# Transfer of CS-Rickart and dual CS-Rickart properties via functors between
abelian categories

Septimiu Crivei  Faculty of Mathematics and Computer Science, Babeş-Bolyai
University, Str. M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania
<EMAIL_ADDRESS>and  Simona Maria Radu  Faculty of Mathematics and
Computer Science, Babeş-Bolyai University, Str. M. Kogălniceanu 1, 400084
Cluj-Napoca, Romania<EMAIL_ADDRESS>
(Date: April 2, 2020)

###### Abstract.

We study the transfer of (dual) relative CS-Rickart properties via functors
between abelian categories. We consider fully faithful functors as well as
adjoint pairs of functors. We give several applications to Grothendieck
categories and, in particular, to (graded) module and comodule categories.

###### Key words and phrases:

Abelian category, (dual) Rickart object, (dual) CS-Rickart object, extending
object, lifting object, module, comodule.

###### 2000 Mathematics Subject Classification:

18E10, 18E15, 16D90, 16T15

##  1\. Introduction

CS-Rickart objects in abelian categories have been introduced and studied by
us in [10] as a common generalization of Rickart objects and extending
objects. We have also considered dual CS-Rickart objects, which generalize
dual Rickart objects and lifting objects. But these dual results follow
automatically by the duality principle in abelian categories, showing the
advantage of working at this level of generality. In order to point out some
previous interest in such topics, we mention Rickart and dual Rickart objects
studied by Crivei, Kör and Olteanu [8, 9], which have been proved to be useful
in the study of regular objects in abelian categories in the sense of
Dăscălescu, Năstăsescu, Tudorache and Dăuş [12]. They generalize the module-
theoretic notions of Rickart and dual Rickart modules in the sense of Lee,
Rizvi and Roman [19, 20], and in particular, Baer and dual Baer modules
studied by Rizvi and Roman [24, 25] and Keskin Tütüncü and Tribak [17]
respectively. Extending modules (also called CS-modules) and lifting modules
have been intensively investigated for the last decades, due to their
important applications to ring and module theory (e.g., see [5, 13]). Let us
also note that CS-Rickart objects and dual CS-Rickart objects extend the
module-theoretic concepts of CS-Rickart and dual CS-Rickart modules studied by
Abyzov, Nhan and Quynh [1, 2]. In our approach we needed to develop specific
categorical techniques in order to generalize results on (dual) CS-Rickart
modules and extending (lifting) modules to abelian categories. The reader is
referred to [10] for further information and motivation on the topic.

The paper [10] contains illustrating examples, and studies direct summands of
(dual) CS-Rickart objects, (co)products of (dual) CS-Rickart objects as well
as classes all of whose objects are (dual) self-CS-Rickart. The present paper
is a companion of [10], and investigates the transfer of (dual) relative CS-
Rickart properties via functors between abelian categories. We consider fully
faithful functors and adjoint pairs of functors between abelian categories,
and we discuss the cases of coreflective and reflective abelian full
subcategories of abelian categories as well as adjoint triples of functors and
recollements between abelian categories. We present several applications to
Grothendieck categories and, in particular, to (graded) module and comodule
categories. Also, we derive consequences for endomorphism rings of (graded)
modules and comodules.

##  2\. Transfer of (dual) CS-Rickart properties via fully faithful functors

Let $\mathcal{A}$ be an abelian category. For every morphism $f:M\to N$ in
$\mathcal{A}$ we denote by ${\rm ker}(f):{\rm Ker}(f)\to M$, ${\rm
coker}(f):N\to{\rm Coker}(f)$, ${\rm coim}(f):M\to{\rm Coim}(f)$ and ${\rm
im}(f):{\rm Im}(f)\to N$ the kernel, the cokernel, the coimage and the image
of $f$ respectively. Since $\mathcal{A}$ is abelian, we have ${\rm
Coim}(f)\cong{\rm Im}(f)$. Recall that a morphism $f:A\to B$ is called a
_section_ (_retraction_) if there is a morphism $f^{\prime}:B\to A$ such that
$f^{\prime}f=1_{A}$ ($ff^{\prime}=1_{B}$).

We also need the following notions.

######  Definition 2.1 ([10, Definition 2.2]).

Let $\mathcal{A}$ be an abelian category.

  1. (1)

A monomorphism $f:M\to N$ in $\mathcal{A}$ is called essential if for every
morphism $h:N\to P$ in $\mathcal{A}$ such that $hf$ is a monomorphism, $h$ is
a monomorphism.

  2. (2)

An epimorphism $f:M\to N$ in $\mathcal{A}$ is called superfluous if for every
morphism $h:P\to M$ in $\mathcal{A}$ such that $fh$ is an epimorphism, $h$ is
an epimorphism.

######  Remark 2.2.

As in module categories [28, 17.2, 19.2], in the definition of essential
monomorphisms one may restrict to epimorphisms $h:N\to P$, while in the
definition of superfluous epimorphisms one may restrict to monomorphisms
$h:P\to M$.

######  Definition 2.3 ([10, Definition 2.4]).

Let $M$ and $N$ be objects of an abelian category $\mathcal{A}$. Then $N$ is
called:

  1. (1)

$M$-CS-Rickart if for every morphism $f:M\to N$ there are an essential
monomorphism $e:{\rm Ker}(f)\to L$ and a section $s:L\to M$ in $\mathcal{A}$
such that ${\rm ker}(f)=se$. Equivalently, $N$ is $M$-CS-Rickart if and only
if for every morphism $f:M\to N$, ${\rm Ker}(f)$ is essential in a direct
summand of $M$.

  2. (2)

dual $M$-CS-Rickart if for every morphism $f:M\to N$ there are a retraction
$r:N\to P$ and a superfluous epimorphism $t:P\to{\rm Coker}(f)$ in
$\mathcal{A}$ such that ${\rm coker}(f)=tr$. Equivalently, $N$ is dual $M$-CS-
Rickart if and only if for every morphism $f:M\to N$, ${\rm Im}(f)$ lies above
a direct summand of $N$, in the sense that ${\rm Im}(f)$ contains a direct
summand $K$ of $M$ such that ${\rm Im}(f)/K$ is superfluous in $M/K$.

  3. (3)

self-CS-Rickart if $N$ is $N$-CS-Rickart.

  4. (4)

dual self-CS-Rickart if $N$ is dual $N$-CS-Rickart.

In general, a functor between abelian categories need not preserve or reflect
CS-Rickart properties, as we may see in the following examples.

######  Example 2.4.

Consider the ring $R=\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}$ and the forgetful
covariant functor $F:{\rm Mod}(R)\to{\rm Mod}(\mathbb{Z})$ between module
categories. Then $F$ is an exact faithful functor. The endomorphism ring of
the right $R$-module $R$ is ${\rm End}_{R}(R_{R})\cong R$, while the
endomorphism ring of the $\mathbb{Z}$-module $(R,+)$ is ${\rm
End}_{\mathbb{Z}}(R,+)\cong\left(\begin{smallmatrix}\mathbb{Z}_{2}&\mathbb{Z}_{2}\\\
\mathbb{Z}_{2}&\mathbb{Z}_{16}\end{smallmatrix}\right)$. Since the rings ${\rm
End}_{R}(R_{R})$ and ${\rm End}_{\mathbb{Z}}(R,+)$ are not isomorphic, it
follows that the functor $F$ is not full. Following [18, p. 64], a ring $R$ is
called right self-injective if the right $R$-module $R$ is self-injective.
Note that $\mathbb{Z}_{2}$ and $\mathbb{Z}_{16}$ are right self-injective
rings [18, Corollary 3.13]. Then $R=\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}$ is
also a right self-injective ring [18, Corollary 3.11B]. Hence the right
$R$-module $R$ is extending [18, Corollary 6.80], and so it is self-CS-
Rickart. But the $\mathbb{Z}$-module
$F(R)=\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}$ is not self-CS-Rickart [10, Example
3.9].

######  Example 2.5.

For every $\mathbb{Z}$-module $M$, denote by $t(M)$ the largest torsion
submodule of $M$ and by $d(M)$ the largest divisible (i.e., injective)
submodule of $M$. Let $\mathcal{A}$ be the category of $\mathbb{Z}$-modules,
and let $\mathcal{B}$ be the category of torsion $\mathbb{Z}$-modules. Note
that both $\mathcal{A}$ and $\mathcal{B}$ are abelian categories. Consider the
covariant functor $F:\mathcal{A}\to\mathcal{B}$ defined by $F(M)=t(d(M))$ on
objects $M$ of $\mathcal{A}$ and accordingly on homomorphisms. Then $F$ is a
left exact functor. Also, $F$ is a full, but not faithful functor [7, Example
4.1]. Consider the $\mathbb{Z}$-module
$M=\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}\oplus\mathbb{Z}(p^{\infty})$ for some
prime $p$. Then $F(M)=\mathbb{Z}(p^{\infty})$ is a torsion injective
$\mathbb{Z}$-module, hence it is an injective object in the category
$\mathcal{B}$. Moreover, it is well known that every object of $\mathcal{B}$
has an injective envelope, namely the injective envelope of a torsion
$\mathbb{Z}$-module $A$ is $t(E(A))$, where $E(A)$ is the injective envelope
of $A$ in $\mathcal{A}$. Then the injective object $F(M)$ of $\mathcal{B}$ is
a self-CS-Rickart object in $\mathcal{B}$ [10, Corollary 5.2]. By [10,
Corollary 3.2] and [10, Example 3.9], the $\mathbb{Z}$-module $M$ is not self-
CS-Rickart, because its direct summand $\mathbb{Z}_{2}\oplus\mathbb{Z}_{16}$
is not self-CS-Rickart.

We continue with a result on preservation and reflection of (dual) relative
CS-Rickart properties via functors. For a covariant functor
$F:\mathcal{A}\to\mathcal{B}$, we denote by ${\rm Im}(F)$ the essential image
of $F$, which consists of all objects $B$ of $\mathcal{B}$ such that $B\cong
F(A)$ for some object $A$ of $\mathcal{A}$.

######  Theorem 2.6.

Let $F:\mathcal{A}\to\mathcal{B}$ be a fully faithful covariant functor
between abelian categories. Let $M$ and $N$ be objects of $\mathcal{A}$.

  1. (i)

Assume that ${\rm Im}(F)$ is closed under kernels or cokernels.

    1. (1)

If $F$ is left exact and $N$ is $M$-CS-Rickart, then $F(N)$ is $F(M)$-CS-
Rickart.

    2. (2)

If $F$ is right exact and $N$ is dual $M$-CS-Rickart, then $F(N)$ is dual
$F(M)$-CS-Rickart.

  2. (ii)

Assume that ${\rm Im}(F)$ is closed under direct summands.

    1. (1)

If $F$ is left exact and $F(N)$ is $F(M)$-CS-Rickart, then $N$ is $M$-CS-
Rickart.

    2. (2)

If $F$ is right exact and $F(N)$ is dual $F(M)$-CS-Rickart, then $N$ is dual
$M$-CS-Rickart.

###### Proof.

(i) (1) Let $g:F(M)\to F(N)$ be a morphism in $\mathcal{B}$. We have $g=F(f)$
for some morphism $f:M\to N$ in $\mathcal{A}$, because $F$ is full. Since $N$
is $M$-CS-Rickart, there are an essential monomorphism $e:{\rm Ker}(f)\to Q$
and a section $s:Q\to M$ in $\mathcal{A}$ such that ${\rm ker}(f)=se$. Since
$F$ is left exact we have ${\rm ker}(g)=F({\rm ker}(f))=F(s)F(e)$, and then
$F(s)$ is a section, and $F(e)$ is a monomorphism. We claim that $F(e)$ is an
essential monomorphism. To this end, denote $K={\rm Ker}(f)$, and let
$h^{\prime}:F(Q)\to Z$ be a morphism in $\mathcal{B}$ such that
$h^{\prime}F(e)$ is a monomorphism.

Assume first that ${\rm Im}(F)$ is closed under kernels. Let $u^{\prime}:{\rm
Ker}(h^{\prime})\to F(Q)$ be the inclusion morphism. Now we have the following
commutative diagram with exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}F(e)}$$\scriptstyle{F(e)}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Ker}(h^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{\prime}}$$\textstyle{F(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\textstyle{Z}$

in which the left square is a pullback. We have ${\rm Ker}(h^{\prime})\cong
F(A)$ for some object $A$ of $\mathcal{A}$, and so $u^{\prime}=F(u)$ for some
monomorphism $u:A\to Q$ in $\mathcal{A}$, because $F$ is fully faithful. Let
$v={\rm coker}(u):Q\to X$. The faithful functor $F$ reflects pullbacks (e.g.,
see [21, Chapter 2, Theorem 7.1]), hence we obtain the following commutative
diagram with exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ve}$$\scriptstyle{e}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{X}$

in which the left square is a pullback. Since ${\rm Ker}(ve)=0$, $ve$ is a
monomorphism. Then $v$ is a monomorphism, because $e$ is an essential
monomorphism. Hence $A=0$, and so ${\rm Ker}(h^{\prime})=0$. Then $h^{\prime}$
is a monomorphism, which shows that $F(e)$ is an essential monomorphism.

Assume now that ${\rm Im}(F)$ is closed under cokernels (epimorphisms). By
Remark 2.2 we may take $h^{\prime}:F(Q)\to L^{\prime}$ to be an epimorphism.
Now $L^{\prime}\cong F(L)$ for some object $L$ of $\mathcal{A}$. Since $F$ is
full, we have $h^{\prime}=F(h)$ for some morphism $h:Q\to L$ in $\mathcal{A}$.
Then $F(he)$ is a monomorphism in $\mathcal{B}$, and so $he$ is a monomorphism
in $\mathcal{A}$, because $F$ is faithful. Since $e$ is an essential
monomorphism, $h$ is a monomorphism. It follows that $h^{\prime}=F(h)$ is a
monomorphism since $F$ is left exact. This shows that $F(e)$ is an essential
monomorphism.

(ii) (1) Let $f:M\to N$ be a morphism in $\mathcal{A}$. Consider the morphism
$F(f):F(M)\to F(N)$ in $\mathcal{B}$. Since $F(N)$ is $F(M)$-CS-Rickart, there
are an essential monomorphism $e^{\prime}:{\rm Ker}(F(f))\to Q^{\prime}$ and a
section $s^{\prime}:Q^{\prime}\to F(M)$ in $\mathcal{B}$ such that ${\rm
ker}(F(f))=s^{\prime}e^{\prime}$. Since ${\rm Im}(F)$ is closed under direct
summands, we have $Q^{\prime}\cong F(Q)$ for some object $Q$ of $\mathcal{A}$.
Since $F$ is left exact and full, it follows that $F({\rm ker}(f))=F(s)F(e)$
for some mophisms $e:{\rm Ker}(f)\to Q$ and $s:Q\to M$ in $\mathcal{A}$. But
$F$ is also faithful, hence we have ${\rm ker}(f)=se$, where $s$ is a section.
We claim that $e$ is an essential monomorphism. To this end, let $h:Q\to L$ be
a morphism in $\mathcal{A}$ such that $he$ is a monomorphism. Since $F$ is
left exact, $F(h)F(e)$ is a monomorphism. But $e^{\prime}=F(e)$ is an
essential monomorphism, hence $F(h)$ is a monomorphism. Since $F$ is faithful,
it follows that $h$ is a monomorphism, which shows that $e$ is an essential
monomorphism. Hence $N$ is $M$-CS-Rickart. ∎

The following result is immediate.

######  Corollary 2.7.

Let $F:\mathcal{A}\to\mathcal{B}$ be an equivalence of abelian categories, and
let $M$, $N$ be objects of $\mathcal{A}$. Then $N$ is (dual) $M$-CS-Rickart if
and only if $F(N)$ is (dual) $F(M)$-CS-Rickart.

##  3\. Transfer of (dual) CS-Rickart properties via adjoint functors

Next we show that (dual) relative CS-Rickart properties transfer via adjoint
functors under reasonable conditions. Let $(L,R)$ be an adjoint pair of
covariant functors $L:\mathcal{A}\to\mathcal{B}$ and
$R:\mathcal{B}\to\mathcal{A}$ between abelian categories. Then $L$ is right
exact and $R$ is left exact. Denote by $\varepsilon:LR\to 1_{\mathcal{B}}$ and
$\eta:1_{\mathcal{A}}\to RL$ the counit and the unit of adjunction
respectively. Following [4], denote by ${\rm Stat}(R)$ the full subcategory of
$\mathcal{B}$ consisting of _$R$ -static_ objects, that is, objects $B$ of
$\mathcal{B}$ such that $\varepsilon_{B}$ is an isomorphism. Also, denote by
${\rm Adst}(R)$ the full subcategory of $\mathcal{A}$ consisting of _$R$
-adstatic_ objects, that is, objects $A$ of $\mathcal{A}$ such that $\eta_{A}$
is an isomorphism.

######  Theorem 3.1.

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories, and let $(L,R)$ be
an adjoint pair of covariant functors $L:\mathcal{A}\to\mathcal{B}$ and
$R:\mathcal{B}\to\mathcal{A}$.

  1. (1)

Assume that $L$ is exact, and let $M,N\in{\rm Stat}(R)$.

    1. (i)

If $N$ is $M$-CS-Rickart in $\mathcal{B}$, then $R(N)$ is $R(M)$-CS-Rickart in
$\mathcal{A}$.

    2. (ii)

If $R$ reflects zero objects (in particular, if $R$ is faithful) and $R(N)$ is
$R(M)$-CS-Rickart in $\mathcal{A}$, then $N$ is $M$-CS-Rickart in
$\mathcal{B}$.

  2. (2)

Assume that $R$ is exact, and let $M,N\in{\rm Adst}(R)$.

    1. (i)

If $N$ is dual $M$-CS-Rickart in $\mathcal{A}$, then $L(N)$ is dual $L(M)$-CS-
Rickart in $\mathcal{B}$.

    2. (ii)

If $L$ reflects zero objects (in particular, if $L$ is faithful) and $L(N)$ is
dual $L(M)$-CS-Rickart in $\mathcal{B}$, then $N$ is dual $M$-CS-Rickart in
$\mathcal{A}$.

###### Proof.

(1) Denote by $\varepsilon:LR\to 1_{\mathcal{B}}$ and $\eta:1_{\mathcal{A}}\to
RL$ the counit and the unit of adjunction respectively. Then
$R(\varepsilon_{B})\eta_{R(B)}=1_{R(B)}$ for every object $B$ of
$\mathcal{B}$, and $\varepsilon_{L(A)}L(\eta_{A})=1_{L(A)}$ for every object
$A$ of $\mathcal{A}$.

(i) Assume that $N$ is $M$-CS-Rickart in $\mathcal{B}$. Let $f:R(M)\to R(N)$
be a morphism in $\mathcal{A}$ with kernel $k={\rm ker}(f):K\to R(M)$.
Consider the morphism $g=\varepsilon_{N}L(f)\varepsilon_{M}^{-1}:M\to N$ in
$\mathcal{B}$. Then there are an essential monomorphism $e:{\rm Ker}(g)\to Q$
and a section $s:Q\to M$ in $\mathcal{B}$ such that ${\rm ker}(g)=se$. Note
that ${\rm Ker}(g)=L(K)$. There is a morphism $r:M\to Q$ such that $rs=1_{Q}$.
Then $LR(s)$ is a section. By naturality we have the following commutative
diagram in $\mathcal{A}$:

$\textstyle{R(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{R(M)}}$$\scriptstyle{f}$$\textstyle{R(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{R(N)}}$$\textstyle{RLR(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RL(f)}$$\textstyle{RLR(N)}$

Since $M,N\in{\rm Stat}(R)$, $R(\varepsilon_{M})$ and $R(\varepsilon_{N})$ are
isomorphisms, and so $\eta_{R(M)}=R(\varepsilon_{M})^{-1}$ and
$\eta_{R(N)}=R(\varepsilon_{N})^{-1}$ are isomorphisms. It follows that:

$f=\eta_{R(N)}^{-1}RL(f)\eta_{R(M)}=R(\varepsilon_{N})RL(f)R(\varepsilon_{M}^{-1})=R(\varepsilon_{N}L(f)\varepsilon_{M}^{-1})=R(g).$

Hence $k={\rm ker}(f)=R({\rm ker}(g))=R(s)R(e)$. Clearly, $R(s)$ is a section
and $R(e)$ is a monomorphism. In order to finish the proof, it is enough to
show that the monomorphism $R(e)$ is essential.

We first claim that the monomorphism $LR(e):LRL(K)\to LR(Q)$ is essential. By
naturality we have the following commutative diagram in $\mathcal{B}$:

$\textstyle{LR(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{M}}$$\scriptstyle{LR(r)}$$\textstyle{LR(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{Q}}$$\scriptstyle{LR(s)}$$\textstyle{LR(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{M}$

Since $\varepsilon_{Q}LR(r)=r\varepsilon_{M}$ is an epimorphism and
$s\varepsilon_{Q}=\varepsilon_{M}LR(s)$ is a monomorphism, $\varepsilon_{Q}$
is an isomorphism, hence $Q\in{\rm Stat}(R)$. By naturality we also have the
following commutative diagram in $\mathcal{B}$:

$\textstyle{LRL(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{L(K)}}$$\scriptstyle{LR(e)}$$\textstyle{LR(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{Q}}$$\textstyle{L(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{Q}$

Then $e\varepsilon_{L(K)}=\varepsilon_{Q}LR(e)$ is a monomorphism, hence so is
$\varepsilon_{L(K)}$. But the equality
$\varepsilon_{L(K)}L(\eta_{K})=1_{L(K)}$ shows that $\varepsilon_{L(K)}$ is
also a retraction, and so $\varepsilon_{L(K)}$ is an isomorphism. Since
$\varepsilon_{Q}LR(e)=e\varepsilon_{L(K)}$ is an essential monomorphism, so is
$LR(e)$, because a composition of monomorphisms is essential if and only if
each of them is essential.

Now we claim that the monomorphism $R(e):RL(K)\to R(Q)$ is essential. To this
end, let $h:R(Q)\to P$ be a morphism in $\mathcal{A}$ such that $hR(e)$ is a
monomorphism. The functor $L$ being exact, $L(h)LR(e)$ is a monomorphism, and
so is $L(h)$, because $LR(e)$ is an essential monomorphism. Thus $RL(h)$ is a
monomorphism. By naturality we have the following commutative diagram in
$\mathcal{A}$:

$\textstyle{R(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{R(Q)}}$$\scriptstyle{h}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{P}}$$\textstyle{RLR(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RL(h)}$$\textstyle{RL(P)}$

Note that $\eta_{R(Q)}$ is an isomorphism, because so is $R(\varepsilon_{Q})$.
Since $\eta_{P}h=RL(h)\eta_{R(Q)}$ is a monomorphism, so is $h$. This shows
that $R(e)$ is an essential monomorphism. Thus $R(N)$ is $R(M)$-CS-Rickart.

(ii) Assume that $R$ reflects zero objects and $R(N)$ is $R(M)$-CS-Rickart in
$\mathcal{A}$. Let $f:M\to N$ be a morphism in $\mathcal{B}$ with kernel
$k={\rm ker}(f):K\to M$. Consider the morphism $R(f):R(M)\to R(N)$ in
$\mathcal{A}$. Then there are an essential monomorphism $e:{\rm Ker}(R(f))\to
Q$ and a section $s:Q\to R(M)$ in $\mathcal{A}$ such that ${\rm
ker}(R(f))=se$.

By naturality we have the following commutative diagram in $\mathcal{B}$:

$\textstyle{LR(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{M}}$$\scriptstyle{LR(f)}$$\textstyle{LR(N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{N}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{N}$

in which $\varepsilon_{M}$ and $\varepsilon_{N}$ are isomorphisms. Since $L$
is exact, it follows that

${\rm ker}(f)={\rm ker}(LR(f))=L({\rm ker}R(f))=L(s)L(e).$

Clearly, $L(s)$ is a section and $L(e)$ is a monomorphism, because $L$ is
exact. In order to finish the proof, it is enough to show that the
monomorphism $L(e)$ is essential.

We first claim that the monomorphism $RL(e):RLR(K)\to RL(Q)$ is essential.
There is a morphism $r:R(M)\to Q$ such that $rs=1_{Q}$. Then $RL(s)$ is a
section and $RL(r)$ is a retraction. By naturality we have the following
commutative diagram in $\mathcal{A}$:

$\textstyle{R(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{R(M)}}$$\scriptstyle{r}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{Q}}$$\scriptstyle{s}$$\textstyle{R(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{R(M)}}$$\textstyle{RLR(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RL(r)}$$\textstyle{RL(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RL(s)}$$\textstyle{RLR(M)}$

Since $M\in{\rm Stat}(R)$, $R(\varepsilon_{M})$ is an isomorphism, and so
$\eta_{R(M)}=R(\varepsilon_{M})^{-1}$ is an isomorphism. Since
$\eta_{Q}r=RL(r)\eta_{R(M)}$ is an epimorphism and
$RL(s)\eta_{Q}=\eta_{R(M)}s$ is a monomorphism, $\eta_{Q}$ is an isomorphism,
hence $Q\in{\rm Adst}(R)$. By naturality we have the following commutative
diagram in $\mathcal{A}$:

$\textstyle{R(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{R(K)}}$$\scriptstyle{e}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{Q}}$$\textstyle{RLR(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{RL(e)}$$\textstyle{RL(Q)}$

in which $\eta_{Q}$ is an isomorphism and $\eta_{R(K)}$ is a section. Since
$RL(e)\eta_{R(K)}=\eta_{Q}e$ is an essential monomorphism, so is $RL(e)$,
because a composition of monomorphisms is essential if and only if each of
them is essential.

Now we claim that the monomorphism $L(e):LR(K)\to L(Q)$ is essential. To this
end, let $h:L(Q)\to P$ be a morphism in $\mathcal{B}$ such that $hL(e)$ is a
monomorphism. Then $R(h)RL(e)$ is a monomorphism, and so $R(h)$ is a
monomorphism, because $RL(e)$ is an essential monomorphism. We have $R({\rm
Ker}(h))={\rm Ker}(R(h))=0$. Since $R$ reflects zero objects, it follows that
${\rm Ker}(h)=0$, and so $h$ is a monomorphism. This shows that $L(e)$ is an
essential monomorphism. Thus $N$ is $M$-CS-Rickart. ∎

We also give the contravariant version of Theorem 3.1, because it will be used
later on. Let $(L,R)$ be a _right adjoint pair_ of contravariant functors
$L:\mathcal{A}\to\mathcal{B}$ and $R:\mathcal{B}\to\mathcal{A}$ between
abelian categories [28, 45.2]. This means that for every objects $A$ of
$\mathcal{A}$ and $B$ of $\mathcal{B}$ there is a bijection

${\rm Hom}_{\mathcal{B}}(B,L(A))\cong{\rm Hom}_{\mathcal{A}}(A,R(B)),$

natural in each variable. Then both $L$ and $R$ are left exact, so they turn
cokernels into kernels. Let $\varepsilon:1_{\mathcal{B}}\to LR$ and
$\eta:1_{\mathcal{A}}\to RL$ be the counit and the unit of adjunction
respectively. Following [3], denote by ${\rm Refl}(R)$ the full subcategory of
$\mathcal{B}$ consisting of _$R$ -reflexive_ objects, that is, objects $B$ of
$\mathcal{B}$ such that $\varepsilon_{B}$ is an isomorphism. Also, denote by
${\rm Refl}(L)$ the full subcategory of $\mathcal{A}$ consisting of _$L$
-reflexive_ objects, that is, objects $A$ of $\mathcal{A}$ such that
$\eta_{A}$ is an isomorphism. Similarly, one may consider a _left adjoint
pair_ of contravariant functors $L:\mathcal{A}\to\mathcal{B}$ and
$R:\mathcal{B}\to\mathcal{A}$ between abelian categories [28, 45.2]. This
means that for every objects $A$ of $\mathcal{A}$ and $B$ of $\mathcal{B}$
there is a bijection

${\rm Hom}_{\mathcal{B}}(L(A),B)\cong{\rm Hom}_{\mathcal{A}}(R(B),A),$

natural in each variable. Then both $L$ and $R$ are right exact, so they turn
kernels into cokernels.

######  Theorem 3.2.

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories, and let $(L,R)$ be
a pair of contravariant functors $L:\mathcal{A}\to\mathcal{B}$ and
$R:\mathcal{B}\to\mathcal{A}$.

  1. (1)

Assume that $(L,R)$ is left adjoint and $L$ is exact, and let $M,N\in{\rm
Refl}(R)$.

    1. (i)

If $N$ is $M$-CS-Rickart in $\mathcal{B}$, then $R(M)$ is dual $R(N)$-CS-
Rickart in $\mathcal{A}$.

    2. (ii)

If $R$ reflects zero objects (in particular, if $R$ is faithful) and $R(M)$ is
dual $R(N)$-CS-Rickart in $\mathcal{A}$, then $N$ is $M$-CS-Rickart in
$\mathcal{B}$.

  2. (2)

Assume that $(L,R)$ is right adjoint and $R$ is exact, and let $M,N\in{\rm
Refl}(L)$.

    1. (i)

If $N$ is dual $M$-CS-Rickart in $\mathcal{A}$, then $L(M)$ is $L(N)$-CS-
Rickart in $\mathcal{B}$.

    2. (ii)

If $L$ reflects zero objects (in particular, if $L$ is faithful) and $L(M)$ is
$L(N)$-CS-Rickart in $\mathcal{B}$, then $N$ is dual $M$-CS-Rickart in
$\mathcal{A}$.

The main consequence of Theorem 3.1 is the following result.

######  Theorem 3.3.

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories, and let $(L,R)$ be
an adjoint pair of covariant functors $L:\mathcal{A}\to\mathcal{B}$ and
$R:\mathcal{B}\to\mathcal{A}$.

  1. (1)

Assume that $L$ is exact and $R$ is fully faithful. Let $M$ and $N$ be objects
of $\mathcal{B}$. Then $N$ is $M$-CS-Rickart in $\mathcal{B}$ if and only if
$R(N)$ is $R(M)$-CS-Rickart in $\mathcal{A}$.

  2. (2)

Assume that $R$ is exact and $L$ is fully faithful. Let $M$ and $N$ be objects
of $\mathcal{A}$. Then $N$ is dual $M$-CS-Rickart in $\mathcal{A}$ if and only
if $L(N)$ is dual $L(M)$-CS-Rickart in $\mathcal{B}$.

###### Proof.

(1) Since $R$ is fully faithful, every object of $\mathcal{B}$ is in ${\rm
Stat}(R)$. Then use Theorem 3.1. ∎

##  4\. Applications

We give several applications of our theorems, showing that their hypotheses
hold in a large number of relevant situations.

Let $\mathcal{A}$ be an abelian category and let $\mathcal{C}$ be a full
subcategory of $\mathcal{A}$. Then $\mathcal{C}$ is called a _reflective_
(_coreflective_) subcategory of $\mathcal{A}$ if the inclusion functor
$i:\mathcal{C}\to\mathcal{A}$ has a left (right) adjoint. In any of the two
cases, $i$ is fully faithful. If $\mathcal{C}$ is reflective, then limits (in
particular, kernels) in $\mathcal{C}$ are the same as limits in $\mathcal{A}$,
while colimits (in particular, cokernels) in $\mathcal{C}$ are taken in
$\mathcal{A}$ and then reflected in $\mathcal{C}$ [26, Chapter X, Proposition
1.2]. For coreflective subcategories one has the dual result. If $\mathcal{C}$
is a reflective (coreflective) subcategory of $\mathcal{A}$ and the left
(right) adjoint of the inclusion functor $i$ preserves kernels (cokernels),
then the subcategory $\mathcal{C}$ is called _Giraud_ (_co-Giraud_). In this
case, the left (right) adjoint of $i$ is exact.

######  Corollary 4.1.

Let $\mathcal{A}$ be an abelian category, $\mathcal{C}$ an abelian full
subcategory of $\mathcal{A}$ and $i:\mathcal{C}\to\mathcal{A}$ the inclusion
functor.

  1. (i)

Assume that $\mathcal{C}$ is a Giraud subcategory of $\mathcal{A}$. Let $M$
and $N$ be objects of $\mathcal{C}$. Then $N$ is $M$-CS-Rickart in
$\mathcal{C}$ if and only if $i(N)$ is $i(M)$-CS-Rickart in $\mathcal{A}$.

  2. (ii)

Assume that $\mathcal{C}$ is a co-Giraud subcategory of $\mathcal{A}$. Let $M$
and $N$ be objects of $\mathcal{C}$. Then $N$ is dual $M$-CS-Rickart in
$\mathcal{C}$ if and only if $i(N)$ is dual $i(M)$-CS-Rickart in
$\mathcal{A}$.

###### Proof.

(1) Since $i$ is fully faithful and its left adjoint is exact, one uses
Theorem 3.3. ∎

For Grothendieck categories we have the following corollary.

######  Corollary 4.2.

Let $\mathcal{A}$ be a Grothendieck category with a generator $U$ with $R={\rm
End}_{\mathcal{A}}(U)$. Let $S={\rm Hom}_{\mathcal{A}}(U,-):\mathcal{A}\to{\rm
Mod}(R)$ and let $T:{\rm Mod}(R)\to\mathcal{A}$ be a left adjoint of $S$. Let
$M$ and $N$ be objects of $\mathcal{A}$. Then $N$ is an $M$-CS-Rickart object
of $\mathcal{A}$ if and only if $S(N)$ is an $S(M)$-CS-Rickart right
$R$-module.

###### Proof.

By the Gabriel-Popescu Theorem [26, Chapter X, Theorem 4.1], $S$ is fully
faithful and has an exact left adjoint $T:{\rm Mod}(R)\to\mathcal{A}$.
Moreover, $\mathcal{A}$ is equivalent to a Giraud subcategory of ${\rm
Mod}(R)$ [26, Chapter X, Proposition 1.5], which is a reflective abelian full
subcategory. Then use Corollary 4.1. ∎

Following [11, Section 2.2], let $C$ be a coalgebra over a field $k$, and let
${}^{C}\mathcal{M}$ be the (Grothendieck) category of left $C$-comodules. Left
$C$-comodules will be identified with rational right $C^{*}$-modules, where
$C^{*}={\rm Hom}_{k}(C,k)$. Let $i:{}^{C}\mathcal{M}\to{\rm Mod}(C^{*})$ be
the inclusion functor, and let ${\rm Rat}:{\rm
Mod}(C^{*})\to{}^{C}\mathcal{M}$ be the functor which associates to every
right $C^{*}$-module its rational $C^{*}$-submodule. Then $(i,{\rm Rat})$ is
an adjoint pair. If $C$ is a right semiperfect coalgebra, then the functor Rat
is exact [11, Corollary 3.2.12], hence ${}^{C}\mathcal{M}$ is a co-Giraud
subcategory of ${\rm Mod}(C^{*})$. Then Corollary 4.1 yields the following
consequence.

######  Corollary 4.3.

Let $C$ be a right semiperfect coalgebra over a field. Let $M$ and $N$ be left
$C$-comodules. Then $N$ is a dual $M$-CS-Rickart left $C$-comodule if and only
if $i(N)$ is a dual $i(M)$-CS-Rickart right $C^{*}$-module.

Recall that an _adjoint triple_ of functors is a triple $(L,F,R)$ of covariant
functors $F:\mathcal{A}\to\mathcal{B}$ and $L,R:\mathcal{B}\to\mathcal{A}$
such that $(L,F)$ and $(F,R)$ are adjoint pairs of functors. Then $F$ is an
exact functor. It is known that $L$ is fully faithful if and only if so is $R$
[14, Lemma 1.3]. Now Theorem 3.3 yields the following consequence.

######  Corollary 4.4.

Let $(L,F,R)$ be an adjoint triple of covariant functors
$F:\mathcal{A}\to\mathcal{B}$ and $L,R:\mathcal{B}\to\mathcal{A}$ between
abelian categories. Let $M$ and $N$ be objects of $\mathcal{B}$, and assume
that $L$ (or $R$) is fully faithful. Then:

  1. (1)

$N$ is $M$-CS-Rickart in $\mathcal{B}$ if and only if $R(N)$ is $R(M)$-CS-
Rickart in $\mathcal{A}$.

  2. (2)

$N$ is dual $M$-CS-Rickart in $\mathcal{B}$ if and only if $L(N)$ is dual
$L(M)$-CS-Rickart in $\mathcal{A}$.

Now let $A,C$ be rings and let ${}_{C}B_{A}$ be a bimodule. Let
$R=\begin{pmatrix}A&0\\\ B&C\end{pmatrix}$ be the formal triangular matrix
ring constructed from $A,B,C$. Following [15, Chapter 4, Section A], for a
right $A$-module $M$, a right $C$-module $N$ and a right $A$-module
homomorphism $f:N\otimes_{C}B\to M$, one may endow the abelian group
$P=M\oplus N$ with a right $R$-module structure as follows. If we write each
pair $(m,n)\in P$ in the form $\left(\begin{smallmatrix}&0\\\ m&\\\ &n\\\
\end{smallmatrix}\right)$, scalar multiplication is defined by

$\left(\begin{smallmatrix}&0\\\ m&\\\ &n\\\
\end{smallmatrix}\right)\begin{pmatrix}a&0\\\
b&c\end{pmatrix}=\left(\begin{smallmatrix}&0\\\ ma+f(n\otimes b)&\\\ &nc\\\
\end{smallmatrix}\right)$

and we write $P=\left(\begin{smallmatrix}&0\\\ M&\\\ &N\\\
\end{smallmatrix}\right)$. Moreover, every right $R$-module has this form.

Following [15, Chapter 4, Section A, Exercises 19 and 20], consider the
covariant functors:

$\displaystyle J_{23}:{\rm Mod}(C)\to{\rm Mod}(R),\quad
J_{23}(N)=\left(\begin{smallmatrix}&0\\\ N\otimes_{C}B&\\\ &N\\\
\end{smallmatrix}\right),$ $\displaystyle J_{3}:{\rm Mod}(C)\to{\rm
Mod}(R),\quad J_{3}(N)=J_{23}(N)/\left(\begin{smallmatrix}&0\\\
N\otimes_{C}B&\\\ &0\\\ \end{smallmatrix}\right),$ $\displaystyle P_{3}:{\rm
Mod}(R)\to{\rm Mod}(C),\quad P_{3}(M)=M\left(\begin{smallmatrix}0&0\\\ 0&C\\\
\end{smallmatrix}\right).$

Then $(J_{23},P_{3},J_{3})$ is an adjoint triple [15, Chapter 4, Section A,
Exercise 22].

######  Corollary 4.5.

Consider the ring $R=\begin{pmatrix}A&0\\\ B&C\end{pmatrix}$, where $A,C$ are
rings and ${}_{C}B_{A}$ is a bimodule. Let $M$ and $N$ be right $C$-modules.
Then:

  1. (1)

$N$ is an $M$-CS-Rickart right $C$-module if and only if $J_{3}(N)$ is a
$J_{3}(M)$-CS-Rickart right $R$-module.

  2. (2)

$N$ is a dual $M$-CS-Rickart right $C$-module if and only if $J_{23}(N)$ is a
dual $J_{23}(M)$-CS-Rickart right $R$-module.

###### Proof.

Since $P_{3}J_{23}\cong 1_{{\rm Mod}(C)}$ and $P_{3}J_{3}\cong 1_{{\rm
Mod}(C)}$ [15, Chapter 4, Section A, Exercise 22], $J_{23}$ and $J_{3}$ are
fully faithful. Then use Corollary 4.4 for the adjoint triple of functors
$(J_{23},P_{3},J_{3})$. ∎

Following [22], we recall some notation and terminology on graded modules. In
what follows $G$ will denote a group with identity element $e$, and $R$ will
be a $G$-graded ring. For a $G$-graded ring $R=\bigoplus_{\sigma\in
G}R_{\sigma}$, denote by ${\rm gr}(R)$ the (Grothendieck) category which has
as objects the $G$-graded unital right $R$-modules and as morphisms the
morphisms of $G$-graded unital right $R$-modules. We consider the following
functors:

  1. (1)

For $\sigma\in G$, the _$\sigma$ -suspension functor_
$T_{\sigma}:\mathrm{gr}(R)\to\mathrm{gr}(R)$ defined by
$T_{\sigma}(M)=M({\sigma})$. Recall that for a graded right $R$-module $M$ and
$\sigma\in G$, the _$\sigma$ -suspension_ $M(\sigma)$ of $M$ is the graded
right $R$-module which is equal to $M$ as a right $R$-module and the gradation
is given by $M({\sigma})_{\tau}=M_{\sigma\tau}$ for every $\tau\in G$.

  2. (2)

For $\sigma\in G$, the functor $(-)_{\sigma}:\mathrm{gr}(R)\to{\rm
Mod}(R_{e})$, which associates to every graded right $R$-module
$M=\bigoplus_{\tau\in G}M_{\tau}$ the right $R_{e}$-module $M_{\sigma}$.

  3. (3)

The _induced functor_ ${\rm Ind}:{\rm Mod}(R_{e})\to{\rm gr}(R)$ defined as
follows: for a right $R_{e}$-module $N$, ${\rm Ind}(N)$ is the graded right
$R$-module $M=N\otimes_{R_{e}}R$, where the gradation of
$M=\bigoplus_{\sigma\in G}M_{\sigma}$ is given by
$M_{\sigma}=N_{\sigma}\otimes_{R_{e}}R$ for every $\sigma\in G$.

  4. (4)

The _coinduced functor_ ${\rm Coind}:{\rm Mod}(R_{e})\to{\rm gr}(R)$ defined
as follows: for a right $R_{e}$-module $N$, ${\rm Coind}(N)$ is the graded
right $R$-module $M^{*}=\bigoplus_{\sigma\in G}M^{\prime}_{\sigma}$, where

$M^{\prime}_{\sigma}=\\{f\in{\rm Hom}_{R_{e}}(R,N)\mid
f(R_{\sigma^{\prime}})=0\textrm{ for every
}\sigma^{\prime}\neq\sigma^{-1}\\}.$

######  Corollary 4.6.

Let $R$ be a $G$-graded ring, and let $M$ and $N$ be right $R_{e}$-modules.
Then:

  1. (1)

$N$ is an $M$-CS-Rickart right $R_{e}$-module if and only if ${\rm Coind}(N)$
is a ${\rm Coind}(M)$-CS-Rickart graded right $R$-module.

  2. (2)

$N$ is a dual $M$-CS-Rickart right $R_{e}$-module if and only if ${\rm
Ind}(N)$ is a dual ${\rm Ind}(M)$-CS-Rickart graded right $R$-module.

###### Proof.

By [22, Theorem 2.5.5], for $\sigma\in G$, $(T_{\sigma^{-1}}\circ{\rm
Ind},(-)_{\sigma},T_{\sigma^{-1}}\circ{\rm Coind})$ is an adjoint triple, and
we have $(-)_{\sigma}\circ T_{\sigma^{-1}}\circ{\rm Ind}\cong 1_{{\rm
Mod}(R_{e})}$ and $(-)_{\sigma}\circ T_{\sigma^{-1}}\circ{\rm Coind}\cong
1_{{\rm Mod}(R_{e})}$, that is, the functors $T_{\sigma^{-1}}\circ{\rm Ind}$
and $T_{\sigma^{-1}}\circ{\rm Coind}$ are fully faithful. In particular,
$({\rm Ind},(-)_{e},{\rm Coind})$ is an adjoint triple, and ${\rm Ind}$ and
${\rm Coind}$ are fully faithful. Now use Corollary 4.4. ∎

Now let us recall the concept of recollement of abelian categories, following
[23]. For an additive functor $F:\mathcal{A}\to\mathcal{B}$ between abelian
categories, we denote by ${\rm Ker}(F)$ the kernel of $F$, which consists of
all objects $A$ of $\mathcal{A}$ such that $F(A)=0$. A _recollement_ of
abelian categories $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$, denoted by
$(\mathcal{A},\mathcal{B},\mathcal{C})$, is a diagram of functors:

$\textstyle{\mathcal{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{p}$$\scriptstyle{q}$$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{l}$

which satisfy the following conditions: (i) $(l,e,r)$ is an adjoint triple;
(ii) $(q,i,p)$ is an adjoint triple; (iii) $i$, $l$ and $r$ are fully
faithful; (iv) ${\rm Im}(i)={\rm Ker}(e)$.

Now Corollary 4.4 yields the following consequence.

######  Corollary 4.7.

Let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of abelian
categories as above. Let $M$ and $N$ be objects of $\mathcal{C}$. Then:

  1. (1)

$N$ is $M$-CS-Rickart in $\mathcal{C}$ if and only if $r(N)$ is $r(M)$-CS-
Rickart in $\mathcal{B}$.

  2. (2)

$N$ is dual $M$-CS-Rickart in $\mathcal{C}$ if and only if $l(N)$ is dual
$l(M)$-CS-Rickart in $\mathcal{B}$.

We recall a specific example of recollement of abelian categories from [23],
where further relevant examples are presented. Let $R$ be a unitary ring and
let $e=e^{2}\in R$ be an idempotent. Then the following diagram is a
recollement of abelian categories [23, Example 2.7]:

$\textstyle{{\rm
Mod}(R/ReR)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{\rm
Mod}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e(-)}$$\scriptstyle{{\rm
Hom}_{R}(R/ReR,-)}$$\scriptstyle{(R/ReR)\otimes_{R}-}$$\textstyle{{\rm
Mod}(eRe)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Hom}_{eRe}(eR,-)}$$\scriptstyle{Re\otimes_{eRe}-}$

where the module categories are categories of left modules, $i$ is the
inclusion functor and $e(-)$ is the functor given by left multiplication by
$e$. Now we have the following consequence of Corollary 4.7.

######  Corollary 4.8.

Let $R$ be a unitary ring and let $e=e^{2}\in R$ be an idempotent. Let $M$ and
$N$ be objects of ${\rm Mod}(eRe)$. Then:

  1. (1)

$N$ is $M$-CS-Rickart in ${\rm Mod}(eRe)$ if and only if ${\rm
Hom}_{eRe}(eR,N)$ is ${\rm Hom}_{eRe}(eR,M)$-CS-Rickart in ${\rm Mod}(R)$.

  2. (2)

$N$ is dual $M$-CS-Rickart in ${\rm Mod}(eRe)$ if and only if
$Re\otimes_{eRe}N$ is dual $Re\otimes_{eRe}M$-CS-Rickart in ${\rm Mod}(R)$.

##  5\. Endomorphism rings

In this section we give some applications to endomorphism rings of (graded)
modules and comodules. We start with a specific result for modules, for which
we need the following concepts.

######  Definition 5.1.

A right $R$-module $M$ is called:

  1. (i)

_im-local-retractable_ if for every monomorphism $k:K\to M$ and for every
$x\in K$, there exists a homomorphism $h:M\to K$ such that $x\in{\rm Im}(hk)$.

  2. (ii)

_im-local-coretractable_ if for every epimorphism $c:M\to C$ and for every
$z\in C$, there exists a homomorphism $h:C\to M$ such that $z\in{\rm Im}(ch)$.

######  Remark 5.2.

(i) Clearly, every im-local-retractable right $R$-module $M$ is $k$-local-
retractable, in the sense that for every endomorphism $f:M\to M$ with $k={\rm
ker}(f):K\to M$ and for every $x\in K$, there exists a homomorphism $h:M\to K$
such that $x\in{\rm Im}(hk)$ [19, Definition 3.6]. Dually, every im-local-
coretractable right $R$-module $M$ is $c$-local-coretractable, in the sense
that for every endomorphism $f:M\to M$ with $c={\rm coker}(f):M\to C$ and for
every $z\in C$, there exists a homomorphism $h:C\to M$ such that $z\in{\rm
Im}(ch)$ [6, Definition 4.5].

(ii) The class of im-local-retractable right $R$-modules and the class of im-
local-coretractable right $R$-modules are closed under direct summands.
Indeed, for the former, let $M$ be a im-local-retractable right $R$-module and
let $N$ be a direct summand of $M$. Let $k:K\to N$ be a monomorphism, $i:N\to
M$ the canonical inclusion and $x\in K$. Then there is a homomorphism $h:M\to
K$, and so there is the homomorphism $hi:N\to K$ such that $x\in{\rm
Im}(hik)$. Hence $N$ is im-local-retractable.

######  Corollary 5.3.

Let $M$ be a right $R$-module, and let $S={\rm End}_{R}(M)$.

  1. (1)

If $M$ is im-local-retractable and $S$ is a self-CS-Rickart right $S$-module,
then $M$ is a self-CS-Rickart right $R$-module.

  2. (2)

If $M$ is im-local-coretractable and $S$ is a dual self-CS-Rickart left
$S$-module, then $M$ is a dual self-CS-Rickart right $R$-module.

###### Proof.

(1) Consider the adjoint pair of functors $(T,H)$, where

$T=-\otimes_{S}M:{\rm Mod}(S)\to{\rm Mod}(R),$ $H={\rm Hom}_{R}(M,-):{\rm
Mod}(R)\to{\rm Mod}(S).$

Assume that $M$ is im-local-retractable and $S$ is a self-CS-Rickart right
$S$-module. Let $f:M\to M$ be a homomorphism with kernel $k={\rm ker}(f):K\to
M$. Since $H(M)=S$ is a self-CS-Rickart right $S$-module, $H(k)={\rm
ker}(H(f))=sm$ for some essential monomorphism $m:H(K)\to Q$ and section
$s:Q\to H(M)$. Let $r:H(M)\to Q$ be a homomorphism such that $rs=1_{Q}$.
Denote $t=\varepsilon_{M}T(s):T(Q)\to M$ and $e=T(r)\varepsilon_{M}^{-1}k:K\to
T(Q)$. Then $t$ is a section.

We prove that $k=te$. To this end, let $x\in K$. Since $M$ is im-local-
retractable, there exists a homomorphism $h:M\to K$ such that $x=hk(y)$ for
some $y\in K$. By naturality we have the following commutative diagram:

$\textstyle{TH(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{M}}$$\scriptstyle{TH(kh)}$$\textstyle{TH(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{M}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{kh}$$\textstyle{M}$

It follows that:

$\displaystyle\begin{split}te(x)&=\varepsilon_{M}T(s)T(r)\varepsilon_{M}^{-1}k(x)=\varepsilon_{M}T(s)T(r)\varepsilon_{M}^{-1}khk(y)\\\
&=\varepsilon_{M}T(s)T(r)\varepsilon_{M}^{-1}kh\varepsilon_{M}\varepsilon_{M}^{-1}k(y)=\varepsilon_{M}T(s)T(r)\varepsilon_{M}^{-1}\varepsilon_{M}TH(kh)\varepsilon_{M}^{-1}k(y)\\\
&=\varepsilon_{M}T(s)T(r)TH(kh)\varepsilon_{M}^{-1}k(y)=\varepsilon_{M}T(s)T(r)TH(k)TH(h)\varepsilon_{M}^{-1}k(y)\\\
&=\varepsilon_{M}T(s)T(r)T(s)T(m)TH(h)\varepsilon_{M}^{-1}k(y)=\varepsilon_{M}T(s)T(m)TH(h)\varepsilon_{M}^{-1}k(y)\\\
&=\varepsilon_{M}TH(k)TH(h)\varepsilon_{M}^{-1}k(y)=\varepsilon_{M}TH(kh)\varepsilon_{M}^{-1}k(y)\\\
&=kh\varepsilon_{M}\varepsilon_{M}^{-1}k(y)=khk(y)=k(x).\end{split}$

Hence $k=te$, which also shows that $e$ is a monomorphism.

Finally, let us prove that $e$ is an essential monomorphism. To this end, let
$v:T(Q)\to V$ be a homomorphism such that $ve:K\to V$ is a monomorphism. As in
the proof of Theorem 3.1 (ii), $\eta_{Q}:Q\to HT(Q)$ is an isomorphism. Then
we have:

$\displaystyle H(ve)$
$\displaystyle=H(v)HT(r)H(\varepsilon_{M}^{-1})H(k)=H(v)HT(r)\eta_{H(M)}H(k)$
$\displaystyle=H(v)\eta_{Q}rH(k)=H(v)\eta_{Q}rsm=H(v)\eta_{Q}m.$

and this is a monomorphism. Since $\eta_{Q}m$ is an essential monomorphism,
$H(v)$ must be a monomorphism. Let $a={\rm ker}(v):A\to T(Q)$ and $z\in A$.
Since $M$ is im-local-retractable, so is its direct summand $T(Q)$ by Remark
5.2. Hence there exists a homomorphism $g:T(Q)\to A$ such that $z=ga(u)$ for
some $u\in A$. By naturality we have the following commutative diagram:

$\textstyle{THT(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{T(Q)}}$$\scriptstyle{TH(ag)}$$\textstyle{THT(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{T(Q)}}$$\textstyle{T(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ag}$$\textstyle{T(Q)}$

Since $H(a)={\rm ker}(H(v))=0$, it follows that:

$a(z)=aga(u)=ag\varepsilon_{T(Q)}\varepsilon_{T(Q)}^{-1}a(u)=\varepsilon_{T(Q)}TH(ag)\varepsilon_{T(Q)}^{-1}a(u)=0.$

Hence ${\rm ker}(v)=0$, and so $v$ is a monomorphism. This shows that $e$ is
an essential monomorphism, and thus $M$ is a self-CS-Rickart right $R$-module.
∎

Following [16], a covariant functor $F:\mathcal{A}\to\mathcal{B}$ between
abelian categories is called _faithfully exact_ provided the sequence
$F(A)\stackrel{{\scriptstyle
F(f)}}{{\longrightarrow}}F(B)\stackrel{{\scriptstyle
F(g)}}{{\longrightarrow}}F(C)$ is exact if and only if the sequence
$A\stackrel{{\scriptstyle f}}{{\longrightarrow}}B\stackrel{{\scriptstyle
g}}{{\longrightarrow}}C$ is exact. Let ${}_{S}M_{R}$ be a bimodule. The right
$R$-module $M$ is called _faithfully projective_ if the functor ${\rm
Hom}_{R}(M,-):{\rm Mod}(R)\to{\rm Mod}(S)$ is faithfully exact. The right
$R$-module $M$ is called _faithfully injective_ if the functor ${\rm
Hom}_{R}(-,M):{\rm Mod}(R)\to{\rm Mod}(S^{\rm op})$ is faithfully exact. The
left $S$-module $M$ is called _faithfully flat_ if the functor
$-\otimes_{S}M:{\rm Mod}(S)\to{\rm Mod}(R)$ is faithfully exact.

######  Corollary 5.4.

Let $M$ be a right $R$-module and $S={\rm End}_{R}(M)$.

  1. (1)

Assume that $M$ is a flat left $S$-module.

    1. (i)

If $M$ is a self-$CS$-Rickart right $R$-module, then $S$ is a
self-$CS$-Rickart right $S$-module.

    2. (ii)

If $M$ is a faithfully projective right $R$-module and $S$ is a
self-$CS$-Rickart right $S$-module, then $M$ is a self-$CS$-Rickart right
$R$-module.

  2. (2)

Assume that $M$ is a projective right $R$-module.

    1. (i)

If $S$ is a dual self-$CS$-Rickart right $S$-module, then $M$ is a dual
self-$CS$-Rickart right $R$-module.

    2. (ii)

If $M$ is a faithfully flat left $S$-module and $M$ is a dual
self-$CS$-Rickart right $R$-module, then $S$ is a dual self-$CS$-Rickart right
$S$-module.

  3. (3)

Assume that $M$ is an injective left $S$-module.

    1. (i)

If $M$ is a dual self-CS-Rickart right $R$-module, then $S$ is a self-CS-
Rickart left $S$-module.

    2. (ii)

If $M$ is a faithfully injective right $R$-module and $S$ is a self-CS-Rickart
left $S$-module, then $M$ is a dual self-CS-Rickart right $R$-module.

###### Proof.

Consider the adjoint pair of covariant functors $(T,H)$, where

$T=-\otimes_{S}M:{\rm Mod}(S)\to{\rm Mod}(R),$ $H={\rm Hom}_{R}(M,-):{\rm
Mod}(R)\to{\rm Mod}(S),$

and the right adjoint pair of contravariant functors $(H_{1},H_{2})$, where

$H_{1}={\rm Hom}_{R}(-,M):{\rm Mod}(R)\to{\rm Mod}(S^{\rm op}),$ $H_{2}={\rm
Hom}_{S}(-,M):{\rm Mod}(S^{\rm op})\to{\rm Mod}(R).$

(1) We have $TH(M)\cong M$, whence $M\in{\rm Stat}(H)$. Since $M$ is a flat
left $S$-module, $T$ is exact. If $M$ is a faithfully projective right
$R$-module, then $H$ reflects zero objects [16, Theorem 2.2]. Finally, use
Theorem 3.1 (1).

(2) We have $HT(S)\cong S$, whence $S\in{\rm Adst}(H)$. Since $M$ is a
projective right $R$-module, $H$ is exact. If $M$ is a faithfully flat left
$S$-module, then $T$ reflects zero objects [16, Theorem 2.1]. Finally, use
Theorem 3.1 (2).

(3) We have $H_{2}H_{1}(M)\cong M$, whence $M\in{\rm Refl}(H_{1})$. Since $M$
is an injective left $S$-module, $H_{2}$ is exact. If $M$ is a faithfully
injective right $R$-module, then $H_{1}$ reflects zero objects [16, Theorem
3.1]. Finally, use Theorem 3.2 (2). ∎

Following [22], for a $G$-graded ring $R$ and two graded right $R$-modules $M$
and $N$, we may consider the graded abelian group $\mathrm{HOM}_{R}(M,N)$,
whose $\sigma$-th homogeneous component is

$\mathrm{HOM}_{R}(M,N)_{\sigma}=\\{f\in{\rm
Hom}_{R}(M,N)\,\mid\,f(M_{\lambda})\subseteq N_{\lambda\sigma}\mbox{ for all
}\lambda\in G\\}.$

Then $S={\rm END}_{R}(M)=\mathrm{HOM}_{R}(M,M)$ is a $G$-graded ring (where
the multiplication is the map composition) and $M$ is a graded
$(S,R)$-bimodule, that is, $S_{\tau}\cdot M_{\sigma}\cdot R_{\lambda}\subseteq
M_{\tau\sigma\lambda}$ for every $\tau,\sigma,\lambda\in G$. For a graded
right $S$-module $N$, the right $R$-module $N\otimes_{S}M$ may be graded by

$(N\otimes_{S}M)_{\tau}=\left\\{\sum_{\sigma\lambda=\tau}n_{\sigma}\otimes
m_{\lambda}\mid n_{\sigma}\in N_{\sigma},m_{\lambda}\in M_{\lambda}\right\\}.$

Following the terminology from modules, one may naturally define the concepts
of _faithfully flat_ , _faithfully projective_ and _faithfully injective_
graded modules.

######  Corollary 5.5.

Let $M$ be a graded right $R$-module and $S={\rm END}_{R}(M)$.

  1. (1)

Assume that $M$ is a flat graded left $S$-module.

    1. (i)

If $M$ is a self-$CS$-Rickart graded right $R$-module, then $S$ is a
self-$CS$-Rickart graded right $S$-module.

    2. (ii)

If $M$ is a faithfully projective graded right $R$-module and $S$ is a
self-$CS$-Rickart graded right $S$-module, then $M$ is a self-$CS$-Rickart
graded right $R$-module.

  2. (2)

Assume that $M$ is a projective graded right $R$-module.

    1. (i)

If $S$ is a dual self-$CS$-Rickart graded right $S$-module, then $M$ is a dual
self-$CS$-Rickart graded right $R$-module.

    2. (ii)

If $M$ is a faithfully flat graded left $S$-module and $M$ is a dual
self-$CS$-Rickart graded right $R$-module, then $S$ is a dual
self-$CS$-Rickart graded right $S$-module.

  3. (3)

Assume that $M$ is an injective graded left $S$-module.

    1. (i)

If $M$ is a dual self-CS-Rickart graded right $R$-module, then $S$ is a self-
CS-Rickart graded left $S$-module.

    2. (ii)

If $M$ is a faithfully injective graded right $R$-module and $S$ is a self-CS-
Rickart graded left $S$-module, then $M$ is a dual self-CS-Rickart graded
right $R$-module.

###### Proof.

Consider the adjoint pair of covariant functors $(T,H)$, where

$T=-\otimes_{S}M:{\rm gr}(S)\to{\rm gr}(R),$ $H=\mathrm{HOM}_{R}(M,-):{\rm
gr}(R)\to{\rm gr}(S),$

and the right adjoint pair of contravariant functors $(H_{1},H_{2})$, where

$H_{1}=\mathrm{HOM}_{R}(-,M):{\rm gr}(R)\to{\rm gr}(S^{\rm op}),$
$H_{2}=\mathrm{HOM}_{S}(-,M):{\rm gr}(S^{\rm op})\to{\rm gr}(R).$

(1) We have $TH(M)\cong M$, whence $M\in{\rm Stat}(H)$. Since $M$ is a flat
graded left $S$-module, $T$ is exact. If $M$ is a faithfully projective graded
right $R$-module, then $H$ reflects zero objects [16, Theorem 1.1]. Finally,
use Theorem 3.1 (1).

(2) We have $HT(S)\cong S$, whence $S\in{\rm Adst}(H)$. Since $M$ is a
projective graded right $R$-module, $H$ is exact. If $M$ is a faithfully flat
graded left $S$-module, then $T$ reflects zero objects [16, Theorem 1.1].
Finally, use Theorem 3.1 (2).

(3) We have $H_{2}H_{1}(M)\cong M$, whence $M\in{\rm Refl}(H_{1})$. Since $M$
is an injective graded left $S$-module, $H_{2}$ is exact. If $M$ is a
faithfully injective graded right $R$-module, then $H_{1}$ reflects zero
objects [16, Theorem 3.1]. Finally, use Theorem 3.2 (2). ∎

Let $C$ be a coalgebra over a field. A left $C$-comodule $Q$ is called _quasi-
finite_ if ${\rm Hom}_{C}(M,Q)$ is finite dimensional for every finite
dimensional left $C$-comodule $M$ [27, Definition 1.1]. For a
$C$-$D$-bicomodule $Q$, consider the _cotensor functor_
$-\square_{C}Q:\mathcal{M}^{C}\to\mathcal{M}^{D}$ (see [11, p. 87]). If $Q$ is
quasi-finite as a right $D$-comodule, then the cotensor functor has a left
adjoint denoted by $h_{D}(Q,-):\mathcal{M}^{D}\to\mathcal{M}^{C}$ [27,
Proposition 1.10], and called the _cohom functor_.

######  Corollary 5.6.

Let $D$ be a coalgebra over a field, let $Q$ be a quasi-finite injective right
$D$-comodule, and let $C={\rm h}_{D}(Q,Q)$. Then:

  1. (i)

If $Q$ is a dual self-CS-Rickart right $D$-comodule, then $C$ is a dual self-
CS-Rickart right $C$-comodule.

  2. (ii)

If $h_{D}(Q,-)$ reflects zero objects and $C$ is a dual self-CS-Rickart right
$C$-comodule, then $Q$ is a dual self-CS-Rickart right $D$-comodule.

###### Proof.

Consider the adjoint pair of covariant functors $(L,R)$, where

$L=h_{D}(Q,-):\mathcal{M}^{D}\to\mathcal{M}^{C},$
$R=-\square_{C}Q:\mathcal{M}^{C}\to\mathcal{M}^{D}.$

We have $RL(Q)\cong Q$, whence $Q\in{\rm Adst}(R)$. Since $Q$ is an injective
right $D$-comodule, $R$ is exact [11, Theorem 2.4.17]. Finally, use Theorem
3.1 (2). ∎

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# CT-ICP: Real-time Elastic LiDAR Odometry with Loop Closure

Pierre Dellenbach1,2, Jean-Emmanuel Deschaud1, Bastien Jacquet2, and François
Goulette1 1 MINES ParisTech, PSL University, Centre for Robotics, 75006 Paris,
France<EMAIL_ADDRESS>Kitware, Computer Vision Team,
69100 Villeurbanne, France<EMAIL_ADDRESS>
###### Abstract

Multi-beam LiDAR sensors are increasingly used in robotics, particularly with
autonomous cars for localization and perception tasks, both relying on the
ability to build a precise map of the environment. For this, we propose a new
real-time LiDAR-only odometry method called CT-ICP (for Continuous-Time ICP),
completed into a full SLAM with a novel loop detection procedure. The core of
this method, is the introduction of the combined continuity in the scan
matching, and discontinuity between scans. It allows both the elastic
distortion of the scan during the registration for increased precision, and
the increased robustness to high frequency motions from the discontinuity.

We build a complete SLAM on top of this odometry, using a fast pure LiDAR loop
detection based on elevation image 2D matching, providing a pose graph with
loop constraints. To show the robustness of the method, we tested it on seven
datasets: KITTI, KITTI-raw, KITTI-360, KITTI-CARLA, ParisLuco, Newer College,
and NCLT in driving and high-frequency motion scenarios. Both the CT-ICP
odometry and the loop detection are made available online. CT-ICP is currently
first, among those giving access to a public code, on the KITTI odometry
leaderboard, with an average Relative Translation Error (RTE) of 0.59% and an
average time per scan of 60$ms$ on a CPU with a single thread.

##  I INTRODUCTION

Many LiDARs are based on the principle of several laser fibers that fire off
continuously out of a rotating unit. We call a “scan” the limited time-span
aggregation of points covering enough field of view (360 degrees for LiDARs
used in autonomous vehicles like in [1, 2]).

To take continuous acquisition into account, LiDAR odometry methods like in
[3] or [4] distort the current scan with a constant velocity motion
assumption. However, this assumption does not take into account large
direction or velocity changes.

On an other hand, recent works like [5, 6] propose real-time LiDAR odometries
with continuous-time trajectories. A pose can be calculated for each LiDAR
point according to its timestamp from control poses (direct poses or using
splines). However, [5] requires an IMU to integrate high-frequency motions,
while the continuity constraints in [6] simply smoothes out these motions, we
believe, at the cost of precision.

We propose a new elastic formulation of the trajectory with a continuity of
poses intra-scan and discontinuity between adjacent scans. In practice, this
is defined by solving an elastic scan-to-map registration, parametrized by two
poses per scan (for the beginning and end of the scan) with a proximity
constraint between the end pose of the previous scan and the beginning pose of
the current scan.

Figure 1:  Top, in color, one LiDAR scan; the color depends on the timestamp
of each point (from the oldest in blue to newest in red). The scan is deformed
elastically to align with the map (white points) by the joint optimization of
two poses at the start and end of the scan and interpolation according to the
timestamp, hence creating a continuous-time scan-to-map odometry. Below, the
formulation of our trajectory with a continuity of poses intra-scan and
discontinuity between scans.

Fig. 1 shows the registration of a scan (points are in color) to the map
(points are in white). The color ranges from blue to red, representing the
relative timestamp ($\alpha_{i}$) for each point used by our CT-ICP method.
Our odometry elastically adjusts the new scan to the building.

As our main contribution, we propose:

  * •

A new elastic LIDAR odometry based on the continuity of poses intra-scan and
discontinuity between scans.

We also present as secondary contributions:

  * •

A local map based on a dense point cloud stored in a sparse voxel structure to
obtain real-time processing speed.

  * •

A large campaign of experiments on 7 datasets in driving and high-frequency
motion scenarios, all reproducible with public and permissive open-source code
111https://github.com/jedeschaud/ct_icp.

  * •

A fast method of loop detection integrated with a pose graph back-end to build
a complete SLAM, integrated into pyLiDAR-SLAM
222https://github.com/Kitware/pyLiDAR-SLAM.

##  II RELATED WORK

Numerous LiDAR odometry methods are based on the Iterative Closest Point (ICP)
method [7] and its more efficient point-to-plane variant [8, 9]. KinectFusion
[10] for RGB-D sensors has shown great progress in moving from frame-to-frame
to frame-to-model registration, likewise, LiDAR odometry methods use scan-to-
map registration.

SuMa [11] and SuMa++ [12] represent a scan as an image (range image) and the
map in the form of a set of surfels. The registration aligns the current scan
with a rendered image obtained by projecting the map of surfels on the GPU.
LOAM [13] detects keypoints of different classes (edges, planes) in the range
image and registers the detected keypoints into a voxel grid, but the
neighborhood search uses separate kd-trees for each class of keypoint. The
keypoint map is sufficiently sparse for the search to be carried out in real
time. LeGO-LOAM [14] improves on the previous method by separating the
keypoints from the ground. F-LOAM [3] has optimized the registration to be
faster and runs at more than 20 Hz. More recently, MULLS [4] has improved this
approach by detecting the keypoints in the scan in 3D with many different
types of keypoints (ground, facade, roof, pillar, beam, and vertex).
Differently, IMLS-SLAM [15] represents the map in the form of a dense point
cloud, [16] in the form of a TSDF, and PUMA [17] with a mesh, but these last
three methods do not work in real time.

To account for sensor movement during scanning, most previous methods like [3,
4, 15] first distort the current scan with a constant velocity motion model
from previous poses. Then, the scan distortion is kept fixed during ICP
iterations. Although this approach works well in most driving scenarios, it is
not robust enough to sudden changes of orientation or fast acceleration from
one scan to another.

Another approach takes into account the motion during scanning by defining a
continuous-time trajectory with control poses (by linear interpolation or
B-splines). CT-SLAM [18] defines the trajectory by using multiple poses per
scan; [19] also uses a continuous-time trajectory with six poses per scan
(using B-splines basis to represent the trajectory). However, both methods are
not in real time. More recently, Elastic LiDAR Fusion in [5] and MARS LiDAR
odometry in [6] proposed a continuous-time formulation of the trajectory that
operates in real time. [5] uses a linear interpolation of poses with a map of
sparse surfels for odometry and dense 2D disk surfels for mapping, while [6]
leverages B-splines for the continous-time trajectory with a multi-resolution
surfel map to get real-time speed. These methods tend to smooth the
trajectory, but in many real-life acquisitions, the motion of the sensor can
be shaky (notably due to terrain irregularities) and produce high-frequency
movements (with respect to the control point frequencies), which are not taken
into account by these methods.

By contrast, CT-ICP defines a continous-time trajectory during the scan and
discontinuous between scans. During a scan, the trajectory is parameterized by
two poses for the beginning and end of the scan. However, conversely to
methods like [6], the final trajectory of CT-ICP is discontinuous, the pose at
the beginning of a scan is not equal to the pose at the end of the previous
scan. We claim that this approach compensates for the motion irregularities
that cannot be compensated for by interpolation.

As precise as it is, LiDAR odometry still accumulates errors in an open
environment, which leads to drift in trajectories. A loop closure procedure
can correct the trajectories globally, but loop detection is still an open
problem for LiDAR SLAM. Currently, most SLAM solutions rely principally on
registration methods to directly close loops [20, 11, 13], which only works
for small trajectories and low drift. Different place recognition methods have
been proposed, operating on individual scans [21, 22, 23, 24], which is
sensitive to environment changes and are more adapted for driving scenarios.
[4] uses a global registration procedure [25] before ICP-based refinement, but
each alignment attempt is very costly, thus restricting the chances to
discover more loops. Recently, deep learning methods have been proposed [23,
24], but are not adapted to new environments due to the training requirements.
By contrast, we propose a new loop closure procedure that operates on
aggregated point clouds projected onto an elevation image. This procedure
requires the motion of the sensor to be mostly 2D and to estimate the gravity
vector; thus, it can be integrated into any LiDAR odometry for which these
conditions are met. Closest to our work is [22], which constructs elevation
images but operates on scans, whereas our procedure runs on local maps and,
therefore, is not required to test each scan against the previous observed
locations, making it more efficient for online SLAM scenarios.

##  III CT-ICP ODOMETRY

Figure 2:  Aggregated point clouds for NCLT dataset (top left), KITTI-CARLA
(top-right), Newer College Dataset (bottom left), and ParisLuco (bottom right)
show the quality of the maps obtained with CT-ICP.

###  III-A Odometry formulation

Figure 3:  Trajectories estimated with CT-ICP odometry, and our Loop Closure
correction (CT-ICP+LC) for the sequences 00 of KITTI-raw (4541 scans), 06 of
KITTI-360 (9698 scans), Town01 of KITTI-CARLA (5000 scans),
01_short_experiment of NCD (15301 scans) and 2012-01-08 of NCLT (42764 scans).

Our CT-ICP odometry is parameterized for the current scan by two poses: a pose
for the beginning of the scan $\Omega_{b}^{n}=(R^{n}_{b},t^{n}_{b})$ ($R$ for
rotation, $t$ for translation, and $b$ for beginning) and a pose for the end
of the scan $\Omega_{e}^{n}=(R^{n}_{e},t^{n}_{e})$ ($e$ for end). To simplify
the notations, in the following, we omit the $n$ referring to the poses of the
current scan. For each sensor measurement captured at a time
$\tau\in[\tau_{b},\tau_{e}]$ between the first $\tau_{b}$ and last $\tau_{e}$
timestamp of the scan, the pose of the sensor is estimated by interpolating
between the two poses of the scan. These poses transform a point from the
LiDAR frame $L(\tau)$ to the world frame $W=L(0)$. Unlike other continuous-
time trajectory methods, the pose $\Omega_{b}$ of the new scan does not match
the end pose $\Omega_{e}^{n-1}$ of the previous scan. We add a proximity
constraint in the optimization to force the two poses to remain close. Our
formulation allows our odometry to be more robust to high-frequency motions of
the sensor.

For each new scan $\mathbb{S}^{n}$, we first extract a sample of keypoints
from $\mathbb{S}^{n}$ indexed by $\mathbb{I}^{n}$:
$\\{p_{i}\in\mathbb{S}^{n}|i\in\mathbb{I}^{n}\\}$ (using a simple grid
sampling of points in the scan), that we register into the local map. This map
is a dense point cloud $\mathbb{M}^{n}=\\{q_{i}^{W}\\}$ built from all the
previously registered scans, and stored in a sparse voxel grid. Its
construction is detailed in section III-B below. Our scan matching then
estimates the two optimal poses $\Omega_{b}^{*}$ and $\Omega_{e}^{*}$, thus
handling the distortion of the scan during the optimization, and proceeds to
transform the points to the world frame before adding them to the local map.

These optimal poses are given by solving the following problem with parameters
$\mathbf{X}=(\,\bm{\Omega_{b}},\bm{\Omega_{e}}\,)\in SE(3)^{2}$ in bold:

$\small\operatorname*{arg\,min}_{\mathbf{X}\in
SE(3)^{2}}\;F_{\text{ICP}}(\,\mathbf{X}\,)+\beta_{l}\,C_{\text{loc}}(\mathbf{X})+\beta_{v}\,C_{\text{vel}}(\mathbf{X})$
(1)

where $F_{\text{ICP}}$ is the scan-to-map continuous-time ICP:

$\displaystyle
F_{\text{ICP}}(\,\mathbf{X}\,)=\frac{1}{|\mathbb{I}^{n}|}\sum_{i\in\mathbb{I}^{n}}\rho(r_{i}^{2}[\mathbf{X}])$
(2)

And, for each $i\in\mathbb{I}^{n}$,

$\displaystyle r_{i}[$
$\displaystyle\mathbf{X}]=a_{i}\;(\,p_{i}^{W}[\mathbf{X}]-q_{i}^{W}\,)\cdot
n_{i}$ (3) $\displaystyle p_{i}^{W}[$
$\displaystyle\mathbf{X}]=R^{\alpha_{i}}[\mathbf{X}]*p_{i}^{L}+t^{\alpha_{i}}[\mathbf{X}]$
(4) $\displaystyle R^{\alpha_{i}}[$
$\displaystyle\mathbf{X}]=\text{slerp}(\,\mathbf{R_{b}},\,\mathbf{R_{e}},\,\alpha_{i}\,)$
(5) $\displaystyle t^{\alpha_{i}}[$
$\displaystyle\mathbf{X}]=(1-\alpha_{i})\,\mathbf{t_{b}}+\alpha_{i}\,\mathbf{t_{e}}$
(6)

$\rho(s)$ is a robust loss function to minimize the influence of outliers, and
$r_{i}$ are the residuals of the point-to-plane distance between a sample
point $p_{i}^{W}$ and its closest neighbor in the map (as classically
formulated by the ICP). $p_{i}^{W}[\mathbf{X}]$ is the point $p_{i}$ expressed
in the world frame, $n_{i}$ is the normal of $p_{i}^{W}$’s neighborhood in the
local map, and $p_{i}^{L}$ is the sensor measurement (in the LiDAR frame).
$\bm{\Omega}^{\alpha_{i}}[\mathbf{X}]=(\mathbf{R^{\alpha_{i}}},\mathbf{t^{\alpha_{i}}})\in
SE(3)$ is the transformation from the LiDAR frame at time $\tau_{i}$,
$L(\tau_{i})$ to the world $W$. It is estimated by an interpolation between
$\bm{\Omega}_{b}$ and $\bm{\Omega}_{e}$ by defining
$\alpha_{i}=(\tau_{i}-\tau_{b})/(\tau_{e}-\tau_{b})$. For rotation
interpolation, we use the standard spherical linear interpolation (slerp).

We also introduce weights to favor planar neighborhoods:
$a_{i}=a_{2D}=(\sigma_{2}-\sigma_{3})/\sigma_{1}$ as defined by [15] (where
$\sigma_{i}$ are the square roots of the eigen values of the neighborhood’s
covariance), is the planarity of the neighborhood of $p_{i}^{W}$. Note that
unlike [15] and most LiDAR odometry methods, we calculate the normals $n_{i}$
and planarity weights $a_{i}$ using the dense point cloud of the local map
instead of the current scan, leading to much richer neighborhoods. This
computation is done at each iteration of the ICP and for each sample point
$p_{i}$, as the refinement on their position leads to more precise
neighborhoods.

Additionally in equation 1, we have introduced the two constraints $C_{loc}$
(location consistency constraint), and $C_{vel}$ (constant velocity
constraint), with respective weights $\beta_{l}$ and $\beta_{v}$, which is
defined as follows:

$\displaystyle\small C_{loc}(\mathbf{t_{b}})$
$\displaystyle=||\mathbf{t_{b}}-t_{e}^{n-1}||^{2}$ (7) $\displaystyle
C_{vel}(\mathbf{t_{b}},\mathbf{t_{e}})$
$\displaystyle=||(\mathbf{t_{e}}-\mathbf{t_{b}})-(t_{e}^{n-1}-t_{b}^{n-1})||^{2}$
(8)

$C_{loc}$ forces the beginning and end location of the sensor to be consistent
(limiting the discontinuity), and $C_{vel}$ limits too rapid accelerations.
These constraints are sufficient to force the elasticity of CT-ICP to be
consistent with the motion model defined implicitly by the weights $\beta_{l}$
and $\beta_{v}$ (taken at 0.001 for all experiments).

CT-ICP performs iterations until a threshold on the norm of parameter’s step
is met (typically 0.1 cm on translation and 0.01° in rotation) or a number of
iterations is reached (5 to ensure real time). By default, CT-ICP runs on a
single thread, but parallel processing (for the construction of the
neighborhoods, or while solving the linear system) can be leveraged for more
challenging scenarios requiring more iterations and sample points.

###  III-B Local map and robust profile

As a local map, we use a point cloud from previous scans (like IMLS-SLAM
[15]), but in contrast, the points in the world frame ($W$) are stored in a
sparse data structure of voxels for faster neighborhood access than kd-trees
(constant time access instead of logarithmic). The voxel size of the map
controls the radius of the neighborhood search, and the level of detail of the
stored point cloud. For driving scenarios, it is set to 1.0 m and 0.80 m for
high-frequency motion scenarios, providing satisfying balance between these
two aspects.

The voxel size defining the grid of the map is important because it defines
the neighborhood search radius, as well as the level of detail of our local
map.

Each voxel stores up to 20 points, such that no two points are closer than 10
cm from each other, to limit the redundancy due to the density of measurements
along the scan lines. Once a voxel is full, no more points will be inserted
into it.

To construct the neighborhood of a point $p_{i}^{W}$ (needed for the
computation of $n_{i}$ and $a_{i}$), we select the $k=20$ nearest neighbors
among all points in the map in the 27 neighboring voxels from the current
point. After running CT-ICP for the current scan $n$, the points are added to
the local map. Note that in contrast with pyLiDAR’s F2M, the local map is not
based on a sliding window of last scans, but voxels are removed from the map
based only on their distance to the center of the last inserted scan.

Odometries with these types of maps are highly sensitive to bad registration,
and cannot recover from map pollutions from bad scan insertions. This is
particularly problematic in datasets with rapid orientation changes. For these
types of datasets, we introduce a robust profile that detects hard cases (fast
orientation changes) and registration failures (location inconsistencies or
large number of new keypoints falling in empty voxels) and attemps a new
registration of the current scan with a more conservative set of parameters
(most notably a larger number of sampled keypoints and a larger neighborhood
search); for important modifications in the orientation ($\geq 5$ ∘), we do
not insert the new scan in the map, which has a higher probability of being
misaligned. The gain in robustness is paid by increased runtime.

##  IV LOOP CLOSURE AND BACK-END

Figure 4:  Qualitative results of loop closure on the sequence 00 of KITTI-360
(11501 scans). The top left is an elevation image built by projecting the
local map. The top right shows both the CT-ICP odometry’s trajectory and the
one corrected using the computed Loop Closure constraints (CT-ICP+LC). The
bottom shows the different loop closure constraints (green) found for the same
turn as the local map at the top left.

Our loop closure algorithm maintains in its memory a window of the last scans
registered by the odometry. When the window reaches a size of $N_{map}$ scans,
the points are aggregated into a point cloud that is placed in the coordinate
frame at the center of the window.

Every point of this map is then inserted into a 2D elevation grid, keeping for
each pixel the point at a maximum elevation. From this 2D grid, an elevation
image is obtained by clipping the $z$ coordinate of each pixel between
$z_{min}$ and $z_{max}$. Rotation invariant 2D features [26] are then
extracted and saved in memory with the elevation grid. All scans except the
last $N_{overlap}$ are dropped from the window.

Every time a new elevation image is built (every $N_{map}-N_{overlap}$ scans),
it is matched against the elevation images saved in its memory. A 2D rigid
transformation between the two features sets is robustly fit with a RANSAC,
and a threshold on the number of inliers is used to validate the
correspondences. When a match is validated, an ICP refinement of the initial
2D transform (using Open3D’s ICP [27]) is performed on the elevation grids’
point clouds, producing an accurate 6-DoF loop closure constraint. To reduce
the number of candidates, an initial filter selects the $n_{candidates}=10$
candidates closest to the current grid.

As a back-end, our SLAM uses a standard pose graph (PG) implemented with g2o
[28] much like in [20]. The PG regularly adds new poses when new odometry
constraints are added with the respective odometry constraints, but the
trajectory is only globally optimized when a new loop constraint is detected,
in which case the trajectory of the loop closure module is also updated. Fig.
4 summarizes our loop closure procedure.

Currently, our procedure requires the sensor’s motion to be mostly planar on
the ground, and for the extrinsic calibration to align the $z$ axis with the
normal of the ground plane. This limits the supported sensor setups, as well
as the authorized motion. Note that this can be addressed if the gravity
vector or the local ground plane are known, in which case the elevation image
can be correctly projected. However, we show in section V that for outdoor
scenarios with a sensor mostly pointing up, our loop closure detection
succeeds in detecting many loops, allowing our back-end to successfully
correct the drift errors from the odometry.

##  V EXPERIMENTS

We demonstrate the efficiency and versatility of our method by conducting
experiments on a large variety of datasets: KITTI, KITTI-raw, KITTI-360,
KITTI-CARLA, ParisLuco, Newer College Dataset (NCD), and NCLT. Our method
requires only the geometry (xyz fields) and the timestamps for each point to
elastically distort the scans (they are estimated linearly based on azimuth
angles for KITTI-raw and KITTI-360).

###  V-A Datasets

####  V-A1 Driving scenarios

KITTI [1] proposes 11 sequences of LiDAR scans acquired from a Velodyne HDL64
mounted on a car and the corresponding ground truth poses from GPS/IMU
measurements. The odometry benchmark proposes motion-corrected scans (using
the GPS/IMU trajectory to distort the scans): we call this KITTI-corrected.
Most odometry methods [13, 11, 15, 4] compare their results on the motion-
corrected version of KITTI. However, this does not allow for testing their
performance with raw LiDAR data. All odometry sequences (except sequence 03)
are available with uncorrected raw scans. We call this KITTI-raw. The same
acquisition setup was later used to generate KITTI-360 [29], which contains
eight sequences that are much longer than KITTI’s (ranging from 3000 to 15000
scans by sequence) in the same environment. These scans are not motion-
corrected, and the timestamps are estimated as described above. For KITTI-
corrected, KITTI-raw, and KITTI-360, we applied for all scans an intrinsic
angle correction of 0.205 ∘, as mentioned in [15].

KITTI-CARLA [30] simulates a similar sensor suite as KITTI in a synthetic
environment generated using the CARLA simulator [31]. The dataset consists of
seven sequences of 5000 scans using a simulated 64-channel LiDAR sensor. The
relative motion of the vehicule during the acquisition of the scan is
simulated, and each scan provides precise ground truth and timestamps.

ParisLuco is one sequence of 4 km acquisition (12751 scans) in the center of
Paris using our own vehicle with a Velodyne HDL32 sensor in a vertical
position. We generated the ground truth with post-processing of GPS/IMU
measurements (only translations are available).

####  V-A2 High-frequency motion scenarios

Cars accelerate slowly with respect to the speed of acquisition of the sensor;
thus, a constant velocity model is valid. Much more challenging are scenarios
were the sensor quickly changes orientations. This is the case for NCLT [32],
a dataset containing 27 long sequences ($\geq 20000$ scans per sequence) of
scans acquired at Michigan University using a Velodyne HDL32 mounted on a two-
wheeled Segway. This dataset is particularly challenging, because the vehicule
introduces abrupt rotations to the LiDAR’s own rotation axis, which is
problematic for classical ICP-based odometry methods, as shown in [33]

NCD [34], the Newer College Dataset, contains two sequences ($\sim$15000 and
$\sim$26000 scans) of a handheld Ouster 64-channel LiDAR mounted on a stick,
that is carried accross the Oxford campus.

###  V-B Odometry Experiments

TABLE I:  Relative Translation Error (RTE) [%] with the Driving profile on KITTI-corrected, KITTI-raw, KITTI-360, KITTI-CARLA, and ParisLuco and with the High-Frequency Motion profile on the NCD and NCLT datasets. AVG is the RTE averaged over all segments of all sequences, $\Delta T$ the average running time per scan. KITTI-corrected is the only dataset whose scans have been motion-corrected $(\star)$; all other datasets have uncorrected raw point clouds as scans. DRIVING PROFILE |  |  |
---|---|---|---
KITTI-corrected⋆ | 00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | AVG | $\Delta T$
IMLS-SLAM [15] | 0.50 | 0.82 | 0.53 | 0.68 | 0.33 | 0.32 | 0.33 | 0.33 | 0.80 | 0.55 | 0.53 | 0.55 | 1250 ms
MULLS [4] | 0.56 | 0.64 | 0.55 | 0.71 | 0.41 | 0.30 | 0.30 | 0.38 | 0.78 | 0.48 | 0.59 | 0.55 | 80 ms
pyLiDAR F2M [33] | 0.51 | 0.79 | 0.51 | 0.64 | 0.36 | 0.29 | 0.29 | 0.32 | 0.78 | 0.46 | 0.57 | 0.53 | 175 ms
CT-ICP (ours) | 0.49 | 0.76 | 0.52 | 0.72 | 0.39 | 0.25 | 0.27 | 0.31 | 0.81 | 0.49 | 0.48 | 0.53 | 60 ms
KITTI-raw | 00 | 01 | 02 | X | 04 | 05 | 06 | 07 | 08 | 09 | 10 | AVG | $\Delta T$
IMLS-SLAM [15] | 0.79 | 0.86 | 0.76 |  | 0.51 | 0.48 | 0.62 | 0.67 | 0.97 | 0.70 | 0.75 | 0.71 | 1070 ms
MULLS [4] | 1.43 | 3.12 | 1.01 |  | 0.57 | 1.93 | 1.60 | 0.69 | 1.28 | 1.49 | 0.71 | 1.41 | 80 ms
pyLiDAR F2M [33] | 2.20 | 0.98 | 1.55 |  | 0.45 | 1.46 | 0.69 | 1.72 | 1.60 | 1.28 | 1.18 | 1.61 | 530 ms
CT-ICP (ours) | 0.51 | 0.81 | 0.55 |  | 0.43 | 0.27 | 0.28 | 0.35 | 0.80 | 0.47 | 0.49 | 0.55 | 65 ms
KITTI-360 | 00 | 02 | 03 | 04 | 05 | 06 | 07 | 09 | 10 |  |  |  |  | AVG | $\Delta T$
IMLS-SLAM [15] | 0.65 | 0.63 | 0.64 | 0.89 | 0.63 | 0.70 | 0.54 | 0.67 | 0.79 |  | 0.68 | 1060 ms
MULLS [4] | 1.60 | 1.29 | 0.87 | 1.64 | 1.27 | 1.48 | 6.25 | 1.28 | 0.88 |  | 1.55 | 90 ms
pyLiDAR F2M [33] | 1.79 | 1.25 | 0.90 | 1.60 | 1.26 | 1.38 | 0.69 | 1.72 | 1.39 |  | 1.46 | 475 ms
CT-ICP (ours) | 0.41 | 0.38 | 0.34 | 0.65 | 0.39 | 0.42 | 0.34 | 0.45 | 0.69 |  |  |  |  | 0.45 | 70 ms
KITTI-CARLA | Town01 | Town02 | Town03 | Town04 | Town05 | Town06 | Town07 |  | AVG | $\Delta T$
IMLS-SLAM [15] | 0.03 | 0.05 | 0.16 | 0.20 | 0.06 | 4.90 | 0.25 |  | 0.81 | 780 ms
MULLS [4] | 1.39 | 0.77 | 0.69 | 1.24 | 1.13 | 0.98 | 0.97 |  | 1.04 | 70 ms
pyLiDAR F2M [33] | 16.25 | 7.92 | 37.16 | 10.06 | 9.11 | 73.29 | 2.69 |  | 23.84 | 530 ms
CT-ICP (ours) | 0.03 | 0.04 | 0.03 | 0.03 | 0.02 | 0.04 | 0.41 |  | 0.09 | 65 ms
ParisLuco | LuxembourgGarden |  |  |  | AVG | $\Delta T$
IMLS-SLAM [15] | 1.26 |  |  |  | 1.26 | 770 ms
MULLS [4] | 3.03 |  |  |  | 3.03 | 55 ms
pyLiDAR F2M [33] | 4.90 |  |  |  | 4.90 | 355 ms
CT-ICP (ours) | 1.11 |  |  |  | 1.11 | 80 ms
HIGH-FREQUENCY MOTION PROFILE |  |  |  |  |  |  |  |  |
NCD | 01_short_experiment | 02_long_experiment |  |  | AVG | $\Delta T$
pyLiDAR F2M [33] | 1.37 | 2.63 |  |  | 2.20 | 1065 ms
CT-ICP (ours) | 0.48 | 0.58 |  |  | 0.55 | 430 ms
NCLT | 2012-01-08 |  |  |  | AVG | $\Delta T$
pyLiDAR F2M [33] | 19.80 |  | 19.80 | 340 ms
CT-ICP (ours) | 1.17 |  | 1.17 | 180 ms

For quantitative evaluations, as in [11, 15, 4, 33], we use the KITTI Relative
Translation Error (RTE), which averages the trajectory drift over segments of
lengths ranging from 100 m to 800 m. When computing the score on multiple
sequences, the average (AVG) is computed over all the segments of all
sequences, and mirrors KITTI’s benchmark evaluation.

To compare our method, we evaluated three other LiDAR odometries on the
datasets presented: MULLS [4], IMLS-SLAM [15], and pyLIDAR-SLAM F2M [33]. For
each of these methods, we searched for the set of parameters offering the best
results, but once the parameters are found, we kept the same for all the
driving datasets. MULLS and IMLS-SLAM are odometries specialized for driving
datasets, which is why we only compared our odometry to pyLIDAR-SLAM F2M [33]
on the high-frequency motion datasets NCD and NCLT.

We can see from Table I that all the methods are very close on the KITTI-
corrected dataset. However, when we use the uncorrected scans (KITTI-raw and
KITTI-360), all the other methods strongly degrade their performance. On the
other hand, CT-ICP manages to obtain results very close to those of KITTI-
corrected. The corrected scans prevent CT-ICP from using its elastic
formulation, but this shows that our map representation and our scan-to-map
algorithm are efficient. We also launched our odometry on the KITTI-corrected
test sequences (from 11 to 21) and submitted the results online on the KITTI
benchmark: we are first among those who put their code online, with a score of
0.59% RTE.

It should be noted that in Table I, we do not obtain the same results
published by MULLS on the KITTI-corrected dataset because we are using a
single set of parameters for all sequences, whereas MULLS had three different
ones (urban, highway, and country).

For KITTI-CARLA, a synthetic dataset with scans comprising greater velocity
variations than the KITTI datasets (due to the simulated dynamics of the
vehicle), the difference between CT-ICP and the other odometries is more
obvious: 0.09% RTE for CT-ICP and 0.81% for the best of the three other
odometries. This shows the interest of our elastic formulation and
discontinuity of poses between scans and how it is able to compensate for
small errors in the poses of previous scans.

ParisLuco is a dataset with a different sensor (32 fibers instead of 64 for
KITTI’s) and low inertia because it was acquired in the center of a city. CT-
ICP can overtake IMLS-SLAM, despite using a much sparser map.

NCD and NCLT are challenging datasets due to the platforms’ unstability (a
stick and segway), their lengths (26000 scans for sequence 02_long_experiment,
and 46257 scans for sequence 2012-01-08), and the variety of the environments
(vegetation, interior, roads, outdoor), and the large number of challenging
edge cases. Table I shows that CT-ICP obtains excellent results on NCD (with
comparable error rates to KITTI). Further results on NCLT show the robustness
of the method because it is able to properly handle all edge cases, resulting
in a much lower RTE than pyLiDAR F2M.

To isolate our main contribution, the elastic formulation of CT-ICP, we
modified the scan matching of our odometry by distorting the scan with a
constant velocity model prior the optimization, and estimated a single pose
(while CT-ICP refines the distortion by estimating two poses). We went from
0.55% RTE to 0.79% for KITTI-raw and from 0.45% RTE to 0.60% for KITTI-360,
showing a significant loss in precision.

###  V-C Loop closure experiments

TABLE II:  Loop Closure (LC) metrics on one sequence of each dataset. ATE = mean Absolute Trajectory Error [m] after rigid transform is fitted between the ground truth and estimated trajectory, $N_{loop}$ the number of loops detected, LO = LiDAR Odometry Dataset | Sequence | $N_{loop}$ | ATE (LO) | ATE (LO+LC)
---|---|---|---|---
KITTI-raw | 00 | 69 | 6.22 | 0.66
KITTI-360 | 00 | 234 | 29.87 | 1.07
KITTI-CARLA | Town01 | 78 | 0.21 | 0.26
ParisLuco | LuxembourgG. | 292 | 29.16 | 9.65
NCD | 01_short_exp. | 59 | 0.22 | 0.36
NCLT | 2012-01-08 | 520 | 2.97 | 2.58

For all datasets, we run CT-ICP with the loop closure module (CT-ICP+LC) with
$N_{map}=100$ and $N_{overlap}=30$. For every dataset, we set $z_{min}$
slightly below the ground (using an approximate extrinsic calibration of the
sensor) and $z_{max}=z_{min}+10$ m. Finally, for NCLT, we apply a rigid
transform to have the default $z$ axis pointing up (instead of down). The
computation of each elevation grid and matching of previous ones requires 1.1
s on average, and the pose graph optimization (only launched when a loop
constraint is detected) requires an additional 1.2 s on average.

We present qualitative results of our loop closure in Fig. 3 and Fig. 4. To
quantitavely evaluate its quality, the RTE is not well suited and tends to
deteriorate after loop closure, as noted by [4]. To prove global trajectory
improvements, we use the standard Absolute Trajectory Error (ATE) to evaluate
our method. However, to decouple the quality assessment from orientation
errors in the initial poses, we first estimate the best rigid transform
between the ground truth and the estimated trajectory before computing the
ATE.

Table II shows the results for a sequence of each dataset. First, we note the
great improvement of the ATE after loop closure for KITTI-raw, KITTI-360, and
ParisLuco; more so on KITTI-360 because the sequence is much longer. On KITTI-
CARLA however, the ATE is almost the same before or after loop closure. This
dataset has a very simple geometry, with large and perfect planes, so the
challenge to the scan-matching is principally the motion of the sensor during
acquisition. CT-ICP manages near-perfect alignments and already leads to 21 cm
of precision on the absolute trajectory estimation; this is visualized in Fig.
3, where the ground truth and estimated trajectories are indistinguishable.

Finally, we note the relative large number of loop closure constraints
detected for each sequence. This number depends on the frequency of the
construction of elevation grids, here as controlled by the size of the overlap
$N_{overlap}=30$ and the size of the map $N_{map}=100$. When arriving at an
intersection, multiple loop constraints can be computed, as shown in Fig. 4,
and with a larger overlap, our method could detect more loop constraints.

##  VI CONCLUSION

We presented a new real-time LiDAR-only odometry method that goes beyond the
state of the art on seven datasets with different profiles, from driving to
high-frequency motion scenarios. At the core of our method is the continuous
scan matching CT-ICP, which elastically distorts a new scan during the
optimization to compensate for the motion during the acquisition. We publish
both the code and datasets to allow for the reproduction of all the results
presented above. In future work, we will focus on improving our back-end to
extend the proposed continuous formulation beyond the scan matching and to
fully leverage our loop closure procedure.

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