3D Geometric Primitive Alignment Revisited

Trung-Thien Tran, Van-Toan Cao and Denis Laurendeau

Computer Vision and Systems Laboratory, Laval University, 1065, avenue de la Medecine, Quebec, G1V 0A6, Canada

Keywords:

Plane, Cylinder, Sphere, Primitive-based Alignment, Rigid Registration, Coarse Registration.

Abstract:

Rigid registration is an important step in 3D scanning and modeling of manufactured objects that are gener-

ally composed of a combination of basic geometric primitives such as planes, spheres, cylinders, etc. In this

paper, an efﬁcient and robust method is proposed to align two basic geometric primitives. The transformation

between two primitives is found by minimizing the parameter error between primitive correspondences. The

approach applies an interior-point method and a new objective function to achieve good results. Compared

to previous primitive-based alignment approach proposed by Rabbani et al. (Rabbani et al., 2007), the pre-

sented approach achieves better results in terms of convergence and accuracy. Finally, the proposed method is

used in various applications such as data completion and primitive-based registration for quality control and

inspection.

1 INTRODUCTION

3D sensors are being used more and more widely in

several applications, especially in industrial manufac-

turing. Since the ﬁeld of view of these sensors is

limited, it is necessary to scan an object from dif-

ferent viewpoints in order to cover its complete sur-

face. These local scans are then used in the mod-

eling and reconstruction process. Among common

problems, a registration step is inevitable to transform

all of the scans into a common coordinate reference

frame. Registration is also referred to as alignment

or matching, so these terms are used interchangeably

throughout the paper.

The 3D registration problem can be classiﬁed in

several ways. On one hand, according to the rigidity

of the objects, registration is categorized as rigid and

non-rigid as summarized in (Tam et al., 2013). On the

other hand, following the quality of the result, regis-

tration methods can also be classiﬁed as coarse regis-

tration (Planitz et al., 2005; D

ıez et al., 2015) and ﬁne

registration (Besl and McKay, 1992; Rusinkiewicz

and Levoy, 2001; Salvi et al., 2007). In this paper,

we investigate the problem of coarse registration in

which the parts are aligned roughly with each other.

From now on, the word registration will refer to the

coarse registration problem.

Currently, registration methods often address the

problem of aligning meshes or point clouds having

similar densities of data points. 3D descriptors (D

ıez

et al., 2015) are proposed and keypoints (Tombari

et al., 2013) are extracted from 3D data for correspon-

dence and shape matching. However, most existing

descriptors and keypoints fail to register two scans of

man-made objects, because the main challenge is to

propose a descriptor that is able to deal with symmet-

rical and similar geometries from the primitive sur-

faces.

Some research has been conducted to align

datasets using primitives as descriptors. The primi-

tives could be linear and circular features as in (Sta-

mos et al., 2008) or planes in (Bosch

e, 2012). Lin-

ear features (Chao and Stamos, 2005) between two

planes and circular features (Stamos et al., 2008)

from 2D and range images of buildings and urban

scenes are used. Bosche’s approach (Bosch

e, 2012)

use planes that are extracted at a single point se-

lected by the user. Iterative Closest Point (ICP) is

then used to compute the transformation using three

pairs of plane correspondences. Rabbani et al. (Rab-

bani et al., 2007) have also used geometric primi-

tives such as planes, cylinders and spheres for regis-

tration and modeling. The method minimizes the sum

of squares of the differences of parameters by using

the Levenberg-Marquardt optimization (LM) (Mar-

quardt, 1963). The proposed approach exploits this

idea and brings many improvements in terms of con-

vergence and accuracy.

The contributions of the paper are summarized as

follows.

Tran, T-T., Cao, V-T. and Laurendeau, D.

3D Geometric Primitive Alignment Revisited.

DOI: 10.5220/0005715800890096

In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2016) - Volume 1: GRAPP, pages 91-98

ISBN: 978-989-758-175-5

• A robust approach is proposed to align geometric

primitives. Each type of primitive (plane, sphere,

cylinder) is described in detail in Sec. 2.3.

• A new minimization technique is exploited to

achieve better results than other techniques with

respect to convergence and error of descriptive pa-

rameters.

• The proposed method is used in primitive-based

alignment and quality control applications, which

registers two scans to create an error map or com-

plete model of the object of interest.

2 PRIMITIVE ALIGNMENT

2.1 Quaternion Representation

Rigid registration consists of ﬁnding a unique trans-

formation T : R

7→ R

, which brings two datasets as

close as possible. This transformation is expressed

as a 3x3 rotation matrix R and a translation vector

t = {t

}. Horn (Horn, 1987) has proposed a

closed-form solution for determining a rotation ma-

trix using a quaternion representation. With a given

quaternion q = {q

}, the rotation matrix R

is computed as

R =







+ 2q

− 1 2(q

− q

) 2(q

+ q

)

2(q

+ q

) 2q

+ 2q

− 1 2(q

− q

)

2(q

− q

) 2(q

+ q

) 2q

+ 2q

− 1







with the constraint kqk =

+ q

1 which guarantees a rotation operator in R

having

only three degrees of freedom.

2.2 Rabbani’s Approach and Associated

Drawbacks

Rabbani et al. (Rabbani et al., 2007) have proposed

an approach that combines registration and modeling

in one framework. At the registration step, quater-

nion and geometric primitives are used to align two

scans called S

and S

, respectively. The transforma-

tion T is calculated using a least-squares algorithm in

the parameter space of the primitives. In other words,

this step minimizes the sum of the square of errors of

primitive parameters as follows:

min

{R(q),t}

|C|

∑

∆

i j

(1)

where ∆

i j

= u

i j

−T(u

i j

) and u

i j

is the j-th parameter

of the i-th correspondence that has |M

| criteria of pa-

rameter convergence. C is the set of correspondences.

For example, when the ﬁrst correspondence pair is

plane-plane, |M

| is equal to 2 since there is a cri-

terion for the error on the normal vector and another

criterion for the error on the distance to the origin.

The mathematical equations for each single primitive

are expressed later in Sec. 2.3.

The Levenberg-Marquardt optimization (Mar-

quardt, 1963) is used to solve the non-linear least-

squares problem in Eq. 1. Partial derivatives of ∆

w.r.t of rotation R and translation t are required and

computed as

∂∆

∂q

∂∆

∂q

∂∆

∂q

∂∆

∂q

∂∆

∂t

∂∆

∂t

∂∆

∂t

These derivatives are used to compute the Jaco-

bian matrix J and update the transformation parame-

ter Γ = {q, t} as follows:

(a) Plane represented by normal vector~n

and the distance to the origin ρ.

(b) Cylinder described by axis vector

~a, radius r and a closest point to the ori-

gin in axis p.

radius r

Figure 1: Representative parameter of primitives: plane, cylinder and sphere.

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

Γ = {q, t} = {q

} (2)

t+1

= Γ

− (J

J + λI)

−1

J∆ (3)

J =

∂∆

∂Γ

; (4)

where t is the iteration step, and J is the Jacobian

matrix. Observing Eq.3, the Levenberg-Marquardt

optimization consists of Gauss-Newton and Steep-

est descent methods by using a non-negative scalar

λ, which is called the Levenberg-Marquardt param-

eter. The selection of λ and its impact on conver-

gence are discussed in (Press et al., 2007). Gen-

erally, Levenberg-Marquardt optimization achieves

better convergence compared to Gauss-Newton and

Steepest descent methods taken separately.

Rabbani’s approach faces many problems that

have an impact on the convergence of the result.

• First, the LM optimization generally provides a

solution to an unconstrained problem. Therefore,

the method does not guarantee that the solution

provides a unit quaternion kqk = 1 for each iter-

ation of the optimization loop. This leads to poor

results for the rotation matrix R. This behavior

has been tested in the experiments reported in Sec.

3. The method proposed in this paper uses an inte-

rior point technique with constraints to guarantee

that condition (kqk = 1) is satisﬁed.

• Second, the objective function for cylinders lacks

a robust criterion for alignment, which has an ad-

verse effect on convergence. The proposed ap-

proach complements the objective function with

a new criterion that guarantees good convergence

(Eq. 8).

2.3 Proposed Approach for Primitive

Alignment

In this section, the detailed framework for registering

two primitives is presented. These primitives can be

planes, cylinders and spheres. We assume the corre-

spondence pairs C = {S

} between primitives are

already determined. The transformation T = {R, t} is

computed by minimizing the following cost function.

min

{R(q),t}

|C|

∑

∆

i j

subject to:kqk =

+ q

= 1

(5)

where the error is deﬁned as ∆

i j

= u

i j

− T(u

i j

To solve the non-linear least-squares problem in

Eq. 5, the interior-point method (Forsgren et al.,

2002) is used. The interior point method guarantees

that the constraint kqk = 1 is always satisﬁed at each

iteration of the optimization loop. The mathematical

equations for aligning each type of primitive are de-

scribed in the following.

2.3.1 Plane

Planes are geometric primitives frequently found on

manufactured objects. A plane is represented by a

normal vector

n and a distance from the closest point

to the origin ρ, as shown in Fig. 1(a). Assume that

there are two planes with parameters {

,ρ

} and

{

,ρ

}, respectively.

To align plane 2 to plane 1, the error on the normal

vector and the error on the distance to the origin are

minimized as follows:

− R

(6)

= ρ

− ρ

+ (R

)

t (7)

Eq. 6-7 are fed to Eq. 5 for the minimization step.

2.3.2 Cylinder

A cylinder is described by three parameters: its axis

vector

−→

a , its radius r and a closest point to the origin

on the cylinder axis p, as shown in Fig. 1(b). To bring

a cylinder C

= {

−→

;r} as close as possible to a

cylinder C

= {

−→

;r}, two points p

and p

and

two vectors

−→

and

−→

need to coincide.

− R

(8)

= R

+ t − (

∗ t

)

−

(9)

where

= R

Eq. 8-9 are fed to Eq. 5 for the minimization step.

2.3.3 Sphere

A sphere is simply represented by its center c and ra-

dius r, as shown in Fig. 1(c). To align a sphere 2

,r} to sphere 1 {c

,r}, the two centers need to co-

incide (c

= 0).

= Rc

+ t − c

(10)

Eq. 10 is fed to Eq. 5 for the minimization step.

The framework is applied to align a set of sev-

eral multiple primitives simultaneously. The objec-

tive functions of each primitive pair are combined and

minimized in Eq. 5. When the minimization ends, the

transformation between the datasets is available.

3 RESULTS AND DISCUSSION

In this section, the results of the proposed approach

are compared to the ones reported by Rabbani’s in

3D Geometric Primitive Alignment Revisited

(Rabbani et al., 2007). Both methods use geomet-

ric primitives for alignment. The sets of primitives

= {P

} and their parameters are known a

priori. Dataset Ψ

is transformed by 100 sets of trans-

formation T randomly generated to create new sets of

primitives Ψ

= {P

}. Then four experiments

are conducted.

• Experiment 1: The transformation T

needed to

transform plane P

to plane P

is estimated. The

convergence is evaluated by

cos(∠(T

),n

)) == 1 (11)

• Experiment 2: The transformation T

needed to

align cylinder C

to cylinder C

is estimated. The

convergence is evaluated by

cos(∠(T

),a

)) == 1 (12)

• Experiment 3: The transformation T

to align a

sphere S

to sphere S

is also computed. The con-

vergence is then evaluated by

− T

== 0 (13)

• Experiment 4: All types of primitives are com-

bined simultaneously into the framework. The

convergence is reached if Eq. 11-13 are satisﬁed

simultaneously.

The proposed framework is implemented on

MATLAB on a 3.2 GHz Intel Core i7 platform.

In the implementation of primitive alignment, Rab-

bani et al. (Rabbani et al., 2007) use Levenberg-

Maquardt non-linear optimization which is imple-

mented in the lsqnonlin function in MATLAB. The

proposed framework rather uses the fmincon function

which is implemented for the interior-point method.

The percentage of success and the error on the pa-

rameters are investigated for both methods.

3.1 Success Rate of the Proposed

Method

With 100 randomly generated sets of transformations,

four alignment experiments are carried out and the

conditions in Eq. 11-13 are checked sequentially or

simultaneously. Fig. 2 illustrates the success rate of

both methods LM Rabbani and IP OUR in the four

experiments.

Rabbani’s method is able to align planar and

sphere primitives, but fails with cylinder primitives,

as shown in Fig. 2. Therefore, in general, Rabbani’s

method cannot align a set of multiple primitives. The

approach proposed in this paper achieves high success

rates for all cases (> 90%).

100

Plane Cylinder Sphere Primitives

LM_Rabbani

IP_OUR

Success Rate

Figure 2: Success rate for Rabbani’s method (LM Rabbani)

and the proposed method (IP OUR).

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

Plane Cylinder Sphere Primitives

LM_Rabbani

IP_OUR

Mean Absolute Error

Figure 3: Logarithmic graph of mean absolute errors

for Rabbani’s method (LM Rabbani) and our method

(IP OUR).

3.2 Errors on Primitive Parameters

The mean absolute error (MAE) values in Eq. 11-

13 are computed for each of the 100 random sets of

transformation.

• Experiment 1: The error on the plane normal after

transformation is computed as follows:

MAE

100

∑

i=1



1 − |cos(∠(T

),n

))|



(14)

• Experiment 2: The error on axis directions of the

cylinders after transformation is computed as:

MAE

100

∑

i=1



1 − |cos(∠(T

),a

))|



(15)

• Experiment 3: The error of the sphere centers af-

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

(a) CAD model designed

with 3DS Max.

(b) The plastic object

printed with a Stratasys

Dimension 1200es printer.

tic model scanned with the

GO!SCAN 3D sensor.

(d) Complete scan of the

plastic model.

Figure 4: The plastic object is printed and scanned to create partial and complete scan datasets.

ter transformation is computed:

MAE

100

∑

i=1



− T



(16)

• Experiment 4: The error on combined primitives

taken altogether is computed as:

MAE

∑

= MAE

+ MAE

(17)

A logarithm plot of the MAE for the four exper-

iments is shown in Fig. 3. The proposed method

clearly outperforms Rabbani’s method in aligning

primitives.

From the results of the experiments, it is con-

cluded that the proposed method, which combines a

new optimization technique and a new objective func-

tion, solves the problem of primitive-based alignment

with better results with respect to convergence and er-

ror of descriptive parameters.

One more experiment has been conducted to show

the performance of the proposed method. The esti-

mated transformation T

in the 4th experiment is com-

pared to the ground-truth transformation T as follows:

• The error of rotation matrix is deﬁned as:

100

∑

i=1

kR − R

= 6.85e

−7

(18)

• Translation vector is commutative, so the error of

translation is computed by:

100

∑

i=1



ktk − kt



= 2.68e

−6

(19)

The values of E

and E

prove that the proposed

method estimates the transformation between datasets

accurately.

3.3 Application of the Proposed

Approach on Real 3D Scans

The ﬁrst application is to register the scanned data of

a printed 3D object with its CAD model. The object

is printed with a Stratasys Dimension 1200es printer

(Fig. 4(b)) and scanned by the Creaform GO!SCAN

3D scanner. Markers are used by the scanner for self

positioning. This part-to-CAD problem is frequently

used in quality control and inspection applications.

The proposed framework for primitive alignment is

applied as follows in this case. First, primitives from

the point cloud are extracted using the method in

(Tran et al., 2015b), and primitives from the CAD

model are extracted by the approached proposed in

(Attene et al., 2006). The correspondence pairs of

primitives are deﬁned manually. After being aligned,

the distances between points and the CAD model are

computed to generate an error map, as shown in Fig.

5. In this experiment, the scanned object’s pose is

close to the one of the CAD model.

Observing the results in Fig. 5, it is observed

that our method outperforms Rabbani’s approach in

terms of alignment errors (Fig. 5(b)-5(c)). On the his-

tograms of the error maps, the error of our method is

smaller than the one of Rabbani’s approach. The rea-

son is that Rabbani’s approach does not guarantee the

condition (kqk = 1) when LM optimization updates

the parameters in the minimization process. More-

over, the lack of objective function for cylinders has

a signiﬁcant inﬂuence on convergence. The proposed

method solves these two problems and achieves better

alignment result.

A more challenging experiment has been con-

ducted in which the pose of the scanned object is in

the opposite direction of the one of the CAD model

(Fig 6). Rabbani’s method fails to align the datasets

while the proposed method achieves a good alignment

result with an error map similar to the one in Fig 5(c).

The second application of the proposed approach

is data completion. The complete model of an object

is often built from multiple partial scans. Separate

scans are needed and must be expressed in a common

coordinate frame using registration. In this section,

the proposed method is used to register two real scans.

In Fig. 7, two scans of the object in Fig. 7(a) are

3D Geometric Primitive Alignment Revisited

(a) Scanned data and CAD model. (b) Error map of the result for Rabbani’s

approach.

posed approach.

(d) Histogram of the error map created from Rab-

bani’s approach, largest value (1.55) mean error

value (0.31).

(e) Histogram of the error map generated from our

proposed method, largest value (0.98) and mean er-

ror value (0.23).

Figure 5: Alignment of scanned data with its CAD model. Error maps of both methods are created and analyzed. The distance

error of our method is smaller than the one of Rabbani’s approach.

(a) (b) (c)

Figure 6: Comparison between two methods in the case where the pose of the scanned object is in the opposite direction of

that of the CAD model. (a) Datasets with opposite pose between the scan and its CAD model. (b) Rabbani’s method fails to

align the datasets. (c) Good alignment result is still achieved by the proposed method.

shown. First, planar, cylindrical and spherical prim-

itives are extracted from the scans using the method

reported in (Tran et al., 2015a; Tran et al., 2016; Tran

et al., 2015b). These primitives are then fed to the

alignment approach presented in Sec. 2.3 to build a

complete model of the object. The resulting model is

shown in Fig. 7(b). The same strategy is used for the

planes on two point clouds of the church in Fig. 7(c)

with the resulting model in Fig. 7(d) after alignment

using planar primitives.

Although primitive correspondences are selected

manually by the user, this operation is not difﬁcult

even for a novice. In addition, this operation is easier

than choosing the points used in manual ICP (Besl

and McKay, 1992) and the other approach proposed

in Bosche et al. (Bosch

e, 2012).

GRAPP 2016 - International Conference on Computer Graphics Theory and Applications

(a) Two partial scans of the plastic model. (b) Complete model created using the extracted primitives

for alignment.

LansleVillard (France). The church is scanned with a LE-

ICA Cyrax scanner.

(d) The church model is built by aligning the extracted

planes from partial data.

Figure 7: Complete models are created from partial data using the proposed method of primitive alignment.

4 CONCLUSION

An efﬁcient and robust method for registering two

basic primitives is proposed in this paper. Using a

new optimization technique and with the addition of

a complementary objective function, the convergence

is guaranteed to achieve accurate results. The pro-

posed method can beneﬁt other applications such as

scan alignment, quality control and inspection.

Currently, the framework of primitive-based regis-

tration manually assigns the correspondence pairs of

primitives between datasets. An automatic detection

of the correspondence between primitives will be the

object of future work.

ACKNOWLEDGEMENT

This research project was supported by the

NSERC/Creaform Industrial Research Chair on

3-D Scanning. The authors thank to AIM@SHAPE

for providing the scanned data of the church in Fig.

7. We are grateful to Annette Schwerdtfeger for

proofreading the manuscript.

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