# A new simple family of Cantor sets in $\mathbb{R}^{3}$ all of whose
projections are one-dimensional

Olga Frolkina111Supported by Russian Foundation of Basic Research; Grant No.
19–01–00169.
Chair of General Topology and Geometry
Faculty of Mechanics and Mathematics
M.V. Lomonosov Moscow State University
and
Moscow Center for Fundamental and Applied Mathematics
Leninskie Gory 1, GSP-1
Moscow 119991, Russia
E-mail<EMAIL_ADDRESS>
###### Abstract

In 1994, J. Cobb described a Cantor set in $\mathbb{R}^{3}$ each of whose
projections into $2$-planes is one-dimensional. A series of works by other
authors developing this field followed. We present another very simple series
of Cantor sets in $\mathbb{R}^{3}$ all of whose projections are connected and
one-dimensional. These are self-similar Cantor sets which go back to the work
of Louis Antoine, and we celebrate their centenary birthday in 2020–2021.

Keywords: Euclidean space, projection, Cantor set, embedding, topological
dimension, Antoine’s necklace, self-similarity, Hausdorff measure, Hausdorff
dimension, Lebesgue measure, demension.

MSC: Primary 54F45; Secondary 57N12, 28A80.

##  1 Introduction

In 1994, J. Cobb constructed a Cantor set in $\mathbb{R}^{3}$ such that its
projection into every $2$-plane is one-dimensional [13]. In the same paper, he
stated a general question: “Given $n>m>k>0$, does there exist a Cantor set in
$\mathbb{R}^{n}$ such that each of its projections into $m$-planes is exactly
$k$-dimensional?” Such sets will be called $(n,m,k)$ Cantor sets, or briefly
$(n,m,k)$-sets.

The case $m=k$ was already solved. L. Antoine constructed a Cantor set in
$\mathbb{R}^{2}$ whose projections coincide with those of a regular hexagon
[5, 9, p. 272; and fig. 2 on p. 273]. (Another $(2,1,1)$ set is described in
[13, p. 124]; for a different example and further results on planar sets with
prescribed projections, see [15].) K. Borsuk constructed an $(n,m,m)$ Cantor
set in [12]; as a corollary, he obtained a positive answer to the question of
R.H. Fox: there exists a simple closed curve in $\mathbb{R}^{3}$ such that
each of its plane projections is two-dimensional.

Later, examples for the following triples appeared: for $(n,m,m-1)$ in [17]
and for $(n,n-1,k)$ in [9] — both generalized Cobb’s idea; and a different
construction for $(n,n-1,n-2)$ in [18], where further references can be found.
Interestingly, all projections of a _typical_ Cantor set are Cantor sets [19].

Cobb’s construction of a $(3,2,1)$ Cantor set is rather complicated; the
resulting set is tame. I suggested a different approach observing that there
exist Antoine’s necklaces all of whose projections are one-dimensional [18,
Thm. 1]. To obtain a necklace with this property, I controlled the location
and thickness of solid tori at prelimit levels of construction; such control
can also be generalized to all Cantor sets in all dimensions: For each Cantor
set $K\subset\mathbb{R}^{n}$, $n\geqslant 2$, there is an arbitrarily small
isotopy $\\{h_{t}\\}:\mathbb{R}^{n}\cong\mathbb{R}^{n}$ such that the
projection of $h_{1}(K)$ into each $(n-1)$-plane has dimension $(n-2)$ [18,
Thm. 2].

The control process is not trivial. It is based on the plank theorem (for the
planar case, see English translation of A. Tarski’s and H. Moese’s papers
(1931–32) in [26, Chapter 7] where the history of the problem is also
discussed; the high-dimensional case is treated in [8]).

The control process is necessary. In fact: For each Cantor set
$K\subset\mathbb{R}^{n}$, $n\geqslant 2$, there exists an arbitrarily small
isotopy $\\{h_{t}\\}:\mathbb{R}^{n}\cong\mathbb{R}^{n}$ such that for each
proper subspace $\Pi\subset\mathbb{R}^{n}$ the dimension of the projection of
$h_{1}(K)$ into $\Pi$ equals $\dim\Pi$ [18, Stat. 12].

In this article, I restore the original definition of Antoine’s necklaces from
[4]. These sets are self-similar. In Theorem 2.6, we formulate simple
additional conditions which ensure that all projections of these sets are one-
dimensional. Theorems 2.7 and 2.8 imply that these conditions can actually be
satisfied. Self-similarity eliminates the need for thickness control: one-
dimensionality is now derived from properties of planar Lebesgue measure or
Hausdorff measure.

We also consider a high-dimensional case (Theorem 3.2). Using Rushing’s idea
to combine results of Shtan’ko and Väisälä [29, p. 599], we get a new shorter
— but less elementary — proof of the above proposition concerning
$(n,n-1,n-2)$-sets, see Corollary 3.6; the usage of the plank theorem is
eliminated from the proof. Finally, we observe that the sets $C^{s}$ from [29]
are $(3,2,1)$-sets assuming that $1\leqslant s<2$ (Remark 3.4).

Let us underline that the purpose of this note is to propose a possibly
simplest example of a $(3,2,1)$ Cantor set in a self-contained presentation.
By this reason we first give an “elementary” exposition, and thereafter state
some generalizations.

Finishing the introduction, I would like to recall some other results of L.
Antoine.

Antoine’s necklace is a first example of a wild Cantor set in
$\mathbb{R}^{3}$; its description [4, 78, p. 91–92], briefly presented in [2],
is clear, in contrast to vague descriptions with chains of “curved solid tori”
that are often found in later retelling. (Note that in order to get a Cantor
set, the diameters of prelimit tori should tend to zero. For self-similar
constructions, this holds automatically. Constructions with e.g. “four curved
tori” must be provided with a possibility proof.) Antoine constructed first
wild arcs in $\mathbb{R}^{3}$. His first example [2], [4, 54–58, p. 65–70]
consists of an arc with a sequence of trefoils tied in it and convergent to
the end point, united with a straight line segment. The second wild arc [2],
[4, 83, p. 97] contains an Antoine’s necklace. Antoine constructed an
everywhere wild arc in $\mathbb{R}^{3}$ [5]. In [3], Antoine announced and in
[5] described in detail an embedded $2$-sphere in $\mathbb{R}^{3}$ which
contains an Antoine’s necklace, this is the first example of a wild surface;
see [27, Thm. 18.7]. I would recommend the whole volume [6] for acquaintance
with Antoine’s destiny and results; and especially [23], [24], and [27,
Preface, Sec. 18, and p. 256] concerning the importance of his topological
discoveries.

### Notation and Conventions

A plane in $\mathbb{R}^{3}$ is a 2-dimensional affine subspace.

For a plane $\Pi\subset\mathbb{R}^{3}$, the orthogonal projection map
$\mathbb{R}^{3}\to\Pi$ is denoted by $p_{\Pi}$.

For a set $X\subset\mathbb{R}^{3}$, its affine hull is denoted by
$\operatorname{Aff}X$, and its diameter by $\operatorname{diam}X$.

For topological dimension theory, see [21]; for definition and properties of
Lebesgue measure, refer to e.g. [11]. Definitions and basic properties of
Hausdorff measure and Hausdorff dimension can be found in [21, Chap. VII] of
[14, 3.6]. For Shtan’ko demension theory, see [31]; its presentation is given
in [16] and [14, 3.4].

##  2 Main construction and results: “elementary” approach

There are several (more or less restrictive) definitions of Antoine’s
necklaces, compare e.g. [30], [20], [35], [18]. They depend on allowed shapes,
numbers and positions of solid tori on prelimit stages. For our purpose, we
will only restrict ourselves by the class of self-similar Antoine’s necklaces.
This construction comes back to L. Antoine [2], [4].

###### Definition 2.1

Let $\Pi$ be a plane in $\mathbb{R}^{3}$. Let $D\subset\Pi$ be a disk of
radius $r>0$ with center $Q$, and let $\ell\subset\Pi$ be a straight line such
that $d(Q,\ell)=R>r$. A _standard solid torus_ $T$ is the solid torus of
revolution generated by revolving $D$ in $\mathbb{R}^{3}$ about $\ell$. The
_central circle_ of $T$ is the circle generated by rotating the point $Q$. The
_center_ of $T$ is the center of its central circle. For a given standard
solid torus $T$, the corresponding radii will be denoted by $r_{T}$ and
$R_{T}$, the central circle by $C_{T}$, and the center by $Q_{T}$.

###### Definition 2.2

_A simple chain_ in a standard solid torus $T\subset\mathbb{R}^{3}$ is a
finite family $T_{1},\ldots,T_{k}$, $k\geqslant 3$, of pairwise disjoint
standard solid tori such that

1) $T_{1}\cup\ldots\cup T_{k}\subset T$;

2) centers of $T_{1},\ldots,T_{k}$ are subsequent vertices of a regular convex
$k$-gon inscribed in the central circle of $T$;

3) $T_{i}$ and $T_{j}$ are linked for $|i-j|\equiv 1\mod k$, and are not
linked otherwise.

See Fig. 1(a).

A simple chain $T_{1},\ldots,T_{k}$ looks like a “usual necklace” which winds
once around the hole of $T$; no one of the tori $T_{i}$ embraces the hole of
$T$.

(a) A simple chain.

(b) A regular simple chain.

Figure 1: Simple chains with $2m=24$ links.

###### Definition 2.3

A simple chain $T_{1},\ldots,T_{2m}$, $m\geqslant 2$ in a standard solid torus
$T\subset\mathbb{R}^{3}$ will be called _regular_ if

1) all $T_{i}$’s are congruent,

2)
$\operatorname{Aff}C_{T_{1}}=\operatorname{Aff}C_{T_{3}}=\ldots=\operatorname{Aff}C_{T_{2m-1}}=\operatorname{Aff}C_{T}$,

3) $\operatorname{Aff}C_{T_{2i}}\bot\operatorname{Aff}C_{T}$ for
$i=1,\ldots,m$, and

4) each straight line
$\operatorname{Aff}C_{T_{2i}}\cap\operatorname{Aff}C_{T}$, $i=1,\ldots,m$,
touches the circle $C_{T}$.

See Fig. 1(b).

###### Definition 2.4

Let $T\subset\mathbb{R}^{3}$ be a standard solid torus, and let
$S_{1},\ldots,S_{k}:\mathbb{R}^{3}\cong\mathbb{R}^{3}$ be similarity
transformations such that $T_{1}:=S_{1}(T),\ldots,T_{k}:=S_{k}(T)$ is a simple
chain in $T$. For each $\lambda\in\mathbb{N}$ take

$M_{\lambda}=\bigcup\limits_{(i_{1},\ldots,i_{\lambda})\in\\{1,\ldots,k\\}^{\lambda}}S_{i_{1}}\circ
S_{i_{2}}\circ\ldots\circ S_{i_{\lambda}}(T).$

We have $T\supset M_{1}\supset M_{2}\supset M_{3}\supset\ldots$. The
intersection
$\mathcal{A}(T;S_{1},\ldots,S_{k}):=\bigcap\limits_{\lambda=1}^{\infty}M_{\lambda}$
can be easily seen to be a Cantor set; we call it _a self-similar Antoine’s
Necklace generated by $(T;S_{1},\ldots,S_{k})$_. For each $i=1,\ldots,k$ the
piece $\mathcal{A}(T;S_{1},\ldots,S_{k})\cap T_{i}$ equals
$S_{i}(\mathcal{A}(T;S_{1},\ldots,S_{k}))$, hence it is geometrically similar
to $\mathcal{A}(T;S_{1},\ldots,S_{k})$.

A self-similar Antoine’s Necklace will be called _regular_ if the simple chain
$T_{1},\ldots,T_{k}$ in $T$ is regular. (In particular, $k$ is even, and the
similarity coefficients of all $S_{i}$’s are equal.) See Fig. 2.

Figure 2: A piece of a self-similar Antoine’s necklace: first stages of
construction.

###### Remark 2.5

The necklace $\mathcal{A}(T;S_{1},\ldots,S_{k})$ is the invariant set with
respect to the set of similarities $\\{S_{1},\ldots,S_{k}\\}$; we thus may
omit $T$ from the notation. For details, see [22].

L. Antoine writes: “Numérotons de $1$ à $k$ les $k$ tores d’ordre $\lambda+1$
intérieurs à un tore d’ordre $\lambda$. Je supposerai que deux tores ayant des
numeros consécutifs, ainsi que les tores $k$ et $1$, sont entlaces, alors que
deux tores non consécutifs ne sont pas enlacés. Je suppose enfin que les
centres de ces $k$ tores sont sur la circonférence d’ordre $\lambda$ et qu’en
joignant ces points dans l’ordre de leurs numéros, on forme un polygone
régulier non étoilé. Ces conditions sont possibles à réaliser. Il suffit pour
cela que le rayon du méridien d’un tore $T$ soit assez petit, par rapport au
rayon de la circonférence $C$ de ce tore, et que $k$ soit assez grand. On peut
alors prendre pour tores d’ordre $\lambda+1$ des tores semblables au tore
d’ordre $\lambda$. La possibilité de la construction a alors lieu quel que
soit $\lambda$, $k$ étant fixe. Pour simplifier quelques raisonnements
ultérieurs, je supposerai que ce nombre $k$ est pair” [4, 78, p. 92].
Antoine’s description does not explicitly indicate similarity transformations,
in contrast to what is done in Definition 2.4. But I believe that, taking
similar tori, he had in mind precisely this. In [2, I] and [5] he carries out
several more constructions using the idea of similarity. Antoine assumed
neither that chains have a special position in space, nor he demanded that the
solid tori which constitute a chain are congruent to each other. To obtain
examples with a greater degree of symmetry, we added some restrictive
conditions (regularity), as is done in [25], [20], [35], [28].

Self-similar Antoine’s necklaces with different $k$’s are inequivalently
embedded, this follows from [30, Thm. 2]. For a regular self-similar Antoine’s
necklace, the inequality $k\geqslant 20$ holds [35]. A schematic picture of
two inequivalently embedded self-similar Antoine’s necklaces one of which is
regular and the other is not is given in [20]; in that example, $k=60$.

###### Theorem 2.6

Let $\mathcal{A}(T;S_{1},\ldots,S_{k})\subset\mathbb{R}^{3}$ be a self-similar
Antoine’s necklace, and let $s_{i}$ be the similarity coefficient of $S_{i}$
for $i=1,\ldots,k$. If $s_{1}^{2}+\ldots+s_{k}^{2}<1$, then
$p_{\Pi}\bigl{(}\mathcal{A}(T;S_{1},\ldots,S_{k})\bigr{)}$ is a connected one-
dimensional set for each plane $\Pi\subset\mathbb{R}^{3}$.

In particular, a regular self-similar Antoine’s necklace
$\mathcal{A}(T;S_{1},\ldots,S_{2m})$ with $2ms^{2}<1$ is a $(3,2,1)$-set; here
$s$ is the similitude coefficient of $S_{i}$’s.

Below we will return to this result from a more general point of view, see
Theorem 3.2.

Two points should be discussed. The first one: do regular self-similar
Antoine’s necklaces exist? (Equivalently, can one construct a standard solid
torus $T\subset\mathbb{R}^{3}$ together with a regular simple chain
$T_{1},\ldots,T_{2m}$ in it such that each $T_{i}$ is similar to $T$ ?) If
“yes”, the second one: do regular self-similar Antoine’s necklaces with
$2ms^{2}<1$ exist?

If $T_{1},\ldots,T_{2m}$ is a regular simple chain in $T$, and $T_{i}$’s are
similar to $T$, then the numbers $\frac{r_{T}}{R_{T}}$, $m$, and $s$ should
satisfy several inequalities, see [35], [28]; it seems that no exact
description of solutions exist. A picture representing a regular self-similar
Antoine’s necklace with $2m=60$ can be found in [20, p. 294]. Using numerical
methods, it is shown in [35] that $2m\geqslant 20$ for any regular self-
similar Antoine’s necklace; and a picture for $2m=20$ is given. For
$2m=20,22,24,26,28$, existence of regular self-similar Antoine’s necklaces
follows from [28, p. 146], where computer software was also used. Given a
standard solid torus $T$ with $\frac{r_{T}}{R_{T}}$ sufficiently small, it
seems plausible that for all sufficiently large integers $m$ there exists in
$T$ a regular simple chain of $2m$ tori similar to $T$. I have not find an
exact statement in the literature. More or less close assertions can be found
without proofs in [25, p. 274] (“It can be verified by elementary calculations
that for all sufficiently big numbers $n$, there exist values of $\lambda$ \-
not too big and not too small - and corresponding thicknesses of $X$ such
that…”; formulaes and a picture follow saying that the standard solid torus
$X$ contains a regular simple chain of $n$ tori similar to $X$ with the
similarity coefficient $\lambda$) and [28, p. 132] (“…It seems clear that once
this minimum is achieved, one can simply add more tori, in specific quantities
derived below, to obtain another viable construction of the Antoine
necklace”). We give a rigorous statement and complete it by a purely
mathematical proof.

###### Theorem 2.7

For a standard solid torus $B\subset\mathbb{R}^{3}$, the following are
equivalent:

(1) $R_{B}>3r_{B}$;

(2) for each sufficiently large integer $m$ there exist a standard solid torus
$T$ and a regular simple chain $T_{1},\ldots,T_{2m}$ in $T$ such that each
$T_{i}$ ($i=1,\ldots,2m$) is congruent to $B$;

(3) for each sufficiently large integer $m$ there exist a standard solid torus
$\mathbf{T}$ and a regular simple chain $T_{1},\ldots,T_{2m}$ in $\mathbf{T}$
such that each $T_{i}$ ($i=1,\ldots,2m$) is congruent to $B$, and $\mathbf{T}$
is similar to $B$.

Now let us discuss the condition $2ms^{2}<1$ from Theorem 2.6. Are we possible
to satisfy it? Yes. Briefly speaking, the similarity coefficient $s$ tends to
zero as $m\to\infty$; but the product $ms$ is bounded. Hence necklaces which
satisfy the assumption $2ms^{2}<1$ should exist. We get several simple
estimates:

###### Theorem 2.8

Let $\mathcal{A}(T;S_{1},\ldots,S_{2m})\subset\mathbb{R}^{3}$ be a regular
self-similar Antoine’s necklace, and let $s$ be the similarity coefficient of
$S_{i}$’s.

1) If $s<\frac{1}{2\pi}$, then $2ms^{2}<1$.

2) If $\frac{r_{T}}{R_{T}}<\frac{1}{2\pi-1}$, then $s<\frac{1}{2\pi}$.

3) If $2m\geqslant 40$, then $s<\frac{1}{2\pi}$.

##  3 A generalization using Hausdorff measure

###### Definition 3.1

A zero-dimensional compact set $X\subset\mathbb{R}^{n}$ is called _tame_ if
there exists a homeomorphism $h$ of $\mathbb{R}^{n}$ onto itself such that
$h(X)$ is a subset of a straight line in $\mathbb{R}^{n}$; otherwise, $X$ is
called _wild_.

In $\mathbb{R}^{2}$ each zero-dimensional compactum is tame [4, 75, p. 87–89]
(one may also refer to [27, Chap. 13]).

Antoine proved that the necklaces he described are wild [4, 80, p. 93]. A
later proof using fundamental group can be found in [27, Sect. 18] or [14,
2.1].

Antoine’s construction was generalized to higher dimensions by W.A.
Blankinship and by A.A. Ivanov, see [18] for references; hence
$\mathbb{R}^{n}$ contains wild Cantor sets provided that $n\geqslant 3$.

###### Theorem 3.2

Let $X\subset\mathbb{R}^{n}$ be a wild Cantor set of a zero
$(n-1)$-dimensional Hausdorff measure: $m_{n-1}(X)=0$. (In particular, this
holds if the Hausdorff dimension $\dim_{H}X<n-1$.) Then $X$ is an
$(n,n-1,n-2)$-set.

For examples to this theorem, see Remark 3.4 and Corollary 3.6.

Theorem 3.2 covers Theorem 2.6 except connectedness of projections because of
the following:

###### Statement 3.3

Let $\mathcal{A}=\mathcal{A}(T;S_{1},\ldots,S_{k})\subset\mathbb{R}^{3}$ be a
self-similar Antoine’s necklace, and let $s_{i}$ be the similarity coefficient
of $S_{i}$ for $i=1,\ldots,k$. If $s_{1}^{2}+\ldots+s_{k}^{2}<1$, then
$m_{2}(\mathcal{A})=0$.

Statement 3.3 follows from Moran’s theorem, see [29, p. 600], [22, 5.3(1)]; we
give a short proof for our simple case.

Proof of Statement 1. For each $\lambda\in\mathbb{N}$, we have
$\mathcal{A}\subset M_{\lambda}$, see Definition 2.4. We can assume (zooming
if necessary) that $\operatorname{diam}T=1$. Now $\mathcal{A}$ is covered by
the union of $k^{\lambda}$ sets $S_{i_{1}}\circ S_{i_{2}}\circ\ldots\circ
S_{i_{\lambda}}(T)$, where
$(i_{1},\ldots,i_{\lambda})\in\\{1,\ldots,k\\}^{\lambda}$, and

$\operatorname{diam}S_{i_{1}}\circ S_{i_{2}}\circ\ldots\circ
S_{i_{\lambda}}(T)=s_{i_{1}}\cdot s_{i_{2}}\cdot\ldots\cdot s_{i_{\lambda}}.$

We have

$\sum\limits_{(i_{1},\ldots,i_{\lambda})\in\\{1,\ldots,k\\}^{\lambda}}\Bigl{(}\operatorname{diam}S_{i_{1}}\circ
S_{i_{2}}\circ\ldots\circ S_{i_{\lambda}}(T)\Bigr{)}^{2}\leqslant$
$\leqslant\sum\limits_{(i_{1},\ldots,i_{\lambda})\in\\{1,\ldots,k\\}^{\lambda}}\bigl{(}s_{i_{1}}\cdot
s_{i_{2}}\cdot\ldots\cdot
s_{i_{\lambda}}\bigr{)}^{2}=\bigl{(}s_{1}^{2}+\ldots+s_{k}^{2}\bigr{)}^{\lambda}.$

This tends to zero as $\lambda\to\infty$, hence the $2$-dimensional Hausdorff
measure $m_{2}(\mathcal{A})=0$. $\square$

###### Remark 3.4

For each $s$ with $1\leqslant s\leqslant 3$, T.B. Rushing [29, Thm. 1, 2]
described a wild Cantor set $C^{s}\subset\mathbb{R}^{3}$ such that
$\dim_{H}C^{s}=s$. Theorem 3.2 implies that the sets $C^{s}$ are $(3,2,1)$
Cantor sets, provided that $1\leqslant s<2$. The sets $C^{s}$ are Antoine-type
necklaces; their prelimit chains constructed under very careful control have a
rather complex arrangement in space; they do not satisfy our Definition 2.4.
Rushing’s scheme is too complicated for our purpose; the aim of this paper is
to give an example which is as simple as possible.

###### Remark 3.5

It is apparently an open question, whether wild _self-similar_ Cantor sets in
$\mathbb{R}^{n}$ for $n\geqslant 4$ do exist [20, p. 298, Question]. T.B.
Rushing, without detailed proof, indicates the possibility of this [29, p.
611].

As another application of Theorem 3.2, we get a new proof of our earlier
result:

###### Corollary 3.6

[18, Thm. 2] Let $X\subset\mathbb{R}^{n}$, $n\geqslant 2$, be any Cantor set.
For each $\varepsilon>0$ there exists an $\varepsilon$-isotopy
$\\{h_{t}\\}:\mathbb{R}^{n}\cong\mathbb{R}^{n}$ such that $h_{1}(X)$ is an
$(n,n-1,n-2)$-set.

The new proof is shorter but less elementary since it refers to strong results
of Shtan’ko, Edwards and Väisälä.

Proof of Corollary 1. The case of a tame $X$ reduces to known results, see
[18] for details.

Let $X$ be wild. We start essentially as in Rushing’s remark [29, p. 599].
Suppose we are given an arbitrary $\varepsilon>0$. According to Väisälä’s
theorem, there exists an isotopy
$\\{h_{t}\\}:\mathbb{R}^{n}\cong\mathbb{R}^{n}$ such that the Hausdorff
dimension $\dim_{H}(h_{1}(X))$ equals the embedding dimension
$\operatorname{dem}X$ [32], [14, Thm. 3.6.2]. By the Edwards construction [16,
p. 208–209], [32, p. 168], this isotopy can be chosen to be an
$\varepsilon$-isotopy. For a wild Cantor set $X\subset\mathbb{R}^{n}$, we have
$\operatorname{dem}X\leqslant n-2$ (for $n\geqslant 4$, see [31, Remark 2,
Prop. 1]; for $n=3$ use [7, Lemma 4]). Consequently
$\dim_{H}(h_{1}(X))\leqslant n-2$, so $h_{1}(X)$ is an $(n,n-1,n-2)$-set by
Theorem 3.2.

(Actually, we have $\operatorname{dem}X=n-2$ for a wild Cantor set
$X\subset\mathbb{R}^{n}$. This is a special case of strong results from
geometric topology due to several authors, see [31], [16, p. 195; Thm. 1.4 and
its proof], [29, p. 598], [14, Thm. 3.4.11, Exer. 3.4.1], [10, p. 5] for
statements, historical details and references.) $\square$

##  4 Proofs of Theorems 2.6–3.2

Proof of Theorem 1. We already derived Theorem 2.6 (except connectedness of
projections) from Theorem 3.2. Here we give an independent “elementary” proof.

For brevity, denote $\mathcal{A}=\mathcal{A}(T;S_{1},\ldots,S_{k})$. By [4,
80-82, p. 93-96], no topological $2$-sphere separates a necklace. Hence all
projections of a necklace are connected (see [18, Stat. 2] for details).

Let $\Pi\subset\mathbb{R}^{3}$ be any plane. It is clear that
$p_{\Pi}\mathcal{A}$ contains more than one point; so it remains to show that
$\dim p_{\Pi}\mathcal{A}\neq 2$.

A subset $X\subset\mathbb{R}^{n}$ is $n$-dimensional iff it contains an
$n$-ball [21, Thm. IV 3]. Therefore it suffices to prove that
$\lambda_{2}(p_{\Pi}\mathcal{A})=0$, where $\lambda_{2}$ denotes planar
Lebesgue measure. Suppose that $\lambda_{2}(p_{\Pi}\mathcal{A})>0$. (In
$\mathbb{R}^{n}$, $n$-dimensional Lebesgue and Hausdorff measures differ by a
constant multiple. We thus may use $m_{2}$ instead of $\lambda_{2}$. Here I
decided to use $\lambda_{2}$ due to its wider popularity.)

For each non-empty compact subset $X\subset\mathbb{R}^{3}$ define a real
number

$\alpha(X)=\sup\\{\lambda_{2}(p_{L}X)\ |\ L\text{ is a plane in
}\mathbb{R}^{3}\\}.$

Any similarity transformation of $\mathbb{R}^{n}$ is a composition of a
scaling and an isometry. As a corollary, for any similarity transformation
$S:\mathbb{R}^{3}\to\mathbb{R}^{3}$ and any non-empty compactum
$X\subset\mathbb{R}^{3}$ we have $\alpha(S(X))=s^{2}\cdot\alpha(X)$, where $s$
is the similarity coefficient of $S$.

By assumption, $\alpha(\mathcal{A})>0$. Take a positive number

$\varepsilon\leqslant\alpha(\mathcal{A})-\bigl{(}s_{1}^{2}+\ldots+s_{k}^{2}\bigr{)}\cdot\alpha(\mathcal{A}).$

Let $L(\varepsilon)$ be any plane such that
$\alpha(\mathcal{A})-\varepsilon<\lambda_{2}(p_{L(\varepsilon)}\mathcal{A})\leqslant\alpha(\mathcal{A})$.
Recall that

$\mathcal{A}=S_{1}(\mathcal{A})\cup\ldots\cup S_{k}(\mathcal{A}).$

We have

$\alpha(\mathcal{A})-\varepsilon<\lambda_{2}(p_{L(\varepsilon)}\mathcal{A})=\lambda_{2}\Bigl{(}p_{L(\varepsilon)}\bigl{(}S_{1}(\mathcal{A})\cup\ldots\cup
S_{k}(\mathcal{A})\bigr{)}\Bigr{)}\leqslant$
$\leqslant\lambda_{2}\Bigl{(}p_{L(\varepsilon)}\bigl{(}S_{1}(\mathcal{A})\bigr{)}\Bigr{)}+\ldots+\lambda_{2}\Bigl{(}p_{L(\varepsilon)}\bigl{(}S_{k}(\mathcal{A})\bigr{)}\Bigr{)}\leqslant$
$\leqslant\alpha\bigl{(}S_{1}(\mathcal{A})\bigr{)}+\ldots+\alpha\bigl{(}S_{k}(\mathcal{A})\bigr{)}=\bigl{(}s_{1}^{2}+\ldots+s_{k}^{2}\bigr{)}\cdot\alpha(\mathcal{A}).$

This contradicts the choice of $\varepsilon$. Thus
$\lambda_{2}(p_{\Pi}\mathcal{A})=0$, and the proof is finished. $\square$

Proof of Theorem 2. Evidently, $(2)$ follows from $(3)$. Also, $(2)$ easily
implies $(1)$, see [29, p. 604, (1)] or [28, p. 136, (2)].

$(1)\Rightarrow(2)$ is intuitively clear, but its rigorous proof is rather
long.

_Step 1: construct a special link._ Take any number
$A\in\bigl{(}R_{B}+r_{B},2(R_{B}-r_{B})\bigr{)}$; the interval is non-empty
since $R_{B}>3r_{B}$. Define the circles
$C_{B_{1}},C_{B_{2}}\subset\mathbb{R}^{3}$ by their parametric equations:

$C_{B_{1}}:\quad x=R_{B}\cos u,\ y=R_{B}\sin u,\ z=0$

and

$C_{B_{2}}:\quad x=A+R_{B}\cos v,\ y=0,\ z=R_{B}\sin v,$

where $u$ and $v$ run over the reals (see Fig. 3(a)).

(a) Initial pair of circles.

(b) Rotation scheme.

(c) After rotations.

Figure 3: Illustration for the proof of the implication $(1)\Rightarrow(2)$ of
Theorem 2.

Let us show that the distance $d(C_{B_{1}},C_{B_{2}})>2r_{B}$. Take any points
$P_{1}\in C_{B_{1}}$, $P_{2}\in C_{B_{2}}$; let $P_{1}=(R_{B}\cos u,R_{B}\sin
u,0)$ and $P_{2}=(A+R_{B}\cos v,0,R_{B}\sin v)$. We have

$d^{2}(P_{1},P_{2})=(A+R_{B}\cos v-R_{B}\cos u)^{2}+(R_{B}\sin
u)^{2}+(R_{B}\sin v)^{2}=$ $=A^{2}+2AR_{B}\cos v-2AR_{B}\cos u-2R_{B}^{2}\cos
u\cos v+2R_{B}^{2}=:F(u,v).$

Now

$\left\\{\begin{array}[]{ll}F^{\prime}_{u}=0\\\ F^{\prime}_{v}=0\\\
\end{array}\right.\\\ \iff\left\\{\begin{array}[]{ll}(A+R_{B}\cos v)\sin
u=0\\\ (A-R_{B}\cos u)\sin v=0\\\ \end{array}\right.\\\
\iff\left\\{\begin{array}[]{ll}\sin u=0\\\ \sin v=0\\\ \end{array}\right.\\\ $

(the last equivalence follows from the assumption $A>R_{B}+r_{B}>R_{B}$). It
remains to compute the distances $d(P_{1},P_{2})$ for $4$ pairs of points
corresponding to $u,v\in\\{0,\pi\\}$; the required inequality is easily
obtained.

It is also easy to verify that $C_{B_{1}}$, $C_{B_{2}}$ are linked.

Now let $B_{1},B_{2}\subset\mathbb{R}^{3}$ be closed $r_{B}$-neighborhoods of
$C_{B_{1}}$, $C_{B_{2}}$; they are disjoint linked standard solid tori
congruent to $B$.

_Step 2: bend the link, complete the new link to a regular simple chain._ For
a straight line $\ell\|Oz$ and an angle $\psi$ we denote by $\rho_{\ell,\psi}$
the rotation transformation of $\mathbb{R}^{3}$ with axis $\ell$ on angle
$\psi$; we rotate in a counterclockwise direction when looking towards the
plane $Oxy$ from the half-line $Oz_{+}$ (the coordinate system $Oxyz$ is
assumed to be a right-handed one).

Let $a_{2}\|Oz$ and $b_{2}\|Oy$ be the straight lines passing through
$Q_{B_{2}}$. For $\psi\in(0,\pi)$ let $Q_{\psi}$ be the (unique) common point
of the lines $\rho_{Oz,\psi}(Oy)$ and $\rho_{a_{2},-\psi}(b_{2})$, see Fig.
3(b).

By continuity, there is a number $\psi_{0}\in(0,\pi)$ such that for each
$\psi\in(0,\psi_{0})$ we have $OQ_{\psi}>R_{B}+r_{B}$, and the set
$B_{1}\cap\rho_{a_{2},-\psi}(B_{2})=\rho_{Oz,\psi}(B_{1})\cap\rho_{a_{2},-\psi}(B_{2})$
is empty.

Now let $m$ be any integer such that $m>\frac{\pi}{2\psi_{0}}$. Fix the angle
$\psi:=\frac{\pi}{2m}<\psi_{0}$. Solid tori
$T_{1}:=B_{1}=\rho_{Oz,\psi}(B_{1})$ and $T_{2}:=\rho_{a_{2},-\psi}(B_{2})$
are congruent to $B$. (These tori depend on $\psi$, but we do not reflect this
in the notation for brevity.) By the above conditions, $T_{1}$ and $T_{2}$ are
disjoint and linked. Their central circles
$C_{T_{1}}=\rho_{Oz,\psi}(C_{B_{1}})$ and
$C_{T_{2}}=\rho_{a_{2},-\psi}(C_{B_{2}})$, the way we see them from the axis
$Oz_{+}$, are shown in Fig. 3(c). For each $k=1,\ldots,m-1$ define

$T_{2k+1}:=\rho_{\ell_{\psi},-\frac{2\pi k}{m}}(T_{1})\quad\text{ and }\quad
T_{2k+2}:=\rho_{\ell_{\psi},-\frac{2\pi k}{m}}(T_{2}),$

where $\ell_{\psi}$ is the straight line passing through $Q_{\psi}$ parallel
to $Oz$. Let $C_{\psi}$ be the circle in the $Oxy$-plane having center
$Q_{\psi}$ and radius $OQ_{\psi}$; it contains the centers of
$T_{1},\ldots,T_{2m}$. Define a standard solid torus $T$ as a closed
$R_{T}$-neighborhood of $C_{\psi}$ for an arbirtary
$R_{T}\in\bigl{(}0,OQ_{\psi}-(R_{B}+r_{B})\bigr{)}$. Now $T_{1},\ldots,T_{2m}$
is a regular simple chain in $T$, as desired.

$(2)\Rightarrow(3)$. Let $T_{1},\ldots,T_{2m}$ be a regular simple chain in
$T$, where each $T_{i}$ is congruent to $B$; let $\mathbf{T}$ be the standard
solid torus similar to $B$ with $C_{\mathbf{T}}=C_{T}$, see Fig. 4; here two
bold solid (resp. dashed) arcs denote the boundary of the solid torus $T$
(resp. $\mathbf{T}$), and the thin dashed arc represents the circle
$C_{\mathbf{T}}=C_{T}$. Assuming that $\sin\frac{\pi}{m}<\frac{r_{B}}{R_{B}}$,
we will prove that $r_{\mathbf{T}}>R_{B}+r_{B}$ (hence
$T_{i}\subset\mathbf{T}$ for each $i=1,\ldots,2m$).

Figure 4: Illustration for the proof of the implication $(2)\Rightarrow(3)$ of
Theorem 2.

Since $T_{1}$ and $T_{3}$ are disjoint, the distance
$Q_{T_{1}}Q_{T_{3}}=2R_{T}\sin\frac{\pi}{m}$ is greater than
$\operatorname{diam}T_{1}+\operatorname{diam}T_{3}=2(R_{B}+r_{B})$. This
together with the assumption $\sin\frac{\pi}{m}<\frac{r_{B}}{R_{B}}$ implies
that

$R_{T}>\frac{R_{B}+r_{B}}{\sin\frac{\pi}{m}}>\frac{R_{B}(R_{B}+r_{B})}{r_{B}}.$

From the similarity of $\mathbf{T}$ and $B$, the equality
$C_{\mathbf{T}}=C_{T}$, and the above inequality, we get

$r_{\mathbf{T}}=\frac{r_{B}R_{\mathbf{T}}}{R_{B}}=\frac{r_{B}R_{T}}{R_{B}}>R_{B}+r_{B}.$

$\square$

Proof of Theorem 3. 1) Intersect $T_{1},T_{3},\ldots,T_{2m-1}$ by the plane
$\operatorname{Aff}C_{T}$. We obtain circular rings with centers
$Q_{T_{1}},Q_{T_{3}},\ldots,Q_{2m-1}\in C_{T}$; the radii of their outer
circles equal $s(R_{T}+r_{T})$, see Fig. 5.

Clearly $Q_{T_{1}}Q_{T_{3}}>2s(R_{T}+r_{T})$. On the other side, the length of
the closed broken line $Q_{T_{1}}Q_{T_{3}}\ldots Q_{2m-1}$ is smaller than the
length of the circle $C_{T}$, and we get

$m\cdot 2s(R_{T}+r_{T})<m\cdot
Q_{T_{1}}Q_{T_{3}}=Q_{T_{1}}Q_{T_{3}}+Q_{T_{3}}Q_{T_{5}}+\ldots+Q_{2m-1}Q_{1}<2\pi
R_{T}.$

Hence $ms<\frac{\pi R_{T}}{R_{T}+r_{T}}<\pi$ and $2ms^{2}=2s\cdot ms<2\pi s$,
which proves 1).

Figure 5: Illustration for the proof of part 1 of Theorem 3.

2) The inclusion $T_{1}\subset T$ implies that $s(R_{T}+r_{T})\leqslant
r_{T}$, and $s\leqslant\frac{r_{T}}{R_{T}+r_{T}}$; this inequality was already
noticed in [29, p. 603], [28, p. 136], [35, (2.1)]. Now

$\frac{r_{T}}{R_{T}+r_{T}}<\frac{1}{2\pi}\iff\frac{r_{T}}{R_{T}}<\frac{1}{2\pi-1}.$

3) We have

$2s(R_{T}+r_{T})<Q_{T_{1}}Q_{T_{3}}=2R_{T}\sin\frac{\pi}{m},$

consequently

$s<\frac{R_{T}\sin\frac{\pi}{m}}{R_{T}+r_{T}}<\sin\frac{\pi}{m}<\frac{\pi}{m}.$

Since $2\pi^{2}<20$, we get $\frac{\pi}{m}<\frac{1}{2\pi}$ for each
$m\geqslant 20$. $\square$

Proof of Theorem 4. Take any hyperplane $L\subset\mathbb{R}^{n}$. The
projection $p_{L}:\mathbb{R}^{n}\to L$ does not increase distances, hence
$m_{n-1}(p_{L}X)=0$. By the Szpilrajn theorem [21, Thm. VII 3] we have $\dim
p_{L}X\leqslant n-2$.

Now suppose that $\dim p_{L}X\leqslant n-3$; by [33], [34] (for $n=3$ by [4,
75, p. 87–89]) $X$ is tame, a contradiction. This proves the theorem.
$\square$

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# Analyzing and Mitigating Bias for Vulnerable Classes: Towards Balanced
Representation in Dataset

Dewant Katare§, David Solans Noguero†, Souneil Park†, Nicolas
Kourtellis$\dagger$, Marijn Janssen§, Aaron Yi Ding§, §Delft University of
Technology, † Telefónica Research Corresponding e-mails<EMAIL_ADDRESS>
###### Abstract

The accuracy and fairness of perception systems in autonomous driving are
crucial, particularly for vulnerable road users. Mainstream research has
looked into improving the performance metrics for classification accuracy.
However, the hidden traits of bias inheritance in the AI models, class
imbalances and disparities in the datasets are often overlooked. In this
context, our study examines the class imbalances for vulnerable road users by
focusing on class distribution analysis, performance evaluation, and bias
impact assessment. We identify the concern of imbalances in class
representation, leading to potential biases in detection accuracy. Utilizing
popular CNN models and Vision Transformers (ViTs) with the nuScenes dataset,
our performance evaluation reveals detection disparities for underrepresented
classes. We propose a methodology for model optimization and bias mitigation,
which includes data augmentation, resampling, and metric-specific learning.
Using the proposed mitigation approaches, we see improvement in IoU(%) and
NDS(%) metrics from 71.3 to 75.6 and 80.6 to 83.7 respectively, for the CNN
model. Similarly, for ViT, we observe improvement in IoU and NDS metrics from
74.9 to 79.2 and 83.8 to 87.1 respectively. This research contributes to
developing more reliable models and datasets, enhancing inclusiveness for
minority classes.

##  I Introduction

The advent of autonomous driving technology marks a significant leap forward
in the automotive industry. While the benefits of autonomous vehicles (AVs)
are manifold, including potential reductions in traffic accidents and
increased mobility for various demographics, the technology’s reliability is
highly contingent on the accuracy and fairness of its perception systems [12]
[14]. These systems, particularly those involving object detection, are
critical for identifying and appropriately reacting to various entities on the
road, such as pedestrians, cyclists, motorcyclists, and mobility-impaired
persons [8] [31]. However, recent studies have underscored the existence of
biases in autonomous driving datasets, which could lead to discriminatory
treatment of certain road users [17] [25]. Our study delves into two prominent
datasets in this domain, nuScenes [4] and Waymo [27], aiming to uncover and
address these biases. We explore the representation of vulnerable classes on
object detection models’ accuracy and fairness through class distribution
analysis, model’s performance evaluation, and bias impact assessment. Class
imbalance refers to the disproportionate representation of different
categories or classes within a dataset [23]. In the context of autonomous
driving, this often manifests as an overrepresentation of certain types of
vehicles, while pedestrians, cyclists, motorcyclists, and especially mobility-
impaired individuals are underrepresented [17]. The imbalance is not only a
statistical concern but a practical one, as it can lead to perception systems
that are less adept at recognizing minor, yet crucial classes. Studies have
shown that this imbalance can result in biased learning outcomes, where models
preferentially detect overrepresented classes [29]. The consequences of such
an imbalance in the dataset cannot be overlooked, as evidenced by incidents
like the Uber self-driving car collision [26]. In this dataset imbalance
context, our paper explores:

  1. 1.

How do the representation and disparities of classes in datasets impact the
accuracy of AI models?

  2. 2.

How do biases in AI systems for detecting under-represented classes manifest
in visual application, and what implications do they have on diverse models?

To counter biases towards learning, our research explores a framework for
balanced class representation and AI model optimization. We emphasize the need
for diverse and representative datasets that include a wide range of
scenarios, reflecting the complexity of real-world conditions. The proposed
framework includes bias assessment and model re-calibration to ensure model
fairness and effectiveness with the retraining of models with the inclusion of
new classes/sub-samples. We also highlight critical gaps in current vision
datasets used in autonomous driving through metric-specific learning to
develop more equitable and reliable systems. The aim is to contribute to the
generalization of AI algorithms and respective datasets.

##  II Background and Related Work

Perception technology in autonomous driving has evolved through extensive
research and development, with significant milestones achieved in sensor
technology, algorithmic processing, and machine learning models. Early systems
relied heavily on rule-based algorithms and limited sensor input, but the
advancements in deep learning and improvements in LiDAR, radar, and camera
technologies have provided next-generation solutions [15]. Despite these
advancements, the challenge of developing perception systems that can reliably
interpret diverse and unpredictable road scenarios remains a significant
research focus. Datasets play a crucial role in training and validating
autonomous driving systems. Datasets like KITTI [9] and Cityscapes [6] have
provided foundations for advancing the field. More recently, datasets such as
nuScenes and Waymo have offered more diverse and complex environments for
training and testing AI models. These datasets include various annotations for
vehicles, pedestrians, cyclists, and other entities, providing a rich
foundation for developing and evaluating perception algorithms.

Class Imbalance: Efforts to address class imbalance and bias in autonomous
driving datasets have included techniques like data augmentation, re-sampling,
and advanced algorithmic approaches like cost-sensitive learning [11].
Research in this area seeks to balance the representation of classes in
training data [28] and develop fair and accurate models across all categories
[29] [31]. For instance, in research paper[10], the author highlights the use
of scenario simulations to test and improve model performance in detecting
imbalanced classes. The ethical implications of biased autonomous driving
systems are significant. There is a growing discourse on the need for
equitable AI systems that ensure the safety and fairness of all road users
[24]. Furthermore, regulatory bodies are increasingly focusing on how these
technologies adhere to safety standards, particularly in terms of their
ability to detect and respond to diverse and representative [5] scenarios.
Targeting imbalanced representation, a novel approach that combines visual
codebook generation with deep features and a non-linear $Chi^{2}$ SVM
classifier is porposed in [23]. This method tackles the issue of imbalanced
datasets, where traditional algorithms often fail to detect minority classes
effectively. The approach involves extracting low-level deep features using
transfer learning with the ResNet-50 pre-trained network and k-means
clustering to create a visual codebook. Each image is then represented as a
Bag-of-Visual-Words (BOVW), derived from the histogram of visual words in the
vocabulary. The $Chi^{2}$ SVM classifier is used for classifying these
features, demonstrating optimal performance in empirical analysis. This method
shows superior accuracy, F1-score, and AUC metrics results compared to state-
of-the-art methods, as validated on two challenging datasets: Graz-02 and TF-
Flowers [23].

With a focus on training methods to address the challenges of learning from
imbalanced data, a new loss function that mitigates the impact of samples
leading to overfitted decision boundaries is proposed in [21]. This loss
function has been shown to enhance the performance of various imbalance
learning methods. The proposed approach is versatile and can be integrated
with existing resampling, meta-learning, and cost-sensitive learning methods
to tackle class imbalance problems. The paper validates this method with
experiments on multiple benchmark datasets, demonstrating its superiority over
state-of-the-art cost-sensitive loss methods. Additionally, the authors have
made their code publicly available, facilitating further research and
application [21]. A novel approach for predicting driver behaviour, which is
crucial for safely integrating autonomous vehicles into human-dominated
traffic, is presented in [3]. The proposed method addresses the shortcomings
of existing predictive models, which either lack transparency (deep neural
networks) or are not expressive enough (rule-based models). The authors
introduce a model that embeds the Intelligent Driver Model (IDM), a rule-based
approach, into deep neural networks. This hybrid model combines the long-term
coherence and interpretability of rule-based models with the expressiveness of
deep learning, aiming to accurately predict driver behaviour in complex
scenarios like merging. The method is an attempt to bridge the gap between two
modelling paradigms. It enhances the interpretability of neural network
predictions while maintaining accuracy, a critical factor for real-time
decision-making in autonomous driving. The model’s transparency is
particularly beneficial for debugging and understanding its predictions.

Model learning and Representation: The paper by Wang et al. [30] introduces a
deep attention-based method for imbalanced image classification (DAIIC),
utilizing an attention mechanism within a logistic regression framework to
prioritize minority classes in prediction and feature representation. It
automatically determines misclassification costs to aid discriminative feature
learning. Robust across different networks and datasets, the DAIIC method has
proven effective, surpassing several benchmarks in single-label and multi-
label contexts [30]. The background and related work in autonomous driving
technologies, datasets, and the challenges of class imbalance and bias
underscore the importance of this research area. Our study builds upon these
foundations, aiming to contribute to the development of more equitable and
reliable autonomous driving systems. By addressing class imbalance and bias in
key datasets, we seek to enhance the safety and fairness of these technologies
for all road users.

##  III Methodology

When an imbalanced dataset $D$ with an unjustified distribution of classes
${C_{1},C_{2},.....,C_{n}}$ is used to train machine learning models,
specifically neural networks and Vision Transformers (ViTs), the models are
likely to inherit biases because of class representation in the dataset. These
biases may arise due to imbalanced class representation and a predominance of
normal conditions as compared to challenging real-world scenarios. For neural
networks, we anticipate biases to be reflected in neurons’ sensitivity and
selectivity scores, while for vision transformers, biases may manifest in the
attention allocation. Therefore, we propose a methodology consisting of
statistical analysis for analyzing and mitigating biases using dataset
analysis, bias impact assessment, bias mitigation, metrics behaviour analysis,
and bias assessment and model re-calibration, as shown in Figure 1. The
following subsections detail and discuss the concepts and terminologies used
in our methods and tests.

Figure 1: Proposed methodology for analyzing and mitigating biases

###  III-A Dataset and Classes

In this paper, we use nuScenes as the case study, focusing on the Pedestrian,
Cyclist, and Motorcyclist classes. However, the approach can be tested and
validated on other driving datasets, e.g., the Waymo dataset [27], where the
samples can be Vehicles, Pedestrians, and Cyclists.

####  III-A1 3.1.1 Class Distribution Analysis

We perform statistical analysis on the frequency of the specified classes in
the nuScenes and Waymo datasets to identify class imbalances. The datasets are
scrutinized to understand the context (urban versus rural settings, diverse
weather and lighting conditions) in which these classes are represented,
highlighting any underrepresented scenarios.

Pedestrian Class: nuScenes dataset includes a significant proportion of
pedestrian annotations, with 149,921 instances labelled as
“human.pedestrian.adult” accounting for 21.61% of all annotations. This
substantial representation highlights the importance of pedestrian detection
in autonomous driving systems. However, other pedestrian sub-categories like
children (1,934 annotations, 0.28%), construction workers (13,582 annotations,
1.96%), and individuals with personal mobility devices (2,281 annotations,
0.33%) are less represented. These differences underscore potential gaps in
the dataset’s and subsamples diversity.

Cyclist Class: It is categorized as “vehicle.bicycle” and has around 17,060
annotations, comprising 2.46% of the total dataset. This relatively lower
representation compared to pedestrians impacts the class weight calculation
and therefore the model’s ability to accurately detect cyclists, which might
become a critical safety concern in urban driving scenarios.

Motorcyclist Class: It is annotated as “vehicle.motorcycle” and has around
16,779 annotations, making up 2.42% of the dataset. This class is slightly
underrepresented compared to pedestrians but is on par with cyclists. The
lesser representation of cyclists and motorcyclists indicates an area for
improvement in class balance, especially considering the different dynamics
and associated risks.

The common metrics used for model performance evaluation in nuScenes are, mean
average Precision (mAP), Intersection over Union (IoU), and nuScenes Detection
Score (NDS). mAP and IoU are well-known metrics used in 3d object detection
and segmentation tasks [4], and are calculated using the fundamentals of
precision, recall, area of overlap and intersection over union respectively.
Similarly, NDS combines several metrics, including the mAP, to provide a
comprehensive performance score for the object detection models. The NDS
metric is calculated as:

$NDS=\frac{1}{10}[5\cdot mAP+\sum_{mTP\in TP}(1-\min(1,mTP))]$ (1)

####  III-A2 3.1.2 Data Augmentation/Re-sampling

Given the class representation imbalances, particularly in the less
represented sub-categories of pedestrians and the overall lower representation
of cyclists and motorcyclists, specific strategies can be employed for model
training and evaluation.

Resampling Technique: Addressing class imbalances in the datasets,
particularly for underrepresented classes like cyclists and motorcyclists, is
crucial for unbiased model training in autonomous driving systems. We apply
both “Random Oversampling” and “Undersampling ” from resampling techniques”.
Random Oversampling duplicates instances from minority classes, ensuring
models have sufficient data to learn from these critical yet less frequent
road users. Conversely, Undersampling reduces the predominance of over-
represented classes, like pedestrians, to prevent bias inheritance in models.
Both methods are carefully balanced to maintain data diversity [20].

Data Augmentation: Another key approach is the use of geometric
transformations, such as rotating and flipping images. This technique
represents various class orientations, which is crucial for autonomous driving
where objects are encountered from multiple directions. These transformations
not only prevent the model from developing orientation-specific biases but
also enhance its ability to adapt to diverse real-world scenarios.
Additionally, this approach contributes to increasing the dataset’s diversity
and size, thereby improving the model’s robustness and generalization
capabilities for more accurate and reliable perception tasks [22].

Combination of Strategies: To further have varied training and testing and to
increase the samples/sub-classes diversity, we combine resampling and data
augmentation to balance the training dataset. This approach integrates the
strengths of both Resampling (Random undersampling/Oversampling) and
Transformations, such as rotating and flipping images, to optimize the
representation and variability of data.

###  III-B Models and Behavioural Metrics

We strategically use popular models such as ResNet18, SqueezeNet from the CNN
and ViT from the transformer family for evaluation. The selection of these
models is intended to provide an understanding of how different architectures
perform on biased datasets and to compare their ability to generalize across
various classes, especially after the implementation of bias mitigation. Using
these models, we further gain insights across different neural network
architectures, ensuring robustness and reliability in our findings.

Model Selection: ResNet18 [18] is a well-known architecture and performs
fairly in perception tasks due to its ability to capture spatial hierarchies,
but it can show a tendency of bias towards common textures and patterns.
SqueezeNet [13], is a compact model and focuses on global image features but
risks inheriting biases from non-diverse datasets [14]. Vision Transformers
(ViTs) process images by analyzing sequences of patches, adeptly focusing on
relevant image parts due to their attention mechanisms [7]. While ViTs are
proficient at capturing dependencies, they may struggle to adequately focus on
underrepresented classes/unique scenarios. Each model’s inherent strengths and
learning mechanisms influence classification and learning, underscoring the
need for tailored bias detection and mitigation strategies.

Behaviour Metrics: Sensitivity and selectivity scores of neurons are used to
evaluate biases in models like ResNet and SqueezeNet. Sensitivity scores
measure a neuron’s response to changes in input from specific classes,
indicating potential biases. For example, a ResNet neuron more sensitive to
vehicles than cyclists might show a vehicle-detection bias. Selectivity
scores, on the other hand, assess the specificity of a neuron’s response to a
class, with high selectivity indicating specialized recognition capabilities.
This helps identify whether models effectively differentiate between classes
or exhibit biases, such as a score showing higher selectivity for pedestrians
over motorcyclists, potentially indicating a feature recognition bias. The
formula is described as:

$\text{Sensitivity Score}=\partial a/\partial x\text{\;}$

where $a$ is activation of neuron, and $x$ is the input feature.

$\text{Selectivity Score}=\frac{a_{c}-a_{avg}}{\max(a_{c},a_{avg})}\text{\;}$

where $a_{c}$ is activation of the neuron for the target class and $a_{avg}$
is the average activation for all classes.

In Vision Transformers (ViTs), attention map analysis is key for identifying
model focus areas and potential decision-making biases. This involves
extracting attention weights from each Transformer layer, which indicates the
model’s focus on different image parts. These heatmaps created using the
scaled dot-product attention mechanism, help determine if the model
disproportionately focuses on certain features, potentially signaling bias.
Attention weights are calculated as
$A=\text{softmax}\left(\frac{QK^{T}}{\sqrt{d_{k}}}\right)V$, where $Q$
(Query), $K$ (Key), and $V$ (Value) are matrices derived from the input, and
$d_{k}$ is the dimension of the Key vectors. By analyzing these attention maps
across different layers and attention heads, we can identify whether the model
disproportionately focuses on certain features or misses critical aspects,
potentially indicating bias.

Layer-wise Relevance Propagation (LRP) offers another approach, starting from
the output layer and backpropagating the relevance score of the predicted
class through the network. LRP redistributes each layer’s relevance to its
inputs, particularly in self-attention layers, where relevance among patches
is allocated based on attention weights [1]. The relevance for a patch in a
layer, highlighting the model’s decision-making basis is calculated as:

$R_{j}^{(l)}=\sum_{i}A_{ij}^{(l)}R_{i}^{(l+1)}$ (2)

Where $R_{j}^{(l)}$ is the relevance of patch $j$ in layer $l$, $A_{ij}^{(l)}$
are the attention weights from patch $j$ to patch $i$ in the self-attention
mechanism, and $R_{i}^{(l+1)}$ is the relevance in the next layer.

###  III-C Bias Impact Assessment

This section describes model errors (false positives and negatives) and
examines learning patterns and bias inheritance, crucial for identifying
potential biases in model performance.

####  III-C1 3.3.1 Error Analysis

We focus on false positives (e.g., incorrectly identifying objects as
belonging to a target class) and false negatives (e.g., failing to detect
cyclists). We further use Class-specific error analysis, which includes
evaluating how well the model detects pedestrians, cyclists, and motorcyclists
under various scenarios. We also examine if there is a tendency to overlook
certain classes and how this impacts overall model performance. The
correlation between these errors and previously identified biases, such as
neurons’ sensitivity or selectivity scores, is also explored to understand the
underlying causes and guide bias mitigation strategies using bias correlation.

####  III-C2 3.3.2 Learning Patterns and Bias Inheritance

Different model architectures lead to varied learning behaviours and potential
biases. CNNs like ResNet may focus on textures and local patterns, which could
introduce biases if such features are not uniformly present across all
classes. This can result in misclassification, especially when distinguishing
features are missing. In contrast, Vision Transformers (ViTs) may develop
biases based on how attention is distributed across an image.

Bias Inheritance from Data: A model’s performance can vary depending on the
diversity/samples encountered in training. For instance, a model trained
predominantly on urban pedestrian images might underperform in rural settings.
Understanding this bias inheritance is crucial for ensuring that perception
systems are adaptable to diverse real-world conditions and can guide necessary
adjustments in data representation and model training.

Comparative Analysis: By comparing how different models like ResNet,
SqueezeNet, and ViTs respond to the same dataset, we can uncover specific
biases inherent in each architecture. This comparative approach helps identify
which models are more prone to biases, helping categorise the most suitable
model for bias-sensitive applications.

Interpretation and Visualization: Layer-wise Relevance Propagation (LRP)
provides insights into what drives a model’s decisions by highlighting
influential input features. For example, if a model consistently focuses on
irrelevant background features for decision-making, it indicates a bias that
needs correction [19]. Analysis of these scores, using visual tools like
heatmaps and attention maps, helps in interpreting these complex models,
offering an understanding of their decision-making processes, and adjusting
the training process to mitigate identified biases.

Overall, this process involving analysis of learning patterns, data bias
inheritance comparative model analysis, and visualization can help discover
bias occurrence/inheritance in different model architectures.

###  III-D Bias Mitigation Techniques

To mitigate biases inherited from sample representation, we explore the
strategies mentioned below to address these class disparities. These
techniques are crucial for addressing the disparities, ensuring that the
model’s performance is equitable and reliable across all categories.

####  III-D1 3.4.1 Cost-Sensitive Learning

Customized cost-sensitive learning offers a robust approach to mitigating
these issues by adjusting the model’s loss function according to the
representation of each class. The class weights calculation and loss function
are discussed below.

Class Weights Calculation: Based on the dataset statistics, we calculate the
weights for the pedestrian ${w_{p}}$, cyclist ${w_{c}}$, and motorcyclist
${w_{m}}$ classes using the inverse of their respective representation
percentages. This approach assigns higher weights to underrepresented classes,
emphasizing their importance during training. The calculated weights are:

${w_{p}}=\frac{1}{21.61{\%}};\hskip 4.26773pt{w_{c}}=\frac{1}{2.46{\%}};\hskip
4.26773pt{w_{m}}=\frac{1}{2.422{\%}}$ (3)

These weights are then normalized to ensure they contribute proportionally and
maintain stability during training.

Loss Function: We integrate these calculated weights into the model’s loss
function, utilizing a weighted multi-class cross-entropy loss. For each
instance i, the loss is defined:

$\small
L_{i}=-w_{p}y_{ip}\log(p_{ip})-w_{c}y_{ic}\log(p_{ic})-w_{m}y_{im}\log(p_{im})\text{\;
(4)}$

Here ${w_{p}}$, ${w_{c}}$, and ${w_{m}}$ are the weights for pedestrians,
cyclists, and motorcyclists. ${y_{ix}}$ is a binary indicator of whether
instance i belongs to a class x. ${p_{ix}}$ is the model’s predicted
probability for instance i belongs to a class x.

Dynamic Adjustment Evaluation: To ensure continual model adaptability we
propose dynamically adjusting these weights based on performance metrics. This
approach can be tested using a validation set to monitor performance
improvements for underrepresented classes and to prevent overfitting.

By employing this customized cost-sensitive learning approach, we aim to
reduce the biases inherent in the AI model from the imbalanced dataset,
ensuring a more equitable and robust model performance across all classes,
which is crucial for the reliability and safety of perception applications.

####  III-D2 3.4.2 Data Augmentation Based on Model Analysis

We use a targeted data augmentation strategy to address biases identified in
ViTs through attention map analysis and Layer-wise Relevance Propagation
(LRP). This approach specifically addresses under-representation issues in
cyclists and motorcyclists within the nuScenes dataset.

Attention-Guided Augmentation: Insights from attention maps guide this
augmentation strategy. For instance, if the model frequently overlooks
pedestrians under night or certain lighting conditions, then there is a need
for more class samples highlighting these scenarios. Techniques such as
zooming or adjusting brightness/contrast are used to emphasize these aspects.
Similarly, additional variations of poses or orientations for motorcyclists
are incorporated to improve model recognition abilities in these contexts.

LRP-Informed Sampling: Layer-wise Relevance Propagation provides a deeper
understanding of which features contribute most significantly to the model’s
decisions. Using insights from LRP, we can selectively augment our dataset to
enhance the representation of features critical for correct classifications.
Augmentation for this case includes scenarios where the model typically
misclassifies images, like partially occluded subjects or specific textures
and patterns, enhancing the model’s ability to differentiate between relevant
and misleading features.

Implementation:This strategy involves iterative dataset refinement based on
continuous model analysis. By aligning augmentation closely with model needs
and performance, we aim to mitigate biases, effectively ensuring balanced and
fair model performance. This is critical for perception systems, where the
accurate detection of all road users, particularly those underrepresented like
cyclists and motorcyclists.

###  III-E Bias Assessment and Model Re-calibration

Effective bias mitigation requires continuous evaluation and adjustment of
models. This section describes our approach to bias assessment and model re-
calibration.

Bias Assessment: In this step, we use behaviour metrics, specifically
sensitivity and selectivity scores for each class, to measure the impact of
our mitigation efforts. Comparing these metrics before and after mitigation
measures provides insights into changes in model behaviour. Additionally, we
analyze error rates, particularly false positives and negatives, to assess
improvements in the model’s predictive accuracy across different classes. For
Vision Transformers, attention map analysis is used to verify if attention
distribution is now more balanced across various classes and scenarios.

Model Re-calibration: This involves dynamically adjusting class weights within
the loss function, guided by real-time performance metrics. This ensures the
model stays optimized to any shifts in class representation or emerging
imbalances. When new class samples or data become available, the model
undergoes re-training to stay aligned with the evolving operational context.
The last step further includes an iterative refinement, monitoring and
updating bias mitigation strategies based on the latest performance
assessments and re-calibrations.

##  IV Experimental Evaluation

###  IV-A Test Setup

Test Set: The overall dataset comprises approximately 1.4 million images, with
annotations across several classes. Our focus is on the Pedestrian (149,921),
Cyclist(17,060), and Motorcyclist (16,779) classes, as it provides a rich
source for analyzing biases in object detection.

Preprocessing Steps: Data preprocessing involved normalization of image pixel
values to the [0,1] range, aligning annotations to ensure consistency, and
resizing images to standard dimensions suitable for input requirements of
chosen models.

Model Architectures: The study uses SqueezeNet and ResNet18 from the CNN
family alongside Vision Transformer: ViT. SqueezeNet, known for its compressed
size and robustness in feature extraction, and ResNet18, recognized for its
efficiency in learning from residual connections, are expected to provide
insights into traditional CNN performance. ViT, representing the Transformer
family, is included to evaluate the effectiveness of its attention-based
mechanism in handling class imbalances.

###  IV-B Hyperparameter Optimization

Initial Hyperparameters: Were set as follows: learning rate of 0.001, batch
size of 32, and weight decay of 0.0001 for CNN models. For ViT we configured
the training with a batch size of 32 and an initial learning rate of 1e-3,
which follows a linear decay to 1e-5. The model was trained for 30 epochs
using the Adam optimizer. A dropout rate of 0.1 and a weight decay of 0.03
were applied to avoid overfitting.

Optimization Techniques: Hyperparameter tuning is performed using a grid
search approach, systematically varying learning rates, batch sizes, and
dropout rates to identify the optimal combination [2]. The optimal set of
hyperparameters was chosen based on the highest average scores across these
metrics as followed in [4], focusing on improvements in detecting classes.

###  IV-C Training Process

Training Details: As our sample size is small, CNN models were trained for 50
epochs using an Adam optimizer. A learning rate scheduler was used to reduce
the learning rate by 10% every 10 epochs. Regularization techniques such as
dropout and data augmentation (rotations and flipping) were used to prevent
overfitting [16].

Data Splits: The dataset is divided into 70% training, 15% validation, and 15%
test splits, ensuring a diverse representation of scenarios in each set.

Computational Resources: Training is conducted on a high-performance computing
cluster with a combination of NVIDIA Tesla V100 GPUs using Open MPI and
PyTorch as the deep learning framework.

###  IV-D Model Evaluation

Performance Metrics: Models were evaluated based on mAP, %IoU and NDS
(Intersection over Union, NuScenes Detection Score), alongside sensitivity and
selectivity scores for class-specific analysis.

Baseline Comparison: Model performances pre and post-implementation of bias
mitigation strategies were compared. Baseline models were trained with respect
to data conditions.

Class-specific Analysis: Performance metrics for each class were analyzed to
assess improvements in detecting pedestrians, cyclists, and motorcyclists.
Attention was paid to false positive or negative rate changes for these
classes.

###  IV-E Bias Impact Analysis

Error Analysis: Figure 2 shows error analysis for the discussed classes. The
ResNet18 model exhibited higher false positive rates across all classes than
the ViT model, indicating a tendency towards over-prediction. Conversely, the
ViT model maintained lower false positives but demonstrated higher false
negatives, particularly for Pedestrians and Cyclists, suggesting a cautious
but potentially under-detecting approach. Notably, the ViT model was more
balanced for the Motorcyclist class, with significantly lower false negatives
than ResNet18, indicating more reliable detection.

Visualizations: Figure 3 shows heatmap visualization to illustrate the models’
focus areas and any changes before the mitigation strategies. When trained
with unbalanced classes, the tests show that both CNN models falsely classify
motorcyclist and cyclist classes.

Figure 2: Figure shows class-wise error rate for ResNet and ViT according to
False Negatives and False Positives in model validation

###  IV-F Discussion of Results

Figure 3: Figure showing heatmap for all three classes TABLE I: NDS and %IoU for classes with ResNet18 %IoU of Classes | NDS |  Data Condition
---|---|---
Pedestrian | Cyclist | Motorcyclist
86.1 | 53.6 | 76.4 | 72.0 | Normal
78.5 | 8.3 | 40.7 | 42.5 | Night
67.3 | 7.5 | 18.9 | 31.2 | Weather
85.7 | 52.8 | 76.1 | 71.5 | Rotated
72.1 | 12.4 | 21.4 | 35.3 | Mixed

For measuring IoU, mAP and NDS scores mentioned in Tables 1 and 2, we used a
subset approach, where the representation of one class (e.g., pedestrian) is
kept at 67% and the other two classes have equal representation. This ratio is
adjusted during iterations until all classes have the same representation. The
purpose is to capture metrics in a scenario where the weighted class functions
are normalized. Table I shows the Intersection over Union (%IoU) for different
classes and the NuScenes Detection Score (NDS) across various conditions for
ResNet18. Pedestrian detection remains relatively high in all conditions, with
%IoU above 70% except when combined adverse conditions are present. Cyclist
detection suffers significantly in all but normal conditions, with %IoU
dropping to as low as 7.5% during adverse weather. Motorcyclist detection is
also affected by these conditions but to a lesser extent. The overall NDS
reflects these trends, with the highest score of 72.0 in normal conditions and
the lowest of 31.2 during adverse weather. The rotated condition only slightly
impacts the %IoU for pedestrians and motorcyclists but more so for cyclists.
Mixed conditions present a challenging scenario with a noticeable decrease in
%IoU across all classes and a resultant NDS of 35.3.

TABLE II: Mean Average Precision (mAP) of different classes With SqueezeNet under various conditions. Pedestrian | Cyclist | Motorcyclist | Total | Data Condition
---|---|---|---|---
86.3 | 54.7 | 77.8 | 72.93 | Normal
78.1 | 7.5 | 38.3 | 41.3 | Night
67.0 | 8.3 | 19.7 | 31.6 | Weather
85.9 | 52.3 | 75.2 | 71.1 | Rotated
73.2 | 14.8 | 22.1 | 36.7 | Mixed

Table II shows the mean average precision (mAP) of pedestrian, cyclist, and
motorcyclist detection under various conditions. Under normal conditions,
pedestrian detection is high at 86.3%, while cyclist detection lags at 54.7%.
Motorcyclist detection is reasonably high at 77.8%, leading to an overall mAP
of 72.93%. However, performance drops significantly at night and during
adverse weather, with the lowest cyclist mAP at 7.5% and 8.3%, respectively.
Interestingly, rotating images in normal conditions slightly reduces
pedestrian and motorcyclist detection by about 1%, but notably impacts cyclist
detection, dropping to 52.3%. The combination of all adverse conditions
drastically reduces overall performance to a mAP of 36.7%, showcasing the
challenges faced by the models in less-than-ideal scenarios.

TABLE III: Class-wise Performance Metrics on the CNN models Class | Model | %IoU | NDS | Sens. | Sel.
---|---|---|---|---|---
Pedestrian | ResNet18 | 75.3 | 82.1% | 0.78 | 0.80
Cyclist | ResNet18 | 68.4 | 79.5% | 0.65 | 0.67
Motorcyclist | ResNet18 | 70.2 | 80.3% | 0.70 | 0.72
Pedestrian | SqueezeNet | 78.1 | 85.4% | 0.81 | 0.83
Cyclist | SqueezeNet | 72.6 | 82.7% | 0.73 | 0.75
Motorcyclist | SqueezeNet | 74.0 | 83.2% | 0.76 | 0.77
Sens. = Sensitivity Score, Sel = Selectivity Score |
TABLE IV: Baseline vs Post-Mitigation Performance Model | Metric (Avg) | Baseline | Post Mitigation
---|---|---|---
ResNet18 | %IoU | 71.3% | 75.6%
ViT | %IoU | 74.9% | 79.2%
ResNet18 | NDS | 80.6% | 83.7%
ViT | NDS | 83.8% | 87.1%

Class-wise Performance Metrics: Table III shows class-wise performance metrics
for two models, ResNet18 and SqueezeNet. For the pedestrian class, SqueezeNet
outperforms ResNet18 with a higher %IoU and NDS. Cyclists and motorcyclists
also see better %IoU and NDS scores with SqueezeNet. Sensitivity and
selectivity scores, which measure the models’ ability to identify and
differentiate classes, are consistently higher for SqueezeNet across all
classes, indicating a more refined recognition capability. These metrics show
overall better performance from SqueezeNet models in distinguishing and
correctly identifying the classes.

Baseline vs Post-Mitigation Performance: Table IV shows average Intersection
over Union (IoU) and NuScenes Detection Score (NDS) before and after applying
bias mitigation techniques for ResNet18 and Vision Transformer (ViT). Both
models show an improvement in %IoU and NDS after mitigation, with ViT
exhibiting a larger increase in NDS. ResNet18’s IoU improved by 4.3%, and its
NDS by 3.1%, while ViT’s IoU and NDS improved by 4.3% and 3.3%, respectively.
This demonstrates the effectiveness of the bias mitigation strategies.

###  IV-G Key Takeaways

  * •

For CNN models, error analysis shows ResNet18 has higher false positive rates
than SqueezeNet.

  * •

Post-mitigation performance, which involves resampling approaches, shows
improvements in both CNN (ResNet18) and ViT models.

  * •

Under all conditions, the pedestrian class had higher detection. However,
overall performance metrics show a noticeable decrease, seeking robust model
development.

  * •

Equal number of samples from classes in a subset, does not necessarily ensure
an equal representation of behaviour metrics during model learning.

##  V Conclusion and Future Work

This study explored the critical issue of class imbalance in driving dataset,
specifically targeting the under-represented classes and its impact on the
performance of AI models. Through a comprehensive methodology encompassing
dataset analysis, model evaluation, and bias impact assessment, we detect
disparities in the representation and detection of vulnerable road users. Our
tests with popular CNN models and Vision Transformers have revealed how
dataset biases can lead to alternate learning outcomes, adversely affecting
the accuracy and correctness of perception systems. Further, we have
implemented and evaluated bias mitigation techniques, which include cost-
sensitive learning and targeted data augmentation. These methods have shown
promising results in improving model performance, especially in IoU and NDS
metrics. Furthermore, introducing a dynamic framework for ongoing bias
assessment and model recalibration signifies an important step towards
developing more equitable AI systems in autonomous driving.

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