Range Data Fusion for Accurate Surface Generation from

Heterogeneous Range Scanners

Mahesh Kr. Singh

, K. S. Venkatesh

and Ashish Dutta

Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, India

Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India

Keywords:

Gaussian Mixture Model, Laser Range Scanner, Kinect, RGB-D image, Delaunay Triangulation.

Abstract:

In this paper, we present a new method for range data fusion from two heterogeneous range scanners for

accurate surface modeling of rough and highly unstructured terrain. First, we present the segmentation of

RGB-D images using the new framework of the GMM by employing the convex relaxation technique. After

segmentation of RGB-D images, we transform both the range data to a common reference frame using PCA

algorithm and apply the ICP algorithm to align both data in the reference frame. Based on a threshold criterion,

we fuse the range data in such a way that the coarser regions are obtained from Kinect sensor and ﬁner regions

of plane are obtained from the Laser range sensor. After fusion, we apply Delaunay triangulation algorithm to

generate the highly accurate surface model of the terrain. Finally, the experimental results show the robustness

of the proposed approach.

1 INTRODUCTION

The multi-range sensor data fusion is the process

of combining the range information from redundant

and/or complementary sensors, to produce a complete

and accurate description of the targeting region. The

range data fusion has a special signiﬁcance to gen-

erate the good quality surface. Nowadays, the gen-

eration of dense 3D representations of the environ-

ment has gained more attention. Some of the ﬁrst

work focused on the fusion of range data by mak-

ing an implicit function (Wheeler et al., 1998) and

then polygonizing it using the marching cubes algo-

rithm for high resolution surface reconstruction. In

(Trevor et al., 2012), the combination of 2D lines and

3D planes with a high level representation and easy

to be annotated with semantic data have generated an

accurate 3D map with its high level features. As dis-

cussed in (An et al., 2012), the authors have presented

a fast incremental method of extracting planes using

2D lines from 3D point clouds acquired sequentially

from a tilted LRF over mobile robot. In (Kl¨aß et al.,

2012), the authors have built the 3D surface element

grid maps and present Monte Carlo localization with

the probabilistic observation models for 2D and 3D

sensors in this map. In (Newcombe et al., 2011), the

authors have presented a new method for real-time 3D

modeling of complex and arbitrary indoor scenes in

variable lighting conditions using a Kinect sensor and

commodity graphics hardware. They have fused all

the streamed depth data into a single global implicit

surface model of the observed scene. In (Lai et al.,

2011), the authors have presented a new method for

RGB-D based object recognition and detection using

color and depth information. In (Johnson and Man-

duchi, 2002), the authors have proposed a probabilis-

tic rule for adaptive resolution integration of 3D data

which has collected from multiple distributed sensors.

In (Singh et al., 2014), the authors have proposed a

new method for range data fusion from two heteroge-

neous range scanners. They have exploited the terrain

characteristic (i.e. coarser and ﬁner region) to fuse the

range data and generated accurate 3D fused surface of

the planner environment.

In this paper, ﬁrst we present the segmentation of

RGB images using the new framework of the Gaus-

sian mixture model by applying the convex relaxation

technique. After segmentation of RGB-images, we

extract the corresponding location in Depth images

by calibration RGB and depth images (Herrera et al.,

2012). Now we are able to detect the ﬁner location in

Kinect frame. Also, we obtain the range data of the

same environment from the Laser range scanner. Us-

ing the PCA algorithm, we transform both the range

data into a common reference frame, and apply the

ICP algorithm to align both range data. Based on a

444

Singh M., Venkatesh K. and Dutta A..

Range Data Fusion for Accurate Surface Generation from Heterogeneous Range Scanners.

DOI: 10.5220/0005574504440449

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 444-449

ISBN: 978-989-758-123-6

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

threshold criterion, we have fused the range data in

such a way that the coarser regions are obtained from

Kinect sensor and ﬁner regions of plane are obtained

from the Laser range sensor. After fusion, we ap-

ply the Delaunay triangulation method to generate the

highly accurate surface model of the terrain.

The remainder of this paper is organized as fol-

lows: Section II describes the proposed method in

detail. Section III presents the experimental results.

Finally, we conclude this paper in Section IV.

2 THE PROPOSED METHOD

The steps of the proposed method are described as

follows:

2.1 Range Data Acquisition Systems

For the fusion of range data, we have used two het-

erogeneous range sensor, i.e. Laser range scanner and

Microsoft Kinect. The ﬁgure 1(a) shows the Laser

range scanner which is deigned at our robotics lab.

In the Laser scanner system, the Laser projects laser

line on the plane and camera captures the laser line

proﬁle. When range scanner moves over the object

surface, the camera acquires images of the distorted

pattern which are reﬂected by the object surface. The

height of the objects are obtained by taking into ac-

count the distortion of the laser light stripe caused

by their shapes. The designed Laser range scanner

gives accurate range measurements of the large an-

gular ﬁeld with angular resolution 0.1125

. The ac-

curacy of the range scanner is approximately ±2-3

mm throughout its range. The major advantage of de-

signed range scanner: it gives accurate result, very

high angular resolution, no correspondence issue be-

cause the camera acquires the illuminated scene to

obtain the dense 3D geometric information in a sin-

gle exposure. The disadvantage is its high scanning

time due to sequentially scan the terrain. The ﬁg.1(b)

shows the Kinect sensor that was introduced in Nov

2010 by Microsoft for the Xbox-360 video game sys-

tem. The detail description of Microsoft Kinect is de-

scribed in the papers (Zhang, 2012; Khoshelham and

Elberink, 2012). In(Khoshelham and Elberink, 2012),

the authors have investigated the accuracy and resolu-

tion of Kinect depth data for indoor mapping applica-

tions. They have presented that the random error of

depth measurement increase quadratic-ally with in-

creasing the distance from the sensor and it ranges

from few millimeters up to 4 cm at the maximum

range of the sensor. For the mapping application, the

working range should be within 1-3 meter distance

(a) (b)

Figure 1: (a) Laser Range sensor designed at our robotics

lab (b) The Microsoft Kinect system.

from the sensor otherwise the quality of data is dete-

riorated by noise and low resolution.

2.2 RGB-D Segmentation

In this section, we describe the method to unsu-

pervised segmentation of RGB-D images using new

framework of GMM using by employing convex re-

laxation approach.

2.2.1 Gaussian Mixture Model and EM

Algorithm

A Gaussian mixture model is a probabilistic model

that presumes all the sample points are generated

from a mixture of a ﬁxed number of Gaussian distri-

butions with unknown parameters. Therefore a Gaus-

sian mixture model is a weighted sum of M Gaussian

component densities of x which is a D-dimensional

measurement vector as given by the equation,

p(x/µ

,σ

) =

∑

i=1



x;µ

,σ



(1)

where ω denote the mixture ratio, p is the Gaus-

sian pdf parameterized by mean µ

and variance σ

Given data, the parameters Θ =



ω,µ,σ



can be ef-

ﬁciently estimated through maximum likelihood es-

timation (MLE) using the EM algorithm, then clus-

ters will be obtained through estimated parameters.

The log-likelihood function for GMM is given by

(McLachlan and Krishnan, 2007)

L (Θ) =

Ω

log

∑

i=1

√

2πσ

exp

(

−

[ f(x) −µ

]

2σ

)

(2)

we drive the conclusion from EM algorithm that the

E-step and M-step can guarantee L



t+1



≥ L (Θ

)

during the updating process (i.e. t → t+1), which de-

notes the local convergence of the EM algorithm.

2.2.2 GMM using Convex Relaxation Approach

In this section, we present GMM algorithm us-

ing a convex relaxation approach. The optimiza-

tion of logarithm and summation function like

RangeDataFusionforAccurateSurfaceGenerationfromHeterogeneousRangeScanners

445

equation (2) is difﬁcult task because the opera-

tions of these functions are not non-commutative.

However, the solution of these function could

be achieved by adding a convex relaxation term.

(u,v) represent the a vector valued function such

as v(x) = (v

(x),v

(x).......v

(x)) and ∆ =



v|0 < v

< 1,

∑

i=1

= 1



. The symbol ∆ is deﬁned

as the convex relaxation term for non binary vector

space (v) =



v|v

= {0, 1},

∑

i=1

= 1



. For commu-

tativity of log-sum function, we have used a deduction

of convex analysis (Rockafellar, 1997)

Lemma 1. The Log-sum Commutativity operations of

given a function α

(x) > 0, for any function β

(x) > 0,

we have

−log

∑

i=1

(x)exp[−β

(x)] = min

v(x)∈∆

(

∑

i=1

[β

(x) −α

(x)]v

(x)

∑

i=1

(x)logv

(x)

)

(3)

Now we set

√

2piσ

= α

(x),

kf(x)−µ

2σ

= β

(x) and

apply lemma in equation (2). The optimization prob-

lem becomes

Θ = argmax

L (Θ) = −argmin

L (Θ)

= argmin

{

Ω

min

v(x)∈∆

(

∑

i=1

[β

(x) −logα

(x)]v

(x)

∑

i=1

(x)logv

(x)

)

dx} (4)

Now we introduce a functional

ε(Θ,v) with two vari-

ables Θ,v.

ε(Θ,v) =

Ω

∑

i=1



kf(x) −µ

2σ

−log

√

2πσ



(x)dx

Ω

∑

i=1

(x)logv

(x)dx (5)

Then we compute the minimizer of

ε(Θ,v) via the

following alternating algorithm:

t+1

= argmin

u∈ξ





t+1

= argmin



Θ,v

t+1



(6)

Where t = 1,2... is the iteration number and Θ

is an

initial guess.

2.2.3 The Basic Model

Now we consider the non-uniform intensity problem

because Kinect sensor capture the RGB-D images

of unstructured terrain, which can be mathematically

modeled as:

f(x) = γ(x)g(x) (7)

where g(x) is the ground truth image, f (x) refers the

observed data and γ(x) refers to a smooth varying bias

ﬁeld. From (Li et al., 2008), we have taken assump-

tions that the bias ﬁeld is non-negative and smoothly

varying. In the nearest neighborhood circle centered

at x i.e. γ(y) = γ(x), for all y ∈O

Here we describe method in (Li et al., 2008) with

statistical interpretation. Let us ﬁrst focus on the

neighborhood O

centered at y , all the intensity f(y)

within neighborhood O

have same pfd p(x) with

the parameter µ

,σ

,β(y). If we deal with different

contributions to the cost functional ε(Θ,v) in terms

of distance to centering point, then we consider to

add some weights for each pixel. We have taken

the Gaussian function with std σ, G

(x) ≈ 0 when

x 6∈ O

. Therefore, integral domain of O

is expanded

to whole domain Ω. i.e.,

(Θ) = −

Ω

(y−x)log

∑

i=1

√

2πσ

exp

(

−

[ f (x) −µ

]

2σ

)

(8)

For the desirable segmentation result, we have

taken the global information and the total cost func-

tional becomes

ε(Θ) =

Ω

(Θ)dy (9)

The segmentation problem can be solve by the mini-

mization problem

Θ = argmin

ε(Θ) (10)

However, the above optimization problem is very dif-

ﬁcult to solve due to the presence of the log-sum func-

tion. As a result, we construct the above cost function

with two variables mention in previous section. Ac-

cording to lemma 1, the ﬁnal data term becomes

ε(Θ,v) =

Ω

∑

i=1

(y−x)



kf(x) −µ

2σ

−log

√

2πσ



(x)dxdy

Ω

∑

i=1

(x)logω

(x)dx +

Ω

∑

i=1

(x)logσ

Ω

logβ(y)dy+

Ω

∑

i=1

(x)logv

(x)

(11)

we have to minimize

ε(Θ,v) under the constraint

v ∈ ∆ to get the optimized v

∗

,Θ

∗

. Inspired by (Wang

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

446

et al., 2009), we have introduced another regulariza-

tion term which is guarantee the precise close-form

solution of v. The regularization term deﬁne as

R (v) =

Ω

∑

i=1

(x)e(x)

N(x;η)

[1−v

(y)]dydx (12)

where e is the edge detector function, N(x;η)give the

neighborhood centered at x with radius η. In the pa-

per, we take e =

1+k▽G

∗fk

. Therefore combining the

data term and regularization term, we introduce our

new model as follows

(Θ

∗

) = arg min

Θ,v∈∆

L :=

ε(Θ,v) + λR (v) (13)

where λ > 0 is a regularization parameter that con-

trols the trade-off between these two function. The

parameter set Θ becomes

Θ =



...ω

,,µ

...µ

,σ

...σ



[

{β(y)}

(14)

This new model is inherently different from clas-

sical GMM. Firstly, the classical GMM is very sen-

sitive to noise and lacks in spatial smoothness con-

straint while the above new model has a controlling

parameter that makes it robust to noise. Secondly,

the data term in traditional GMM is only works well

for images that have almost piece-wise constant and

it can not handle the images with non-uniform inten-

sity. However, the new model incorporates local bias

function information, global intensity and edges in-

formation. Therefore, it works well on the images

with non-uniform intensity. The minimizing eq. (15),

starting from a given initial value Θ

such that

t+1

= argmin

u∈ξ





t+1

= argmin



Θ,v

t+1



(15)

The stopping criteria of above proposed method is

−L

t+1

< δk L

. In this way, we segment the

RGB-images obtained from Kinect. Now our aim to

correlate the segmented the RGB-images to their cor-

responding Depth images. For calibration of as dis-

cussed in (Herrera et al., 2012), the calibration of

Depth and the color image pair is done using planar

surface and a simple checkerboard pattern. The same

calibration method is used to establish the relation be-

tween two range sensors. After calibration, we can

easily locate the objects in the environment.

2.3 Data Fusion

The time cost of data acquisition from Laser range

scanner is high, but it provides a very high quality

range data i.e. 2 mm precision throughout the range.

On the other hand the random depth error in Kinect

(Khoshelham and Elberink, 2012) increases with in-

creasing distance from the sensor, but it is low cost,

compact range sensor and very fast relative to de-

signed Laser range sensor. The terrain characteristic

is determine in terms of surface elevation. The ﬁner

regions of terrain are determined as the regions whose

elevation is greater than 4 cm. Therefore, the thresh-

old criterion for ﬁner regions is determined in terms

of elevation. We use Kinect to scan the coarser re-

gions of terrain and the ﬁner regions are scanned from

a Laser range scanner. Since the range data obtain

from both the range scanners are in different coordi-

nate systems. Thus, it is necessary to transform both

the range data into a common referenceframe. We ap-

ply the Principal component analysis (PCA)(Jolliffe,

2005) to both range data sets that orthogonality trans-

form the data set to the new coordinate system such

that the largest variance of the data is deﬁned as ﬁrst

coordinate (i.e. ﬁrst principle component) and so on.

This new coordinate system is deﬁned as a common

reference frame. Both range data are transformed

into this reference coordinate system. With the help

of segmented depth data, we ﬁnd their correspond-

ing points in the transformed reference frame. The

ﬁner detailed regions are identiﬁed in the reference

frame. Now we apply the ICP algorithm (Elseberg

et al., 2012) to align both the data set in this frame.

This alignment of the two heterogeneous range data

in reference frame is much faster than alignment of

the range data in different coordinate system. Based

on a threshold criterion, we fuse the ﬁner region of

data which is obtained from Laser range scanner to

Kinect’s coarser regions data. The fused range data

eliminate the demerits of the range scanners by com-

plementing each other. To generate the surface, we

apply the Delaunay algorithm to the fused range data.

In this way, we reconstruct the accurate, realistic sur-

face of the terrain.

3 EXPERIMENTAL RESULTS

The proposed fusion method is tested on real world

data by creating different types of environment in our

lab. In the experiment, we have used the two hetero-

geneous Laser range scanner and Kinect shown in Fig

(1). The purpose of the range data fusion is to gen-

erate the accurate, realistic, and a fast 3D surface of

the terrain and eliminates the demerits of both range

scanners by complementing each other. In the experi-

ments, two range data sets of the same environment

are obtained from both scanners (i.e. Laser range

scanner and Kinect). First Kinect captures the RGB

RangeDataFusionforAccurateSurfaceGenerationfromHeterogeneousRangeScanners

447

(a) (b) (c) (d)

Figure 2: (a) Figure of different types of object placed in plane (b) Active contour map (c) Segmented blocks of RGB image

and(d) Corresponding depth image.

(a) (b) (c) (d) (e)

Figure 3: (a) The 3D surface of plane isobtained from Kinect (b) After applying ICP algorithm the aligned surface from Kinect

and (c) from Laser range scanner (d) The Segmented ﬁne region of surface from Laser range scanner (e) The accurately fused

3D surface model from both range sensors.

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4: (a) Figure of different objects are placed on sandy plane (b) Figure of rectangular aluminum log (c) The plywood

board consist of several stick’s is placed (d) The different objects are placed (e) Shows the fused accurate 3D model of different

objects (f) Shows the fused 3D model of aluminum log (g) Shows the fused 3D surface of plywood board (h) Shows the fused

3D surface of different objects.

and Depth image of the environment. The Fig 2(a)

shows the RGB-image, The Fig 2(b) shows the ac-

tive contour with new approach of GMM and Fig 2(c)

shows the segmented objects in the image. To estab-

lish the relation between rgb image and depth image,

we have used the calibration method given by (Her-

rera et al., 2012). The Fig 2(d) shows the segmented

objects in the depth image. The ﬁner regions of terrain

are deﬁned as the points of range data whose elevation

is greater than 4 cm which is the threshold criterion.

The range data acquired from both the range sensors

are in different coordinate system. Therefore, it is

necessary transform both range data into one common

reference frame. We transform the both range data to

a new coordinate system such that new set of uncorre-

lated variables axis, called principle components. The

axis’s of common reference frame are deﬁned as the

largest variance after transformation of data set to ﬁrst

coordinate system and so on. The ﬁgure 3(b-c) shows

the aligned 3D surface of the terrain after applying the

ICP algorithm. The alignment of both range data in

the common reference frame using ICP algorithm is

much faster than directly apply the ICP algorithm for

alignment of the range data in two different frames.

We have deﬁned the threshold for selecting ﬁne re-

gion based on height data variation. Since Laser scan-

ner time cost is high, therefore we have scanned the

ﬁner region of terrain using Laser scanner based on

segmented depth data obtained from Kinect and rest

regions are taken from Kinect i.e. we have retained

the coarser detailed regions and erase the ﬁne detailed

region of Kinect range data. Now we have fused ﬁne

region range data acquired from Laser scanners to

the coarser regions range data obtained from Kinect.

Using Delaunay algorithm, we have generated sur-

face of the terrain. The Fig. 3(e) shows the ﬁnally

fused surface. In 3D fusion experiment, the relative

sensor disparity of Laser range sensor relative to the

Kinect in the reference frame is as the rotation matrix

R=[ 0.9704 0.2418 0;-0.2418 0.9704 0; 0 0 1.0000];

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

448

and translation vector t=[119.4117 127.9851 0]. The

alignment root mean square error of the fused data is

approximately 3.2 mm. Many range data fusion ex-

periments have been performed with different objects

in the environment. Fig 4(a) shows the different kind

of objects place in the plane and its accurate fused 3D

terrain model is shown in Fig 4(e). Again the rectan-

gular aluminum log is placed in plane as shown in Fig

4(b) and its fused terrain model is shown in Fig 4(f).

Similarly, now we place the plywood board that has

15 rectangular log and different objects in the plane

as Fig. 4(c-d). Fig. 4 (g-h) shows the accurately fused

3D model of the terrain. The resulting fused surface

shows, the proposed method is applied to accurately,

realistically and rapidly represent the real-world envi-

ronment.

4 CONCLUSIONS

In this paper, we have presented a new approach for

range data fusion from two heterogeneous range scan-

ners (i.e. Laser range scanner and Microsoft Kinect)

in order to integrate the merits of both scanners for

the generation of accurate, realistic surface of the ter-

rain. First, we have presented a new framework of

the GMM using convex relaxation approach for seg-

mentation of RGB-D images having inhomogeneous

intensity. After transforming both the range data to

common reference frame, we have applied the ICP

algorithm to align these range data. The alignment

method of two different range data in a common refer-

ence frame is much faster than directly apply the only

ICP algorithm to their scanner coordinate system. In

the fusion process, we have selected the coarser de-

tailed region from Kinect and ﬁner region from Laser

scanner. The fused surface of the terrain is recon-

structed using the Delaunay triangulation algorithm.

In this way, we have generated a seamless integra-

tion of the terrain surface from two overlapping range

data. The experimental results have shown the accu-

rate 3D model of terrain from fused range data. The

alignment rms error of the fused data has approxi-

mately 3-5 mm. So the main contribution of this pa-

per is to present a range data fusion approach that

eliminates the limitation of both the range sensors and

generate the accurate surface modeling of rough and

highly unstructured terrain.

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