1}$. Since $S$ is flat, we have an isomorphism. $H(A_{S})\cong H(A)\otimes S\
.$ Since the homology $H(A)$ is of finite presentation, it induces an
isomorphism,

$\tilde{\mathfrak{g}}_{H(A_{S})}\cong\tilde{\mathfrak{g}}\otimes S\ ,$

which leads to an homology isomorphism

(8) $H_{-1}\left(\tilde{\mathfrak{g}}_{H(A_{S})}^{\varphi_{*}\otimes
1}\right)\cong H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)\otimes S\
.$

By faithful flatness, since $K^{1}_{\varphi_{t}\otimes 1}$ is zero on the left
hand-side, so does $K_{\varphi_{t}}^{1}$. Suppose that

$K^{k-1}_{\varphi_{t}}=0\quad\mbox{for}\quad 2\leqslant k\leqslant n\ .$

By Theorem 2.44, there exists an $\Omega\mathcal{C}$-algebra $(H(A),\psi)$
such that

$\psi^{(1)}=\varphi_{*}\quad\mbox{and}\quad\psi^{(i)}=0\quad\mbox{for all
}2\leqslant i\leqslant k$

and an $\infty$-isotopy $f:\varphi_{t}\rightsquigarrow\psi$. Point (2) of
Lemma 2.38 implies that

$K_{\psi}^{k}=\mathrm{Ad}_{\mathfrak{D}(f)}^{k}\left(K_{\varphi_{t}}^{k}\right)\quad\mbox{and}\quad
K_{\psi\otimes 1}^{k}=\mathrm{Ad}_{\mathfrak{D}(f\otimes
1)}^{k}\left(K_{\varphi_{t}\otimes 1}^{k}\right)=0\ .$

By construction, we have

$K_{\psi}^{k}=\left[k\psi^{(k+1)}\hbar^{k-1}\right]\in
H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}[\\![\hbar]\\!]/(\hbar^{k})\right)\quad\mbox{and}\quad
K_{\psi\otimes 1}^{k}=0\in
H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}[\\![\hbar]\\!]/(\hbar^{k})\right)\otimes
S\ .$

The isomorphism (8) again implies that $K_{\psi}^{k}$ is zero, so does
$K_{\varphi_{t}}^{k}$. By induction, the $n^{\text{th}}$-truncated Kaledin
class is zero. Finally, the $\Omega\mathcal{C}$-algebra $(A,\varphi)$ is gauge
$n$-formal by Theorem 2.44. Suppose now that $R$ is a $\mathbb{Q}$-algebra and
$(A_{S},\varphi\otimes 1)$ is gauge formal. The same induction can be
performed higher-up and proves that $K^{n}_{\varphi_{t}}$ are zero for all
$n\geqslant 1$. Thus, the Kaledin class $K_{\varphi_{t}}$ is zero by
Proposition 2.39 and the $(A,\varphi)$ is also gauge formal by Theorem 2.44. ∎

For example, suppose that $(R,\mathfrak{m})$ is a Noetherian local ring and
let $\mathfrak{I}\subset\mathfrak{m}$ be an ideal. We denote $\hat{R}$ the
completion of $R$ with respect to $\mathfrak{I}$. Then the ring morphism
$R\to\hat{R}$ is faithfully flat and the result follows from Theorem 4.1. Let
$p$ be a prime number. Let us consider the commutative ring $\mathbb{Z}_{p}$
of $p$-adic integers. It is the completion of the local ring

$R=\mathbb{Z}_{(p)}=\\{{\tfrac{n}{d}}\mid n,d\in\mathbb{Z},\,d\not\in
p\mathbb{Z}\\},$

with respect to $\mathfrak{m}=\mathfrak{I}=(p)$. In order to establish for the
first time formality results with coefficients in $\mathbb{Z}_{(p)}$, we apply
Theorem 4.1 as follows.

######  Theorem 4.2.

Let $p$ be a prime number. Let $X$ be a complement of a hyperplane arrangement
over $\mathbb{C}$. Let $\mathbb{Q}_{p}\hookrightarrow K$ be a finite
extension. We denote by $q$ the order of the residue field of the ring of
integers of $K$. Suppose that $X$ is defined over $K$, i.e. there exists an
embedding $K\hookrightarrow\mathbb{C}$ and there exists $\mathcal{X}$ a
complement of a hyperplane arrangement over $K$ such that

$\mathcal{X}\times_{K}\mathbb{C}\cong X\ .$

Let $\ell$ be a prime different from $p$. The algebra of singular cochains
$C(X_{\mathrm{an}},\mathbb{Z}_{(\ell)})$ is gauge $(s-1)$-formal, where
$X_{\mathrm{an}}$ is the complex analytic space underlying
$X_{\mathbb{C}}=X\times_{K}\mathbb{C}$ and where $s$ is the order of $q$ in
$\mathbb{F}_{\ell}^{\times}$ .

###### Proof.

Drummond-Cole and Horel proved in [DCH21, Theorem 4.7] that the algebra of
singular cochains $C(X_{\mathrm{an}},\mathbb{Z}_{\ell})$ is gauge
$(s-1)$-formal, see Point (4) of Example 4.13 for more details. The results
then follows directly from Theorem 4.1. ∎

###  4.2. Formality in families

Kaledin was motivated by establishing formality criteria, for which the
earlier methods do not apply. The work of Kaledin was extended by Lunts in
[Lun07] who established Theorem 4.3 and Theorem 4.4 in the context of any
$A_{\infty}$-algebras over a ring of characteristic zero. Using the present
Kaledin classes, we obtain a straightforward generalization of these results
for any algebra over a colored operad or properad. We refer the reader to
[Emp24] for applications.

######  Theorem 4.3.

Under Assumptions 3, suppose that $R$ is an integral domain, that $H(A)$ is an
$R$-module of finite presentation and that the following $R$-module is torsion
free:

$H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)\ .$

Let $\eta\in\mathrm{Spec}R$ be a generic point. The
$\Omega\mathcal{C}$-algebra $(A,\varphi)$ is gauge $n$-formal if and only if

$A_{\eta}=(A\otimes\kappa(\eta),\varphi\otimes 1)$

is gauge $n$-formal. In particular, $(A,\varphi)$ is gauge $n$-formal if and
only if $A_{x}$ is gauge $n$-formal, for all points $x\in\mathrm{Spec}R$. If
$R$ is a $\mathbb{Q}$-algebra, the same result holds for gauge formality.

###### Proof.

The proof is the same as the one of Theorem 4.1, replacing $S$ by the residue
field $\kappa(\eta)$, the faithful flatness by the fact
$H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)$ is torsion free. ∎

######  Theorem 4.4.

Under Assumptions 3, suppose that $R$ is a noetherian ring, that $H(A)$ is an
$R$-module of finite presentation and that the following $R$-module is
projective:

$H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)\ .$

  1. (1)

The following subset is closed under specialization:

$F\left(A\right)\coloneqq\\{x\in\mathrm{Spec}R\mid
A_{x}=(A,\varphi)\otimes_{R}\kappa(x)\mbox{ is gauge $n$-formal }\\}.$

  2. (2)

Let $I\subset R$ be an ideal such that $\cap_{k}I^{k}=0$ . The
$\Omega\mathcal{C}$-algebra $(A,\varphi)$ is gauge $n$-formal if and only if

$A/I^{k}=A\otimes_{R}R/I^{k}$

is $n$-gauge formal, for all $k\geqslant 1$ .

  3. (3)

Assume that $R$ has trivial radical. The $\Omega\mathcal{C}$-algebra
$(A,\varphi)$ is gauge $n$-formal if and only if $A_{x}$ is gauge $n$-formal,
for all closed point $x\in\mathrm{Spec}R$.

If $R$ is a $\mathbb{Q}$-algebra, the same result holds for gauge formality.

###### Proof.

Let us prove Point (1). If $F\left(A\right)$ is empty the result is immediate.
Otherwise, let us fix $\eta\in F\left(A\right)$. We denote by
$\bar{\eta}=\mathrm{Spec}\bar{R}$ its closure. The ring $\bar{R}$ is an
integral domain and the following $\bar{R}$-module is torsion free

$H_{-1}\left(\tilde{\mathfrak{g}}_{H(A_{\bar{R}})}^{\varphi_{*}\otimes
1}\right)\cong
H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)\otimes\bar{R}\ .$

By Theorem 4.3, the $\Omega\mathcal{C}$-algebra $(A_{\bar{R}},\varphi\otimes
1)$ is gauge $n$-formal, so does $A_{x}$ for all $x\in\mathrm{Spec}\bar{R}$.

The proof of Point (2) is similar to the one of Theorem 4.1, replacing $S$ by
$R/I^{k}$ and the faithful flatness by the facts that $\cap_{k}I^{k}=0$ and
$H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)$ is projective. For
Point (3), let $\mathfrak{m}\subset R$ be a maximal ideal. The proof is again
the same as the one of Theorem 4.1, replacing $S$ by $R/\mathfrak{m}$ and the
faithful flatness by the facts that the radical of $\mathfrak{m}$ is trivial
and the projectivity of
$H_{-1}\left(\tilde{\mathfrak{g}}^{\varphi_{*}}\right)$. ∎

###  4.3. Intrinsic formality

######  Definition 4.5 (Intrinsic formality).

Let $H$ be a graded $R$-module and let $(H,\varphi_{*})$ be a
$\Omega\mathcal{C}$-algebra structure concentrated in weight $1$. It is said
_intrinsically formal_ if for any $\Omega\mathcal{C}$-algebra structure
$(H,\psi)$ such that $\psi^{(1)}=\varphi_{*}$, there exists an
$\infty$-isotopy

${(H,\psi)}$${(H,\varphi_{*})\ .}$$\scriptstyle{\sim}$

######  Theorem 4.6.

Under Assumptions 3, let $H$ be a graded $R$-module and let $(H,\varphi_{*})$
be a $\Omega\mathcal{C}$-algebra structure concentrated in weight $1$. Let
$(H,\psi)$ be any $\Omega\mathcal{C}$-algebra such that
$\psi^{(1)}=\varphi_{*}$.

  1. (1)

If
$H_{-1}\left(\mathfrak{g}_{H}^{\varphi_{*}}/\mathcal{F}^{n}\mathfrak{g}_{H}\right)=0$,
then $(H,\psi)$ is gauge $n$-formal.

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra and if
$H_{-1}(\mathfrak{g}^{\varphi_{*}}_{H})=0$, then $(H,\varphi_{*})$ is
intrinsically formal.

###### Proof.

Point (2) is an immediate corollary of Point (2) of Theorem 1.31. The proof of
Point (1) makes use of the Kaledin classes and Proposition 2.32 in a way
similar to the proofs of Theorem 1.26. ∎

######  Example 4.7 (Tamarkin’s proof of Kontsevich formality theorem).

Theorem 4.6 generalizes Theorem 4.1.3 of [Hin03], established for a
characteristic zero field. In his paper, Hinich presents Tamarkin’s proof of
Kontsevich formality theorem that can be stated as follows. For every
polynomial algebra $A$ over a field of characteristic zero, the shifted
Hochschild cochain complex $C(A;A)[1]$ is formal as a dg Lie algebra.

Recall that the Hochschild cohomology $H\coloneqq HH^{\bullet}(A)$ admits a
natural Gerstenhaber algebra structure denoted $\varphi_{*}$, after [Ger63].
Let us denote by $Gerst$ the Koszul operad encoding the Gerstenhaber algebras.
Tamarkin’s proof relies on the following two facts,

  * $\centerdot$

the existence of a $Gerst_{\infty}$-algebra structure on the Hochschild
cochain complex of an associative algebra, which induces the $Gerst$-algebra
structure on $HH^{\bullet}(A)$ , see [Tam98];

  * $\centerdot$

the intrinsic formality as a $Gerst$-algebra of the Hochschild cohomology
$HH^{\bullet}(A)$ .

Using exactly the same strategy as in [Hin03, Section 5.4], the last point we
can proved by checking that $H_{-1}(\mathfrak{g}^{\varphi_{*}})=0$ where

$\mathfrak{g}=\mathrm{Hom}_{\mathbb{S}}\left(\overline{Gerst}^{\text{\raisebox{0.0pt}{\textexclamdown}}},\mathrm{End}_{H}\right)\
.$

######  Example 4.8 (The intrinsic formality of $E_{n}$-operads).

In [FW20], Fresse and Willwacher establish an intrinsic formality result for
$E_{n}$-operads. More precisely, let $Pois_{n}^{c}$ be the dual cooperad of
the $n$-Poisson operad $Pois_{n}$. When $n\geqslant 3$, the authors establish
that a Hopf cooperad $\mathcal{K}$ is related to $Pois_{n}^{c}$ by a zig-zag
of quasi-isomorphisms of Hopf cooperads as soon as there is an isomorphism

$H(\mathcal{K})\cong Pois_{n}^{c}\ .$

When $4$ divides $n$, an extra assumption that $\mathcal{K}$ is equipped with
an involutive isomorphism mimicking the action of a hyperplane reflection is
also required. Using Bousfield obstruction theory, they prove that the result
boils down to the vanishing of the cohomology group of degree $1$ of a certain
deformation complex. This deformation complex is very likely to be isomorphic
to the twisted convolution dg Lie algebra $\mathfrak{g}^{\varphi_{*}}$ where,

$\mathfrak{g}\coloneqq\mathrm{Hom}_{\mathbb{S}}(\overline{\mathcal{O}}^{\text{\raisebox{0.0pt}{\textexclamdown}}},\mathrm{End}_{Pois_{n}})\
,$

where $\mathcal{O}$ is the colored properad encoding Hopf cooperad. One would
conclude with a colored properadic version of Theorem 4.6.

###  4.4. Chain lifts of homology automorphisms

Motivated by Weil’s conjectures insights, there has been a long tradition of
using purity in order to prove formality. This goes back to the introduction
of [DGMS75] and the following observation. Let $\alpha$ be a unit of infinite
order in $R$ and let $\sigma_{\alpha}$ be the grading automorphism defined as
the multiplication by $\alpha^{k}$ in homological degree $k$. If the grading
automorphism admits a chain level lift, this prevents the existence of non-
trivial higher structures and thus implies formality, see [Sul77], [GSNPR05],
[Pet14], [Dup16], [CH20], [DCH21] and [CH22]. In this section, we use Kaledin
classes to generalize this result to properads and answer the following
questions:

  * $\centerdot$

Is grading automorphism the only homology automorphism satisfying this
property?

  * $\centerdot$

Under which condition on an homology automorphism, the existence of a lift
implies formality?

######  Definition 4.9.

Under Assumptions 3, let $u\in\mathrm{Aut}\left(H(A),\varphi_{*}\right)$ be a
strict automorphism of the homology. We see it as an element of
$\mathfrak{a}_{H(A)}$ through the map

$u:\mathcal{I}\to\mathrm{End}_{H(A)}\ ,\quad 1\mapsto u\ .$

  1. (1)

We say that it admits a _chain level lift_ if there exists an $\infty$-quasi-
isomorphism $v$ of $(A,\varphi)$ such that $H\left(v^{(0)}\right)=u$.

  2. (2)

We denote by $\mathrm{Ad}_{u}$ the dg Lie algebra automorphism of
$\mathfrak{g}^{\varphi_{*}}$ defined by

$\mathrm{Ad}_{u}(x)=(u\circ x)\circledcirc
u^{-1}\quad\left(\mbox{resp.}\;\;\mathrm{Ad}_{u}(x)=\left(u\rhd x\right)\lhd
u^{-1}\;\right)$

in the operadic case (resp. properadic case).

######  Theorem 4.10.

Under Assumptions 3, the following propositions hold true. Let
$u\in\mathrm{Aut}\left(H(A),\varphi_{*}\right)$ be an automorphism that admits
a chain level lift.

  1. (1)

If $\mathrm{Ad}_{u}-\mathrm{id}$ is invertible modulo
$\mathcal{F}^{n+2}\mathfrak{g}$, the $\Omega\mathcal{C}$-algebra $(A,\varphi)$
is gauge $n$-formal.

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra and $\mathrm{Ad}_{u}-\mathrm{id}$ is
invertible, then

     * $\centerdot$

the $\Omega\mathcal{C}$-algebra structure $(A,\varphi)$ is gauge formal;

     * $\centerdot$

every automorphism of $\left(H(A),\varphi_{*}\right)$ admits a chain level
lift.

The proof of Theorem 4.10 relies on the following lemma.

######  Lemma 4.11.

Under Assumptions 3, let $(H(A),\phi)$ be an $\Omega\mathcal{C}$-algebra
structure such that

$\phi\equiv\varphi_{*}\pmod{\mathcal{F}^{k+1}\mathfrak{g}}\ ,$

where $k\leqslant n$ is an integer. Let $s$ be an $\infty$-automorphism of
$(H(A),\phi)$ such that

$s^{(0)}=u\quad\mbox{and}\quad s^{(i)}=0\quad\mbox{for all}\ 1\leqslant
i\leqslant k-1\ .$

Let $u\in\mathrm{Aut}\left(H(A),\varphi_{*}\right)$ that admits a chain level
lift. If $\mathrm{Ad}_{u}-\mathrm{id}$ is invertible modulo
$\mathcal{F}^{k+2}\mathfrak{g}$,

  1. (1)

there exists an $\infty$-isotopy $\phi\rightsquigarrow\psi$ where
$\psi\in\mathrm{MC}(\mathfrak{g})$ is $\Omega\mathcal{C}$-algebra structure
such that

$\psi\equiv\varphi_{*}\pmod{\mathcal{F}^{k+2}\mathfrak{g}}\ ,$

  2. (2)

there exists an $\infty$-automorphism $\tilde{s}$ of $(H,\psi)$ such that

$\tilde{s}^{(0)}=u\quad\mbox{and}\quad\tilde{s}^{(i)}=0\quad\mbox{for all}\
1\leqslant i\leqslant k\ .$

###### Proof.

Let us prove the result in the operadic case. By Lemma 2.29, we have

$\left(\mathrm{Ad}_{u}-\mathrm{id}\right)\left(\phi^{(k+1)}\right)=d^{\varphi_{*}}\left(s^{(k)}\circledcirc
u^{-1}\right)\ ,$

since $s$ is an $\infty$-automorphism of $\phi$. Since
$\mathrm{Ad}_{u}-\mathrm{id}$ is invertible modulo
$\mathcal{F}^{k+2}\mathfrak{g}$, we have

$\phi^{(k+1)}\equiv
d^{\varphi_{*}}\left(\left(\mathrm{Ad}_{u}-\mathrm{id}\right)^{-1}\left(s^{(k)}\circledcirc
u^{-1}\right)\right)\ ,\pmod{\mathcal{F}^{k+2}\mathfrak{g}}$

and $K^{k}_{\phi}=\left[k\phi^{(k+1)}\hbar^{k-1}\right]=0$ . By Proposition
2.32, there exist $\psi\in\mathrm{MC}(\mathfrak{g}_{H})$ of the desired form
and an $\infty$-isotopy

$f:(H,\phi)\rightsquigarrow(H,\psi)\ .$

More precisely, we can take $f$ of the form $f=1+f^{(k)}$ such that

$\left(\mathrm{Ad}_{u}-\mathrm{id}\right)\left(f^{(k)}\right)=s^{(k)}\circledcirc
u^{-1}\ ,$

by the proof of Proposition 2.32. Finally, let us set $\tilde{s}\coloneqq
f\circledcirc s\circledcirc f^{-1}\leavevmode\nobreak\ .$ The inverse $f^{-1}$
can be explicitly computed using [LV12, Theorem 10.4.2]. It is an
$\infty$-isotopy of the form

$f^{-1}=1-f^{(k)}+\cdots$

This implies that $\tilde{s}^{(0)}=u$ and $\tilde{s}^{(i)}=0$ for $1\leqslant
i\leqslant k-1$. Furthermore, we have

$\tilde{s}^{(k)}=s^{(k)}+f^{(k)}\circledcirc u-u\star
f^{(k)}=\left(s^{(k)}\circledcirc
u^{-1}-\left(\mathrm{Ad}_{u}-\mathrm{id}\right)\left(f^{(k)}\right)\right)\circledcirc
u=0\ .$

The proof is similar in the properadic case, using Lemma 3.22 and the fact
that

$\left(\mathrm{Ad}_{u}-\mathrm{id}\right)\left(\phi^{(k+1)}\right)=d^{\varphi_{*}}\left(s^{(k)}\lhd
u^{-1}\right)\ .\qed$

###### Proof of Theorem 4.10.

The transferred structure comes equipped with $\infty$-quasi-isomorphisms

$(A,d,\varphi)$$(H(A),0,\varphi_{t})\ .$$p_{\infty}$$i_{\infty}$

The $\infty$-quasi-isomorphism $p_{\infty}$ extends the projection $p:A\to
H(A)$ and $i_{\infty}$ extends the inclusion $H(A)\to A$ . Consider the
$\infty$-automorphism

$s\coloneqq p_{\infty}\circledcirc v\circledcirc i_{\infty}\ .$

By definition of a contraction, we have $s^{(0)}=p\circ v^{(0)}\circ i=u\ .$
Let us prove the point (1). Starting from $(H(A),\varphi_{t})$ and the
$\infty$-automorphism $s$ , one can apply Lemma 4.11 recursively for
$k\leqslant n$. We obtain an $\Omega\mathcal{C}$-algebra structure
$\psi\in\mathrm{MC}(\mathfrak{g})$ such that

$\psi^{(1)}=\varphi_{*}\quad\mbox{and}\quad\psi^{(i)}=0\quad\mbox{for}\quad
2\leqslant i\leqslant n+1\ ,$

and an $\infty$-isotopy $f:(H,\varphi_{t})\rightsquigarrow(H,\psi)$. Thus, the
$\Omega\mathcal{C}$-algebra $(A,\varphi)$ is gauge $n$-formal.

Let us prove the point (2). As before, starting from $(H(A),\varphi_{t})$ and
the $\infty$-automorphism $s=p_{\infty}\circledcirc v\circledcirc i_{\infty}$
, one can apply Lemma 4.11 recursively. For all $k$, we obtain an
$\Omega\mathcal{C}$-algebra structure $\psi_{n}\in\mathrm{MC}(\mathfrak{g})$
such that

$\psi_{n}^{(1)}=\varphi_{*}\quad\mbox{and}\quad\psi^{(k)}_{n}=0\quad\mbox{for}\quad
2\leqslant k\leqslant n+1\ ,$

and an $\infty$-isotopy $f_{n}:(H,\varphi_{t})\rightsquigarrow(H,\psi_{n})$.
In the operadic case, all the truncations $K^{n}_{\varphi_{t}}$ are zero by
Proposition 2.32. This implies that $K_{\varphi_{t}}=0$ by Proposition 2.39
and $(A,\varphi)$ is formal by Theorem 2.44. In the properadic case, all the
truncations $K^{n}_{\varphi_{t}}$ are zero by Theorem 3.18. This implies that
$K_{\varphi_{t}}=0$ by Proposition 1.24 and $(A,\varphi)$ is gauge formal by
Theorem 3.18. Let $f:\varphi_{t}\rightsquigarrow\varphi_{*}$ be an
$\infty$-isotopy. For all
$\tilde{\omega}\in\mathrm{Aut}\left(H(A),\varphi_{*}\right)$, we have
$H\left(\omega^{(0)}\right)=\tilde{\omega}$ with

$\omega\coloneqq i_{\infty}\circledcirc f^{-1}\circledcirc u\circledcirc
f\circledcirc p_{\infty}\ .\qed$

######  Corollary 4.12.

Under Assumptions 3, suppose that the weight-grading of $\mathcal{C}$ is given
by the homological degree. Let $\alpha$ be a unit in $R$ and let $\vartheta$
be a non-zero rational number. Let $\sigma_{(\alpha,\vartheta)}$ be the
automorphism of $(H(A),\varphi_{*})$ defined by

$\sigma_{(\alpha,\vartheta)}(x)=\alpha^{\vartheta k}x\quad\mbox{for all }x\in
H_{k}(A)\ .$

Suppose that $(A,\varphi)$ is $\vartheta$-pure, i.e. that
$\sigma_{(\alpha,\vartheta)}$ admits a chain level lift.

  1. (1)

If $\alpha^{\vartheta k}-1$ is a unit in $R$ for all $k\leqslant n+1$, then
$(A,\varphi)$ is gauge $n$-formal;

  2. (2)

If $R$ is a $\mathbb{Q}$-algebra and if $\alpha^{\vartheta k}-1$ is a unit in
$R$ for all $k$, then $(A,\varphi)$ is gauge formal.

###### Proof.

For all $\psi$ of degree $k$ in $\mathfrak{g}^{\varphi_{*}}$, we have
$\mathrm{Ad}_{\sigma_{(\alpha,\vartheta)}}(\psi)=\alpha^{\vartheta k}\psi\ .$

  * $\centerdot$

If $\alpha^{\vartheta k}-1$ is a unit in $R$ for all $k\leqslant n$, then
$\mathrm{Ad}_{\sigma_{(\alpha,\vartheta)}}-\mathrm{id}$ is invertible modulo
$\mathcal{F}^{n+2}\mathfrak{g}$.

  * $\centerdot$

If $\alpha^{\vartheta k}-1$ is a unit in $R$ for all $k$, then
$\mathrm{Ad}_{\sigma_{(\alpha,\vartheta)}}-\mathrm{id}$ is invertible.

The result now follows from Theorem 4.10. ∎

######  Examples 4.13.

Let us present examples where the formality is established using $1$-purity,
i.e. the existence of a chain level lift of the grading automorphism
$\sigma_{(\alpha,1)}$.

  1. (1)

Little disks operad. The above type of result is used by Petersen in [Pet14]
to give a proof of the formality of the little $2$-disks operad. He use the
action of the Grothendieck-Teichmüller group at the chain level to exhibit a
lift of the grading automorphism . Similar arguments where used in [BdBH21]
and [CH22] to prove the formality of higher dimensional little disks operads
and with torsion coefficients respectively.

  2. (2)

Formality of cochains on BG. In [BG23], Benson and Greenlees prove that for a
compact Lie group $G$ with maximal torus $T$, if $|N_{G}(T)/T|$ is invertible
in the field $k$ then the algebra of cochains $C^{*}(BG;k)$ is formal as an
$A_{\infty}$-algebra. The spirit of their proof with the approximations of
[BG23, Section 3] boils down to finding a lift of a grading automorphism and
can be rephrased in terms of Corollary 4.12.

  3. (3)

Modular Koszul duality. In [RSW14], Riche, Soergel, and Williamson prove an
analogue of Koszul duality for the category $\mathcal{O}$ of a reductive group
$G$ in certain positive characteristic. Their proof is base on a formality
result of the dg associative algebra of extensions of parity sheaves on the
flag variety, which is established by constructing a lift of the grading
automorphism.

  4. (4)

Complement of hyperplane arrangements. Let $p$ and $\ell$ be two different
prime numbers. Let $X$ be a complement of a hyperplane arrangement over
$\mathbb{C}$. Let $\mathbb{Q}_{p}\hookrightarrow K$ be a finite extension. We
denote by $q$ order of the residue field of the ring of integers of $K$.
Suppose that $X$ is defined over $K$, i.e. that there exists an embedding
$K\hookrightarrow\mathbb{C}$ and that there exists $\mathcal{X}$ a complement
of a hyperplane arrangement over $K$ such that
$\mathcal{X}\times_{K}\mathbb{C}\cong X\ .$ Drummond-Cole and Horel prove in
[DCH21] that the algebra

$C^{*}_{\mathrm{sing}}(X_{\mathrm{an}},\mathbb{Z}_{\ell})$

is $(s-1)$-formal, where $X_{\mathrm{an}}$ denotes the complex analytic space
underlying $X_{\mathbb{C}}=X\times_{K}\mathbb{C}$ and where $s$ is the order
of $q$ in $\mathbb{F}_{\ell}^{\times}$ . The idea is the following one. There
exists a zig-zag of quasi-isomorphisms of dg associative algebras

$C^{*}_{\mathrm{sing}}(X_{\mathrm{an}},\mathbb{Z}_{\ell})\xleftarrow{\sim}C^{*}_{\mathrm{\acute{e}t}}(\mathcal{X}_{\mathbb{C}},\mathbb{Z}_{\ell})\xrightarrow{\sim}C^{*}_{\mathrm{\acute{e}t}}(\mathcal{X}_{\overline{K}},\mathbb{Z}_{\ell})\
,$

see the proof of Theorem 4.16 for more details. It was proved in [Kim94,
Theorem 1’] that the action of a Frobenius lift on
$H_{\mathrm{\acute{e}t}}^{*}(\mathcal{X}_{\overline{K}},\mathbb{Z}_{\ell})$ is
given by $\sigma_{(q,1)}$. Thus the grading automorphism admits a chain level
lift and we get the desired formality result. This result is really specific
to complement of a hyperplane arrangements: in general, a Frobenius lift on
the étale cohomology of a $K$-scheme is not a grading automorphism.
Nonetheless, using the work of Deligne in the context of Weil conjectures, we
prove bellow in Theorem 4.16 that the action of a Frobenius lift on the étale
cohomology of any smooth and proper $K$-scheme satisfies Theorem 4.10.

######  Examples 4.14.

Let us present formality examples relying on $\vartheta$-purity with
$\vartheta\neq 1$.

  1. (5)

Gravity operad In [DH18], Dupont and Horel prove the formality of the gravity
operad of Getzler, which is an operad structure on the collection

$\\{H_{*}(\mathcal{M}_{0,n+1})\\}_{n\in\mathbb{N}}\ ,$

where $\mathcal{M}_{0,n+1}$ denotes the moduli spaces of stable algebraic
curves of genus zero with $n+1$ marked points. To do so, they exhibit a model
$\mathcal{G}rav^{W^{\prime}}$ which comes with an action of the Grothendieck-
Teichmüller group. By exploiting this action, one can prove that
$\mathcal{G}rav^{W^{\prime}}$ is $2$-pure, i.e. that $\sigma_{(\alpha,2)}$
admits a chain level lift, see [CH20, Section 7.4]. Thus it is formal, so does
the gravity operad.

  2. (6)

Good arrangement of codimension $c$ subspaces Let $p$ be a prime number. Let
$\mathbb{Q}_{p}\hookrightarrow K$ be a finite extension. We denote by $q$
order of the residue field of the ring of integers of $K$. Let $V$ be a
$d$-dimensional $K$-vector space and let $\\{W_{i}\\}_{i\in I}$ be a good
arrangement of codimension $c$ subspaces, i.e. a family of subspaces of $V$
such that for all $i\in I$, the subspace $W_{i}$ is of codimension $c$ and the
subspaces of $W_{i}$

$\\{W_{i}\cap W_{j}\\}_{i\neq j}$

form a good arrangement of codimension $c$ subspaces. By [BE97], the étale
cochains

$C^{*}_{\mathrm{\acute{e}t}}\left(V-\cup_{i}W_{i},\mathbb{F}_{\ell}\right)$

is $(c/(2c-1))$-pure and thus gauge $n$-formal by Corollary 4.12 with

$n\coloneqq\lfloor(s-1)(2c-1)/c\rfloor\ ,$

where $s$ is the order of $q$ in $\mathbb{F}_{\ell}^{\times}$ . In [CH22],
Horel and Cirici establish the $n$-formality of this algebra in the case
$c\geqslant 1$.

The following corollary of Theorem 4.10 offers a more manageable criterion
over a field.

######  Corollary 4.15.

Under Assumptions 3, suppose that $R$ is a field and that for all $k\geqslant
0$, the vector space $H_{k}(A)$ is finite dimensional. Suppose that the
weight-grading on $\mathcal{C}$ is the homological degree. Suppose that there
exists

$u\in\mathrm{Aut}\left(H(A),\varphi_{*}\right)$

that admits a chain level lift. For all $k$, we denote by $u_{k}$ the
restriction of $u$ to $H_{k}(A)$ and $\mathrm{Spec}(u_{k})$ its set of
eigenvalues. If we have

(9) $\mathrm{Spec}(u_{\alpha_{1}}\otimes\dots\otimes
u_{\alpha_{p}})\cap\mathrm{Spec}(u_{\beta_{1}}\otimes\dots\otimes
u_{\beta_{q}})=\varnothing$

for all $k$ (resp. $k\leqslant n+1$), for all $p$-tuples
$(\alpha_{1},\dots,\alpha_{p})$ and all $q$-tuples
$(\beta_{1},\dots,\beta_{q})$ such that

$k=\alpha_{1}+\cdots+\alpha_{p}-\beta_{1}-\cdots-\beta_{q}\ ,$

then $(A,\varphi)$ is gauge formal (resp gauge $n$-formal).

###### Proof.

The eigenvalues of the restrictions of $\mathrm{Ad}_{u}$ to degree $k$
elements are given by quotients

$\frac{\lambda_{\alpha_{1}}\dots\lambda_{\alpha_{p}}}{\lambda_{\beta_{1}}\dots\lambda_{\beta_{q}}}\
,\quad\lambda_{i}\in\mathrm{Spec}(u_{i})$

where $k=\alpha_{1}+\cdots+\alpha_{p}-\beta_{1}-\cdots-\beta_{q}$. By
hypothesis (9), none of these quotients is equal to 1. Thus
$\mathrm{Ad}_{u}-\mathrm{id}$ is invertible and the result follows from
Theorem 4.10. The case of gauge $n$-formality if similar. ∎

In [Del71], Deligne proved the formality with coefficients in
$\mathbb{Q}_{\ell}$ of smooth proper varieties defined over a finite field,
using the weights of the Frobenius action in the $\ell$-adic cohomology and
his solution of the Riemann hypothesis. This result arises from Kaledin
classes theory using Corollary 4.15 and admits the following generalization:

######  Theorem 4.16.

Let $X$ be a smooth and proper scheme over $\mathbb{C}$ and let
$X_{\mathrm{an}}$ denotes the associated complex analytic space. The dg
associative algebra of singular cochains
$C^{*}_{\mathrm{sing}}(X_{\mathrm{an}},\mathbb{Q})$ is formal.

###### Proof.

Let $\ell$ be a fixed prime number. Let $R\subset\mathbb{C}$ be the ring of
finite type generated by the coefficients of equations defining $X$. By
inverting elements $R$, there exists a smooth and proper model $X_{R}$ over
$R$ such that $X=X_{R}\otimes_{R}\mathbb{C}$ and $R$ is a local ring. Let
$\mathfrak{m}$ be the maximal ideal of $R$. The field $R/\mathfrak{m}$ is of
the form $\mathbb{F}_{q}$ where $q=p^{r}$. Without loss of generality, we can
suppose that $p$ is different from $\ell$, even if it means also inverting
$\ell$ in $R$. There exists a finite extension $K$ of $\mathbb{Q}_{p}$ and an
embedding $R\hookrightarrow\mathcal{O}_{K}$, such that

$\mathfrak{m}_{\mathcal{O}_{K}}\cap R=\mathfrak{m}\ .$

Let us consider the smooth and proper model $\mathcal{X}\coloneqq
X_{R}\otimes_{R}\mathcal{O}_{K}$. By construction, $X_{R}$ can be seen as a
$K$-scheme of _good reduction_ i.e. for which there exists a smooth and proper
model $\mathcal{X}$ over ${\mathcal{O}_{K}}$. There exists a zig-zag of quasi-
isomorphisms of dg associative algebras

$C_{\mathrm{sing}}^{*}(X_{\mathrm{an}},\mathbb{Q}_{\ell})\overset{\sim}{\longleftarrow}C_{\mathrm{\acute{e}t}}^{*}(X_{\mathbb{C}},\mathbb{Q}_{\ell})\overset{\sim}{\longrightarrow}C_{\mathrm{\acute{e}t}}^{*}(X_{\overline{K}},\mathbb{Q}_{\ell})\
,$

where the left quasi-isomorphism is given by Artin’s Theorem [SGA, XI, Theorem
4.4, (iii)] and the right one follows from invariance of étale cochains by
extension of algebraically closed fields [SGA, VIII, Corollary 1.6]. Since $K$
is a $p$-adic field with residue field $\mathbb{F}_{q}$, there exists a
surjective map

$\mathrm{Gal}(\overline{K}/K)\twoheadrightarrow\mathrm{Gal}(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})\
.$

The Galois group $\mathrm{Gal}(\overline{\mathbb{F}_{q}}/\mathbb{F}_{q})$ is
isomorphic to $\hat{\mathbb{Z}}$ generated by the Frobenius. Let us consider
the smooth and proper $\mathbb{F}_{q}$-scheme given by
$Y\coloneqq\mathcal{X}\otimes_{\mathcal{O}_{K}}\mathbb{F}_{q}$. By smooth and
proper base change [SGA, XVI, Corollary 2.2], there exists a quasi-isomorphism

$C_{\mathrm{\acute{e}t}}^{*}(Y_{\overline{\mathbb{F}}_{q}},\mathbb{Q}_{\ell})\overset{\sim}{\longrightarrow}C_{\mathrm{\acute{e}t}}^{*}(X_{\overline{K}},\mathbb{Q}_{\ell})$

compatible with the Galois group actions on both sides. Let $u_{k}$ be the
endomorphism of

$H_{\mathrm{\acute{e}t}}^{k}(Y_{\overline{\mathbb{F}}_{q}}\mathbb{Q}_{\ell})$

induced by the Frobenius. By Deligne’s theorem [Del74], every eigenvalue $x$
of $u_{k}$ is a Weil number of weight $k$ i.e. then for any embedding
$\iota:\overline{\mathbb{Q}}_{\ell}\hookrightarrow\mathbb{C}$ , we have
$|\iota(x)|=q^{k/2}\ .$ In particular, an eigenvalue $x$ of
$u_{\alpha_{1}}\otimes\dots\otimes u_{\alpha_{p}}$ satisfies
$|\iota(x)|=q^{k/2}$ where $k=\alpha_{1}+\cdots+\alpha_{p}$. Thus, the
Frobenius satisfies the incompatibility condition (9) and it follows from
Corollary 4.15 that
$C_{\mathrm{\acute{e}t}}^{*}(Y_{\overline{\mathbb{F}}_{q}},\mathbb{Q}_{\ell})$
is formal and so is
$C_{\mathrm{sing}}^{*}(X_{\mathrm{an}},\mathbb{Q}_{\ell})$. The result with
coefficients in $\mathbb{Q}$ follows from Theorem 4.1, using the fact that
$X_{\mathrm{an}}$ has finite dimensional cohomology, see e.g. [Voi02,
Corollary 5.25]. ∎

The previous results extends to prove the formality of algebraic structures in
smooth and proper schemes. This leads to the following theorem.

######  Theorem 4.17.

Let $\mathbb{V}$ be a groupoid and let $\mathcal{P}$ be a $\mathbb{V}$-colored
operad in sets. Let $p$ be a prime number. Let $K$ be a finite extension of
$\mathbb{Q}_{p}$ and let $K\hookrightarrow\mathbb{C}$ be an embedding. Let $X$
be a $\mathcal{P}$-algebra in the category of smooth and proper schemes over
$K$ of good reduction, i.e. for which there exists a smooth and proper model
$\mathcal{X}$ over the ring of integers $\mathcal{O}_{K}$. The dg
$\mathcal{P}$-algebra of singular chains
$C_{*}^{\mathrm{sing}}(X_{\mathrm{an}},\mathbb{Q})$ is formal.

###### Proof.

The proof is similar to the one of Theorem 4.16, using the same strategy than
[CH22, Section 3]. The functor
$C_{\mathrm{\acute{e}t}}^{*}(-,\mathbb{Q}_{\ell})$ descends to a symmetric
monoidal $\infty$-functor
$\mathbf{C_{\mathrm{\acute{e}t}}^{*}}(-,\mathbb{Q}_{\ell})$ from the nerve
$\mathbf{N}(\mathrm{Sch}_{K})$ to the $\infty$-category of chain complexes.
Since finite type schemes have finitely generated étale cohomology, we can
compose with the duality symmetric monoidal $\infty$-functor. This leads to a
covariant symmetric monoidal $\infty$-functor of étale chains
$\mathbf{C^{\mathrm{\acute{e}t}}_{*}}(-,\mathbb{Q}_{\ell})$, which is
equivalent to $\mathbf{C^{\mathrm{sing}}_{*}}(-,\mathbb{Q}_{\ell})$ as lax
monoidal functors $\infty$-functor. The same arguments than before now apply
to prove that for any $\mathcal{P}$-algebra $X$ in the category of smooth and
proper schemes over $K$ of good reduction. Thus, the dg $\mathcal{P}$-algebra
in the $\infty$-category of chain complexes
$\mathbf{C^{\mathrm{sing}}_{*}}(X_{\mathrm{an}},\mathbb{Q}_{\ell})$ is formal.
By [CH20, Corollary 2.4], this implies that
$C^{\mathrm{sing}}_{*}(X_{\mathrm{an}},\mathbb{Q}_{\ell})$ is formal as a dg
$\mathcal{P}$-algebra, so does
$C^{\mathrm{sing}}_{*}(X_{\mathrm{an}},\mathbb{Q})$ by Theorem 4.1. ∎

######  Examples 4.18.

Moduli spaces of stable algebraic curves. Let us consider the cyclic operad
$\overline{\mathcal{M}}$ of moduli spaces of stable algebraic curves of genus
$0$ with marked points. This is a cyclic operad in smooth and proper
$\mathbb{Z}$-schemes. By Theorem 4.17, the cyclic operad

$C_{*}^{\mathrm{sing}}(\overline{\mathcal{M}};\mathbb{Q})$

is formal, which allows us to recover [GSNPR05, Corollary 7.2.1].

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11institutetext: Birla Institute of Technology and Science Pilani, India
22institutetext: Independent Researcher, India. Previously with Institute of
Mathematics and Image Computing, University of Lübeck, Germany
33institutetext: Gurudwara Singh Sabha Charitable Dispensary, Rajpura, India

# Urine Microscopic Image Dataset

Dipam Goswami 11 Hari Om Aggrawal 22 Rajiv Gupta 33 Vinti Agarwal 11

###### Abstract

Urinalysis is a standard diagnostic test to detect urinary system related
problems. The automation of urinalysis will reduce the overall diagnostic
time. Recent studies used urine microscopic datasets for designing deep
learning based algorithms to classify and detect urine cells. But these
datasets are not publicly available for further research. To alleviate the
need for urine datsets, we prepare our urine sediment microscopic image (UMID)
dataset comprising of around 3700 cell annotations and 3 categories of cells
namely RBC, pus and epithelial cells. We discuss the several challenges
involved in preparing the dataset and the annotations. We make the dataset
publicly available111https://github.com/dipamgoswami/UMID-Urine-Microscopic-
Image-Dataset.

###### Keywords:

Urinalysis RBCs Pus cells Epithelial cells.

##  1 Introduction

Analysis of urine sediment particles from microscopic images play a vital role
in the diagnosis of urinary and kidney diseases. The constituents of urine
samples are Red Blood Cells (RBCs), White Blood Cells (WBCs) referred to as
pus cells, epithelial cells, casts, bacteria, crystals and other artifacts. A
routine urine test involves chemical, physical and microscopic analysis of the
sample. We are concerned with the microscopic analysis of the urine sediment.
Routine microscopic analysis generally looks for the presence of RBCs, pus
cells and epithelial cells. A urine sample may also contain some indeterminate
objects as well as some out-of-focus objects and lens artifacts.

The classification of cells is vital for the detection of diseases and its
diagnosis. This work is aimed at providing medical support to medical centres
across rural areas where there is a shortage of skilled lab technicians to
analyze urine samples of patients. To the best of our knowledge, our UMID
dataset is the first urine images dataset to be made publicly available to all
researchers. Our dataset has been annotated under the supervision of medical
professionals and have been cross-verified. We expect that the availability of
the dataset will accelerate research in the urine sediment cell detection
domain.

##  2 Related Work

Urine Sediment Datasets Various datasets have been used to study cell
detection from urine sediments though not publicly available. Some of these
studies like [1] are focused on classifying only RBC and WBC cells while other
works [2] classified objects into erythrocyte, leukocyte, epithelial cell,
crystal, cast, mycete, epithelial nuclei and noise, [3] used sub-classes of
epithelial cells like low-transitional epithelium and squamous epithelial
cells and [4] also considered bacteria and sperms. Some papers [4, 5] have
highlighted the regular occurrence of WBC and RBC clusters in urine datasets
but are limited to detecting the clusters [4] or predicting segmentation mask
boundaries for RBC and WBC clumps [5].

Detecting individual cells from these cell clusters have not been studied yet
and our work focuses to detect the densely packed cells using weak instance-
level information. With improved precision in detection of cluster cells, the
urine test reports can give a better count of the different cells leading to
better diagnosis of diseases. In our dataset, we have observed RBCs, pus cells
and epithelial cells as the main components. We have labelled the confusing
and poorly focused components into a missedlabel category and we avoid these
instances while training the deep learning models.

##  3 UMID Dataset

A physician at our university hospital collected all urine microscopic images
in the course of his patient diagnosis. He followed the normal routine
procedure for data collection. His primary objective was to differentiate and
count urine sediments manually from the images for diagnosis instead of
acquiring good quality images for the dataset. Hence, the acquired images is
not a best set of images that one would like to have to train a machine
learning model in view of obtaining a remarkable performance. But, the
acquired dataset aligns quite well with our objective. In this study, our goal
is to build a neural network model that generalizes well and work
satisfactorily even with low quality images. We explain image quality in our
context later in this section.

In practice, doctors and lab technicians does not follow the recommended
protocols [6] very precisely due to the shortage of time specially in highly
populated countries where the daily influx of patients is quite high.
Moreover, microscopes does not get servicing regularly that leads to many
flaws in the images. Instead of assuming ideal situation, we should design a
robust learning framework that could sustain these issues up to some extent.
All the automatic urine analyzers comes with a special microscope that is
attached to the analyzers. Hence, they works well in practice. However, our
long term goal is to develop a system that works with existing lab setup with
only a minor addition of camera or a smartphone to one of the eyepiece.

Due to the unavailability of proper infrastructure, the patient information is
not stored anywhere. Hence, images are inherently anonymized at the source.
Images are subject to bias of the doctor towards his acquisition habits; it is
one of the limitation of our study.

###  3.1 Urine sediments and their clinical significance

A routine urine diagnostic report mention the numbers of erythrocytes/RBC,
leukocytes/pus, epithelial cells, and cast per high power field of a urine
sample. Other urine sediment constituents such as crystals, bacteria, yeast
cells, salts, and spermatozoa are not counted, but instead given as crosses
[7]. Casts appears very rarely in a urine sample [7]. During our data
collection duration, we did not get any urine sample with cast. Hence, in this
study our focus is mainly to detect and classify only three types of urine
sediments, RBC, pus, and epithelial cell. These three classes can be also sub
classified. For example, the epithelial cells can further be classified into
squamous and transitional epithelial cells. Though, these sub-classifications
have clinical relevance, but they are hardly reported in a routine urine
diagnostic report.

####  3.1.1 Morphology

RBC are round cells, contains no nucleus or granules, smaller than pus cells,
and has relatively sharp boundaries. Based on the pH value of the urine, it is
either biconcave, disc-shaped (pH = 6), thorn apple-shaped (pH $<$ 6), or a
light disc (pH $>$ 6). They also sometimes appears as ring-shaped. Pus cells
are generally larger than RBC cells. They are round in shape and has dark and
granular surface. Pus cells could also have segmented nucleus. Pus cells lie
individually and in clusters. Epithelial cells are irregular in shape,
generally larger than pus cells, and has a small size nucleus. Sometimes,
epithelial cells have two nucleus as well. These cells could also lie
individually and clusters.

####  3.1.2 Clinical significance

The normal range for RBCs are 0-1/HPF. Higher values of RBC indicates
hematuria in kideny tumors, urinary tract tumors, urinary tract infections, or
trauma. The normal range of pus cells are 1-4/HPF. Excess of it could
indicates inflammation in the urinary tract system and inflammatory renal
disease. The normal range for epithelial cells are 0-15/HPF. A higher value
indicates inflammation of urinary tract system.

###  3.2 Image acquisition -
recommended vs in practice

We used two bright-field microscopes for acquiring images. Images from one
microscope was of a lower quality than the other. Majorly, we observe a half
circular ring artifact in the images. We could not find the exact reason
behind the artifact, but we suspect that it is mainly because of misalignment
between lens and the light source. In our dataset, the ratio of images are
approximately 10:2 from old to new microscopes. We could acquire only a
limited amount of images from the hospital due to the limited patients
visiting the hospital due to the COVID-19 restrictions.

The urine sample on a glass slide has a certain depth and the cells are
present at different depths. Hence, we say that the urine slide has a multi-
layer structure. Due to that, all the cells are not clearly visible within the
depth of field of a lens focused at a particular focal plane. Hence, numerous
focus adjustments are required to identify every cell in the sample. Moreover,
microscope could examine only a certain area of the whole urine sample. Hence,
it is recommended to examine 20-30 high power fields (HPF) for one urine
sample and accumulate the findings to build the final diagnostic report.
Certainly, it is a very time consuming task.

We observed in practice that the doctors follow the recommendations only for a
few sensitive cases. Otherwise, doctors spend only a few seconds to analyse
one urine slide that covers approximately 10 HPF in a urine slide and adjust
focus of the lens up to a certain level that is enough to differentiate the
cell features and its size. Hence, the acquired images majorly have a certain
amount of blur and hence, object edges are not very sharp and cell granularity
deteriorates.

There a few more limitations that originates from urine multi-layer structure.
We observe that cells in an image can have a different degree of blur based on
their depth with respect to the distance from the lens. Few cells could be
highly out of focus that lies beyond the depth-of-field of the lens. This
irregular blurring effect makes it very hard sometimes to differentiate cells
even for an experienced doctor. To handle such cases, we introduced a missed
label class.

Both microscopes are the product of MAGNUS OPTO SYSTEMS INDIA PVT. LTD, model
number CH20iBIMF. Eyepieces are with 10x magnification and 18mm field number
and the achromatic objective lens are with 40x magnification. Each high power
field is examined at 400×magnification (corresponding to an 10× eyepiece and a
40× objective). We connected a COSLAB 5 megapixel microscope digital camera
(model number COSUSB5000) to one of the eyepiece of the microscope to acquire
a digital image of the urine sample and store it on the desktop. The images
are acquired at the lower resolution of $1280\times 720$. The video feed to
the desktop from the camera was quite slow at higher resolution and hence
avoided by the doctor for image acquisition. If camera is attached to the
microscope, doctors prefer to use video feed on the desktop to differentiate
cells rather than directly observing through the eyepiece. The video feed
provides more flexibility and easiness to scan the entire urine slides.

###  3.3 Novel data annotation strategies

Annotating a urine microscopic image is a challenging task. Optical blurring,
overlapping of cells, low contrast, small cell size are some sources of
difficulties. It is easy to overlook RBC cells due to their small size and
very smooth surface that leads to very low contrast in the surrounding of the
cell.

Three authors of this paper including a doctor jointly annotated the cells in
the images. We used Microsoft VoTT software for annotations. We primarily
followed three strategies to annotate cells from which cluster and missed
label strategies are novel approaches that we proposed in this work. These
novel approaches are based on the properties of urine microscopic images that
we describe in the next paragraph. Moreover, these approaches greatly reduces
the overall annotation time. Doctors are generally over occupied with patients
and have very busy schedule. Hence, reducing the annotation time greatly
helped to get feedback from the doctors.

####  3.3.1 Standard annotation strategy

For a object detection and classification task, the standard approach is to
train the neural network with respect to the ground truth of all the
foreground objects present in the image. Each object is assigned a class and a
bounding box that encodes the location and size of the object in a given
image. We follow the same annotation strategy in our work, but only for a few
cells that lie individually in the image.

####  3.3.2 Novel missed label class

In some circumstances, doctors could not make a reliable guess about the
appropriate class of a cell. Hence, in theory, we can not place them neither
in main classes nor in the background. If we treat them as a background, in
principle, we are forcing network to learn a wrong classification. Instead of
that, we place such cells in a separate missed label class and do not include
members of this class during training and leave it to the network to predict
the class for such cells.

####  3.3.3 Point annotations for clusters

In urine images, cells are also found in clusters. In clusters, cells are
densely packed and overlapping with each other, hence, annotation is a
difficult, puzzling, and a time-consuming task inside a cluster. The
annotations can result in inconsistent bounding boxes. Hence, we avoid
annotating each cell inside the clusters.

We propose to specify a single point on the cell (near to its center
approximately) instead of deciding boundaries and drawing boxes to localize
the cells in a cluster. The size of cells can be estimated from points and
these generated pseudo boxes can be used during training. Annotating a point
is a much simpler task and reduces time, efforts, and dilemma.

###  3.4 Data statistics and analysis

####  3.4.1 Training, testing, and validation set

In the UMID dataset, we have 366 urine images of resolution $1280\times 720$
that we divided into three sets for training ($\approx 76\%$), validation
($\approx 10\%$), and testing ($\approx 14\%$); see Table
LABEL:tab:cell_stats. The division is based on the total 3733 annotated cells
in the dataset instead of the images. The cell features are the primary source
of learning.

As discussed in the last section, cells inside the cluster are difficult to
annotate; hence we annotate cells with boxes (outside clusters) and points
(inside clusters) in the training dataset. But for testing and validation
dataset, we purposely annotate cells lies inside clusters with boxes to
evaluate the detection performance of the ML model. Generally, pus and
epithelial cells form clusters; hence, we have more points annotations
compared to the RBC cells. In total, $20\%$ cells are annotated with points
for which pseudo ground truth boxes are required for training.

The dataset has approximately $44\%$ RBC, $33\%$ pus, and $23\%$ epithelial
cells. Though, the classes are slightly imbalanced, the performance of the ML
model is not biased towards one class.

We have approximately $11\%$ cells for which the annotators were unsure about
the correct class. For these cells only bounding boxes are drawn and label
them to missed label class. These structures are not used during training the
models.

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