Gaussian Curvature Criterion based Random Sample Matching for

Improved 3D Registration

Faisal Azhar, Stephen Pollard and Guy Ad ams

HP Labs, Bristol, U.K.

Keywords:

Gaussian Curvature, 3D Registration, Matching, Point Cloud, Hash Table.

Abstract:

We propose a novel Gaussian Curvature (GC) based criterion to discard false point correspondences within the

RANdom SAmple Matching (RANSAM) framework to improve the 3D registration. The RANSAM method

is used to ﬁnd a point pair correspondence match between two surfaces and GC is used to verify whether this

point pair is a correct (similar curvatures) or false (dissimilar curvatures) match. The point pairs which pass

the curvature t est are used to compute a transformation which aligns the two overlapping surfaces. The results

on shape alignment benchmarks show improved accuracy of the GRANSAM versus RANSAM and six other

registration methods while maintaining efﬁciency.

1 INTRODUCTION

Three-dimensional (3D) point cloud registration is

used to align CAD o r full 3D scans with partial

scan to perform robotic bin picking, 2D and 3D in-

spection or authentication, and full 3D scene recon-

struction. Partial views, clutter, illumination varia-

tion and noise in scan from the 3D sensor present a

signiﬁcant c hallenge to any 3D registration approach.

Hence, Random Sample Consensus (RANSAC) (Fis-

chler and Bolles, 1981) model ﬁtting schem e , which

ﬁnds a small nu mber of correct point corresponden-

ces to align the scene to th e model, is often used as a

standard approach to deal with such challenges. In ad-

dition, extra geometric features or descriptors (invari-

ant under rigid transformation) are used in the RAN-

SAC scheme to prune out mismatc hed points in a ma-

tching and registration scenario. Such geometric fe-

atures often re quire computationally expensive accu-

mulation or differientiation of local info rmation, e. g.,

volumetric invariant (Gelfand et al., 2005), mean cur-

vature, G aussian curvature (GC), etc (Besl and Jain,

1986). Hence, we are mo tivated to propose a sim-

ple and e asy imp le mentation of a Gaussian curvature

check in RANSAM (Winkelbach et al., 2006) frame-

work to prune misma tc hed points in order to improve

accuracy and maintain efﬁciency of 3D registration.

We show that an efﬁciently computed GC descriptor

used for pruning can bring improvement to the exi-

siting RANSAM approach. However, comparison of

different geometric descriptors is not within the scope

of this paper.

The key observation is that Gaussian curvature

provides a simple but powerful descriptor, so that

points with different curvatures are unlikely to contri-

bute to a correct correspondence, and thus can be d is-

carded. We evaluate this idea on a number of shape

alignment benchmarks and show improvement over

several of them. The main contribution of our work is

the inclusion of the discrete Gaussian cu rvature check

into the RANSAM framework for discarding un likely

feature point corresp ondence s during the shape align-

ment. The advantages o f our approach are: (a) impro-

vement in accuracy, (b) minimal computational cost

and (c) simple and easy integration into existing re-

gistration approaches.

This paper is organised as f ollows. Section 2

presents literature review. Section 3 presents our

GRANSAM method. Sec tion 4 co mpares the perfor-

mance of the GRANSAM with RANSAM and six ot-

her global registration metho ds. Section 5 concludes

the paper.

2 RELATED WORK

Most popular 3D feature descriptors include point fe-

ature histogram (PFH) (Rusu et al., 2008), fast point

feature histogram (FPFH) (Rusu et al., 2009), signa-

ture of histogram of orien ta tions (SHOT) (Tombari

et al., 2010) , etc. RANSAC is used to repeated ly es-

Azhar, F., Pollard, S. and Adams, G.

Gaussian Curvature Criterion based Random Sample Matching for Improved 3D Registration.

DOI: 10.5220/0007343403190325

In Proceedings of the 14th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2019), pages 319-325

ISBN: 978-989-758-354-4

319

timate an alignment f or a randomly chosen subset of

correspo ndences which is validated on the entire or

subset point cloud. PCL (Holz et al., 2015; Rusu

et al., 2009) is the sample co nsensus initial align-

ment algorithm, from the Point Cloud Library, which

uses histogram o f point pair featu res, i.e., FPFH, to

obtain a global registration. CZK (Choi et al., 2015)

is a method which combines geometric registration

of scene fragments with ro bust global optimization

based on line processes for 3D scene reconstruction.

Noisy data and partially overlapping point clouds cre-

ate a signiﬁcant problem to these methods because

they require many r e petitions to ﬁnd a go od point cor-

respond ence set. In contrast, the FGR (Zhou et al. ,

2016) method uses a ﬁxed subset of corresponding

points, some cor rect and some not, which are con-

sidered together. It aims to iterate towards a solution

that selects the c orrect point matches while discarding

the noise using a German McClure objective func tion

within a grad uated non-convexity framework. The

MFR method in (Faisal Azhar and Adams, 2019)

builds upon FGR method but uses weighted median

in a re-weighted least squares approac h with gradua-

ted M-Estimation to rapidly converge to optimal re-

gistration. These me thods may totally fail to align e n-

gineering parts (with sig niﬁcant planar surfaces) be-

cause of lack of unique local features and do not al-

ways perform well w ith different v iews of more gene-

ral (non-planar surfaces) parts because the descriptor

based matching does not always ﬁnd good point cor-

respond ence match suitable for 3D registration. Re-

cently, (Rantoson et al., 201 6) introduced registration

which uses Gaussian curvature constrain t on the rand-

omly sampled point pair matches to impr ove registra-

tion. This method uses DARCES algorithm, which is

a computation a lly expensive procedure, to randomly

compare three scene points to all the points in the mo-

del to ﬁnd point ma tc hes. Also, redundant compari-

sons migh t produce many bad matches in the initial

step which can not be resolved using Gaussian cur-

vature con stra int in a later step. Also, it uses com-

putationally expensive Eigen value decomposition to

compute the Gaussian curvature.

Super4PCS (Mellado et al., 2014) is a n optimal

linear time output-sensitive algorithm which uses an

efﬁcient data struc ture to obtain a global alignment.

OpenCV ( D rost et al., 2010) provides a surface regis-

tration algorithm wh ic h uses point pair features with

hash table lookup and voting with p ose clustering to

obtain a global registration. The RANSAM method

(Winkelbach et al., 2006) uses a point pair feature

with hash tables to compute an initial alignment. We

are motivated by the RANSAM approach but additi-

onally use a surface curvature test to prune early the

Figure 1: Illustration of 4D (α

,β

,δ

,) relation fe-

atures for a point pair p

and p

(a) (b)

Figure 2: Efﬁcient Gaussian curvature from discrete trian-

gularization of the mesh. ∆

is the angle deﬁcit computed

using the length a

of the sides of the i

triangle. (a)

3D view and (b) planar view.

potential point pair hits in the hash table which im-

proves robustness (see Section 3.1).

3 MATHEMATICAL DETAIL

Let P

= [p

,..., p

] and N

= [n

,...,n

] be 3D points

and corresponding surface normals of a surface A re-

spectively. The combination of point with normal is

referred as an oriented point (see 1). L et the set of

oriented points A of surface A and oriented points B

of surface B be



u = [p

]|p

∈P

and n

∈ N





v = [p

]|p

∈ P

and n

∈N



(1)

Consider pairs of candidate matching points a,c ∈

A and b,d ∈ B, two pre-deﬁned frames ca n be used

to determine a homogeneo us 4 × 4 transformation

T = [R

3×3

3×1

; 0

1×3

that optima lly and

robustly aligns the dipoles (a,c) and (b,d) such that

T = F(a,c)

−1

.F(b,d) (2)

where the function F(u,v) or F represents a coordi-

nate system lying between oriented points u and v



×n

×p

×n

×p

0 0 0 1



(3)

where p

−p

)

/kp

−p

k and n

= n

. Singu-

lar frames are avoided by ensuring the length of p

and n

is not zero.

VISAPP 2019 - 14th International Conference on Computer Vision Theory and Applications

320

3.1 GRANSAM

Gaussian curvature is an intrinsic surface property,

i.e., isometrically invariant under rigid transformation

(Besl and Jain, 1986). The standard Gaussian cur-

vature computation (with der ivatives) over the mesh

requires computationally expensive per vertex local

neighbourho od operations such as Eigen value de-

composition of the covariance matrix (Besl and Jain,

1986; Rantoson et al., 2016). Instead, w e use discrete

triangularization of the surface to efﬁciently compute

Gaussian curvature over the mesh (Besl and Jain,

1986). Consider a point x

(Red circle) on the mesh

which is a vertex for N different triangles as shown

in the Figure 2. The Gaussian curvature for a point x

can be co mputed as

K(x

) =

2π −

∑

i=1

∆

∑

i=1

(4)

where Λ

s(s −a

)(s −b

)(s −c

) is th e area of

triangle, with semiperimeter s =

/2, ∆

is the

angle deﬁcit

∆

= arccos



+ b

−c



(5)

and a

are the leng th of the sides of the i

tri-

angle. In practise, we use an approximated Gaussian

curvature as

K(x

) = 2π−

∑

i=1

∆

(6)

Our GRANSAM uses the same 4D relation table

as used in RANSAM (Winkelbach et al., 2006) con-

sisting of the Euclidean distance d

between points,

the angle of inclination α

and β

between the nor-

mals n

and n

, the line connecting p

and p

, and

the rotation angle δ

between the normals around the

connection line as shown in Figure 1. The rel(u,v) or

rel is computed as

rel







cosα

cosβ













− p

· p

tan

−1

(

·(p

×n

)

×p

)·(p

×n

)







(7)

We also use the same match quality metric Ω as

used by RANSAM (Winkelbach et al., 2006) to eva-

luate the transformation T computed using the pair of

matches,

Ω =

∑

u,v=1

(

1 minkTp

− p

k < ε

0 else

1.96

√

(8)

Algorithm 1: Our GRANSAM method.

Input:

Set of oriented po ints A of surface A and oriented

points B of surface B.

Threshold ξ=0.1.

Output: Transformation matrix T which aligns A to

1. Compute Gaussian curvature K for surface A and

B using Equation 6.

2. Random ly choose or iented point pair a,c ∈A and

calculate rel(a,c) using Equation 7.

3. Insert it into the relation table for surface A: suc h

as R

[rel(a,c)] = (a, c).

4. Read the same position in the relation table for

surface B: (b,d) = R

[rel(a,c)].

5. If entry exists, verify if |K(a) −K(b)| < ξ and

|K(c) −K(d)| < ξ then compute T for (a, b,c, d)

using Equatio n 2 and 3, and estimate match qua-

lity Ω using Equation 8.

6. Random ly choose orien te d point pair b,d ∈ B and

calculate rel(b,d) using E quation 7.

7. Insert it into the relation table for surface B: such

as R

[rel(b,d)] = (b,d).

8. Read the same position in the relation table for

surface A: (a, c) = R

[rel(b,d)].

9. If entry exists, verify if |K(a) −K(b)| < ξ and

|K(c) −K(d)| < ξ then compute T for (a, b,c, d)

using Equatio n 2 and 3, and estimate match qua-

lity using Equation 8.

10. Repeat steps 2-9 until ma tc h quality is good

enoug h or iteration limit is exceeds.

where k is the numb er of points, subscript u = 1,..., k

and v = 1,...,k. (see our Algorithm 1)

4 EXPERIMENTAL RESULTS

We have conducted a series of experim e nts to com-

pare the performance of our GRANSAM method

against state-of-the-art globa l registration methods

RANSAM, FGR and MFR on two real world data

sets, i.e., our 3D printed part dataset and the publi-

cly available UWA be nchmark dataset (Mian et al.,

2006). Synthe tic experiments have not been conside-

red as 3D registration m ethods operating well on si-

mulated da ta set do not necessarily perform well on

real world dataset. Our 3D printed part dataset is

used to test the performance of 3D registration m e t-

hods to align individual views of a single part to a

referenc e multi-v iew mode l or CAD model. It con-

tains partial v iews of challenging engineer ing parts

with pla nar surfaces. The UWA dataset tests the abi-

lity to deal with multiple objects, c lutter, occ lusion

Gaussian Curvature Criterion based Random Sample Matching for Improved 3D Registration

321

(a) (b) (c)

(d) (e) (f)

Figure 3: RANSAM (Blue points) versus GRANSAM (Red points) registration on model (Green points). (a, b, c) v.s (d, e, f)

for Extruder, Dinosaur and Slide lock respectively. Note misalignment in top versus bottom row (zoom in PDF t o see detail).

(a) (b)

Figure 4: RANSAM (Black line) versus GRANSAM

(Green line) 3D registration on Extruder (top row) and

Bunny ( bottom row). (a)-(c) RMS point-to-point error in

mm and (b)-(d) corresponding overlap in percentage.

and low overlap. Our GRANSAM method which

outperforms FGR an d MFR can also be considered

to surpass the performance of six other global regis-

tration methods (termed as Go ICP, GoICP-Trim, Su-

per4PCS, OpenCV PCL and CZK).

4.1 3D Printed Part Dataset

This dataset consists of six models (Extruder, Dino-

saur, Slide lock, Bunny, Focus housing and Camera

mount) wh ic h a re captured from 10 different views,

i.e., a total of 60 scans for testing GRANSAM versus

RANSAM r egistration. The Figure 3 shows the accu-

rate GRANSAM (Red points) registration versus mi-

saligned RANSAM (Blue points) registration on mo-

del (Green points). In our experiments, GRANSAM’s

Gaussian curvature pruning consistently obtained bet-

ter registration than RANSAM.

We applied the GRANSAM and RANSAM met-

hod 1 00 times on the 3D printed part dataset to

compute the average RMS error, overlap percen-

tage, and computational time score. We use a thres-

hold of 1 mm to compute the RMS error and over-

lap percenta ge. The Figu re 4 shows the reduced

average RMS and increased overlap in percentage o f

the GRANSAM versus RANSAM along with 25

and 75

percentiles of the population (wh ich are

shown as vertical bars) for each view of the Ex tru-

der and Bunny. GRANSAM’s considerable impro-

vement over RANSAM can be seen in cer ta in views,

for example, view 2, 3, 4 of the E xtruder ha s r e du-

ced RMS error with up to 4.5% increase in overlap

and view 1, 2, 8 of the Bunny has reduced RMS

error with up to 12.5% in c rease in overlap. In our

experiments, th e reduction in RMS error and impro-

vement in overlap perce ntage was observed in many

views of the 3D printed part dataset. Note that as ex-

pected GRANSAM performance was better for ob-

jects with r ough surface or crest and trough and was

not improved for objects with planar surface. Ho-

wever, there are numerous real world shapes which

have sufﬁcient su rface curvature where GRANSAM’s

Gaussian curvature ba sed pruning will beneﬁt any re-

gistration scheme.

In Table 1, we summarize the average RMS er-

ror in mm and overlap in percentage across all views

VISAPP 2019 - 14th International Conference on Computer Vision Theory and Applications

322

Table 1: Average across all views of the 3D printed part dataset: (1) Extruder, (2) Bunny, (3) Camera mount, (4) Dinosaur,

(5) Focus housing, (6) Slide lock. Bold shows winner per category.

RMS error in mm

1 2 3 4 5 6

RANSAM 0.46 0.42 0.21 0.53 0.26 0.37

GRANSAM 0.45 0.41 0.21 0.52 0.25 0.36

Overlap in percentage

1 2 3 4 5 6

RANSAM 78.8 69.5 96.1 59.2 96.2 82.9

GRANSAM 80.5 69.6 96.3 61.8 96.5 83.1

(a) (b) (c)

(d) (e) (f)

Figure 5: Comparison of the computational time in seconds for 3D registration obtained using RANSAM (Black line) and

GRANSAM (Green line). (a) Extruder, (b) Bunny, (c) Camera mount, (d) Dinosaur, (e) Focus housing and (f ) Slide Lock.

of the 3D printed part dataset using RANSAM and

GRANSAM. The Figure 3, 4 and Table 1 show that

our GRANSAM obtains more consistent registration

than RANSAM.

Figure 5 shows that GRANSAM (Green line) has

similar computational time as RANSAM (Black line)

on the 3D printed dataset for a maximum of 5 000

iterations per r un. GRANSAM introduces minima l

computational cost of up to 0.1 seconds (the cost of

pre-computing the Gaussian curvature is very small,

i.e., 1-10 m illiseconds).

4.2 UWA Benchmark Dataset

This dataset consists of four models (Cheff, Chicken,

Parasaurolophus, T-rex) w ith 50 scenes of m ultiple

objects, i.e., all f our objects are present in most of

the scenes. Similar to (Zhou et al., 2016), a total of

188 model and scene pairs for testing registration. It

is a challenging data set due to clutter, occlu sio n and

low overlap between model an d scene.

The Figure 6 shows the GRANSAM (Red

points) registration versus misalign ed RANSAM

(Blue points) on model ( Green points). It can be

seen in Figure 6a-6b versus 6d-6e that RANSAM fails

while GRANSAM correctly aligns the scene to the

model.

In Figure 7, we summarize the average RMS er-

ror and overlap percentage across the 188 pairwise

registration of the UWA dataset using FGR, MFR,

RANSAM and GRANSAM method. It can be seen

from Figure 7a that on average our GRANSAM met-

hod obtains a lower RMS error than FGR, MFR and

RANSAM methods while maintain ing good overlap

percentage as shown in Figure 7b.

Table 2 summarizes the percenta ge reduction in

RMS error using GRANSAM versus other methods

on UWA dataset. Note that on average GRANSAM

has a 37.73%, 37.03%, 7.84% reduction in RMS error

versus FGR, MFR and RANSAM method re specti-

vely. Hence, GRANSAM is able to deal with low

overlap, m ultiple objects and high occlusion UWA da-

Gaussian Curvature Criterion based Random Sample Matching for Improved 3D Registration

323

(a) (b) (c)

(d) (e) (f)

Figure 6: RANSAM (Blue points) versus GRANSAM (Red points) registration on model (Green points). Cheff, Chicken and

T-rex left-to-right. Note misalignment i n top versus bottom row (zoom in PDF to see detail).

(a)

-4pt]

(b)

Figure 7: GR A NSAM versus FGR, MFR, and RANSAM

3D registration on UWA dataset. (a) Mean RMS error and

(b) mean overlap percentage.

taset and also surpasses the performance of six global

registration methods (termed as GoICP, GoICP-Trim,

Super4PCS, OpenCV PCL and CZK).

4.3 Computational Speed

GRANSAM has a low complexity of O(k) (Winkel-

bach et al., 2006). The computational time was me-

asured in the Microsof t Visual Studio 2015 on a HP

Z mobile work statio n with an Intel i7 2.66 GHz pro-

cessor with 16 GB RAM. GRANSAM and RANSAM

both take in total about 3-6 seconds to compute the 3D

registration between a scene and m odel. In contrast,

the descriptor based FGR and MFR metho ds take in

total about 30 seconds to a few minutes to compute

the 3D registration depending on the nu mber of points

in the scene and mode l.

5 CONCLUSION

We have pre sented a discrete Gaussian cu rva-

ture criterion based RANdom SAmple Ma tc hing

(RANSAM) method to remove false matches to im-

prove the accuracy of 3D registration while maintai-

ning efﬁciency. The main novelty of our work is the

inclusion of Gaussian curvature check to prune out

mismatched point cor responde nces during sh ape alig-

nment within the RANSAM framework. We avoid

geometric features requiring computationally expen-

sive accumulatio n or differentiation of loc a l infor-

mation and show that a simple and efﬁcient discrete

Gaussian curvature feature provide s a powerful and

easy to integrate descriptor to improve any 3D regis-

tration approach. The results on a numbe r of shape

alignment benchmar k showed the considerable im-

provement in 3D r egistration by using the Gaussian

curvature criterion .

VISAPP 2019 - 14th International Conference on Computer Vision Theory and Applications

324

Table 2: Percentage reduction in RMS error of GRANSAM

versus other methods on UWA dataset.

FGR MFR RANSAM

GRANSAM 37.7 3 37.03 7.84

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