articol science

7/21/2019 articol science

1/9


2/9

When the object has a known shape, for example a perfect

plane, 3D reconstruction is not really necessary. However, laser-

based triangulation can be used to characterize vibrations [19,20].

These methods estimate vibrations from the relative depth varia-

tions. A similar approach can be used with laser Doppler vibro-

metry [21,22]. In this case, vibrations are estimated from the

Doppler shift of the reected laser beam frequency. All these

methods assume a static shape of the object, i.e., the shape does

not change over time. Thus, only vibrations are responsible forvariations in measurements. When the shape of the object

changes, these methods are not applicable.

A known shape of the object or a known type of vibrations

could be considered 3D reconstruction invariants. When these

invariants do not exist these procedures to remove vibrations are

not possible. Therefore, in order to remove vibrations new invar-

iants need to be created.

This work proposes a method to remove vibrations in 3D

reconstruction based on using multiple laser stripes. When more

than one laser stripe is used, a new invariant is created: the

displacement from one laser stripe to the next is not dependent on

vibrations, but only on the shape of the object. Therefore, the real

shape of the object can be obtained without the interference of

vibrations. However, the exact number of stripes needed depends

on the complexity of the vibrations. When vibrations only consist

of vertical translations, two laser stripes are required. On the other

hand, when vibrations are a combination of vertical translations

and rotations, three laser stripes are required. More than three

laser stripes produce an over-determined system that delivers a

more robust solution. In this work all these possibilities are

analyzed and the mathematical procedures to remove vibrations

in these cases are developed. Then, the proposed procedure is

applied to synthetic and real data. The accuracy of the results is

discussed with and without noisy data and compared with the

benets of using more than two laser stripes. Results indicate that

a laser-based 3D reconstruction method using two laser stripes

presents a far more cost efcient solution than other approaches,

also with similar performance.

2. Modeling vibrations

Vibrations are periodic or random motion from an equilibrium

position. In the case of laser-based 3D reconstruction methods,

vibrations are an undesirable phenomenon that produces move-

ments of the object as it moves forward. The consequences of

these vibrations are particularly harmful for 3D reconstruction

because the projection of the stripe is not only deformed due to

the shape of the object itself but also due to the movement of the

object caused by vibrations. Therefore, the 3D reconstruction of an

object with a at surface could result in an object with a wavy

surface, as can be seen in Fig. 1. In a different scenario, it is the

laser projector that moves and is affected by vibrations, for

example in a robot arm. However, the result is the same but from

a different point of view. In this work, vibrations are considered

from the point of view of a moving object, although it would be

equivalent from the point of view of a moving laser projector.

The two types of vibrations affecting laser-based 3D reconstruc-

tion methods are translations and rotations. Translations are verticalmovements of the object similar to bouncing, while rotations are

semi-circular movements of the object around a pivot position

similar to rocking. These vibrations occur while the object moves

forward. The two types of vibrations can be present at the same time,

although translations are much more common than rotations.

The type of translations depends on the way the object is

moved. When the object is moved using a conveyor belt, transla-

tions can appear as random vertical movements without a pattern.

On the other hand, semi-sinusoidal translations can appear if the

object hits the conveyor belt periodically. When the object is not

actually touching the ground, for example when moving a sheet of

paper, vibrations can show a sinusoidal pattern while the object

moves forward along a roll path, upwards and downwards.

Fig. 2 shows the resulting effect of translations in 3D recon-

struction when a single laser stripe is used. The gure shows a

simplied example with one single point from a side view. In this

case the object moves forward while it is affected by vibrations, in

particular with translations. In t1 a new point is acquired, but its

position not only depends on the shape of the object but also on

the vertical translation from t0 to t1. In t2 and t3 new points are

acquired that are used to create a reconstruction of the shape of

the object. However, the resulting reconstruction is corrupted due

to the vertical translations, as can be seen in the gure.

Rotations are less common because they only occur when the

object is held at two distant points along the length of the object,

such as steel strips. In these cases, the movement of the object can

be affected by rotations around a pivot position, perpendicular to

the movement.

The two types of vibrations can be described using a mathe-matical model based on geometric transformations. The mathe-

matical principles are described next.

2.1. Geometric transformations

There are four classes of geometric transformations. These

classes can be described in terms of those elements or quantities

that are preserved or invariant, such as for example the distance or

the angle. Also, they form a hierarchy of transformations, where a

class extends the previous class.

Rotations

Translations

Laser projector Camera

3D

reconstruction

Movement

of the object

Fig. 1. General architecture for laser-based 3D reconstruction methods with one single laser stripe.

R. Usamentiaga et al. / Optics and Lasers in Engineering 53 (2014) 515952


3/9

Isometry is the most basic class. This class of transformationpreserves the Euclidean distance. When the isometry also pre-

serves the orientation it is called Euclidean transformation. This

type of isometry is a composition of translation and rotation.

Euclidean transformations model the motion of rigid objects.

The second class of transformation is similarities. A similarity

transformation is an isometry composed with an isotropic scaling

(uniform scaling in all axes). This class of transformation does not

preserve the distance, but the ratio of two lengths.

An afne transformation is an extended version of a similarity

transformation with an anisotropic scaling added (different scaling

for each axis). This class of transformation preserves the ratio of

areas and parallel lines. Angles are not preserved, so that shapes

are skewed under the transformation.

The fourth class is projective transformations. This class gen-eralizes an afne transformation by considering the non-linear

effects of the projection. The projective invariant is the cross ratio

of lengths on a line.

These classes of geometric transformations are composed of

simple transformations, such as translation, rotation, or reection.

Translation is not linear in 2D, i.e., it cannot be represented with

2 2 matrix. Thus, homogeneous coordinates are used by adding a

third coordinate. A 2D point, P x;y, is transformed to homo-

geneous coordinates using

Px

y

!-P

x

y

1

0

B@

1

CA 1

In homogeneous coordinates, a translation can be expressedusing(2), where tx and ty are the horizontal and vertical transla-

tions, respectively

T

1 0 tx

0 1 ty

0 0 1

0B@

1CA 2

Scaling can be expressed using(3), wheresis the scaling factor,

in this case equal for both axes

S

s 0 0

0 s 0

0 0 1

0B@

1CA 3

Rotation can be expressed using (4), where

is the rotationangle

R

cos sin 0

sin cos 0

0 0 1

0B@

1CA 4

These three translations can be combined into a single matrix,

H, using matrix multiplication (5). H represents a similarity

transformation. Removing the scale parameter would result in

Euclidean transformation. These combined transformations are

applied from right to left

H T SR

s cos s sin tx

s sin s cos ty

0 0 1

0

B@

1

CA 5

Height

Length

Height

Length

Height

Length

Height

Length

Movement

Movement

Movement

Verticaltranslation

Vertical

translation

Verticaltranslation

Reconstruction

Realshape

t3

t2

t1

t0

Fig. 2. Effects of vibrations in laser-based 3D reconstruction with one single laser stripe.

R. Usamentiaga et al. / Optics and Lasers in Engineering 53 (2014) 5159 53


4/9

Any point could be transformed using (6), which produces

Eqs.(7) and (8)

P V P-

x

y

1

0B@

1CA

s cos s sin tx

s sin s cos ty

0 0 1

0B@

1CA

x

y

1

0B@

1CA 6

xxs cos ys sin tx 7

yxs sin ys cos ty 8

2.2. Modeling vibrations based on geometric transformations

Two types of vibrations affect objects while they are moved or

transported under the laser stripe: translations and rotations.

Translations are vertical movements of the object while it is being

moved. This type of vibrations could be modeled using the

translation matrix(9), where ty represents the vertical movement

T

1 0 0

0 1 ty

0 0 1

0B@

1CA

9

The other type of vibrations considered is rotations. This type ofvibrations occurs on exible objects, such as steel strips or paper

sheets. Rotations are semi-circular movements of the object

around a pivot position. This type of vibrations could be modeled

using the rotation matrix using (4), where represents the

rotation angle. However, this transformation is only valid when

the pivot position is at the origin. When the rotation is not carried

out around the origin but around a pivot point xp;yp, i t i s

necessary to perform previously a translation from the pivot point

to the origin, a rotation, and nally a translation from the origin

back to the pivot point. The process can be expressed with a

combination of two translations and one rotation matrix using

R

1 0 xp

0 1 yp

0 0 10B@ 1CA

cos sin 0

sin cos 0

0 0 10B@ 1CA

1 0 xp

0 1 yp

0 0 10B@ 1CA 10

A particular type of vibration Vcomposed of translation and

rotation can be modeled as a combination of(9) and(10), using

matrix multiplication

V RT

cos sin xp cos tyyp sin xp

sin cos xp sin tyyp cos yp0 0 1

0B@

1CA11

A point P over the surface of an object with vibrations V is

transformed using(12), which produces Eqs.(13)and (14)

P V P 12

xx cos y sin xp cos tyyp sin xp 13

yx sin y cos xp sin tyyp cos yp 14

These equations create a model for vibrations that determines

the transformation of points in space. The model is parametrized

by four values: , xp,yp, and ty.

3. Estimating vibrations

The model for vibrations is based on four unknown values: ,

xp, yp, andty. Without a priori information about the shape of the

object or about the type of vibrations it is not possible to estimate

these values. The proposed procedure in this work is to use

multiple laser stripes. This provides redundant information that

can be used to estimate vibrations, and thus, to remove them.

The proposed architecture can be seen in Fig. 3. The projections

of the laser stripes on the object are very close. Thus, the move-

ment of the object caused by vibrations affects the deformation of

the laser stripes equally. When the object moves forward, a new

image is acquired. In the new image some of the laser stripes are

projected at the same position as the previous image. The shape of

the object in these two images is the same; the difference is only

caused by vibrations. Therefore, this redundant information can be

used to estimate vibrations.

Fig. 4 shows how two laser stripes can be used to remove

vertical translations. Again, the gure shows a side view with one

longitudinal line, but the idea is the same with more points. In t0two points are acquired. The vertical distance between these two

points,P0t0and P1t0, only depends on the shape of the object.

Later int1, after the object is moved, two new points are acquired,

P0t1andP1t1. The pointP0t1should be in the same position as

P1t0, but a vertical translation has modied it. The vertical

translation can be estimated by subtracting P0t1 from P1t0.

The resulting value, Vt1 is the vertical translation from t0 to t1.

This vertical translation has affected P0t1 and P1t1 equally,

therefore vibrations can be removed in P1t1 by adding Vt1,

which results inPP1t1. Int2a new vertical translation occurs. This

new vertical translation, Vt2, is estimated as P1t1P0t2. Now,

vibrations inP1t2are removed by adding Vt2and alsoVt1, i.e.,PP1t2 P1t2 Vt1 Vt2. IfVt1 Vt2 is called VVt2, then

PP1t3would be calculated asP1t3 VVt2 Vt3. For a general

case, the position of a pointPP1tiwithout vibrations is calculated

as P1ti VVti1 Vti. This procedure completely removes

vertical translations, producing a perfect reconstruction of the

shape of the object. For more complex vibrations that include

rotations more than two laser stripes are required. The mathema-

tical procedure is described next.

3.1. Estimating vibrations with three or more laser stripes

In(12) there are four unknown values: , xp, yp, and ty. These

values dene the vibration. In order to calculate these four values

four equations are needed. These four equations can be created

from two points, P1 x1;y1 and P2 x2;y2, and from the

obtained values after the vibration has transformed their positions

and they are acquired again, P1 x

1;y

1 and P

2 x

2;y

2. This

means that at least three laser stripes are required, so that two

points are measured twice, as can be seen inFig. 3. Each pair of

points produces two equations, one for x and one fory. Therefore,

with the two pairs of points four equations are generated.

Gathering all values, model (15)can be written as

x1 x

2

y1 y

2

1 1

0

B@

1

CA V

x1 x2

y1 y2

1 1

0

B@

1

CA15

Rotations

Translations

Laser projectors Camera

3Dreconstruction

Movementof the object

Fig. 3. Improved architecture for laser-based 3D reconstruction methods.



5/9

The system of equations in model(15)is non-linear and cannotbe solved analytically. Therefore, it is not an appropriate model for

fast and accurate computation. However, it can be transformed

into a linear system by creating a set a intermediate variables. Four

variables are created, V1, V2, V3, andV4, which correspond to

V1 cos 16

V2 sin 17

V3xp cos tyyp sin xp 18

V4 xp sin tyyp cos yp 19

Substituting (16)(19)in (11) produces(20). Using this deni-

tion of V, a point P x;

y over the surface of an object withvibrationsV is transformed using (21), which produces Eqs. (22)

and(23)

V

V1 V2 V3

V2 V1 V4

0 0 1

0B@

1CA 20

P V P-

x

y

1

0B@

1CA

V1 V2 V3

V2 V1 V4

0 0 1

0B@

1CA

x

y

1

0B@

1CA 21

x V1xV2yV3 22

y V2x V1y V4 23

The unknown values in (22) and (23) are V1, V2, V3, and V4.Thus, these equations can be transformed into(24)and(25). These

equations can be expressed in matrix form, as shown in (26)

xV1yV2V3x 24

yV1xV2V4y 25

x y 1 0

y x 0 1

! V1V2

V3

V4

0BBB@

1CCCA

x

y

! 26

The unknown values V1, V2, V3, and V4 can be calculated withtwo points, P1 x1;y1 and P2 x2;y2, and from the obtained

values after the vibration has transformed their positions,

P1 x

1;y

1 and P

2 x

2;y

2. The nal system of equations is

shown

x1 y1 1 0

y1 x1 0 1

x2 y2 1 0

y2 x2 0 1

0BBBB@

1CCCCA

V1

V2

V3

V4

0BBB@

1CCCA

x1y1x2y2

0BBBB@

1CCCCA 27

Gathering all the unknown parameters into a column vector,

model(27)can be rewritten as (28), where X V1; V2; V3; V4T

AX B 28

Height

Length

Height

Length

Height

Length

Height

Length

Movement

Movement

Movement

Verticaltranslation

Vertical

translation

Verticaltranslation

ReconstructionRealshape

t3

t2

t1

t0

Fig. 4. Removing vibrations in laser-based 3D reconstruction using multiple stripes.



6/9

The system of equations in (28)is linear and can be solved as

XA1B 29

When the system is solved, the unknown value can be

determined using (30). The values of xp, yp, and ty cannot be

calculated directly. Instead, global values for translation in both

axes,tx andt

y, are calculated as(31)and (32). Although the nal

values ofxp,yp, andtyare unknown, the model for the vibrations V

is perfectly de

ned. Thus, the transformation of the points in spacedue to vibrations is known, and it can be reversed

tan 1V2V1

30

tx V3 31

ty V4 32

This procedure could also be applied to estimate a similarity

transformation. In this case the scale parameter, s, could be

calculated as

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV11V

22

q 33

To calculate the unknowns in the transformation matrix V, at

least two pair of points are needed. However, more than two

points could also be used. In this case, matrixA would have more

rows than columns and the systems of equations in (28)cannot be

directly solved. With the purpose of minimizing the distances, the

least square solution for the parameter vector Xcan be computed

by

X ATA1ATB 34

Solving the least square problem using (34) could lead to

numerical instabilities[23]. In general, an orthogonal decomposi-

tion is preferred. These methods are slower but more numerically

stable. An alternative decomposition for X is the singular value

decomposition (SVD)[24]. This method is the most computation-

ally intensive, but is particularly useful if round-off errors could

create singularity problems in matrix X.

3.2. Estimating vibrations with only two laser stripes

When only two laser stripes are used, translations and rota-

tions cannot be estimated because there is only one point

measured twice. A single point can be used to create two

equations, which are not enough to calculate the four unknown

values. However, this point can be used to estimate translations

and assume that there are no rotations. In this case, vibrations can

be modeled with a single translation matrix as follows:

V T

1 0 0

0 1 ty

0 0 1

0B@

1CA 35

In order to estimate the translation with P1 x1;y1 and from

the obtained value after the vibration has transformed its position,

P1 x

1;y

1, (36)could be used

tyy1y

1 36

When three or more laser stripes are used there is a possibility

that the system of equations does not have a solution, that is, the

coefcient matrix is singular, for example due to repeated points.

In this case, vibrations could be estimated the same as when using

only two laser stripes, only translations are taken into account.

4. Results and discussion

In order to assess the proposed procedure, several experiments

have been carried out. First, a synthetic object was created. The

object contains a low frequency curve at the center and highfrequency curves at the edges, as can be seen in Fig. 5(a). Different

types of vibrations were added to the object. The resulting 3D

reconstructions using a single laser stripe are shown in Fig. 5(b)

(d).Fig. 5(b) shows the object with uniformly distributed random

translations of 3% of the object height (Rand). Fig. 5(c) shows the

object with rotations of 0.51 around the center of each prole

(Rot). Fig. 5(c) shows the object with a combination of sinusoidal

translations and rotations with same amplitudes as before

(SinRot). These types of vibrations have been used because they

are common in many applications.

In order to test the proposed procedure, it is assumed that

there are multiple laser stripes available for 3D reconstruction.

When only two laser stripes are used, only translations are

estimated and removed. The difference between the original and

the reconstructed shape of the object is assessed with the mean

absolute error (MAE) using(37), wheren is the number of points,

Hi is the height at point i, and Ri is the reconstructed height at

point i. This value is normalized using (37), where Hmax is the

maximum height. This error value is a percentage over the

maximum height of the object, and it is independent from the

specic height of the object

MAE1

n

n

i 1

HiRijj 37

Fig. 5. Synthetic vibrations used for tests. (a) Real shape of the object. (b) Shape of the object after adding random translations. (c) Shape of the object after adding rotations.

(d) Shape of the object after adding sinusoidal translations and rotations.



7/9

Error 100MAE

Hmax38

Table 1shows the error when using two, three and four laser

stripes with different types of vibrations. The error in all cases is

very low. Most errors are around 1e14, which is very close to the

representation error of real numbers and it can be considered as

zero (no error). In the case of random translations, the error is null

in all cases. When rotations are added and three or more stripes

are used the error is also null. However, when only two stripes are

used the error increases. The obtained errors with two stripes are

still very low and can be considered negligible for most applica-

tions. The results show that there is no difference between only

rotations and translations plus rotations.

The obtained errors with two stripes are very low. The reason is

that estimating only translations removes most of the vibrations,

even when there are rotations. This can be clearly seen with an

example. Let us suppose there are two points with these values:

P1 5; 5 and P2 5; 5:1. Then, the position of these points is

measured after vibrations with ty 1 and 0:251 (around the

origin) transform the positions of the points. The result is

P1 4:973; 6:021 and P

2 4:973; 6:121. If only one point is

used to estimate vibrations (P1 and P

1, two laser stripes), the result

would be ty1.021 and 0. Removing this vibration from P

2

results in a height of 5.099, that is, the error compared to P2 is

1:8e05. This error depends on the rotation and also on the

proximity of the height of P1 and P2. However, consecutive points

tend to be very close and the rotations in vibrations are low. Thus,

this indicates that for most applications two laser stripes could be

enough.

Fig. 6 shows the differences between the real object and the

reconstructed object using two and three laser stripes when

vibrations consisting of sinusoidal translations and rotations are

added. As can be seen, with two stripes errors appear at the center

and at the edges, where there are curves in the object. However,

these errors are lower than 1e05, negligible in most cases.

The error when using two laser stripes depends on the degrees

of rotation.Fig. 7shows the error for different degrees. As can beseen, the error increases, but the value for 601 is still low.

Moreover, 601 is an unrealistic high value in real applications.

The proposed procedure assumes that the same points in an

image are acquired again in the next image in a different laser stripe

with accuracy. However, repeatedly measuring points always incur in

measurement errors due to acquisition uncertainties and noise.

Table 2shows the error when using two, three and four laser stripes

with different types of vibrations and when there is a 1% randomly

distributed measurement error. This measurement error is added to

the positions of the transformed points after vibrations. The recon-

struction errors increase notably compared with the previous experi-

ments. Also, there is a major difference: the results when using only

two laser stripes are better than when three or four are used. The

reason is that adding noise can provoke spurious estimations of the

vibrations. This problem nally results in outliers in the reconstruc-

tion of the shape.

Fig. 8 shows the resulting reconstruction of the object using

two and three laser stripes when vibrations consisting of sinusoi-dal translations and rotations are added, and with a 1% randomly

distributed measurement error. The added noise provokes errors

in the reconstruction. When using three laser stripes, outliers can

be clearly seen.

An example can be used to understand the problem with three

laser stripes and outliers. Let us suppose the same two points as in

the previous example:P1 5; 5,P2 5; 5:1,P

1 4:973; 6:021

and P2 4:973; 6:121. Vibrations are also ty1 and 0:251(around the origin). Assuming 1% error in the repeated measure-

ment, the values of the transformed points would be

P1 4:973; 6:081 and P

2 4:973; 6:060. The rst point is mea-

sured 1% above and the second point is measured 1% below (worst

case scenario). These measurement errors will result in an estimation

of vibrations ty

7.13 and 1:1661, very different from the real

vibration. Let us suppose the third point affected by the same

vibrations is P3 5; 5:2. Removing the real vibration would pro-

vide a height of 4.178, but removing the estimated vibrations would

provide a height of 9.11. Therefore, the result is an outlier.

The estimation of vibrations with noisy data can be improved

by increasing the number of laser stripes. Fig. 9 shows the

Table 1

Reconstruction error when using multiple stripes.

Stripes Rand Rot Sin Rot

2 1:9e16 2:7e06 2:7e06

3 2:1e15 4:8e15 4:8e15

4 9:5e14 8:0e14 8:0e14

Fig. 6. Error maps with vibrations consisting of sinusoidal translations and rotations. (a) Using two laser stripes. (b) Using three laser stripes.

0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

Rotation degree

Error

Fig. 7. Error when using two laser stripes and vibrations with different degrees of

rotation.

Table 2

Reconstruction error when using multiple stripes and there is measurement

uncertainty.

Stripes Rand Rot Sin Rot

2 9:7e02 1:1e01 1:1e01

3 6:6e01 5:1e01 5:4e01

4 3:6e01 3:3e01 4:2e01



8/9

resulting error for a different number of stripes. The error

increases sharply from two to three stripes, and it decreases

slowly as the number of stripes is increased. When there are a

large number of stripes the error is low due to the improved

robustness of the estimation of vibrations.

The proposed procedure has been applied to real data. In this

case, steel strips have been used. Strips are generally 2 m wide and

10 km long. 3D reconstruction is required for quality control, such

as atness inspection[25]. The laser source is located at the centerof the roll path making the laser plane perpendicular to the plane

on which the strip lies. The camera is located at 451with respect to

the laser plane, also centered over the roll path of the production

line. The camera used is a Basler A501k, which provides images of

1280 1024 pixels using a CMOS sensor at 74 frames per second

with 8 bit depth.

Vibrations in steel strips consist of high frequency vertical

translations. They have a period around 20 cm with a magnitude

of approximately 5 mm. This type of vibrations can be effectively

removed using low-pass lters. Fig. 10(a) and (c) shows two

examples of steel strips reconstructed using a single laser stripe.As can be appreciated, the shape of the strips is corrupted by

vibrations. Shape quality metrics calculated from this reconstruc-

tion would result in large errors. Vibrations can be estimated from

data, as they are the high frequency components of the shape.

Then, the proposed procedure with two laser stripes can be

applied simulating this particular type of vibrations. Fig. 10

(b) and (d) shows the results. The proposed procedure estimates

and removes all vibrations, producing a 3D reconstruction with a

smooth surface. Manual measurements of the shape of the strips

conrm the success of the resulting 3D reconstruction.

The proposed procedure to remove vibrations using multiple

laser stripes is very likely to nd potential applications in a

number of different areas. Vibrations affect the movement of the

objects to a greater or lesser extent in any 3D reconstruction

application based on laser light. Thus, height proles extracted

with a single laser stripe will contain errors, as the reconstructed

shape will be affected by these vibrations. Therefore, the proposed

procedure provides the opportunity to calculate a much more

accurate 3D reconstruction of the object, regardless of the parti-

cular application.

5. Conclusion

Vibrations are a major issue for laser-based 3D reconstruction

methods. The movement of the object caused by vibrations affects

3D reconstruction and leads to large errors. The proposed proce-

dure in this work is to use multiple laser stripes and estimate and

remove vibrations from 3D reconstruction. First, vibrations aremodeled based on geometrical transformations. In particular,

using translations and rotations, the most common types of

vibrations. Then, a mathematical procedure to estimate vibrations

from data is developed. Using two laser stripes, only translations

can be estimated. Using three laser stripes, both the translations

and the rotations can be estimated. Using more than three laser

stripes produces an overdetermined system that delivers a more

Fig. 8. Error maps with vibrations consisting in sinusoidal translations and

rotations. (a) Using two laser stripes. (b) Using three laser stripes.

Fig. 10. 3D reconstruction of steel strips using the proposed procedure. (a) First strip with a single laser stripe. (b) First strip with two laser stripes. (c) Second strip with a

single laser stripe. (d) Second strip with two laser stripes.

2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

Number of stripes

E

rror

Fig. 9. Error when using multiple laser stripes with noisy data.



9/9

robust solution. Multiple tests have been carried out to evaluate

the performance of the proposed procedure with synthetic and

real data.

Experimental results show excellent performance for the pro-

posed procedure. Using three or more laser stripes completely

removes vibrations from the 3D reconstruction. When only two

laser stripes are used, very low error appears in the results,

negligible for most applications. Furthermore, when tests are

carried out using noisy data, a laser-based 3D reconstructionsystem with only two stripes clearly outperforms systems with

more stripes, unless many stripes are used. The reason is that

adding noise can provoke an incorrect estimation of vibrations

that can nally result in outliers in the reconstruction of the shape.

A laser projector can signicantly increase the cost of a 3D

reconstruction system. Also, using more than two laser stripes is

only a marginal advantage in some cases. When noise is present in

data, more than two laser stripes is a clear disadvantage, as was

demonstrated in the tests. Therefore, unless there are very specic

requirements, a system with only two stripes is recommended. A

laser-based 3D reconstruction method with two laser stripes

presents a far more cost efcient solution than others with similar

or even better performance.

Final tests on real data with two laser stripes indicate that the

obtained 3D reconstruction after applying the proposed procedure

is comparable to manual measurements, effectively removing the

unwanted effects of vibrations on the estimated shape.

Acknowledgments

This work has been partially funded by the Project TIN2001-

24903 of the Spanish National Plan for Research, Development and

Innovation.

References

[1] Szeliski R. Computer vision: algorithms and applications. Springer; 2010.

[2] Frauel Y, Tajahuerce E, Matoba O, Castro A, Javidi B. Comparison of passiveranging integral imaging and active imaging digital holography for three-dimensional object recognition. Appl Opt 2004;43(2):45262.

[3] Tombari F, Mattoccia S, Di Stefano L, Addimanda E. Classication and evalua-tion of cost aggregation methods for stereo correspondence. In: IEEE con-ference on computer vision and pattern recognition, 2008. CVPR 2008. IEEE;2008. p. 18.

[4] Ballan L, Cortelazzo GM. Multimodal 3D shape recovery from texture, silhou-ette and shadow information. In: Proceedings of the third internationalsymposium on 3D data processing, visualization, and transmission(3DPVT'06), 2006. p. 92430.

[5] Salvi J, Pags J, Batlle J. Pattern codication strategies in structured light

systems. Pattern Recognition 2004;37(4):82749.[6] Forest J, Salvi J. A review of laser scanning three-dimensional digitisers. In:

IEEE/RSJ international conference on intelligent robots and system, vol. 1,

2002.[7] Hongjian Y, Shiqiang Z. 3d building reconstruction from aerial CCD image and

sparse laser sample data. Opt Lasers Eng 2006;44(6):55566.[8] Yu S-J, Sukumar SR, Koschan AF, Page DL, Abidi MA. 3d reconstruction of road

surfaces using an integrated multi-sensory approach. Opt Lasers Eng 2007;45

(7):80818.

[9] Molleda J, Usamentiaga R, Garcia DF, Bulnes FG, Ema L. Shape measurement ofsteel strips using a laser-based three-dimensional reconstruction technique.

IEEE Trans Ind Appl 2011;47(4):153644.[10] Liu Z, Sun J, Wang H, Zhang G. Simple and fast rail wear measurement method

based on structured light. Opt Lasers Eng 2011;49(11):134351.[11] Fernandez A, Garcia R, Alvarez E, Campos AM, Garcia D, Usamentiaga R, et al.

Low cost system for weld tracking based on articial vision. IEEE Trans Ind

Appl 2011;47(3):115967.[12] Isheil A, Gonnet JP, Joannic D, Fontaine JF. Systematic error correction of a 3d

laser scanning measurement device. Opt Lasers Eng 2011;49(1):1624.[13] Forest J, Salvi J, Cabruja E, Pous C. Laser stripe peak detector for 3d scanners. a

FIR lter approach. In: Proceedings of the 17th international conference on

pattern recognition. ICPR 2004, vol. 3. IEEE; 2004. p. 6469.[14] Schnee J, Futterlieb J. Laser line segmentation with dynamic line models. In:

Computer analysis of images and patterns. Springer; 2011. p. 12634.[15] Usamentiaga R, Molleda J, Garcia DF. Fast and robust laser stripe extraction for

3D reconstruction in industrial environments. Mach Vision Appl 2012;23:

17996.

[16] Molleda J, Usamentiaga R, Garcia D, Bulnes FG, Espina A, Dieye B, et al. Animproved 3D imaging system for dimensional quality inspection of rolled

products in the metal industry. Comput Ind, http://dx.doi.org/10.1016/j.com

pind.2013.05.002 , in press.[17] Pernkopf F. 3d surface acquisition and reconstruction for inspection of raw

steel products. Comput Ind 2005;56(8):87685.[18] Usamentiaga R, Garcia DF, Molleda J, Bulnes FG, Peregrina S. Vibrations in steel

strips: effects on atness measurement and ltering. In: 43st IAS annual

meeting. Conference record of the 2013 IEEE industry applications conference,

vol. 1. IEEE; 2013. p. 110.[19] Doignon C, Knittel D. A structured light vision system for out-of-plane

vibration frequencies location of a moving web. Mach Vision Appl 2005;16

(5):28997.[20] Maurice X, Doignon C, Knittel D. On-line vibrations characterization of a

moving exible web with a fast structured lighting stereovision system. In:

IEEE Instrumentation and Measurement Technology Conference Proceedings,

2007. IMTC 2007. IEEE; 2007. p. 15.[21] Bell JR, Rothberg S. Rotational vibration measurements using laser doppler

vibrometry: comprehensive theory and practical application. J Sound Vib2000;238(4):67390.[22] Castellini P, Martarelli M, Tomasini E. Laser doppler vibrometry: development

of advanced solutions answering to technology's needs. Mech Syst Signal

Process 2006;20(6):126585.[23] Bjorck A. Numerical methods for least squares problems. Soc Ind Appl Math

1996;51.[24] Lawson CL, Hanson RJ. Solving least squares problems, vol. 161. SIAM; 1974 .[25] Molleda J, Usamentiaga R, Garcia D, Bulnes F. Real-time atness inspection of

rolled products based on optical laser triangulation and three-dimensional

surface reconstruction. J Electron Imaging 2010;19:031206.

http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref1http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref1http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0015http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0015http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0015http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0025http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0025http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0025http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0025http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0030http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0030http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0030http://dx.doi.org/10.1016/j.compind.2013.05.002http://dx.doi.org/10.1016/j.compind.2013.05.002http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref23http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref23http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref23http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref24http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref24http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref25http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref24http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref23http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref23http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref22http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref21http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0040http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref19http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0035http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref17http://dx.doi.org/10.1016/j.compind.2013.05.002http://dx.doi.org/10.1016/j.compind.2013.05.002http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0030http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0030http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0030http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref15http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0025http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0025http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0020http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref12http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref11http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref10http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref9http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref8http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref7http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0015http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0015http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0015http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref5http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0010http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/othref0005http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref2http://refhub.elsevier.com/S0143-8166(13)00244-3/sbref1

articol science

Documents