Uncertainty Hypervolume in Point Feature-Based Visual Odometry

InJun Mun

and Sukhan Lee

2,* b

Intelligent Systems Research Institute, Department of Artificial Intelligence,

Sungkyunkwan University, Suwon 16419, South Korea

Keywords: Localization, Visual Odometry, Essential Matrix, Optimal Feature Selection, Uncertainty Hypervolume,

Bucketing.

Abstract: Visual odometry based on point feature matching has been well-established. Notably, methods based on

essential and fundamental as well as homography matrices have been widely used. It is known that the

accuracy of visual odometry is affected by the choice of matched feature point pairs. However, no

mathematically rigorous formula relating the choice of feature point pairs to the uncertainty involved in visual

odometry is available. Instead, point selection heuristics based on feature point distribution combined with

RANSAC-based refinement are mostly adopted to ensure accuracy. In this paper, we present “Uncertainty

Hypervolume” as a rigorous mathematical formula that relates the selected feature point pairs to the

uncertainty of visual odometry. The uncertainty hypervolume associated with selected feature point pairs

provides a precise metric for evaluating the selected feature point pairs and the resulting visual odometry.

This metric is useful in practice not only for selecting the best feature point pairs but also for selecting poor

feature point pairs available for visual odometry. Furthermore, it accurately identifies the uncertainty in visual

odometry, which helps better manage the performance of visual odometry applications.

1 INTRODUCTION

Visual odometry (

Nistér et al., 2004)

is a fundamental

technique in computer vision and robotics that

facilitates estimating camera movement by analyzing

sequential images. The accuracy of pose estimation,

a critical aspect of visual odometry, depends not only

on the chosen pose estimation method but also

significantly on the quality of the selected

correspondence feature points. S. Poddar, R. Kottath,

and V.Karar (Poddar et al., 2019) conducted a

comprehensive review of feature selection strategies

for visual odometry, outlining key steps including

feature detection, description, inlier/outlier detection,

feature distribution, and consideration of feature

quality. This multi-step process emphasizes the

intricate relationship between the accuracy of pose

estimation and the characteristics of the selected

feature points.

In feature selection, the initial removal of outliers

is paramount, as mismatched feature pairs can lead to

https://orcid.org/0009-0009-8524-0719

https://orcid.org/0000-0002-1281-6889

* Corresponding author

erroneous pose estimation. To solve this issue,

pioneering work by Fischler (Fischler et al., 1981)

introduced the random sample consensus (RANSAC)

algorithm, which uses geometric constraints to

remove outliers from the feature set effectively.

However, the effort to improve accuracy in pose

estimation extends beyond the methodological level.

Researchers have recognized that the distribution and

uniformity of corresponding feature points in space

also play a critical role in determining visual

odometry performance (Cvišić et al., 2015). As

pointed out in Poddar's review, traditional feature

selection methods often result in a non-uniform

distribution of feature points across the image. As a

result, clusters of closely spaced feature points can

lead to suboptimal pose estimation results.

To overcome this limitation and achieve

improved accuracy, an innovative approach has

emerged. This approach, known as the bucketing

technique (Zhang et al., 1995, Kitt et al., 2010)

attempts to achieve uniformity in the distribution of

correspondence feature points. By partitioning the

290

Mun, I. and Lee, S.

Uncertainty Hypervolume in Point Feature-Based Visual Odometry.

DOI: 10.5220/0013019300003822

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024) - Volume 2, pages 290-299

ISBN: 978-989-758-717-7; ISSN: 2184-2809

image into a grid of M × M buckets and selecting only

a small number of features from each bucket, the

bucketing technique ensures a well-distributed

selection of features. In particular, this uniform

distribution of features has the potential to improve

both the accuracy and computational efficiency of

pose estimation.

The previously mentioned approaches for feature

selection aimed to select dependable features through

the removal of outliers or consideration of feature

point distribution. While these methods have proven

effective in enhancing the accuracy of VO, they do

not ensure optimality in terms of mathematical

formalism, which guarantees the minimum

uncertainty of VO.

Recently, the concept of the Orthogonality Index

(Nguyen and Lee, 2019) has been introduced to

analytically derive optimal feature selection. This

approach demonstrates optimal feature selection

through a well-defined mathematical format instead

of random selection. The process increases the

orthogonal exponent of individual equations and

applies constraints to computation to reduce

uncertainty when estimating Essential, Fundamental,

or Homography matrices associated with visual

odometry. However, while the Orthogonality Index

provides a mathematical method for optimal feature

selection, they do not account for uncertainty in

feature points due to measurement or other noise.

This issue must be addressed as it significantly

impacts VO estimation. Therefore, a method that

reflects these factors is necessary to ensure optimal

feature selection.

To this end, our study capitalizes on insights

gained from simulation experiments, which have

shown that the measurement error variance and the

spatial distribution of the extracted feature points

significantly affect pose estimation. We propose a

novel approach that incorporates both of these

factors.

Our approach can be summarized as follows: If

the matched feature point pairs are well-matched with

minimal measurement error and are uniformly

distributed throughout the image, the estimated

essential matrix is expected to be close to the ground

truth essential matrix. However, due to the

uncertainty of the matched feature point pairs used to

estimate the essential matrix and the error of the

equations generated using them, the estimated

essential matrix forms a stochastically distributed

distribution centered on the ground truth(GT). We

experimentally demonstrate that the degree of

dispersion depends on the magnitude of the

uncertainty in estimating the essential matrix. We

found that the spatial distribution they form should be

taken into account when selecting matching feature

point pairs, and present a novel "Uncertainty

Hypervolume" approach that takes both into account.

The estimated essential matrix is stochastically

distributed around the reference ground truth

essential matrix, and we quantify this with

hypervolume. Through experiments, we show a

significant correlation between hypervolume and the

error of the pose derived from the essential matrix.

Based on these results, we propose a mathematically

well-structured Uncertainty Hypervolume based

approach for feature point pair selection to obtain the

optimal solution.

In the following sections, we detail our

methodology, experimental setup, and results,

culminating in a comprehensive analysis of the

interplay between feature selection, spatial

distribution, and pose estimation accuracy.

2 PROBLEM DEFINITION AND

APPROACH

2.1 Preliminary

Figure 1: Epipolar Geometry, A 3D point 𝑃 is projected

onto the normalized image plane of each camera at 𝑝 and 𝑞.

The points 𝑒 and 𝑒′ where the line connecting the two

camera origins and the image plane meet are called epipole,

and the straight lines 𝑙 and 𝑙′ connecting the projection

points and the epipole are called epiline (epipolar line).

In epipolar geometry (Deriche et al., 1994), given a

point 𝑃 in space, cameras 𝐶



and 𝐶



view the point 𝑃

from two different perspectives. The point 𝑃 is then

projected onto the normalized image plane of each

camera 𝐶



and 𝐶



as 𝑝 and 𝑞 ( 𝑝 and 𝑞 are

homogeneous normalized image coordinates). It is

known that there is always a 3x3 essential matrix

(Nistér, 2004), 𝑬 between the projected points 𝑝 and

𝑞 that satisfies the epipolar constraint 𝑝



𝐸 𝑞= 0.

Uncertainty Hypervolume in Point Feature-Based Visual Odometry

291

This essential matrix obeys the following

constraints.

𝑑𝑒𝑡

(

𝐸

)

=0 (1)

2𝐸

𝐸



𝐸

−𝑡𝑟

(

𝐸



)

𝐸

=0 (2)

The second expression is a matrix constraint that

gives nine equations for the elements of 𝑬. However,

only two of them are algebraically independent. Thus,

with the two essential matrix constraints mentioned

above, we can determine the essential matrix with

only five corresponding point pairs (Deriche et al.,

1994).

Once 𝑬 is determined, the rotation matrix 𝑹 and

the translation vector 𝒕 can be obtained by performing

a Singular Value Decomposition (SVD).

While 𝑹 has 3 degrees of freedom and 𝒕 has 3

degrees of freedom, if we consider the essential

matrix as a projection element, it has 5 degrees of

freedom with the scale factor removed. Therefore, we

can estimate 𝑬 with five pairs of corresponding

feature points and the Essential Matrix constraint.

The epipolar constraint 𝑝



𝐸 𝑞 = 0 can be

expressed simply as follows.

𝑣

𝐸



=0 (3)

where,

𝑣=



𝑝



,𝑝



𝑞



,𝑝



𝑞



,𝑝



𝑞



,𝑝



𝑞



,𝑝



𝑞



,𝑝



𝑞



,𝑝



𝑞



,𝑝



𝑞





(4)

𝑎𝑛𝑑 𝐸





𝐸



,𝐸



,𝐸



,𝐸



,𝐸



,𝐸



,𝐸



,𝐸



,𝐸







(5)

𝑬 can be determined based on the five pairs of

corresponding feature points, 𝑝 and 𝑞 that define the

following 5x9 matrix equation:

𝐴 𝐸



=0 (6)

where, 𝐴=



𝑣



𝑣



𝑣



𝑣



𝑣







(7)

Then, VO between the two camera frames can be

derived from E obtained by (1), (2), and (6).

2.2 Problem Definition

The accuracy of the estimated essential matrix

determines the accuracy of the transformation

relationship between the two cameras. This is

equivalent to the performance of Visual Odometry. In

VO, we use pairs of corresponding feature points that

match in both image planes to compute the essential

matrix. In other words, it is obvious that the accuracy

of the estimated essential matrix will increase if the

pairs of corresponding feature points with good

quality and evenly distributed in the image plane are

selected and computed. In this paper, we investigate

how the accuracy of VO is affected by selecting well-

distributed and high-quality corresponding feature

points when estimating the essential matrix. We

propose a new approach to feature point selection

using a canonicalization metric called "Uncertainty

Hypervolume".

2.3 Approach

Corresponding pairs of feature points (𝑝



,𝑞



) have

uncertainties due to measurement error, matching

error, noise error, etc. In the epipolar constraint of (3),

the solution of 𝐸 lies in the space perpendicular to 𝑣



The uncertainty of 𝑣



leads to the uncertainty of 𝐸,

and the estimated 𝐸 is stochastically distributed

around the GT due to the uncertainty of 𝑣



. The

solution subspace formed around the GT changes in

size as a function of the error associated with 𝑣



, i.e.,

the larger the uncertainty of the corresponding pair of

feature points, the more stochastically spread the

solution subspace becomes. This is equivalent to

saying that the volume size of the solution subspace

represents the uncertainty. From now on, we will

refer to the size of the solution subspace in higher

dimensions as the "Hypervolume". Our goal is to

choose (𝑝



,𝑞



) for 𝑣



such that the size of this

hypervolume is minimized (i.e., we choose 𝑣



such

that the uncertainty of the solution is small). To

achieve our goal, we will explore how the uncertainty

of a corresponding pair of feature points affects the

uncertainty of the solution subspace, and more

specifically, we will define and explain the concept of

a hypervolume.

Consider the simplest quadratic form of the

problem (the solution of the Essential matrix we want

to find is high-dimensional, with 9 dimensions. So,

we extend the concepts from lower to higher

dimensions).



𝑎



𝑏



𝑎



𝑏





𝑥



𝑥



= 

𝑐



𝑐



 (8)

Let

𝐴=

𝑎



𝑏



𝑎



𝑏



 , 𝑋=

𝑥



𝑥



 , 𝑐=

𝑐



𝑐



 and the

solution we want to find is 𝑋. Knowing 𝐴 and 𝑐, there

is only one solution ( 𝑎



≠𝑎



,𝑏



≠𝑏



,𝐴≠0).

However, if we consider the case where there is

uncertainty due to the error of 𝐴, it is equivalent to

(9).



𝑎



±∆𝑎



𝑏



±∆𝑏



𝑎



±∆𝑎



𝑏



±∆𝑏





𝑥



𝑥



=

𝑐



𝑐



 (9)

𝐴′



=

𝑎



±∆𝑎



𝑏



±∆𝑏



𝑎



±∆𝑎



𝑏



±∆𝑏



 (10)

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

292

Figure 2: (Left) Graph plotted when the equation of two

linear lines has a negative slope. The intercept is determined

by ±∆𝑎



, ±∆𝑏



, ±∆𝑎



, and ±∆𝑏



, which is the variance due

to uncertainty. (Right) When one linear equation has a

positive slope and another has a negative slope. There are a

total of four possible cases where 𝐴





(

𝑖=1,⋯,4

)

can be

determined, but the equations of this line have four vertices,

and the trapezoid is the hypervolume.

The magnitude of the error is denoted by ∆𝑎



∆𝑏



, ∆𝑎



, and ∆𝑏



for the elements of 𝐴 . The

meaning of ± is the variance of the error, which

corresponds to the uncertainty by having a value in

the range. Fig. 2 shows the graph of (9). In the graph,

the solid line is the equation of the original straight

line before adding the error, and the equation of the

straight line represented by the dashed line is the

equation of 𝐴





(𝑖=1,⋯,4) when the error variance

is maximum.

𝐴





=

∆𝑎



∆𝑏



∆𝑎



∆𝑏



, 𝐴





=

∆𝑎



∆𝑏



−∆𝑎



−∆𝑏





(11)

𝐴′



=

−∆𝑎



−∆𝑏



∆𝑎



∆𝑏



, 𝐴′



=

−∆𝑎



−∆𝑏



−∆𝑎



−∆𝑏





The solution of the original system of equations

without error is the intersection of the equations of the

two straight lines represented by the solid lines and is

determined to be one. However, the solution of the

system of equations with the uncertainty given by the

error will probabilistically lie within a trapezoid

whose vertices are the intersections of the equations

of the dotted lines shown in the graph. The larger the

error, the greater the width of this trapezoid, i.e., the

greater the uncertainty of the solution. The

"Uncertainty hypervolume" defined in the previous

section 2.3 corresponds to the area of the trapezoid in

this problem. Thus, by calculating the size of the

hypervolume, we can quantitatively represent the

uncertainty. The hypervolume we are talking about

forms an n-dimensional hypercube depending on the

dimension of the problem to be solved, which is two-

dimensional in the case of equation (9), so it becomes

a two-dimensional trapezoid, that is, the area of the

shape. In this case, the number of vertices that form

the boundary of the shape is 2



, where 𝑛 is the

number of unknown variables. In the next higher

dimension, three-dimensional, we can think of the

hypervolume as a three-dimensional cube, which is a

crumpled cube with eight vertices (one dimension:

line, two dimensions face, three dimensions: cube, ⋯,

𝑛-dimensional: 𝑛-hypercube).

Figure 3: 𝐸



is the essential matrix computed with five

pairs of corresponding feature points.

𝐸



,⋯𝐸



, obtained

by (13), form the vertices of a hypercube, which is a

clustering of groups that have the maximum variance and

are probabilistically likely to be the solution subspace. Its

hypervolume reflects the uncertainty, and the goal is to find

the corresponding feature point set that minimizes it.

Let's return to our original problem and extend the

concept of hypervolume defined in a low-

dimensional space to a higher-dimensional domain.

The essential matrix we want to find is a 3x3 matrix

with 9 elements, as shown in (5). The solution of 𝐸

exists somewhere in the 9-dimensional space and

must be singularized. However, due to the uncertainty

caused by the errors of 𝑣



, the corresponding feature

point pairs, the estimated solution of 𝐸 will be

stochastically distributed around the ground truth 𝐸.

In the case of the 5-point algorithm we use for VO

estimation, the solution can be obtained using only 5

pairs of corresponding feature points due to the

additional constraints (1) and (2). Therefore, using

equation (6) to express the error variance to represent

the uncertainty in the manner of (9), 2



=32 pairs of





,(𝑖=1,…,32) are generated (13). If the Essential

matrix is estimated using this as input to the 5-point

algorithm, 𝐸



(𝑖=1,…32) is generated. This means

that the manifold formed by the two constraints

projects the solution subspace that exists in the 9-

dimension to a lower dimension in the 5-dimension.

Therefore, the 𝐸



(𝑖=1,…32) calculated using this

method are stochastically distributed around the

Uncertainty Hypervolume in Point Feature-Based Visual Odometry

293

ground truth 𝐸 and form the outermost vertices of the

solution subspace. Fig. 3 shows the projection of 𝐸 in

9-dimensional space into 3-dimensional space (not an

exact projection, but an approximation for illustration

purposes). The uncertainty due to multiple error

factors is represented by a 5-dimensional hypercube

with 32 vertices, and the solution probabilistically

exists inside the hypercube. The volume of the

hypercube is the hypervolume, and its size represents

the degree of uncertainty. We try to minimize the

uncertainty of VO estimation by selecting a set of

feature points with the minimum hypervolume size.

2.4 Hypervolume Calculation Using

Qhull Algorithm

In the previous section, we detailed the procedure for

acquiring 32 vertices 𝐸



(𝑖=1,…,32) that constitute

the hypervolume and its significance. Additionally, it

was proposed that the size of the hypervolume

corresponds to the level of uncertainty, which can be

quantified through its acquisition. We utilized the

MATLAB function convexhulln to obtain the

hypervolume with the 32 vertices 𝐸



that comprise its

outermost layer. The function is founded on the Qhull

algorithm, which functions in the following manner.

a. Point Sorting: First, use the technique of aligning

data points appropriately. Because data points can

be randomly distributed in higher-dimensional

spaces, setting up the ordered order makes it more

efficient in subsequent steps. Sorted data helps

you calculate convex shells.

b. Centrum Location: Calculates the center position

of the data. This is used to clip points based on the

center position and to quickly calculate the

convex shell. The center position can be related to

the mean or median of high-dimensional data.

c. Create Point Clipping: Use the center position to

clip data points and apply techniques to remove

unnecessary points. This reduces unnecessary

calculations and optimizes memory usage. In

high-dimensional data, many points may not

contribute to the formation of convex shells.

d. Convex Hull of Clipped Points: Calculates the

convex shell for clipped points. This forms most

of the final convex shell. The process of

calculating convex shells for clipped data is

efficient.

e. Return Results: Finally, the Qhull algorithm

returns the calculated convex shell. This gives

results as convex polygons or convex polygons

surrounding a given data point in a high-

dimensional space.

Through these various geometric and

computational techniques, the Qhull algorithm

effectively computes the convex shells of high-

dimensional data. And then, The optimal feature

selection based on the proposed hypervolume method

is described as follows:

Algorithm 1: Hypervolume-based optimal feature selection.

Data: 𝐿 correspondence feature point sets

Result: five-

oint sets with the lowest h

pervolume

Step 1. Random selection of five feature point sets,

{

(

𝑝



,𝑞



)

, 𝑖=1,⋯,5} from the 𝐿 feature pairs

detected from the images of two camera views

subject to VO. It is worth noting that a bucketing

approach can be incorporated into this step to

improve the initial selection of five feature point

sets.

Step 2. Generate the corresponding coefficient

vectors, {

𝑣



,𝑖=1,⋯,5 }. Then, compute the

essential matrix 𝐸



using the five points algorithm

with {

𝑣



,𝑖=1,⋯,5 } as input.

Step 3. Generate 32 pairs of

𝑣





= ( 𝑝



′, 𝑞



′), {𝑖 =

1,⋯32} taking into account the error of the

corresponding feature points to { 𝑣



,𝑖=1,⋯,5 }.

With this, the 32-vertex essential matrix is

computed. And, compute the hypervolume of the

hypercube composed of these 32 vertices using the

Convex Hull algorithm.

Step 4. Select feature sets with hypervolume values

less than a threshold 𝐾. Estimate VO using the best

set of selected feature points.

𝑝





=

𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒





𝑝





±𝑒







𝑞





=

𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒





𝑞





±𝑒







𝑣





= (𝑝



′, 𝑞



′), {𝑖=1,⋯32} (12)

In Step 1 of Algorithm 1, partitioning the image

into grids using the bucketing technique and

extracting feature points from each grid region

resulted in improved performance regarding running

time and estimation error. The use of the bucketing

method proves to be more efficient in finding

solutions over multiple iterations. This can be

explained by the fact that, as shown in Section 2.3, we

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

294

empirically verified that the pose estimation error is

minimized when a well-spread distribution of feature

points is used as input. In the following, we showcase

our proposed approach and evaluate its practical

effectiveness.

𝑝







𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒







(13)



𝑝





−𝑒





𝑝





−𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒





𝑝





𝑒









𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒





𝑝





−𝑒







𝑞







𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒









𝑞





−𝑒





𝑞





−𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒





𝑞





𝑒









𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒





𝑞





−𝑒







3 EXPERIMENTS

In this section, we assess the effectiveness of our

suggested methodology using real data. The

subsequent text provides an overview of the

experimental setting. We conducted all experiments

utilizing parallel computation on a computer

equipped with an Intel Core i5-9400F CPU operating

at 2.9 GHz in MATLAB. The experimental data was

evaluated using the public RGB-D TUM datasets

freiburg1_desk. In this evaluation, we evaluate the

relationship of the estimation accuracy of the

Essential matrix with the hypervolume using real

data. The input points used in this evaluation were

extracted as ORB feature points (Rublee et al., 2011)

using the detectORBFeatures function of the

Computer Vision Toolbox, and the matching feature

points used to estimate the essential matrix were

obtained using the matchFeatures function.

3.1 Real Data

To evaluate our proposed approach to real-world

images, we used the TUM-RGBD dataset. This data

provides the RGB and depth image data and the

ground truth trajectory data for evaluating the Visual

Odometry and Visual SLAM systems. All data is at a

full frame rate of 30Hz and the camera sensor, a

Microsoft Kinect sensor, has a resolution of 640x480.

Next section we demonstrate the effectiveness of our

proposed optimal feature selection using the

hypervolume in the following experiments.

Figure 4: (Top) Graph showing rotation error and

correlation when hypervolume is small. (Bottom)

Distribution of selected 5 pairs of corresponding feature

points with small rotation error among cases with small

hypervolume.

3.2 The Effect of Hypervolume Based

Optimal Feature Selection

After completing Step 4 of Algorithm 1, the rotation

matrix calculated from the hypervolume and essential

matrix was evaluated by analyzing the correlation

using the error with the ground truth. The data was

accumulated and analyzed through 100 iterations. Fig.

4(Top) is a graph showing the relationship between

rotation error and hypervolume. For the rotation error,

the essential matrix estimated by the five-point

algorithm and the rotation matrix between the two

images were obtained from the ground truth trajectory

provided by the TUM RGB-D dataset. The x-axis of

the graph is represented by an index based on the size

of the hypervolume, and the y-axis is the rotation

error. The general trend is that the larger the

hypervolume, the larger the rotation error. First, let's

look at the 10 data with the smallest hypervolume.

Among them, No. 1 and No. 2 are the cases with the

Uncertainty Hypervolume in Point Feature-Based Visual Odometry

295

smallest rotation error, and the selected matching

feature point set is shown in Fig. 4(Bottom).

Figure 5: (Top) Graph showing rotation error and

correlation when hypervolume is large. (Bottom)

Distribution of selected 5 pairs of corresponding feature

points in the case with large rotation error among the cases

with large hypervolume.

We can see that the five pairs of corresponding

feature points are evenly distributed among each

other. This is the same result we found experimentally

in Section 3. The second case is when the

hypervolume is large and the rotation error is the

largest. As shown in Fig. 5, the five pairs of

corresponding feature points have a clustered

distribution. This confirms the correlation of different

rotation errors with hypervolume size and proves the

validity of our proposed approach.

3.3 Threshold for Hypervolume

Selection

When estimating the essential matrix using our

proposed method, we used multiple iterations to

select a set of 5 pairs of corresponding feature points

when the hypervolume is less than a threshold 𝐾

value. The criterion for selecting 𝐾 depends on the

number of matching points, which we found through

experimentation. In our case, if the number of

matching points is 150 or less, we selected a set of 5

pairs of corresponding feature points when the value

of logscale applied to the hypervolume is less than -

32, and if the number of matching points is more than

150, we set 𝐾 −36.

3.4 Comparison with RANSAC

Algorithm Using KITTI Odometry

Benchmark Dataset

The following evaluates the performance of the

proposed method by comparing the Absolute Pose

Error (APE) with the commonly used RANSAC

algorithm using the KITTI Odometry Benchmark

dataset (Geiger et al., 2012). In the previous section,

it is the same as the environment that evaluates the

correlation between Uncertainty Hypervolume and

pose estimation using the TUM dataset. The

algorithm for calculating VO used mono_vo, and the

feature point extraction method used the FAST

algorithm. The feature point matching method was

performed using a KLT tracker and a 5-point

Algorithm for motion estimation. The Scale Factor

was extracted and used from the Ground Truth

provided by the KITTI dataset. The method of

selecting the optimal feature point set with the

proposed Uncertainty Hypervolume method is the

same as Algorithm 2.

The difference from Algorithm 1 is that posture

estimation cannot be performed because the optimal

set of feature points cannot be selected depending on

the threshold value during the VO process. Therefore,

in the process of selecting the optimal set of feature

points, as in Algorithm 2, the Uncertainty

Hypervolume measurement index value calculated by

several extracts is sorted, and then VO is used as input

data for estimation by selecting the lower 10% set

with a small value. In addition, in this dataset

experiment, the Uncertainty Hypervolume

measurement index using the Rotation Matrix

obtained using it as Roll, Pitch, Yaw, and its size as

an indicator performed better than the Uncertainty

Hypervolume measurement index using Algorithm 1.

The following are the results of evaluating the KITTI

Odometry Benchmark 00, 02, 03, 04, and 05

sequences.

The results of the 00 sequence are shown in Fig.

6. Although both results have large errors due to drift

and estimation errors by scale factor, it can be seen

that APE's RMSE and Mean are relatively less

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

296

(a) Sequence 00

(b) Sequence 02

(d) Sequence 04

(e) Sequence 05

Figure 6: The KITTI Odometry Benchmark dataset (00, 02, 03, 04, 05) APE Results. Each sequence includes a graph of the

translation part APE, RMSE, Median, Mean, and Std, and a graph of the Error Mapped ontology. The left is when the proposed

Uncertified Hypervolume method is applied, and the right is the RANSAC method. It can be seen that the overall proposed

method has a smaller error than the case of RANSAC. Performance comparisons are summarized in Table 1.

Uncertainty Hypervolume in Point Feature-Based Visual Odometry

297

in the proposed method compared to RANSAC using

all feature points. In VO, the sensitivity to rotation is

high, so when calculating the Uncertainty

Hypervolume, it is better to use it to convert it to

Rotation Matrix, convert it to Roll, Pitch, and Yaw,

and then use the difference from the reference

Rotation Matrix as Uncertainty Hypervolume. In the

simplified part of the Uncertainty Hypervolume

calculation mentioned above, the measurement

indicators obtained using the Rotation Matrix instead

of the Essential Matrix are used. On the 02 sequence,

the same method as the 00 sequence shows slightly

better results. Next, for the 03 sequence, the proposed

method for RMSE shows better results, but for Mean,

RANSAC has a slight advantage. Next, it can be seen

from the graph that the results of the proposed method

for both the 04 and 05 sequences clearly show good

performance.

Table 1: Comparison of the Uncertainty Hypervolume

Method with the RANSAC Method.

Uncertainty

Hypervolume Method

APE (m)

RANSAC

Method APE (m)

uence RMSE Mean RMSE Mean

00 100.66 89.07 126.85 102.07

02 133.74 109.84 220.29 194.16

03 19.73 18.35 19.98 18.27

04 3.54 3.31 8.19 7.81

05 68.93 55.66 91.53 74.92

4 CONCLUSIONS

This study proposes the selection of optimal feature

points based on Uncertainty Hypervolume, a new

approach for estimating the Essential Matrix for

visual odometry. Through the pioneers' previous

research and simulation experiments, it was found

that uncertainty due to various errors in the selected

corresponding feature point pair affects posture

estimation, and that better performance can be

obtained in VO if a set of feature point pairs well

distributed in space is selected without clumping or

forming lines. Based on this, the uncertainty in VO

estimation is quantified by Uncertainty Hypervolume,

a new measurement index that considers the error of

the selected corresponding feature point pair and the

distribution they form. Using actual data, it was

confirmed that selecting a feature point set with a

small measurement index had a smaller error with the

Ground Truth value. The proposed method can work

effectively even though there are many features

extracted with large errors due to low visibility or bad

weather conditions. It can provide robustness in VO

because only features with high quality are collected

and used for VO estimation. The future work includes

the optimization of algorithms to improve the

computational efficiency while maintaining the

performance advantage of the proposed approach

over other conventional methods that rely on inlier

feature points and RANSAC.

Algorithm 2: Uncertainty hypervolume-based optimal

feature selection for visual odometry.

Data: 𝐿 correspondence feature point sets

Result: A set of five-feature point pairs from the

ottom 10% with low Uncertaint

pervolume

Step 1. Randomly select K sets of five-feature point

sets, {

(

𝑝



,𝑞



)

, 𝑖=1,⋯,5} from the feature pairs

detected from the images of two camera views

subject to VO. Incorporating a bucketing approach

into this step to improve the initial selection of the

five feature sets provides a way to select well-

distributed feature sets with few attempts.

Step 2. Generate the corresponding coefficient

vectors, { 𝑣



,𝑖=1,⋯,5 } . Then, compute the

essential matrix using the five-point algorithm with

{𝑣



,𝑖=1,⋯,5 } as input.

Step 3. Generate 32 pairs of 𝑣





= (𝑝



′, 𝑞



′), {𝑖=1

,⋯32} taking into account the error of the

corresponding feature points to {𝑣



,𝑖=1,⋯,5 }.

With this, the 32-vertices Essential matrix is

computed. And, the rotation matrix is estimated

using the essential matrix that constitutes the

vertex. The matrix is then converted to Roll, Pitch,

and Yaw. The difference between this and the

reference rotation, obtained with the reference

essential matrix, is taken as the uncertainty

hypervolume.

Step 4. Select a set of five feature point pairs from

the bottom 10% with low Uncertainty

Hypervolume. The VO is estimated using the best

set of five selected feature point pairs.

ACKNOWLEDGMENTS

This research was funded, in part, by the “Intelligent

Manufacturing Solution under Edge-Brain Framework”

(Grant 2022-0-00067 and IITP-2022-0-00187)

project, in part, by the Artificial Intelligence Graduate

School Program, Grant No. 2019-0-00421, and by ICT

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

298

Consilience Program, IITP-2020-0-01821, of the

Institute of Information and Communication

Technology Planning & Evaluation (IITP), sponsored

by the Korean Ministry of Science and Information

Technology (MSIT).

REFERENCES

D. Nistér, N. Oleg and B. James, "Visual odometry."

Computer Vision and Pattern Recognition, 2004.

CVPR 2004. Proceedings of the 2004 IEEE Computer

Society Conference on. Vol. 1. IEEE, pp I-I, 2004.

S. Poddar, R. Kottath, and V.Karar, “Motion Estimation

Made Easy: Evolution and Trends in Visual Odometry

”. In Recent Advances in Computer Vision (pp. 305-

331). Springer, Cham, 2019.

M.A. Fischler and C.B. Robert, "Random sample

consensus: an automated cartography."

Communications of the ACM 24.6 (1981): 381-395.

Igor Cvišić and Ivan Petrović, “Stereo odometry based on

careful feature selection and tracking,” in Proc. Eur.

Conf. Mobile Robots (ECMR), Sep. 2015.

Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, “A

robust technique for matching two uncalibrated images

through the recovery of the unknown epipolar

geometry,” Artificial Intelligence, vol. 78, no.1–2, pp.

87–119, 1995.

B. Kitt, A. Geiger, H. Lategahn, “Visual odometry based on

stereo image sequences with RANSAC-based outlier

rejection scheme, ” in Intelligent Vehicles

Symposium, 2010, pp. 486-492

H. H. Nguyen, and S. Lee, Orthogonality Index Based

Optimal Feature Selection for Visual Odometry. IEEE

Access, 7, 6228 4-62299, 2019.

R. Deriche, Z. Zhang, Q.-T. Luong and 0. Faugeras, Robust

recovery of the epipolar geometry for an uncalibrated

stereo rig, in: Proceedings Third European Conference

on Computer Vision I, Stockholm, Sweden (1994) 567-

576.

Nister, D. “ An Efficient Solution to the Five-Point

Relative Pose Problem. ” IEEE Transactions on

Pattern Analysis and Machine Intelligence.Volume 26,

Issue 6, June 2004.

Rublee, E., Rabaud, V., Konolige, K., & Bradski, G.

(2011). ORB: An efficient alternative to SIFT or SURF.

2011 International Conference on Computer Vision,

2564-2571.

Geiger, A., Lenz, P., & Urtasun, R. (2012). Are we ready

for autonomous driving? The KITTI vision benchmark

suite. 2012 IEEE Conference on Computer Vision and

Pattern Recognition (CVPR), 3354-3361.

Uncertainty Hypervolume in Point Feature-Based Visual Odometry

299