First, we have chosen a different order for the source categories of the trivalent functor $P \from \cat B^{\op} \times \cat A \times \cat B \to \cat C$ then what is usually the norm.
Second, in its full generality, the `target functor' $Q$ of $\zeta \from P \multimap Q$ could be trivalent as well.
That is, it could be of the form $Q \from \cat D^{\op} \times \cat A \times \cat D \to \cat C$, where $\cat D$ is a category which is possibly distinct from $\cat A$, $\cat B$ and $\cat C$.

Consider an extranatural transformation $\zeta \from P \multimap Q$ from a functor $P\from \cat B^{\op} \times \cat A \times \cat B \to \cat C$ to a functor $Q \from \cat A \to \cat C$.
We call the pair $(Q, \zeta)$ universal if for every other extranatural transformation $\xi \from P \multimap R$ from $P$ to a functor $R \from \cat A \to \cat C$ there exists a unique natural transformation $\nu \from Q \to R$ such that for all $A \in \cat A$ and $f \in \cat B(B,B')$ the following diagram commutes:

\begin{equation}
\begin{tikzcd}
R(A) & & & \\
& Q(A) \arrow[lu, "\exists ! \nu_A" description, dashed]
& & {P(B',A,B')} \arrow[ll, "{\zeta_{B',A}}"'] \arrow[lllu, "{\xi_{B',A}}"', bend right] \\
& {P(B,A,B)} \arrow[u, "{\zeta_{B,A}}"] \arrow[luu, "{\xi_{B,A}}", bend left] & & {P(B',A,B)} \arrow[ll, "{P(f, \id_A, \id_B)}"] \arrow[u, "{P(\id_{B'}, \id_A, f)}"']
\end{tikzcd}
\end{equation}

In this case, we call  $Q(A)$ the coend of $P(\blank, A, \blank) \from \cat B^{\op} \times \cat B \to \cat C$ for any $A \in \cat A$.

It follows from their definition that universal extranatural transformations are unique up to unique natural isomorphisms.

§ A MONADIC PERSPECTIVE ON TWISTED CENTRES

The anti-modules of a finite-dimensional Hopf algebra are a module category over the modules.
Subsequently, they are implemented by a comodule algebra over the Drinfeld double, see [25].
As explained in Section <ref>, we find ourselves in a similar situation.
Our replacement of the anti-modules, the anti-Drinfeld centre,
is a module category over the Drinfeld centre.

We replace finite-dimensional vector spaces by a rigid, possibly pivotal, category $\cat V$ and the underlying Hopf algebra with a Hopf monad  $H \from \cat V \to \cat V$.
In this section we study a Hopf monad $D(H) \from \cat V \to \cat V$ and over it a comodule monad $Q(H)\from \cat V \to \cat V$, which realise the centre and its twisted cousin as their respective modules.
and Virelizier gave a transparent description of $D(H)$ in [11] by extending results of Day and Street, see [17].
The key concept in its construction is the so-called centraliser of the identity functor of $\cat V^H$.
It is used to define a Hopf monad $\mathfrak D(\cat V^H)$ on $\cat V^H$ with $\ZCat{\cat V^H}$ as its category.
From this, one obtains—as an application of Beck's theory of distributive laws—the Drinfeld double $D(H) \from \cat V \to \cat V$.
We apply the same techniques to define the anti-double $Q(H)$ of $H$, whose modules are isomorphic to the `dual' of the anti-Drinfeld centre $\QCat{\cat V^H}$.
This approach is best summarised by the following diagram:

[column sep=1cm,between origins, row sep=0.875cm,between origins]

[rrddd, "U^(Z)" description, bend left]
[lldddd, leftrightarrow, "Σ^(𝔇)" description, bend left=15]

[llddd, "U^(Q)" description, bend left]
[llll, "action" description, no head, dotted]
[rrdddd, leftrightarrow, "Σ^(𝔔)" description, bend right=15]

[rruuu, "F^(Q)" description, bend left]
[lluuu, "F^(Z)" description, bend left]
[lllld, "F_𝔇" description, bend left=20]
[rrrrd, "F_𝔔" description, bend left=20]
[ddd, "U_H" description, bend left]

[rrrru, "U_𝔇" description, bend left=20]
[rrdddd, leftrightarrow, "Σ^(D(H))" description, bend left=15]

[llllu, "U_𝔔" description, bend left=20]
[lldddd, leftrightarrow, "Σ^(Q(H))" description, bend right=15]

[uuu, "F_H" description, bend left]
[rrdd, "F_Q(H)" description, bend left]
[lldd, "F_D(H)" description, bend left]

[rruu, "U_D(H)" description, bend left]

[lluu, "U_Q(H)" description, bend left]
[llll, "action" description, no head, dotted]

figureA cobweb of adjunctions, monads and various versions of the Drinfeld and anti-Drinfeld centre.

The translation of module functors between $\ZCat*{\cat V^H}$ and $\QCat{\cat V^H}$ into morphisms of comodule monads between $Q(H)$ and $D(H)$ yields our desired monadic version of Theorem <ref>, which we prove in Theorem <ref>.
We end our endeavour into the theory of comodule monads with Corollary <ref>.
In it, we explain how pivotal structures on $\cat V^H$ arise from module morphisms between the so-called central Hopf monad $\mathfrak D$ and the anti-central comodule monad $\mathfrak Q$.

§.§ Centralisable functors and the central bimonad

The construction of the double of a Hopf monad $H \from \cat V \to \cat V$ given in [11] relies heavily on an `accessible' left dual of the forgetful functor $U^{(Z)} \from \ZCat*{\cat V^H} \to \cat V^H$.
It is obtained as an application of the coend calculus covered in Section <ref>.

Suppose $\cat C$ to be a rigid category and $T \from \cat C \to \cat C$ to be an endofunctor.
We call $T$ centralisable if there exists a universal extranatural transformation

\begin{equation*}
\zeta_{Y,X} \from \lDual{T(Y)} \otimes X \otimes Y \to Z_T(Y),
\qquad \text{ for } X, Y \in \cat C.
\end{equation*}

A centralisable functor $T \from \cat C \to \cat C$ admits a universal coaction

\begin{equation}\label{eq: universal_coaction}
\chi_{X,Y} \defeq (\id_{T(Y)} \otimes \zeta_{Y,X})(\coev^L_{T(Y)} \otimes \id_{Y}),
\qquad \text{ for } X, Y \in \cat C,
\end{equation}

which is natural in both variables.
We call the pair $(Z_T, \chi)$ a centraliser of $T$.

Graphically, we represent the universal coaction as

\begin{equation}\label{eq: universal_coaction_string_diag}
\inputtikz{universal_coaction}
\end{equation}

It being natural equates to

\begin{equation}\label{eq: universal_coaction_natural}
\inputtikz{universal_coaction_natural}
\end{equation}

The extended factorisation property of universal coactions provides us with a potent tool for constructing bi- and comodule monads.
Its proof is given for example in <cit.>.

Let $(Z_T, \chi)$ be the centraliser of a functor $T \from \cat C \to \cat C$ and suppose that $L, R \from \cat D \to \cat C$ are two functors.
For any $n \in \mathbb N$ and any natural transformation

\begin{equation*}
\phi_{X, Y_1, \dots, Y_n} \from L(X) \otimes  Y_1 \otimes \dots \otimes Y_n \to T(X_1) \otimes \dots \otimes T(X_n) \otimes R(Y),
\end{equation*}

where $X \in \cat D$ and $Y_1, \dots, Y_n \in \cat C$, there exists a unique natural transformation

\begin{equation*}
\nu_V \from Z_T^n L(X) \to R(X) , \qquad \text{ for } V \in \cat D,
\end{equation*}

which satisfies

\begin{equation}
\inputtikz{property_of_univ_coaction}
\end{equation}

Suppose $(T, \Delta^{(T)}, \varepsilon^{(T)}) \from \cat C \to \cat C$ to be an oplax monoidal functor with centraliser $(Z_T, \chi)$.
For all $X \in \cat C$, the counit of $T$ combined with the universal coaction of $Z_T$ gives rise to a natural transformation

\begin{equation}\label{eq: unit_of_centraliser}
\inputtikz{unit_T_central_monad}
\end{equation}

We derive another natural transformation $\mu^{(Z_T)} \from Z_T^2 \to Z_T$ from the comultiplication of $T$.
Due to Lemma <ref> it is uniquely defined by

\begin{equation}\label{eq: mult_of_centraliser}
\inputtikz{mult_T_central_monad}
\end{equation}

The centraliser $(Z_T, \chi)$  of an oplax monoidal endofunctor $T \from \cat C \to \cat C$ is  a monad with multiplication and unit as given in Equations (<ref>) and (<ref>).

The above lemma is proven as the first part of <cit.>.
In it, the authors further consider $T \from \cat C \to \cat C$ to be equipped with a Hopf monad structure and show that in this case $Z_T$ is a Hopf monad as well.
The extended factorisation property given in Lemma <ref> allows us to reconstruct a comultiplication on $Z_T$ from a twofold application of the universal coaction and the multiplication of $T$:

\begin{equation}\label{eq: comult_of_centraliser}
\inputtikz{comult_central_bimonad}
\end{equation}

Likewise, the unit of $T$ induces a counit on $Z_T$ via

\begin{equation}\label{eq: counit_of_centraliser}
\inputtikz{counit_central_bimonad}
\end{equation}

A direct computation now verifies that the centraliser $Z_T$ is a bimonad as well.
For the construction of left and right antipodes, see <cit.>.

We think of $\ZCat[H]{C}$ as the centre of an oplax bimodule category as stated in Remark <ref>, see also <cit.>.
Objects in $\ZCat[H]{C}$ are pairs $(X, \sigma_{X, \blank})$, where $X \in \cat C$ and

\begin{equation*}
\sigma_{X,Y} \from X \otimes Y \to H(Y)\otimes X, \qquad \qquad \text{ for all } Y \in \cat C
\end{equation*}

is a natural transformation satisfying for all $X,Y,W \in \cat C$

\begin{gather}
(\Delta^{(H)}_{Y,W} \otimes \id_X) \sigma_{X, Y \otimes W} = (\id_{H(Y)} \otimes \sigma_{X,W})(\sigma_{X,Y} \otimes \id_{W}) \\
(\varepsilon^{(H)} \otimes \id_{X}) \sigma_{X,1} = \id_{X}.
\end{gather}

Analogous to the centres studied before, the morphisms in $\ZCat[H]{C}$ are those morphisms of $\cat C$ which commute with the respective half-braidings.
As shown in <cit.>, the structure morphisms of a Hopf monad $H \from \cat C \to \cat C$ can be used to define a rigid structure on $\ZCat[H]{C}$.
For example, the tensor product of two modules $(X, \sigma_{X,\blank}), (Y, \sigma_{Y,\blank}) \in \ZCat[H]{C}$ is $X\otimes Y \in \cat C$ together with the half-braiding

\begin{equation}
\inputtikz{tensor_product_half_braidings_Hopf_monad}
\end{equation}

Since centralisers of Hopf monads are Hopf monads themselves, it stands to reason that their modules implement the twisted centres discussed in the previous remark as a rigid category.
This is proven in <cit.>.

Suppose $H \from \cat C \to \cat C$ to be a centralisable Hopf monad.
The modules $\cat C^{Z_H}$ of its centraliser $(Z_H, \chi)$ are isomorphic as a rigid category to $\ZCat[H]{C}$.

Applying the above theorem to the identity functor $\Id \from \cat C \to \cat C$, we obtain a Hopf monadic description of the Drinfeld centre $\ZCat C$ of a rigid category $\cat C$.
The terminology of our next definition is due to Shimizu, see [50].

Let $\Id \from \cat C \to \cat C$ be centralisable with centraliser $(Z, \chi)$.
We call  $\mathfrak D(\cat C) \defeq (Z, \mu^{(Z)}, \eta^{(Z)}, \Delta^{(Z)}, \varepsilon^{(Z)}) \from \cat C \to \cat C$ the central Hopf monad of $\cat C$ and denote the category of its modules by $\cat C^{\mathfrak D}$.

An important step in proving Theorem <ref> is determining an inverse to the comparison functor $\Sigma^{(Z_T)} \from \ZCat[T]{C} \to \cat C^{Z_T}$.
This construction will also play a substantial role in our monadic description of the anti-Drinfeld centre, hence why we recall it in its full generality.
Let $T \from \cat C \to \cat C$ be a centralisable oplax monoidal endofunctor with $(Z_T, \chi)$ as its centraliser.
To every module $(M, \vartheta_M)$ over $Z_T$ we associate a half-braiding $\sigma_{M, \blank} \from M \otimes \blank \to T(\blank) \otimes M$.
For any $X \in \cat C$ it is given by the composition

\begin{equation}\label{eq: free_half_braiding}
\inputtikz{braiding_from_universal_coaction}
\end{equation}

This yields a functor $E^{(Z_T)} \from \cat C^{Z_T} \to \ZCat[T]{C}$ which is the identity on morphisms and on objects is given by

\begin{equation}\label{eq: inverse_comparison_functor}
E^{(Z_T)}(M, \vartheta_{M}) = (M, \sigma_{M,\blank}), \qquad \text{ for all }(M, \vartheta_{M}) \in \cat C^{Z_T}.
\end{equation}

Conversely, we assign to every object $(M, \sigma_{M, \blank}) \in \ZCat[T]{C}$ a module over $Z_T$ whose action $\vartheta_M$ is uniquely defined by

\begin{equation}
\inputtikz{action_from_braiding}
\end{equation}

As it turns out, this yields the comparison functor $\Sigma^{(Z_T)} \from \ZCat[T]{C} \to \cat C^{Z_T}$.

Suppose $T \from \cat C \to \cat C$ to be a centralisable oplax monoidal endofunctor with  $(Z_T, \chi)$ as its centraliser.
Denote the free functor of the adjunction of  $Z_T$ by $F_{Z_T} \from \cat C \to \cat C^{Z_T}$.
The composition

\begin{equation}
\cat C \trightarrow{F_{Z_T}} \cat C^{Z_T} \trightarrow{E^{(Z_T)}}  \ZCat[T]{C}
\end{equation}

defines a left adjoint of the forgetful functor $U^{(T)} \from \ZCat[T]{C} \to \cat C$.

We recall <cit.>, which proves the adjunction $F^{(T)} \adjoint U^{(T)}$  to be monadic.

Assume $(Z_T, \chi)$ to be a centraliser of the oplax monoidal endofunctor  $T \from \cat C \to \cat C$.
The functor $E^{(Z_T)} \from \cat C^{Z_T} \to \ZCat[T]{C}$ is an isomorphism of categories whose inverse is the comparison functor $\Sigma^{(Z_T)} \from \ZCat[T]{C} \to \cat C^{Z_T}$.

§.§ Centralisers and comodule monads

We will now apply the methods of and Virelizier to twisted centres for the purpose of obtaining a comodule monad that implements the anti-Drinfeld centre.
Hereto, we need a generalised version of the concept of modules over a monad.
Our approach is based on [42].

Suppose $B\from \cat C \to \cat C$ to be a bimonad and $F \from \cat D \to \cat C$ an oplax monoidal functor.
An oplax monoidal right action of $B$ on $F$ is an oplax natural transformation
$\alpha \from FB \to F$, such that for all $X \in \cat D$

\begin{equation}\label{eq: oplax_right_action}
\alpha_X\alpha_{B(X)} = \alpha_X\mu_{F(X)} \qquad \text{ and } \qquad
\alpha_XF(\eta_X)= \id_{F(X)}.
\end{equation}

Similarly, we could define oplax monoidal left actions.
A prime example of the latter is given by the forgetful functor $U_B \from \cat C^B \to \cat C$ of a bimonad $B \from \cat C \to \cat C$ together with the action displayed in Diagram (<ref>).

To keep our notation concise, in the following we fix an oplax monoidal functor $L \from \cat C \to \cat C$ with an oplax right action $\alpha \from LB \to L$ by a bimonad $B\from \cat C \to \cat C$ and assume that $L$ and $B$ are centralisable.
Their centralisers will be denoted by $(Q, \xi)$ and $(Z, \chi)$, respectively.

We think of $\ZCat[B]{C}$ as a more general version of the Drinfeld centre
which is supposed to act on  $\ZCat[L]{C}$ from the right.
To emphasise this, and in line with the colouring scheme of Section <ref>, we use black for objects in $\cat C$ or its generalised Drinfeld centre and blue for objects in $\ZCat[L]{C}$.

Consider two objects $(M, \sigma_{M,\blank}) \in \ZCat[L]{C}$ and $(X, \sigma_{X,\blank}) \in \ZCat[B]{C}$.
The action of $B$ on $L$, combined with the half-braidings of $M$ and $X$, yields a natural transformation

\begin{equation}\label{eq: right_action_of_gen_centre}
\inputtikz{right_action_of_H_centre_on_T_centre}
\end{equation}

The centre $\ZCat[B]{C}$ acts on $\ZCat[L]{C}$
from the right by tensoring the underlying objects and gluing together the half-braidings as in Equation (<ref>).
With respect to this action, the forgetful functor $U^{(L)} \from \ZCat[L]{C} \to \cat C $ is a strict comodule functor over $U^{(B)} \from  \ZCat[B]{C} \to \cat C$.

We proceed as in <cit.> and fix objects $(M, \sigma_{M, \blank} ) \in \ZCat[L]{C}$ and $(X, \sigma_{X,\blank}) \in \ZCat[B]{C}$.
The compatibility of the half-braiding of $M \otimes X$ with the unit of $\cat C$ is a short computation:

\begin{equation*}
\inputtikz{unit_identity_from_bimonad}
\end{equation*}

Similarly, we verify the hexagon axiom:

\begin{equation*}
\inputtikz{hexagon_identity_from_bimonad}
\end{equation*}

The compatibility of the action $\alpha \from LB \to L$ with the multiplication and unit of $B$ asserts that $\ZCat[L]{C}$ is a right module of the generalised Drinfeld centre.

By construction, we have for all $(M, \sigma_{M, \blank} ) \in \ZCat[L]{C}$ and $(X, \sigma_{X,\blank}) \in \ZCat[B]{C}$

\begin{equation*}
U^{(L)}((M, \sigma_{M,\blank}) \ract (X, \sigma_{X,\blank}))
= M \otimes X
= U^{(L)}(M, \sigma_{M,\blank}) \otimes U^{(B)} (X, \sigma_{X,\blank}).
\end{equation*}

Thus, $U^{(L)}$ is a strict comodule functor over $U^{(B)}$.

We extend our colouring scheme to universal coactions and write

\begin{equation}\label{eq: colouring_scheme_universal_coactions}
\inputtikz{universal_coaction_centre_and_twisted_centre}
\end{equation}

The identification of $\cat C^Z$ and $\cat C^Q$ with the generalised Drinfeld centre and its twisted cousin suggest that $Q$ is a comodule monad over $Z$.
In analogy with Equation (<ref>), we define a candidate for the coaction of $Q$ by

\begin{equation}\label{eq: coaction_anti_central_monad}
\inputtikz{coaction_anti_central_comodule_monad}
\end{equation}

Let $\alpha \from LB \to L$ be an oplax monoidal right action of a bimonad $B \from \cat C \to \cat C$ on an oplax monoidal functor $L \from \cat C \to \cat C$.
Suppose furthermore that the centralisers $(Q, \xi)$ of $L$  and  $(Z, \chi)$ of $B$ exist.
The coaction  of Equation (<ref>) turns $Q$ into a comodule monad over $Z$ such that $\cat C^Q$ is isomorphic as a right module category over $\cat C^Z$ to $\ZCat[L]{C}$.

By Remark <ref> and Theorem <ref> we have monadic adjunctions

\begin{equation*}
\adj{F^{(B)}}{U^{(B)}}{\cat C}{\ZCat[B]{C}} \qquad \text{ and } \qquad \adj{F^{(L)}}{U^{(L)}}{\cat C}{\ZCat[L]{C}}
\end{equation*}

which, due to <cit.>, give rise to the bimonad $Z$ and monad $Q$, respectively.
Lemma <ref> shows that $U^{(L)}$ is a strict comodule functor over $U^{(B)}$ and therefore, by Theorem <ref>, we obtain that $Q$ is a comodule monad over $B$.
Following Corollary <ref>, the coaction  $\lambda \from Q(\blank \otimes \blank) \to Q(\blank) \otimes Z(\blank)$ implementing the action of $\cat C^{Z}$ on $\cat C^{Q}$ is for all $X, Y \in \cat C$ given by

\begin{equation}
\lambda_{X, Y} = \vartheta_{Q(X) \otimes Z(Y)} Q(\eta^{(Q)}_{X} \otimes \eta^{(Z)}_{Y}).
\end{equation}

By using the relation between universal coactions and half-braidings, explained in Equation (<ref>), and  applying the hexagon identity we compute:

\begin{equation}
\inputtikz{uniqueness_of_coaction}
\end{equation}

The uniqueness property of universal coactions implies $\lambda= \delta^{(Q)}$.

It remains to show that $\cat C^{Q}$ and $\ZCat[L]{C}$ are isomorphic as modules over $\cat C^{Z}$.
Note that by Lemmas <ref> and <ref> as well as Theorem <ref> and Lemma <ref>, the comparison functor
$\Sigma^{(Z)} \from \ZCat[B]{C} \to \cat C^{Z}$ is strong monoidal and
$\Sigma^{(Q)} \from \ZCat[L]{C} \to \cat C^{Q}$ is a strong comodule functor over it.
Furthermore, due to Theorem <ref>, both $\Sigma^{(Z)}$ and $\Sigma^{(Q)}$  admit inverses

\begin{equation*}
E^{(Z)} \from \cat C^{Z} \to \ZCat[B]{C} \qquad \text{ and } \qquad
E^{(Q)} \from \cat C^{Q} \to \ZCat[L]{C}.
\end{equation*}

Using that $E^{(Z)}$ is monoidal as well, we identify the right action of $\ZCat[B]{C}$ on $\ZCat[L]{C}$ with a right action $\bract\from\ZCat[L]{C} \times \cat C^Z \to \ZCat[L]{C} $ of $\cat C^Z$ by setting

\begin{equation*}
\ZCat[L]{C} \times \cat C^Z \trightarrow{\Id \times E^{(Z)}}
\ZCat[L]{C} \times \ZCat[B]{C} \trightarrow{(\blank)\ract(\blank)}
\ZCat[L]{C}.
\end{equation*}

For any $M \in \ZCat[L]{C}$ and $X \in\ZCat[L]{C}$  we have

\begin{equation*}
\Sigma^{(Q)} (M\bract X)
\Sigma^{(Q)} (M \ract E^{(Z)}(X))
\trightarrow{\delta^{\Sigma^{(Q)}}} \Sigma^{(Q)} (M) \ract \Sigma^{(Z)}E^{(Z)}(X) = \Sigma^{(Q)} (M) \ract X
\end{equation*}

and therefore $\Sigma^{(Q)} \from \ZCat[L]{C} \to \cat C^{Q}$ is an isomorphism of module categories.

Let us apply our findings to the identity and biduality functor of a rigid category $\cat C$.
Suppose $(Q, \xi)$ and  $(Z, \chi)$ to be the centralisers of $\lbiDual{(\blank)}$ and $\Id_{\cat C}\from \cat C \to \cat C$, respectively.
There is a trivial right action of the identity of $\cat C$ on its biduality functor,

\begin{equation*}
\id_X \from\lbiDual{(\Id_{\cat C} (X))} \to \lbiDual{X}, \qquad \qquad \text{for all } X \in \cat C.
\end{equation*}

It turns $Q$ into a comodule monad over $Z$ and its modules $\cat C^Q$ are isomorphic to $\QCat C$ as a $\cat C^Z$-module category.
Due to Remark <ref>, we can identity $\QCat C$ with $\ACat C$, justifying our next definition.

Assume $\lbiDual{(\blank)}, \Id_{\cat C}\from \cat C \to \cat C$ to admit centralisers $(Q, \xi)$ and $(Z, \chi)$.
We call $\mathfrak{Q}(\cat C) \defeq (Q, \mu^{(Q)}, \eta^{(Q)}, \delta^{(Q)})$ the anti-central comodule monad of $\cat C$.

§.§ The Drinfeld and anti-Drinfeld double of a Hopf monad

We are now able to untangle the relationship between the various adjunctions and categories displayed in Figure <ref>.
To that end, we fix a Hopf monad $H \from \cat V \to \cat V$ on a rigid category $\cat V$ together with an oplax monoidal functor $L \from \cat V^H \to \cat V^H$, a bimonad $B \from \cat V^H\to \cat V^H$ and an oplax monoidal right action $\alpha \from LB \to B$.
Furthermore, we assume that the cross products $B \rtimes H$ and $L \rtimes H$ have centralisers $(Z_H, \nu)$ and $(Q_H, \tau)$.

We start by extending the action of $B$ on $L$ to an action of the respective cross products.

The action $\alpha \from LB \to B$ induces an oplax monoidal action

\begin{equation}
\inputtikz{extension_of_action_to_cross_products}
\end{equation}

From the pictorial description of the multiplication and unit of $B \rtimes H$, given in Definition <ref>, it becomes apparent that $\alpha_H$ is a right action of $B\rtimes H$ on $L \rtimes H$.
Additionally, as a composite of oplax monoidal natural transformations, it is oplax monoidal itself.

The following variant of <cit.> lies at the heart of our ensuing investigation.

Both $B, L\from \cat V^H\to \cat V^H$ admit centralisers $(Z, \chi)$ and $(Q, \xi)$
such that $Z$ is a lift of $Z_H$ as a bimonad and $Q$ is a lift of $Q_H$ as a comodule monad.

By <cit.>, we know that there are centralisers $(Q, \xi)$ and $(Z, \chi)$ of $L$ and $B$
that satisfy for all $(X, \vartheta_X), (Y, \vartheta_Y) \in \cat V^H$

\begin{align*}
U_H Q (X, \vartheta_X) &= Q_H(X),   &
U_H(\xi_{(X, \vartheta_X), (Y, \vartheta_Y)}) &= (U_HL(\vartheta_Y) \otimes  \id_{Q_H(X)})\tau_{X,Y}, \\
U_H Z (X, \vartheta_X) &= Z_H(X),  &
U_H(\chi_{(X, \vartheta_X), (Y, \vartheta_Y)}) &= (U_HB(\vartheta_Y) \otimes \id_{Z_H(X)})\nu_{X,Y}.
\end{align*}

The second and third part of the above mentioned theorem state that $Q$ is a lift of the monad $Q_H$ and $Z$ is a lift of the bimonad $Z_H$.
It remains for us to show that the coactions of $Q$ and $Q_H$ are compatible with the forgetful functor $U_H \from \cat V^H \to \cat V$.
We fix objects $(X, \vartheta_X), (Y, \vartheta_Y) \in \cat V^H$ and $W \in \cat V$ and compute:

\begin{equation*}
\inputtikz{compatibility_coactions_cross_product}
\end{equation*}

The uniqueness property of universal coactions as given in Lemma <ref> then implies that $U_H(\delta^{(Q)}_{(X, \vartheta_X), (Y, \vartheta_Y)}) = \delta^{(Q_H)}_{X,Y}$.
Since $U_{H}\from \cat V^H \to \cat V$ is a strict comodule functor, the claim follows.

The previous theorem together with Lemma <ref> imply that we obtain a comodule monad $D(L,H)\defeq Q \rtimes H$ over $D(B, H) \defeq Z \rtimes H$.
The correspondence between lifts and monads given in Theorem <ref> yields a unique comodule distributive law $(H Q_H  \trightarrow{\Omega} Q_H H, H Z_H \trightarrow{\Lambda} Z_H H)$ such that

\begin{equation}
D(L, H) = Q_H \circ_{\Omega} H \qquad \text{and} \qquad
D(B, H) = Z_H \circ_{\Lambda} H.
\end{equation}

We call $D(B,H)$ and $D(L,H)$ the double and twisted double of the pairs $(B,H)$ and $(L,H)$.

The relationship between doubles and generalised Drinfeld centres is explained in <cit.>.
Our next result uses the same techniques to prove how twisted doubles parameterise twisted centres.

The twisted double $D(L, H)$ is a comodule monad over $D(B,H)$ and $\cat V^{D(L, H)}$ is isomorphic as a module category over $\cat V^{D(B, H)}$ to
$\ZCat*[L]{\cat V^H}$.

Since $Q$ is a lift of $Q_H$ as a comodule monad, the twisted double $D(L, H)$ is a comodule monad over $D(B,H)$.
By Lemma <ref>, this implies the existence of an isomorphism of $\cat V^{D(B, H)}$-module categories $K^{(\Omega)} \from \cat V^{D(L, H)} \to \left(\cat V^H\right)^Q$.
Due to the proof of Theorem <ref> the comparison functor $\Sigma^{(Q)}\from  \ZCat*[L]{\cat V^H} \to  \left(\cat V^H\right)^Q$ implements an isomorphism of module categories and the statement follows by considering

\begin{equation}
\cat V^{D(L, H)} \trightarrow{K^{(\Omega)}}
\left(\cat V^H\right)^Q \trightarrow{E^{(Q)}}
\ZCat*[L]{\cat V^H}.
\end{equation}

Suppose $B = \Id_{\cat C^K}$ and $L=\lbiDual{(\blank)} \from \cat V^H \to \cat V^H$.
We refer to $D(H) \defeq D(B,H)$ and $Q(H) \defeq D(L,H)$ as the Drinfeld and anti-Drinfeld double of $H$.

Our previous definition can be understood as an extension of the notion of the anti-Drinfeld double given by [25] to the monadic framework.

§.§ Pairs in involution for Hopf monads

For the final step in our investigation, let us consider a Hopf monad $H \from \cat V \to \cat V$ which admits a double and anti-double.
Tracing the various identifications of the centre and anti-centre of a monoidal category given in Figure <ref>, we observe that module functors from $\ZCat*{\cat V^H}$ to $\QCat*{\cat V^H}$ equate bidirectionally to module functors between $\cat V^{D(H)}$ and $\cat V^{Q(H)}$.
In the spirit of viewing $D(H)$ and $Q(H)$ as `coordinate systems' of their respective modules, we want to translate such functors into comodule monad morphisms.
Our main focus here is on pivotal structures of $\cat V^{D(H)}$.

We begin by developing the notion of pairs in involution for a Hopf monad.
Classically, pairs in involution consist of a group-like and character of a Hopf algebra, which implement the square of its antipode by their adjoint actions.

Let $H \from \cat V \to \cat V$ be a Hopf monad.
A character of $H$ is a module $\beta \defeq (1, \vartheta_{\beta}) \in \cat{V}^H$, whose underlying object is the monoidal unit of $\cat{V}$.

A group-like element of $H$ is a natural transformation $g \from \Id_{\cat V} \to H$ satisfying for all $X, Y \in \cat V$

\begin{equation} \label{eq:2}
\Delta^{(H)}_{X,Y}g_{X\otimes Y} = g_X \otimes g_Y \qquad \text{ and } \qquad  \varepsilon^{(H)} g_1= \id_1.
\end{equation}

We write $\Char(H)$ for the characters of $H$ and $\Gr(H)$ for its group-likes.

Note that the characters $\Char(H)$ of a Hopf-monad $H \from \cat V \to \cat V$ form a monoid and, by Lemma <cit.>, the set $\Gr(H)$ of group-like elements bears a group structure.

Furthermore, the group-likes of a Hopf monad $H$ act on it by conjugation.
We recall this construction based on <cit.>.
Given a natural transformation $g \from \Id_{\cat C} \to H$, we define the left and right regular action of $g$ on $H$ to be the natural transformations defined for every $X \in \cat V$ by

\begin{align}\label{eq: left_and_right_regular_action}
L_{g, X} &\defeq H(X) \trightarrow{g_{H(X)}} H^2(X) \trightarrow{\mu^{(H)}_X} H(X), \\
R_{g, X} &\defeq H(X) \trightarrow{H(g_X)} H^2(X) \trightarrow{\mu^{(H)}_X} H(X).
\end{align}

Before we state our next definition, we set for all $X,Y, W \in \cat V$

\begin{equation}\label{eq: iterated_coproducts}
\Delta^{(H)}_{X,Y,W} \defeq(\Delta^{(H)}_{X,Y} \otimes \id_{H(W)}) \Delta^{(H)}_{X\otimes Y, W} = (\id_{H(X)}\otimes \Delta^{(H)}_{Y, W}) \Delta^{(H)}_{X, Y \otimes W}.
\end{equation}

Every group-like $g \in \Gr(H)$ and character $\beta \in \Char(H)$ of a Hopf monad $H \from \cat V \to \cat V$ give rise to natural transformations

\begin{align}\label{eq: adjoint_action}
\Ad_{g, X} &\defeq L_{g,X}R_{g^{-1}, X} \from H(X) \to H(X), &\text{ for all } X \in \cat V, \\
\Ad_{\beta, X} &\defeq (\vartheta_{\beta}\otimes \id_{H(X)} \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1, X,1} \from H(X) \to H(X), &\text{ for all } X \in \cat V.
\end{align}

called the adjoint actions of $g$ and $\beta$ on $H$, respectively.

To define pairs in involution, we need the `square of the antipode'.
This notion was developed in <cit.>.

Suppose $\phi \from \Id_{\cat V} \to \lbiDual{(\blank)}$ to be a pivotal structure on $\cat V$ and let $H \from \cat V \to \cat V$ be a Hopf monad.
The square of the antipode of $H$ is a natural transformation $S^{2}\from H \to H$, which is defined for every $X \in \cat V$ by

\begin{equation}
S^2_X \defeq \phi^{-1}_{H(X)} s^l_{\lDual{H(X)}} H(\lDual{s^l_X}) H(\phi_X).
\end{equation}

Analogous to the Hopf algebraic case, we state the following:

A pair in involution for a Hopf monad $H\from \cat V \to \cat V$ consists of a group-like $g \in \Gr(H)$ and character $\beta \in \Char(H)$ such that for all $X \in \cat{V}$

\begin{equation}\label{eq: antipode_condition}
Ad_{g, X}=  Ad_{\beta,X}S^{2}_{X}.
\end{equation}

To prove that pairs in involution correspond to certain pivotal structures on the Drinfeld centre of $\cat V^H$, we need two technical results.
The first is a special case of <cit.>.

Let $H$ be a monad with associated forgetful functor $U_H \from \cat \cat V^H \to \cat V$.
Then there exists a canonical bijection

\begin{equation}\label{eq: sharp-lift}
(\blank)^{\sharp}\from \Nat(H, H) \to \Nat(HU_H, U_H), \qquad f \mapsto f^{\sharp},
\end{equation}

where $f^{\sharp}_{(M, \vartheta_M)} =\vartheta_M f_M$.

The next lemma is a variant of <cit.>.

Let $\phi \from \Id_{\cat V} \to \lbiDual{(\blank)}$ be a pivotal structure on $\cat V$ and $H \from \cat V \to \cat V$ a Hopf monad.
For any group-like $g \in \Gr(H)$ and character $\beta \in \Char(H)$ the following are equivalent:

* The arrows $g$ and $\beta$ form a pair in involution.

* The natural arrow $\phi g^{\sharp}\in \Nat(U_H, U_H)$ lifts to $\Nat(\Id_{\cat V^H}, \beta \otimes \lbiDual{(\blank)}\otimes \lDual \beta )$.

Consider a module $(M, \vartheta_M) \in \cat V^H$.
By <cit.> and the definition of $S^2$, the action on $\lbiDual M$ is given by

\begin{equation*}
\vartheta_{\lbiDual M}
= \lbiDual{\vartheta_M} s^l_{\lbiDual{H(M)}} H(\lDual{s^l_M})
= \phi_M \vartheta_M S^2_M H(\phi^{-1}_M)
\end{equation*}

and therefore we have

\begin{equation*}
\vartheta_{\beta \otimes \lbiDual M \otimes \lDual \beta}
= (\vartheta_{\beta} \otimes \vartheta_{\lbiDual{M}} \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1}
= (\vartheta_{\beta} \otimes \phi_M \vartheta_M S^2_M H(\phi^{-1}_M) \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1}.
\end{equation*}

By definition $\phi g^{\sharp}$ lifts to a natural transformation from $\Id_{\cat V^H}$ to $\beta \otimes \lbiDual{(\blank)} \otimes \lDual \beta$,  if and only if for any $H$-module $(M, \vartheta_M)$, we have

\begin{equation} \label{eq: sharpbeta-lifts}
(\phi g^{\sharp})_M \vartheta_M = \vartheta_{\beta \otimes \lbiDual{M} \otimes \lDual \beta}H( (\phi g^{\sharp})_M).
\end{equation}

Let us now successively simplify both sides of the equation.
Using the naturality of $g \from \Id_{\cat V} \to H$, the fact that $\vartheta_M$ is an action and the definition of $g^{\sharp}$ as given in Lemma <ref>, we can rewrite the left hand side of the equation as

\begin{align*}
(\phi g^{\sharp})_M \vartheta_M
& = \phi_M \vartheta_M g_M \vartheta_M
= \phi_M \vartheta_M H(\vartheta_M) g_{H(M)} \\
& = \phi_M \vartheta_M \mu^{(H)}_M g_{H(M)}.
\end{align*}

Similarly, we simplify the right-hand side to

\begin{align*}
\vartheta_{\beta \otimes \lbiDual{M} \otimes \lDual \beta}H( (\phi g^{\sharp})_M)
& = (\vartheta_{\beta} \otimes \phi_M \vartheta_M S^2_M H(\phi^{-1}_M) \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1} H( (\phi g^{\sharp})_M) \\
& = (\vartheta_{\beta} \otimes \phi_M \vartheta_M  S^2_M H(\phi^{-1}_M) H( (\phi g^{\sharp})_M) \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1}  \\
& = (\vartheta_{\beta} \otimes \phi_M \vartheta_M  S^2_M H(\vartheta_M g_M ) \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1} \\
& = (\vartheta_{\beta} \otimes \phi_M \vartheta_M   H(\vartheta_M g_M ) S^2_M \otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1} \\
& = \phi_M \vartheta_M   H(\vartheta_M g_M ) (\vartheta_{\beta} \otimes  \id_{H(M)}\otimes \vartheta_{\lDual \beta}) \Delta^{(H)}_{1,M,1}S^2_M  \\
& = \phi_M \vartheta_{M}\mu^{(H)}_M H(g_M) \Ad_{\beta,M} S^2_M .
\end{align*}

Using the fact that $\phi$ is an isomorphism, Equation (<ref>) can thus be restated as

\begin{align*}
& & \vartheta_M \mu^{(H)}_M g_{H(M)} &= \vartheta_M\mu^{(H)}_M H(g_M)  \Ad_{\beta,M} S^2_M \\
\iff & & \vartheta_M L_{g, M} &= \vartheta_M R_{g, M} \Ad_{\beta, M} S^2_M
\end{align*}

Since $(\blank)^{\sharp}$ is a bijection by Lemma <ref>, the above equation is equivalent to $L_{g, M} = R_{g, M} \Ad_{\beta, M} S^2_M$.
We conclude the proof by multiplying both sides with $R_{g^{-1}, M}$.

The previous lemma leads to an identification of pairs in involution of $H$ with certain quasi-pivotal structures on $\cat V^H$.

Suppose $H \from \cat V \to \cat V$ to be a Hopf monad and $\phi \from \Id_{\cat V} \to \lbiDual{(\blank)}$ a pivotal structure on $\cat V$.
Then $H$ admits a pair in involution if and only if there exists a quasi-pivotal structure on $\cat V^{H}$ that is given for any $X \in \cat V^H$ and $\beta \in \Char(H)$ by

\begin{equation}\label{eq: lift_unit_qpiv_struct}
\rho_{\beta, X} \from X \to \beta \otimes\lbiDual{X}\otimes \lbiDual \beta.
\end{equation}

We proceed analogous to <cit.>.
Suppose $g \in \Gr(H)$ and $\beta \in \Char (H)$ to constitute a pair in involution for $H$.
By the previous lemma, $\phi g^{\sharp}$ lifts to a natural isomorphism

\begin{equation*}
\rho_{\beta,X}\from X \to \beta \otimes \lbiDual X \otimes \lDual X, \qquad \text{ for all } X \in \cat V^H.
\end{equation*}

Since $\phi$ is monoidal by definition and $g^{\sharp}$ is monoidal by virtue of $g$ being a group-like, see <cit.>, we obtain a pivotal structure
$\rho_{\beta} \from \Id_{\cat V^H} \to \beta \otimes \lbiDual{(\blank)}\otimes \lDual \beta$.

On the other hand, consider a quasi-pivotal structure $(\beta,\rho_{\beta})$, where $\beta \in \Char(H)$  is a character.
Since the forgetful functor $U_H$ is strong monoidal and thus

\begin{equation*}
U_H(\beta\otimes \lbiDual{(\blank)}\otimes \lDual \beta) = U_H(\lbiDual{(\blank)})= \lbiDual{(U_H(\blank))},
\end{equation*}

there exists a monoidal natural transformation

\begin{equation*}
\phi^{-1}_{U_H(X)}U_H(\rho_{\beta,X}) \from U_H(X) \to U_H(X), \qquad \qquad \text{for all } X \in \cat V^H.
\end{equation*}

Again, we apply <cit.> and obtain a unique group-like $g\in \Gr(H)$ such that $g^{\sharp} = \phi^{-1}_{U_H(X)}U_H(\rho_{\beta,X})$.
As $\phi g^{\sharp} = U_H(\rho_{\beta})$ lifts to the quasi-pivotal structure $(\beta,\rho_{\beta})$ on $\cat V^H$, Lemma <ref> implies that $g$ and $\beta$ form a pair in involution.

Let us now study a variant of <cit.>.

Assume $K, C \from \cat M \to \cat M$  to be two comodule monads over a bimonad $B \from \cat C \to \cat C$.
There is a bijective correspondence between morphisms of comodule monads $f \from K \to C$ and strict module functors $F \from \cat M^C \to \cat M^K$ such that $U_K F = U_C$.

As shown for example in <cit.>, we know that any functor  $F \from \cat M^C \to \cat M^K$ with  $U_K F = U_C$ is `induced' by a unique  morphism of monads $f \from K \to C$.
That is, $F$ is the identity on morphisms and on objects it is defined by

\begin{equation*}
F(M, \vartheta_M) = (M, \vartheta_M f_M), \qquad \qquad \text{ for all } (M, \vartheta_M) \in \cat M^C.
\end{equation*}

It remains to show that $f$ is a morphism of comodules if and only if $F$ is a strict module functor in the sense of Definition <ref>.
Let $(M, \vartheta_M)\in \cat M^C$ and $(X, \vartheta_X) \in \cat C^B$.
We compute

\begin{align*}
F((M, \vartheta_M) \ract (X, \vartheta_X))
&= (M \ract X, (\vartheta_M \ract \vartheta_X)\delta^{(C)}_{M,X} f_{M\ract X}), \\
F(M, \vartheta_M) \ract  (X, \vartheta_X)
&= (M \ract X, (\vartheta_M \ract \vartheta_X) (f_M \ract \id_{B(X)}) \delta^{(K)}_{M, X}).
\end{align*}

According to <cit.>, these modules coincide if and only if

\begin{equation*}
\delta^{(C)}_{M,X} f_{M \ract X} = (f_M \ract \id_{B(X)}) \delta^{(K)}_{M, X},
\end{equation*}

which is exactly the condition for $f$ to be a comodule morphism.

The above result readily implies the desired monadic version of Theorem <ref>.

Let $H \from \cat V \to \cat V$ be a Hopf monad that admits a double $D(H)$ and anti-double $Q(H)$.
The following statements are equivalent:

* The monoidal unit $1 \in \cat V$ lifts to a module over $Q(H)$.

* The Drinfeld double and anti-Drinfeld double of $H$ are isomorphic as comodule monads.

* There is an isomorphism of monads $g \from Q(H) \to D(H)$.

Additionally, if $\cat V$ is pivotal, one of the above statements holds if and only if $H$ admits a pair in involution.

$(i) \!\! \implies \!\! (ii)$: suppose $\omega \in \QCat*{\cat V^H}$ with $U_{Q(H)}(\omega) = 1$.
As shown in Equation (<ref>), it induces a functor of module categories

\begin{equation*}
\blank \otimes \omega \from \cat V^{D(H)} \to \cat V^{Q(H)}.
\end{equation*}

Since $U_{Q(H)}(\omega)= 1\in \cat V$, we can apply Theorem <ref> and obtain that $Q(H)$ and $D(H)$ are isomorphic as comodule monads.

It immediately follows that $(ii)$ implies $(iii)$; we proceed with $(iii) \!\! \implies \!\! (i)$: consider an isomorphism of monads $g \from Q(H) \to D(H)$.
It gives rise to a functor $G \from \cat V^{D(H)} \to \cat V^{Q(H)}$ that, on objects, is defined by

\begin{equation*}
G(M, \vartheta_M) = (M, \vartheta_M g_M), \qquad \qquad \text{ for all } (M, \vartheta_M) \in \cat C^Z.
\end{equation*}

We compose $G$ with the inverse of the comparison functor $E^{(Q(H))} \from \cat V^{Q(H)} \to \QCat C$, defined in Equation (<ref>), and see that there exists an object

\begin{equation*}
1^{(Q)} \defeq E^{(Q(H))}G(1) \in \QCat C
\end{equation*}

whose underlying object is the unit of $\cat V$.

Now let $\cat V$ be pivotal.
By Lemma <ref>, lifts of $1\in \cat V$ to the dual of the anti-center $\QCat*{\cat V^H}$ are in correspondence with quasi-pivotal structures
$(\beta, \rho_{\beta})$, where $\beta \in \Char(H)$.
By Theorem <ref> such a quasi-pivotal structure exists if and only if $H$ admits a pair in involution.

As a corollary, we can determine whether a category is pivotal in terms of monad isomorphisms between the central and anti-central monad

Assume $\cat C$ to admit a central and anti-central monad.
Then $\cat C$ is pivotal if and only if $\mathfrak D(\cat C)$ and $\mathfrak Q(\cat C)$ are isomorphic.

We consider the identity $\Id_{\cat C} \from \cat C \to \cat C$ as a Hopf monad.
Its Drinfeld  and anti-Drinfeld double are $D(\Id_{\cat C}) = \mathfrak D(\cat C) \rtimes \Id_{\cat C}$ and $Q(\Id_{\cat C}) = \mathfrak Q(\cat C) \rtimes \Id_{\cat C}$.
From here it follows that $D(\Id_{\cat C}) = \mathfrak D(\cat C)$ and similarly $Q(\Id_{\cat C}) = \mathfrak Q(\cat C)$.
The proof is concluded by Theorem <ref>.

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∗Equal contribution.

# Revisit Dictionary Learning for Video Compressive Sensing under the Plug-
and-Play Framework

∗Qing Yang  State Key Laboratory of Precision Spectroscopy, East China Normal
University  ∗Yaping Zhao  The University of Hong Kong

###### Abstract

Aiming at high-dimensional (HD) data acquisition and analysis, snapshot
compressive imaging (SCI) obtains the 2D compressed measurement of HD data
with optical imaging systems and reconstructs HD data using compressive
sensing algorithms. While the Plug-and-Play (PnP) framework offers an emerging
solution to SCI reconstruction, its intrinsic denoising process is still a
challenging problem. Unfortunately, existing denoisers in the PnP framework
either suffer limited performance or require extensive training data. In this
paper, we propose an efficient and effective shallow-learning-based algorithm
for video SCI reconstruction. Revisiting dictionary learning methods, we
empower the PnP framework with a new denoiser, the kernel singular value
decomposition (KSVD). Benefited from the advent of KSVD, our algorithm retains
a good trade-off among quality, speed, and training difficulty. On a variety
of datasets, both quantitative and qualitative evaluations of our simulation
results demonstrate the effectiveness of our proposed method. In comparison to
a typical baseline using total variation, our method achieves around $2$ dB
improvement in PSNR and 0.2 in SSIM. We expect that our proposed PnP-KSVD
algorithm can serve as a new baseline for video SCI reconstruction.

###### keywords:

Dictionary Learning, Compressive Sensing, Plug-and-Play Framework,
Computational Imaging

##  1 Introduction

The rapid development in machine learning (ML) and artificial intelligence
(AI) leads to the necessity and urgency of high-dimensional (HD) visual signal
acquisition and analysis. Though optical sensors are diversely growing to
capture visual (and other) signals, most of them are composed of two-
dimensional (2D) arrays. The fundamental issue remains: how to acquire HD
($\geq 3$D) data efficiently. To address this problem, snapshot compressive
imaging (SCI) is proposed to obtain the 2D compressed measurement of HD data
with optical imaging systems (encoder), and then reconstruct HD data using
compressive sensing algorithms (decoder) [1]. During the past twenty years,
both the hardware [2, 3] and algorithms [4, 5] have been improved
significantly for SCI. In this paper, we focus on the algorithmic part, e.i.,
the decoding part in SCI.

The latest trend in SCI is to exploit the plug-and-play (PnP) framework for HD
data reconstruction, which generally leads to a closed-form solution updating
sub-problem (projecting the measurement to the signal space) and an image
denoising sub-task [6]. Such a denoising process is an inevitable step under
the PnP framework, and therefore has a major effect on the final
reconstruction. While the PnP framework has been successfully applied in video
SCI, choosing a proper denoiser is still a challenging problem.

To perform denoising, various existing algorithms could be categorized into
non-learning-based ones, shallow-learning-based ones, and deep-learning-based
ones, shown in Figure 1 . Among non-learning-based algorithms, total variation
(TV) [7] and sparsity-based [reddy2011p2c2] ones exhibit limited performance,
and DeSCI [8] suffers low speed. On the other hand, deep-learning-based ones
[4, 9] need extensive training data and long training time for the sake of
high quality [5, 10].

Figure 1: Investigation of non-learning-based algorithm, shallow-learning-
based algorithm and deep-learning-based algorithm for SCI reconstruction.
Inspired by the PnP framework, a new baseline for video SCI using KSVD is
proposed in this paper.

In contrast, shallow-learning-based algorithms stand out because of their good
trade-off among quality, speed, and training difficulty. This line of research
mostly includes the patch-based learning methods such as Gaussian mixture
models (GMM) [11, 12] and dictionary learning [Hitomi11ICCV]. Compared with
the state-of-the-art optimization-based algorithm, DeSCI (Decompress-Snapshot-
Compressive-Imaging) [8], GMM can only provide limited performance. Though
dictionary learning methods have been used for video SCI in [Hitomi11ICCV],
the algorithm developed therein is fairly complicated and thus not efficient.
Thanks to the PnP framework [13, 14, 15], now various denoisers can be used in
an in-situ manner. As mentioned in [6, 16], DeSCI can be recognized as PnP-
WNNM, where WNNM (weighted nuclear norm minimization) is one of the state-of-
the-art denoising algorithms [17]. However, as mentioned before, DeSCI is very
slow, though it can provide high-quality images, and this precludes its wide
applications in real SCI systems. We aim to address the following question in
this paper:

  * •

Can we build an efficient and effective shallow-learning-based algorithm for
SCI reconstruction?

To answer this question, we revisit the renowned dictionary learning method,
the kernel singular value decomposition (KSVD) [Aharon06TSP] and apply it for
SCI reconstruction. KSVD is much easier to train than convolutional neural
networks (CNN), also with the smaller requirement of training data. In
addition, for task-specific reconstruction, it is easy to train a different
dictionary for different types of data, such as normal images, CT images,
microscopy images, diffraction images, etc. It has been noticed in [18] that
the CNN based denoiser is not flexible for different tasks. For example, the
well-trained FFDnet [19] for visual images cannot perform well for
hyperspectral image reconstruction. Therefore, a new denoiser has to be re-
trained and this takes weeks as well as a large amount of training data and
computational resources. It is very challenging to train a deep model for
various datasets.

Bearing these in mind, in this paper, we apply KSVD based denoiser into the
PnP framework and we show that with a well-trained dictionary, the results of
our proposed PnP-KSVD are superior to those of GAP-TV. Specific contributions
of our work are listed as follows:

  * •

We provide a new baseline for video SCI reconstruction by integrating the KSVD
method into the PnP framework.

  * •

In comparison to others, our results retain a good trade-off among quality,
speed, and training difficulty.

  * •

Extensive experiments on both simulated and real data demonstrate the
robustness of our method.

##  2 related work

###  2.1 Non-learning-based algorithm

Traditional work of non-learning-based algorithms used in SCI usually includes
total variation (TV)[7], sparsity [reddy2011p2c2, 20] and DeSCI[8] based on
rank minimization. From previous experimental data, TV and sparsity-based
algorithms are robust but of limited quality. DeSCI is of high quality but
slow.

###  2.2 Deep-learning-based algorithm

As we all know, recently the most popular algorithms in SCI are based on deep
learning such as E2E-CNN [2], BIRNAT [5], RevSCI[10] and MetaSCI [21].
Complicated deep learning networks perform high-quality reconstruction
efficiently but need extensive training data and long training time.

###  2.3 Shallow-learning-based algorithm

The performance of shallow-learning-based algorithms such as GMM and KSVD
related [Hitomi11ICCV] were between the two algorithms mentioned above.
Considering this fact, we provide a new baseline for video SCI using the KSVD
algorithm inspired by the PnP [6] framework. KSVD is a kind of traditional
iteration algorithm of dictionary learning in the image or video denoising. In
this article, we plug the KSVD algorithm into the PnP framework as denoising
for video SCI systems creatively. And the experimental results show pretty
well. We hope this study would provide a meaningful inspiration for
researchers in promoting the properties of the SCI system.

##  3 Method

Figure 2: Principle and framework of PnP-KSVD for video SCI. (a) Dynamic coded
aperture called masks coding dynamic scenes equivalents differential video in
temporal dimension. Using camera like CCD to integrate coded scene and then
generating compressed measurement (hiding information of time dimension). PnP
framework helps compressed measurement decode. Meanwhile, using algorithm like
KSVD as denoiser to reduce inevitable errors. (b) Put data of measurement and
coded masks into the PnP-KSVD iterative algorithm. Using dictionary
$\mathbf{D}$ generated by KSVD algorithm as priors to accomplish
reconstruction.

###  3.1 Mathematical Model of SCI

As Figure 2 shows, a typical video SCI system compresses $B$ video frames,
denoted as $\mathbf{X}\in{\mathbb{R}}^{n_{x}\times n_{y}\times B}$, into a 2D
measurement $\mathbf{Y}\in{\mathbb{R}}^{n_{x}\times n_{y}}$ by coding masks
$\mathbf{M}\in{\mathbb{R}}^{n_{x}\times n_{y}\times B}$. This process could be
formulated as

$\displaystyle\mathbf{Y}=\sum^{B}_{b=1}\mathbf{M}_{b}\odot\mathbf{X}_{b}+\mathbf{E},$
(1)

where ${\mathbb{E}}^{n_{x}\times n_{y}}$ denotes the noise;
$\mathbf{M}_{b}=\mathbf{M}(:,:,b)$ and
$\mathbf{X}_{b}=\mathbf{X}(:,:,b)\in{\mathbb{R}}^{n_{x}\times n_{y}}$
represent the $b$-th coding mask and the corresponding video frame,
respectively; $\odot$ denotes the Hadamard (element-wise) product.
Mathematically, the measurement in Equation (1) can be rewritten as

$\displaystyle\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{e},$ (2)

where $\mathbf{y}=\text{Vec}(\mathbf{Y})\in{\mathbb{R}}^{n_{x}n_{y}}$,
$\mathbf{e}=\text{Vec}(\mathbf{E})\in{\mathbb{R}}^{n_{x}n_{y}}$,
$\mathbf{x}=\text{Vec}(\mathbf{X})\in{\mathbb{R}}^{n_{x}n_{y}B}$, and thus
$\mathbf{\Phi}\in{\mathbb{R}}^{n_{x}n_{y}\times n_{x}n_{y}B}$. The forward
model of SCI is fairly simple as shown in Eq. (2). The real challenge is the
inverse problem, i.e., the decoder or reconstruction algorithms [22]. More
specifically, given the compressed measurement $\mathbf{y}$ and sensing matrix
$\mathbf{\Phi}$.

###  3.2 PnP framework for SCI

As can be seen from Equation (2), the inverse problem of SCI is ill-posed
($\mathbf{\Phi}$ have more columns than rows) and thus a regularizer is
usually employed. With a regularization function $R$ that could be applied to
$\mathbf{x}$, we could estimate $\mathbf{\hat{x}}$ by solving the optimization
problem

$\displaystyle\mathbf{\hat{x}}=\mathop{\arg\min}_{\mathbf{x}}\frac{1}{2}||\mathbf{y}-\mathbf{\Phi}\mathbf{x}||_{2}^{2}+R(\mathbf{x}),$
(3)

where $\mathbf{\hat{x}}\in{\mathbb{R}}^{n_{x}n_{y}B}$ is an estimate of the
original visual signal $\mathbf{x}$. Using the alternating direction method of
multiplier (ADMM) [23], the unconstrained optimization in Equation (3) can be
converted into

$\displaystyle\begin{aligned}
(\mathbf{\hat{x}},\mathbf{\hat{v}})=\mathop{\arg\min}_{\mathbf{x},\mathbf{v}}\frac{1}{2}||\mathbf{y}-\mathbf{\Phi}\mathbf{x}||_{2}^{2}+R(\mathbf{v}),\\\
~{}{\text{ subject to }}~{}\ \mathbf{x}=\mathbf{v}.\end{aligned}$ (4)

For lower computational workload, we could use the generalized alternating
projection (GAP) [24] as a special case of ADMM. To solve Equation (4), GAP
updates $\mathbf{x}^{k}$ and $\mathbf{v}^{k}$ as follows,

$\displaystyle\mathbf{x}^{(k+1)}$
$\displaystyle=\mathbf{v}^{k}+{\mathbf{\Phi}}^{\top}(\mathbf{\Phi}{\mathbf{\Phi}}^{\top})^{-1}(\mathbf{y}-\mathbf{\Phi}\mathbf{v}^{k}),$
(5) $\displaystyle\mathbf{v}^{(k+1)}$
$\displaystyle=\mathcal{D}(\mathbf{x}^{(k+1)}),$ (6)

where $\mathcal{D}$ is the denoiser (or the denoising prior) being used in the
PnP framework [6].

Different from previous work [16], hereby we tend to use a lightweight
learning method, e.i., the KSVD dictionary learning algorithm as the denoising
prior. In the following, we briefly introduce the KSVD algorithm.

To be concrete, a noisy image $\mathbf{Q}$ can be represented as follows,

$\displaystyle\mathbf{Q}=\mathbf{X}+\mathbf{Z},$ (7)

where $\mathbf{X}$ denotes the original clean image that we want to obtain and
$\mathbf{Z}$ is the noise that we aim to remove; hereby, we abuse the notation
of $\mathbf{X}$ to show both the clean images and videos in the SCI forward
model without confusion. The only information we have is the noisy observation
$\mathbf{Q}$.

Under the assumption that by dividing the $\mathbf{X}$ into small
(overlapping) patches, each (vectorized) patch
$\mathbf{x}_{i}\in{\mathbb{R}}^{p^{2}}$ (with $p\times p$ the patch size and
usually $p=8$) could be represented by an over-complete matrix $\mathbf{D}$
which we call the dictionary, the problem turns to an optimization problem
[25] as follows,

$\displaystyle{\mathbf{x}}_{i}={\mathbf{D}}{\mathbf{\alpha}}_{i},\quad\mathbf{\hat{\alpha}}_{i}=\arg\min_{\mathbf{\alpha}_{i}}||\mathbf{{q}}_{i}-\mathbf{{D}}{\mathbf{\alpha}}_{i}||_{2}^{2}+\mu||\alpha_{i}||_{0},~{}\forall
i=1,\dots N_{p},$ (8)

where ${\alpha}_{i}$ represents the sparse coefficients of the patch
${\mathbf{x}}_{i}$. $\mathbf{{q}}_{i}$ is the corresponding (vectorized) noisy
observation, ${\mu}$ is a weighting parameter and $N_{p}$ is the total number
of patches. Solving norm-minimization equation with constraint condition like
Equation (8) gives the estimation ${\hat{\alpha}}_{i}$. Specifically, given
the dictionary $\mathbf{D}$, ${\hat{\alpha}}_{i}$ can be solved by the
orthogonal matching pursuit (OMP) algorithm [26].

Regarding the dictionary $\mathbf{D}$, it needs to be trained by jointly
considering all the image patches (in the training set) as these patches share
the same dictionary. Furthermore, $\mathbf{D}$ can be pre-trained or learned
during the noise removal process. Please refer to the KSVD paper [25] for
details.

By plugging this KSVD denoising prior into the denoising step (6) in the PnP
framework, we achieve the proposed PnP-KSVD algorithm for SCI reconstruction.
In our work, we use the pre-trained $\mathbf{D}$ and conduct denoising in each
frame of the SCI measurement. An online dictionary learning or adaptively
updating the dictionary during the reconstruction can lead to better results
but takes a longer time. We leave this for future work.

##  4 experiment

###  4.1 Dataset

In our experiments, we use simulation data including Traffic, Runner, Crash,
Kobe [27], and Drop datasets. The original images are resized to $256\times
256$ through down sampling. Following the setting in [28], eight $(B=8)$
consequent frames are modulated by shifting binary masks and then collapsed to
a single measurement.

###  4.2 Compared method

For both quantitative and qualitative evaluations, we compare our method with
GAP-TV [29], which solves the video SCI problem with a total variation
minimization, and serves as a proper baseline for SCI reconstruction.
Reconstruction results on the above-mentioned simulation datasets are taken
into comparison. Since the number of measurements in each simulation data sets
varies, we use the mean value across all testing trials to evaluate the
reconstruction performance. For a comprehensive comparison, we use two image
quality metrics, peak signal to noise ratio (PSNR) and structural similarity
index (SSIM) [30].

Figure 3: SCI reconstruction results comparison. Using PnP-KSVD algorithm and
GAP-TV algorithm to reconstruct videos named Traffic, Runner, Crash, Kobe,
Drop respectively. We could see that PnP-KSVD performs better than GAP-TV in
reconstruction qualitatively.

Figure 4: Real data results. (a) and (b) show real-world reconstruction using GAP-TV and PnP-KSVD, respectively. Results in (a) are more blurry and noisy in comparison to those in (b). Table 1: Performance comparison of PnP-KSVD and GAP-TV on four simulation scenes using PSNR (in dB, left in each cell) and SSIM (right in each cell) as the metrics. Mean values of PSNR and SSIM corresponding to experimental results are shown as follows. We could see that PnP-KSVD performs better than GAP-TV under metrics of PSNR and SSIM quantitatively. Algorithm | Traffic | Runner | Kobe | Drop | Average
---|---|---|---|---|---
PnP-KSVD | 21.7148, 0.801 | 33.2566, 0.925 | 30.9758, 0.888 | 39.7027, 0.974 | 31.4125, 0.897
GAP-TV | 20.7958, 0.707 | 30.6506, 0.921 | 28.8466, 0.895 | 36.1566, 0.976 | 29.1124, 0.875

###  4.3 Experiment results

Examining Table 1, our method outperforms GAP-TV considerably on various
datasets, including Traffic, Runner, Kobe, Drop. And our mean values of PSNR
and SSIM across all the testing trials are also superior to that of GAP-TV.
Figure 4 and Figure 4 presents the visual comparisons of real data results. In
comparison to GAP-TV, our method can provide finer details.

##  5 conclusion

In this paper, we revisit the kernel singular value decomposition (KSVD), a
dictionary learning method, and apply it for SCI reconstruction. By
integrating KSVD-based denoiser into the PnP framework, we provide a new
baseline for video SCI reconstruction. Experiments on various datasets show
that with a well-trained dictionary, the results of our proposed PnP-KSVD are
better than GAP-TV. In addition, our method retains a good trade-off between
quality and training difficulty.

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