013);

before reaching this above long-term goal, we must

nonetheless solve satisfactorily the problem of cali-

brating triangulation Lidars off-line.

In this paper we thus:

• propose and assess a model for the measurement

process of triangulation Lidars (see Section 3 and

model (1)). Our model not only generalizes the

model proposed in (Lima et al., 2015; Campos

et al., 2016), but also motivates it starting from

mechanical and physical interpretations;

• on top of this model, propose and assess a ML

calibration procedure that uses data from a Mo-

tion Capture (MoCap) system. Importantly, our

calibration procedure extends the one proposed

in (Lima et al., 2015): there authors decoupled

the learning process into two separate stages (cor-

responding to estimate two different sets of pa-

rameters), while here the calibration is performed

simultaneously on both sets of parameters;

• propose and assess novel ML and LS strategies for

correcting the measurements from the sensor with

the model inferred during the calibration stage.

As reported in (31) and (32), the overall strategy is

then shown to be capable to improve the normalized

MSE of the raw information from the sensor from

0.0789 to 0.0046.

1.1 Organization of the Manuscript

Section 2 describes the working principles of trian-

gulation Lidars. Based on these working principles,

Section 3 proposes a statistical model of the measure-

ment process of the device. Section 4 then validates

this statistical model using data acquired through a

MoCap system. Section 5 then presents a calibration

algorithm for sensors deployed in a test environment.

Section 7 eventually concludes the paper with the de-

scription of future research issues.

2 The Triangulation Lidar Range

Sensor

We now describe the functioning principle of the tri-

angulation scanners; this discussion will be useful

for explaining why the moments of the measurement

noise depend on the actual measured distance. More

details about the constructive details of triangulation

Lidars can be found in (Blais, 2004; Konolige et al.,

2008).

A prototypical triangulation Lidar is the one in

Figure 1. Its working principles are then explained

with the diagram in Figure 2 and its caption.

This simple working principle helps keeping the

cost of the sensor low

, and making it commercially

Incidentally, the sensor was costing $135.00 as of February

2016 in Ebay. Nonetheless, the original industrial goal was

to reach an end user price of $30.00.

Statistical Modeling and Calibration of Triangulation Lidars

309

Figure 1: Photo of a triangulation Lidar.

laser beam

parallel of

the laser beam

laser

object

pinhole

camera

pinhole

lens

Figure 2: Diagram exemplifying the working principle of a

triangulation Lidar . The laser emits an infra-red laser signal

that is then reﬂected by the object to be detected. The beam

passes through a pinhole lens and hits a CCD camera sensor.

By construction, thus, the triangles deﬁned by (b, d

) and by

, d

) are similar: this means that the distance to the object

is nonlinearly proportional to the angle of the reﬂected light,

and as soon as the camera measures the distance b

one can

estimate the actual distance d

using triangles similarities

concepts.

usable in low-cost devices like robotic vacuum clean-

ers. The low cost of the sensor comes nonetheless

with some well-deﬁned mechanical problems (Kono-

lige et al., 2008):

• low-cost lens, that generate nonlinear distortion

effects;

• imprecise pointing accuracy, that is known of at

best 6 degrees;

• not rigid physical linkages among lens elements,

camera, laser, and laser optics, that may suffer

from distortion effects during the life of the de-

vice.

As it can be seen in Figure 3, all these problems in-

duce measurement errors; more precisely, triangula-

tion Lidars suffer from strong nonlinearities in both

the bias and the standard deviation of the measure-

ment noise. This pushes towards ﬁnding some sig-

nal processing tools that can alleviate these problems,

and keep the sensor cheap while improving its perfor-

mance.

1 2 3 4

0.25

0.5

0.75

actual distance [m]

meaurements error [m]

Figure 3: Dataset.

3 A NOVEL STATISTICAL

MODEL FOR THE Lidar

MEASUREMENTS

Let y

be the k-th measurement returned by the Lidar

when the true distance is d

. Physically, y

is com-

puted by the logic of the sensor through a static trans-

formation of b

in Figure 2; we assume here that this

static transformation is unknown, that b

is not avail-

able, and that we want to improve the estimation for

from just y

Our ansatz for the whole transformation from d

to y

is then

= f (d

) + f (d

)

(1)

where

• f (·) is an unknown non-linear function;

• e

∼ N



0, σ



is a Gaussian and white additive

measurement noise.

In the following Section 3.1 we motivate the presence

of f (·) from mechanical considerations, while in the

following Section 3.2 we motivate the presence of the

f (·)

multiplying the noise e

starting from physical

considerations.

3.1 Explaining the Presence of the

Nonlinear Term f (·) in Model (1)

The nonlinear term f (·) in (1) is related to what is

called the radial distortion in camera calibration liter-

ature (Zhang, 2000; Weng et al., 1992; Brown, 1964;

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

310

Duane, 1971). Indeed camera lenses are notoriously

nonlinear at their borders, with this nonlinearity in-

creasing as the light beam passes closer to the lens

edges. In our settings this thus happens when targets

are very close or very far.

Radial distortions are usually modeled in the cam-

era calibration literature as a series of odd powers, i.e.,

f (d

) =

∑

i=0

2i+1

(2)

where the α

’s are the model parameters.

As numerically shown during the validation of (1)

in Section 4, model (2) does not describe well the ev-

idence collected in our experiments. Indeed the spe-

ciﬁc case of triangulation Lidars lacks of the symme-

tries encountered in computer vision settings (see (4)

and the discussion on that identity), and thus in our

settings there is no need for odd symmetries in the

model (in other words, doubling d does not lead to

doubling b

). We thus propose to remove this con-

straint and use a potentially non-symmetric polyno-

mial, i.e.,

f (d

) =

∑

i=0

. (3)

The numerical validations of model (3) shown in Sec-

tion 4 conﬁrm then our physical intuition.

3.2 Explaining the Presence of the

Multiplicative Term f (d

)

Model (1)

Assume for now that there are no lens-distortion ef-

fects. The similarity between the triangles in Figure 2

then implies

. (4)

In (4) d

and b

are generally time-varying quanti-

ties, while b and d

are constants from the geome-

try of the Lidar. Assume now that the quantity mea-

sured by the CCD at time k is corrupted by a Gaussian

noise, so that z

= b

+ w

with w

∼ N



0, σ

CCD



and σ

CCD

constant and independent of d

. Thus

∼ N



, σ

CCD



; since

, (5)

assuming a Gaussian measurement noise on the CCD

implies that y

is a reciprocal Gaussian r.v. This kind

of variable is notoriously difﬁcult to treat (e.g., their

statistical moments cannot be derived from closed

form expressions starting from the original Gaussian

variables). For this reason we perform a ﬁrst order

Taylor approximation of the nonlinear map (5) above.

In general, if



∼ N



b, σ



= φ(z

)

(6)

then the ﬁrst order Taylor approximation of the distri-

bution of y

is (Gustafsson, 2010, (A.16))

∼ N



φ(b), φ

(b)



(7)

where φ

(·) is the ﬁrst derivative of φ(·) w.r.t. z

. Sub-

stituting the values of our speciﬁc problem into for-

mula (7) leads then to the novel approximated model

∼ N



−bd



CCD

, (8)

or, equivalently,

= d

+ d

∼ N



0, σ



(9)

where σ

CCD

is a scaled version of σ

CCD

inde-

pendent of d

and to be estimated from the data.

Consider now that actually there are some lens

distortion effects that imply the presence of the non-

linear term f (d

). We can then repeat the very same

discussion above, and obtain model (1) by substitut-

ing d

with f (d

) in (9).

4 VALIDATION OF THE

APPROXIMATION (8)

The approximation introduced by the ﬁrst order Tay-

lor expansion in (8) can be seen as arbitrary. Nonethe-

less we show in this section that on the collected

datasets it actually corresponds to the most powerful

approximation in a statistical sense.

To this aim we perform this two-step validation:

1. (check if the noises are independent and identi-

cally distributed (iid) and normal) perform a nor-

mality test on the y

’s assuming that measure-

ments are collected at a ﬁxed distance (i.e., d

constant): indeed e

is approximately Gaussian as

much as y

is;

2. (check the order of the term multiplying e

) com-

pare the following alternative statistical models

for the measurements y

: y

= f (d

) + e

: y

= f (d

) + f (d

: y

= f (d

) + f (d

)

: y

= f (d

) + f (d

)

(10)

and check which one describes better the collected

information.

Statistical Modeling and Calibration of Triangulation Lidars

311

As for point 1 we can use standard iid tests

(like the Wald-Wolfowitz runs (Croarkin and Tobias,

2006)) and standard normality tests (like the Shapiro-

Wilk normality test). These tests performed on our

registered data showed p-values of 0.56 and 0.42, so

we can safely consider the measurement noises to be

iid and Gaussian.

As for point 2, we instead consider the following

strategy: for every model above, assuming that mea-

surements are collected at a ﬁxed distance (i.e., d

constant), we can perform a simple algebraic manip-

ulation of (1) to obtain

− y

k−1

f (d

)

= e

− e

k−1

(11)

where ? indicates the order of the model (that means

? ∈

{

0, . . . , 3

}

). (11) in its turn indicates that, since e

and e

k−1

are assumed iid,

− y

k−1

f (d

)

∼ N



0, 2σ



, ? ∈

{

0, . . . , 3

}

. (12)

Assume now that the dataset is composed by dif-

ferent batches each corresponding to d

’s that are con-

stant in the batch, but different among batches. More-

over assume that each batch is sufﬁciently rich to

make it is possible to estimate with good conﬁdence

the unknown f (d

) through the empirical mean of the

relative to that batch. By combining the informa-

tion from different batches it is then possible to check

which model ? describes better the measured infor-

mation.

Indicate then with B the number of batches in the

dataset, with b = 1, . . . , B the index of each batch, and

with B

the set of k’s that are relative to that speciﬁc

batch b. In formulas, we thus:

1. estimate, for each model batch b = 1, . . . , B, the

distance

∑

k∈B

; (13)

2. estimate, for each model ? = 0, . . . , 3, the variance

of e

∑

b=1





2|B

∑

k,k−1∈B

− y

k−1





(14)

3. compute, for each model ? = 0, . . . , 3, the log-

likelihood of the data as

−log P

y ; d

∑

b=1







|log





∑

k=1



−









(15)

where y

= [y

, . . . , y

]

and d

= [d

, . . . , d

]

In Figure 4 we then show the log-likelihoods for

the different models. As it can be seen, hypothesis H

is the one that best describes the collected evidence.

·10

y ; d

Figure 4: Evaluation of (15) on the collected datasets.

A non rigorous (but graphical and intuitive) argu-

ment supporting H

as the hypothesis best describing

the evidence is then the one offered in Figure 5. The

argument goes as follows: for the exact ? ∈

{

0, . . . , 3

}

the quantities

− y

k−1

f (d

)

? ∈

{

0, . . . , 3

}

. (16)

should be iid independently of d

. This iid-ness is

indeed a necessary condition for iid-ness of the mea-

surement noises (one of our assumptions).

Since f (·) is actually unknown, this iid-ness test

must be performed by means of some estimate of

f (·). In the following we use the estimator deﬁned

in Section 5 over an experiment where we manually

increase the true distance d

. As it can be seen, the

hypothesis H

is the unique one for which the quan-

tities

− y

k−1

f (d

)

are homoscedastic. Thus the normal-

izing factor ? = 2 is the unique one guaranteeing iid-

ness for the measurement noises. Notice that this ar-

gument is a non rigorous wishful thinking, since we

use some estimates as the ground truth; nonetheless

the heteroscedasticity of the noises for ? = 0, 1, 3 in-

dicates that these hypotheses are non-descriptive.

5 CALIBRATING THE Lidar

Our overall goal is not just to propose the statistical

model (1) describing the measurement process of the

Lidar but also to ﬁnd a calibration procedure for es-

timating the unknowns f (·) and σ

from some col-

lected information.

Once again the long term goal is to calibrate (1)

on-line and continuously using information from

other sensors like odometry, ultrasonic sensors, etc.

Instrumental to this future direction we now solve the

ﬁrst step, that is to estimate f (·) and σ

from a dataset

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

312

− y

k−1

− y

k−1

f (d

)

− y

k−1

f (d

)

2 3 4

distance [m]

− y

k−1

f (d

)

Figure 5: Plots of the quantities

− y

k−1

f (d

)

for ? = 0, . . . , 3

and for increasing d

and for

f (·) computed as in Section 5.

The results graphically suggest that

f (d

)

is the unique

normalizing factor for which we obtain homoscedastic sam-

ples.

D =

{

, d

}

in which we know d

(e.g., thanks to a

MoCap system).

Given our Fisherian setting, we seek for the ML

estimate for both f (·) and σ

, where we recall that

(due to the radial distortion hypothesis as the source

of f (·), see Section 3.1) f (·) is modeled as a non-

symmetric polynomial, i.e., as f (d

) =

∑

i=0

in (3). Since now model (1) implies

− f (d

) ∼ N



0, f (d

)



, (17)

it follows immediately that the corresponding nega-

tive log-likelihood is proportional to

= log(det Σ) +



y − f

f (d



−1



y − f

f (d



(18)

where

• y

= [y

, . . . , y

]

;

• d

= [d

, . . . , d

]

;

• f

f (d

= [ f (d

), . . . , f (d

)];

• Σ

= diag



f (d

)

, . . . , f (d

)



Finding the ML estimates in our settings thus

means:

1. solving

argmin

θ∈Θ

L (θ) (19)

for several different n, with



, . . . , α

, σ



(20)

and Θ the set of θ ∈ R

n+1

for which σ

> 0;

2. deciding which n is the best one using some model

order selection criterion, e.g., AIC.

Unfortunately problem (19) is not convex, so it

neither admits a closed form solution nor it can be

easily computed using numerical procedures. Solving

problem (19) is thus numerically difﬁcult. Keeping

in mind that our long-term goal is the development of

on-line calibration procedures, where numerical prob-

lems will be even more complex, we strive for some

alternative calibration procedure.

5.1 An Approximate Calibration

Procedure

We here propose an alternative estimator that trades

off statistical performance for solvability in a closed

form. We indeed propose to seek an estimate for θ

in (20) by using the alternative model

= f (d

) + d

, (21)

that differs from (1) only for the fact that the noise is

multiplied by d

instead of f (d

)

. This approxima-

tion is intuitively meaningful, since f (d

) represents

a distortion term induced by the pinhole lens: ideally,

indeed, f (d

) should be equal to d

Assuming model (21) it is now possible do de-

rive a ML estimator of θ. Indeed dividing both sides

of (21) by d

we get

= g(d

) + e

(22)

where (cf. (3))

g(d

) =

∑

i=0

i−2

. (23)

This means that the estimation problem can be cast

as the problem of estimating the parameters α

[α

, . . . , α

]

and the noise variance σ

describing the

linear system



−2

. . . d

n−2















+ e

, (24)

for which the ML solution is directly

α =





−1



y − H





y − H



(25)

Statistical Modeling and Calibration of Triangulation Lidars

313

with







−2

··· d

n−2

−2

··· d

n−2



















. (26)

Notice that the procedure above does not deter-

mine the model complexity n. For inferring this pa-

rameter we then propose to rely on classical model

order selection criteria such as AIC.

5.2 Using the Calibration Results to

Estimate d

Once the sensor has been calibrated, i.e., a

α and

have been computed, it is possible to invert the pro-

cess and use the learned information for testing pur-

poses. This means that given some measurements y

collected in an unknown environment we can, through

α and

, estimate d

5.2.1 Computing the ML Estimate of d

Rewriting model (3) as

f (d

) = d

α d













(27)

and equating the score of y

parametrized by α

α and

to zero leads to the equation



− d



− d

(I − K)α



= σ

(28)

with

= diag



, . . . ,



. (29)

This means that estimating d

from y

α and

can

be performed by solving (28) in d

after substituting

the real values α

α and σ

with their estimates.

Since polynomial (28) is quartic for n = 0, 1, 2,

and of order at least 6 for any other n, the ML estimate

for d

must then either rely on complex algebraic for-

mulas or numerical roots ﬁnding methods.

5.2.2 Computing the LS Estimate of d

Given our assumption (3) on the structure of f (·), and

given an estimate

f for f , the problem of estimating

from y

is the one of minimizing the squared loss



−

f (d

)



. Once again, the problem is of ﬁnding

the roots of a polynomial, since the solutions of the

LS problem above are directly

∈

d s.t. y

−





= 0

. (30)

Thus if the Lidar has heavy nonlinear radial distor-

tions (that means that it requires high order polynomi-

als f (·)) then one is again required to compute poly-

nomial roots.

Notice also that some of the roots above may not

belong to the measurement range of the sensor (e.g.,

some roots may be negative); these ones can safely

be discarded from the set of plausible solutions. The

other ones, instead, are equally plausible.

This raises a question on how to decide which root

should be selected among the equally plausible ones.

This question is actually non-trivial, and cannot be

solved by means of the frequentist approach used in

this manuscript. We thus leave this question unan-

swered for now, and leave it as a future research ques-

tion. Bayesian formulations will be explored to see if

they solve the multiple plausible roots problem.

6 NUMERICAL EXPERIMENTS

Our experiments consist of a robot with the Lidar

mounted on top moving with piecewise constant

speeds towards a target. We recorded several datasets

for training and testing purposes, consisting of the

Lidar measurements and a ground truth information

collected by a MoCap system (see Figure 6). Training

Figure 6: Experimental setup used for recording the dataset.

The Lidar was mounted over a Pioneer 3AT robot facing an

obstacle; the photo moreover shows some of the cameras of

the MoCap system.

datasets were thus initially used to estimate α

α and σ

as described in Equation (25). As for the model order

selection, we empirically detected that n = 2 was al-

ways the best choice when using AIC measures. E.g.,

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314

0 1 2 3 4

0.25

0.5

0.75

actual distance [m]

measurement error [m]

training set

order 1

order 2

order 3

Figure 7: A typical training set collected in our experiments.

The plotted quantities correspond to the measurement errors

and to the polynomial models ﬁtting these errors.

Table 1: AIC scores for the different models complexities

involved in the training set of Figure 7.

polynomial order AIC score

1 -5.774

2 -7.380

3 -5.824

4 -3.890

for the dataset shown in Figure 7 we obtained the AIC

scores reported in Table 1.

The estimated α

α and σ

were also used for testing

purposes to reﬁne the estimate of the distances d

non-controlled environments. Notice that the selected

model order was always 2, so it was always possible

to solve the LS problem in a closed form and also

discard one of the roots in (30), so that the set of roots

was always a singleton. As shown in Figure 8,

much closer to d

than y

. For example, the empirical

normalized MSEs for the test set in Figure 8 were

∑

k=1



− d



= 0.0046, (31)

∑

k=1

− d

= 0.0789. (32)

7 CONCLUSIONS

We derived, starting from a combination of physical

and statistical considerations, a model that describes

the statistical behavior of the measurements returned

by triangulation Lidars. This statistical model, given

in (1), is based on two assumptions:

1. the effects of radial distortions in the pinhole lens

can be captured by means of a polynomial func-

tion;

0 1 2 3 4

5 6

actual distance [m]

measured distance [m]

test set

MoCap

estimated

Figure 8: Effects of the estimation procedure on the origi-

nal Lidar measurement. It can be noticed how the overall

strategy removes the nonlinearities induced by the pinhole

lens - CCD camera system.

2. the nonlinearities induced by the geometry of the

laser-CCD system can be captured by means of

a heteroscedastic noise which standard deviation

depends in ﬁrst approximation quadratically with

the measured distance.

This model, validated through some experiments

on real devices, allows to build tailored triangulation

Lidars calibration strategies that follow the classical

training-testing paradigm:

• in the training phase, collect information in a con-

trolled environment and use it to estimate through

ML paradigms the parameters deﬁning the statis-

tical behavior of the sensor;

• in the test phase, use this information and some

statistical inference techniques such as ML or LS

to correct the measurements from the sensor when

this is in a non-controlled environment.

It turns then out that both the ML and LS estima-

tion strategies may be numerically demanding, spe-

cially for sensors suffering from strong radial distor-

tions in the pinhole camera. In this case, indeed, the

estimators may require to use numerical root ﬁnding

procedures and lead to some computational disadvan-

tages.

Irrespectively of these issues, that can in any case

be mitigated by limiting the complexity of the polyno-

mials describing the radial distortions, the estimation

strategies above have been proved to be effective in

our tests. Real-life experiments indeed showed that

the techniques allow to reduce the empirical MSE of

the sensor of a factor 17.15.

Despite this promising result, the research asso-

ciated to triangulation Lidars is not ﬁnished. First

Statistical Modeling and Calibration of Triangulation Lidars

315

of all, the techniques should be modiﬁed so to be

implementable using recursive estimation schemes.

Moreover, by following a classical training-testing

approach, the techniques above present some limi-

tations. Different sensors may in fact differ even if

nominally being constructed in the same way. More-

over sensors may change their statistical behavior in

time, due to aging or mechanical shocks. This means

that techniques based on results from a controlled en-

vironment on just one sensor and just once are even-

tually not entirely meaningful.

A robust approach must indeed perform contin-

uous learning for each sensor independently in a

non-controlled environment by performing informa-

tion fusion steps, e.g., combining also information

from other sensors like odometry, ultrasonic and ac-

celerometers.

This information-fusion continuous-learning al-

gorithm nonetheless must be based on some prelim-

inary results on what are the statistical models of tri-

angulation Lidars and on how inference can be per-

formed on them. This paper can thus be seen as the

ﬁrst step towards more evolved strategies.

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