Dedicated to Anna and Jack

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most $1$
(that is, it consists of a matching and some isolated vertices) and the red
subgraph has minimum degree at least $1$ and contains no $3$-edge path.
Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring crumby.
Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky [2] constructed an infinite family refuting the above conjecture.
Their prototype counterexample is $2$-connected, planar, but contains a $K_4$-minor and also a $5$-cycle.
This leaves the above conjecture open for some important graph classes: outerplanar graphs, $K_4$-minor-free graphs, bipartite graphs.
In this regard, we prove that $2$-connected outerplanar graphs, subdivisions of $K_4$ and $1$-subdivisions of cubic graphs admit crumby colorings.
A subdivision of $G$ is genuine if every edge is subdivided at least once.
We show that every genuine subdivision of any subcubic graph admits a crumby coloring.
We slightly generalise some of these results and formulate a few conjectures.

§ INTRODUCTION

Our notations and terminology mostly follow the Graph Theory book by Bondy and Murty [3].
In particular, we call a graph subcubic if it has maximum degree at most 3 and $P_k$ denotes a path on $k$ vertices.

Thomassen [8] gave an intricate inductive proof of the following result, a classical case of Wegner's conjecture [10]: the square of every planar cubic graph is $7$-colorable.
In the same paper, Thomassen formulated an attractive conjecture, which would imply the aforementioned result.

Every $3$-connected cubic graph has a red-blue vertex coloring such that
the blue subgraph has maximum degree at most $1$
(that is, it consists of a matching and some isolated vertices) and the red
subgraph has minimum degree at least $1$ and contains no $3$-edge path.

Since every monochromatic subgraph is small, but the conditions in the two colors are asymmetric, making this coloring somewhat irregular, we call this a crumby coloring.
From a classical graph decomposition point of view, we are seeking graph classes such that every member admits a crumby coloring.
As it was remarked early, the $3$-prism does not have a crumby coloring. For some time, it looked like this is the only counterexample.
Supporting this, Barát [1] showed that every subcubic tree has a crumby coloring.
However, Bellitto et al. [2] found a construction that produces an infinite family of $3$-connected cubic counterexamples and also $2$-connected planar graphs without crumby colorings.
This fact gives evidence that the intricate induction by Thomassen is somewhat unavoidable.
On the other hand, it leaves open the possibility that crumby colorings might exist for some important graph classes. For instance, outerplanar graphs or bipartite graphs.
Indeed, in Section <ref> we show that any $2$-connected subcubic outerplanar graph admits a crumby coloring even if an arbitrary vertex's color is prescribed.

The fact that we can prescribe the color of a vertex is useful in the  following sense. We believe that crumby colorings exist for every subcubic outerplanar graph.
However, there are various difficulties to extend the results on $2$-connected graphs to all outerplanar graphs.
In a general outerplanar graph there might be trees attached to $2$-connected blocks or between them.
Since Conjecture <ref> holds for trees, it gives some hope to combine these two results as building bricks, where having the extra freedom of prescribing the color of a vertex comes into the picture.

The following theorem is a straightforward strengthening of a result of Barát [1]. It is routine to check that the original proof literally holds for this version.

Every subcubic tree admits a crumby coloring such that the color of a leaf is prescribed.

We strengthen this result further in Section <ref>.
This allows us to significantly decrease the number of problematic attached trees.

As a weakening of Conjecture <ref>, we conjecture that every $K_4$-minor-free graph admits a crumby coloring. This class is interesting for several reasons. Since outerplanar graphs are $K_4$- and $K_{2,3}$-minor-free, this would be a natural extension step from outerplanar graphs. It also concurs with the fact that all known counterexamples to Conjecture <ref> contain $K_4$-minors. In contrast, we show a crumby coloring of any subdivision of $K_4$ in Section <ref>.

However, we first prove that a special class of bipartite graphs admit crumby colorings in Section <ref>.
They are the $1$-subdivisions of cubic graphs, that arise from cubic graphs by adding an extra vertex on each edge.
In this way, we form bipartite graphs, where the vertices in one class have degree 2 and in the other class degree 3.
Motivated by these observations, we introduced the notion of genuine subdivision of a graph $G$ in the latter part of Section <ref>. That is a graph $H$ which one get from $G$ by subdividing every edge of $G$ with at least one vertex. As a generalization, we prove that every genuine subdivision of any subcubic graph admits a crumby coloring.
A crucial idea in both proof is to use the maximum matching of the original graph. To this end, we employ the famous Edmonds-Gallai decomposition theorem.

§ BIPARTITE GRAPHS AND SUBDIVISIONS

Despite the infinite family of counterexamples in [2], we still believe that Conjecture <ref> holds for most subcubic graphs.
We pose the following

Every subcubic bipartite graph admits a crumby coloring.

We can prove this for a special class of bipartite graphs, where the degrees are all 2 in one class and 3 in the other class.
In the proof, we apply the Edmonds-Gallai decomposition theorem [4, 5] that gives us information about the structure of the maximum matchings of a graph $G$.
We recall that $P_k$ denotes a path with $k$ vertices and $N(X)$ denotes the set of neighbors of a vertex set $X$.
A graph $G$ is hypomatchable or factor-critical if for every vertex $x$, the graph $G-x$ has a perfect matching.

Let $G$ be a graph and let $A\subseteq V(G)$ be the collection of all vertices $v$ such that there exists a maximum size matching which does not cover $v$.
Set $B=N(A)$ and $C=V(G)\setminus (A \cup B)$. Now

$(i)$ Every odd component $O$ of $G-B$ is hypomatchable and $V(O)\subseteq A$.

$(ii)$ Every even component $Q$ of $G-B$ has a perfect matching and $V(Q)\subseteq C$.

$(iii)$ For every $X\subseteq B$, the set $N(X)$ contains vertices in more than $|X|$ odd components of $G-B$.

In what follows, we study subdivisions of cubic graphs.
If we add precisely one new vertex on each edge, then the resulting graph is a $1$-subdivision.
We support Conjecture <ref> by showing the following

Let $S(G)$ be the $1$-subdivision of a cubic graph $G$.
The bipartite graph $S(G)$ admits a crumby coloring.

The idea of the proof is to color the original vertices (in $G$) red and color the subdivision vertices blue.
If $G$ admits a perfect matching $M$, then we recolor the subdivision vertices on $M$ to red.
This results in a crumby coloring consisting of red $P_3$-s and blue singletons.
We refer to this idea later as the standard process.
For instance, every 2-edge-connected graph $G$ admits a perfect matching by Petersen's Theorem.
If the graph $S(G)$ is the 1-subdivision of such $G$, then the standard process gives a crumby coloring of $S(G)$.

In what follows, we modify this simple idea to the general case, where $G$ is any cubic graph.
If $G$ does not possess a perfect matching, we can still consider a maximum size matching in $G$ and use the Edmonds-Gallai decomposition.

Let $G$ be a cubic graph, and let $B$ be the set given by the Edmonds-Gallai theorem.

Any isolated vertex in $B$ must be connected to at least two odd components of $G-B$.
The third edge might go to a third odd component, an even component or to one of the first two odd components.

Initially, let every vertex of $G$ be red and
every subdivision vertex blue.
We recolor a few vertices as follows.
In every even component, there exists a perfect matching and we recolor the subdivision vertices on the matching edges to red.

Consider the vertex sets $A$ and $B$ of the Edmonds-Gallai decomposition.
Contract the components of $A$ to vertices to get $A^*$.
The bipartite graph $(A^*,B)$ satisfies the Hall-condition by property $(iii)$. Therefore, we find a matching $M$ covering $B$.
We recolor the subdivision vertices of the matching edges in $M$ to red.
We continue with the odd components corresponding to the vertices of $A^*$ saturated by $M$.
In these components, we use property $(i)$ and find an almost perfect matching.
The subdivision vertices on these matching edges are colored red as well.
So far we only created red $P_3$-s separated by blue singletons.
What is left to consider is the union of odd components corresponding to unsaturated vertices of $A^*$.
Let $H$ be such a component.
Let $x$ be an arbitrary vertex and consider a perfect matching in $H-x$ by property $(i)$.
We recolor the subdivision vertices on these matching edges to red.
Let $y$ be a neighbor of $x$ in $H$ and denote the subdivision vertex on the edge $xy$ by $v_{xy}$.
Let $x$ and $v_{xy}$ be red and $y$ blue.
Let us check that the neighborhood of $y$ remains a crumby coloring.
Since there was a matching edge $zy$ and both $z$ and $v_{yz}$ are red, this is appropriate.
There is a third edge $wy$ incident to $y$, and the subdivision vertex on this edge is blue forming a blue $P_2$ together with $y$, this is also appropriate.

After doing this for every remaining odd component $H$ a crumby coloring of $S(G)$ arises.

Next, we complement the previous result.
Here we allow all longer subdivisions.

Let $G$ be a cubic graph.
Let $H$ be an arbitrary subdivision of $G$ such that every edge is subdivided at least twice.
The graph $H$ admits a crumby coloring.

Let us color the original vertices of $G$ blue.
We find that almost any subdivided edge admits a crumby coloring such that the end-vertices are singleton blues.
The only exception is the case with 4 subdivision vertices.
In particular, we use the following colorings for the internal vertices:
$rr$, $rrr$, $rrrb$, $rrbrr$, $rrrbrr$, $rrbbrrr$, $rrbrrbrr$ etc.

Let us use these particular colorings on $H$.
We might create some blue stars with 2 or 3 leaves.
Apart from that, the coloring satisfies the crumby conditions.
Now we recolor the problematic blue centers of these stars red.
If vertex $c$ is such a center, and there was a blue 3-star at $c$, then we recolor the neighbor $n_1$ of $c$ red and recolor the neighbor $n_2$ of $n_1$ blue.
If vertex $c$ was the center of a blue 2-star, then we have to consider two cases according to the red neighbor $v$ of $c$.
If $v$ was the end-vertex of a red $P_3$, then we do the same recoloring as in the previous case, but also recolor $v$ to blue.
If $v$ was the end-vertex of a red $P_2$, then the recoloring of $c$ creates a red $P_3$ and we are done.

The process terminates with a crumby coloring of $H$.

Motivated by the results of this section, we introduce the following notion.
A graph $H$ is a genuine subdivision of a graph $G$ if every edge of $G$ contains at least one subdivision vertex.

Every genuine subdivision $S(G)$ of any subcubic graph $G$ admits a crumby coloring.

We may assume that $G$ is connected (otherwise one can repeat the same argument on the connected components).
Our proof uses the same ideas as the proof of Theorem <ref>. To make the argument more transparent,
we prove a lemma assuming $G$ has a perfect matching.

Let $G$ be a subcubic graph, which has a perfect matching.
Every genuine subdivision $S(G)$ of $G$ admits a crumby coloring.

First, let us color the vertices of $G$ red and suppose that $G$ has a perfect matching $M$. Our aim is to color the vertices on all the edges outside of $M$ satisfying the crumby conditions such that every vertex of $G$ remains red, but none of them is the end of a red $P_3$. In the last step of the proof, we show that the vertices on the edges of $M$ can be colored in such a way that the possible problems (i.e. still isolated red vertices of $G$) disappear.

Since $G$ is a subcubic graph, the subgraph $G-M$ is a disjoint union of cycles and paths.
In both cases, we give a crumby coloring by taking one edge of $G-M$ at a time (which is a path on at least 3 vertices in $S(G)$) and color its vertices such that none of the endpoints become the end of a red $P_3$, but most of them get a red neighbor. In Table <ref>, we summarize crumby colorings of the paths $P_k$ on $k$ vertices for different purposes, which we use later. Note that we highlighted by capital letters those cases, in which the aim is not attainable (for $k\ge 8$ all of the aims are).

\[
\begin{array}{|c|c|c|c|}
\hline
k & \makecell{\mathrm{all~of~the} \\ \mathrm{endpoints~are} \\ \mathrm{singleton~red}} & \makecell{\mathrm{all~of~the} \\ \mathrm{endpoints~are} \\ \mathrm{in~a~red~} K_2} & \makecell{\mathrm{one ~endpoint~is} \\ \mathrm{a~singleton~red,} \\ \mathrm{and~the~other}\\ \mathrm{is~in~a~red~} K_2} \\
\hline
3 & rbr & RBR & RBR \\
\hline
4 & rbbr & RRBR & rrbr \\
\hline
5 & RRBBR & rrbrr & rrbbr \\
\hline
6 & rbrrbr & rrbbrr & RRBBRR \\
\hline
7 & rbrrbbr & RRBRRBR & rrbrrbr \\
\hline
8 & rbbrrbbr & rrbrrbrr & rrbbrrbr \\
\hline
\end{array}
\]

Crumby colorings of $P_k$ for different purposes

In the first step of the proof, we are going through the edges of all the path components (starting from a leaf) and all cycles (in a cyclic order) and use the second column of Table <ref> on every edge in order to leave as few red singletons as possible. Thus a red singleton either has degree 3 and an edge outside $M$ with exactly one subdivision vertex or it has degree 2 and the only edge of its path component has 1, 2 or 5 subdivision vertices, or it has degree 1 and hence we have not considered it yet.

In the final correction step, we color the vertices along the edges of $M$ using the second column of Table <ref>. We emphasize that after this, there is still no red vertex of $G$, which is an end of a red $P_3$ (some of them can be a middle vertex on a red $P_3$). There are four situations, in which this does not eliminate all the singleton red vertices. In the sequel, we solve each of these situations.

Case 1: Both $u$ and $v$ are red singletons and there is exactly 1 subdivision vertex $x$ along $uv$. If $d(u)=2$, then we color $u$ to blue and $x$ to red.
If $d(u)=3$, then there is an incident edge with 1 subdivision vertex $y$, hence we can recolor $y$ and $x$ red and $u$ blue.

Case 2: Both $u$ and $v$ are red singletons and there are exactly 5 subdivision vertices $x_1,x_2,x_3,x_4,x_5$ along $uv$. If $d(u)=2$, then we color $u$, $x_3$ and $x_4$ blue and $x_1$, $x_2$ and $x_5$ red. If $d(u)=3$, then there is an incident edge with 1 subdivision vertex $y$, hence we can recolor $y$ red, and use the same coloring along $uv$ as before.

Case 3: Both $u$ and $v$ are red singletons and there are exactly 2 subdivision vertices $x_1,x_2$ along $uv$. We color $u$ and $v$ blue and $x_1$ and $x_2$ red. If $u$ or $v$ have degree 3, then there is an incident edge with 1 subdivision vertex $y$, hence we can recolor $y$ red.

Case 4: Suppose that $v$ is a red singleton and $u$ has red neighbors and there is exactly 1 subdivision vertex $x$ along $uv$. We color $v$ blue and $x$ red. Again, if $u$ or $v$ have degree 3, then there is an incident edge with 1 subdivision vertex $y$, hence we can recolor $y$ red.

Notice that the recolorings in the previous cases cannot create a large blue component, since the other edges incident to $u$ cannot have two blue vertices next to $u$. Lastly, if $d(u)=d(v)=1$, then by the connectivity assumption, this path is the whole $G$ hence it has a crumby coloring. This concludes the case that $G$ has a perfect matching.

By Lemma <ref>, we may assume that $G$ does not have a perfect matching, and we use the Edmonds-Gallai decomposition. The idea is to fix a maximal matching $M$ of $G$ and color the vertices along the edges of the subgraph $G-M$ first. Let $A' \subset A$ consist of those odd components, which have exactly one vertex unsaturated by $M$. For every $O_i\in A'$, denote the uncovered vertex by $z_i$. We can perform the first step of the proof of Lemma <ref> on the restricted subgraph of $S(G)$ on the vertex set $V' = V(S(G)) \setminus \bigcup_{O_i \in A'} z_i$. This results in an almost crumby coloring, the only exceptions can be some red singletons of $G$. Note that there may be some red $P_3$ components, but the vertices of $G$ can never be their endpoints.

In the second step, we color the vertices along the edges incident to each $z_i$. As before, we use the second column of Table <ref> to color those vertices, but make sure that $z_i$ surely gets a red neighbor. It can be done, except when all of the edges incident with $z_i$ have exactly 1 subdivision vertex. But in that case, we recolor these subdivison vertices to red and $z_i$ to blue.

Now only the vertices along the edges of $M$ remained uncolored. It is time to perform the final correction step of the proof of Lemma <ref>, which concludes the proof.

One can observe that in the proof of Theorem <ref> we actually do not rely on the existence of subdivision vertices on those edges of the original subcubic graph $G$, which have an endpoint of degree $1$.

§ OUTERPLANAR GRAPHS

We know that Conjecture <ref> holds for trees and fails in general for $2$-connected planar graphs.
A natural minor-closed class between the aformentioned classes is the class of outerplanar graphs.
As the first step, we prove the following.

Let $G$ be a $2$-connected subcubic outerplanar graph and let $v$ be a vertex of $G$. We may prescribe the color of $v$ and find a crumby coloring of $G$.

We consider $G$ together with its outerplanar embedding. An ear decomposition of a $2$-connected graph $G$ is a series of graphs $G_0, G_1, \dots, G_k=G$ such that $G_0$ is a cycle and $G_i=G_{i-1}\cup E_i$, where each $E_i$ is a path that have its two leaves in common with $G_{i-1}$. We may assume that $G_0$ is a bounded face containing the vertex $v$, and if $d(v)=3$ then let $v$ be an endpoint of $E_1$.
Since $G$ is a 2-connected outerplanar graph, it has an open ear decomposition such that on each ear the attachment vertices (endpoints) are adjacent.
The endpoints of the ears are different by the subcubic property.

In general, we start the coloring process with $G_0$.
There is an exceptional situation though.
If $d(v)=3$, then we immediately add the other bounded face containing $v$ as the first ear to form $G_1$.
We first show that the starting subgraph ($G_0$ or $G_1$ depending on the degree of $v$) of $G$ has a crumby coloring.
Secondly, we show that if $G_i$ has a crumby coloring, then $G_{i+1}$ also admits a crumby coloring in which the colors of the vertices of $G_i$ are unchanged except possibly the endpoints of the ear $E_{i+1}$.
This procedure leads to a crumby coloring of $G$.
During the coloring process, we establish and maintain a significant property of our crumby coloring.
Namely, we never color two adjacent vertices of degree 2 (with respect to the current subgraph) blue, unless we know that there is no later ear with this pair of endpoints. Let us call it .

We use the shorthand $r$ for red and $b$ for blue.

Starting the procedure: If $d(v)=2$, then the only bounded face of $G$ containing vertex $v$ is a cycle of length $k\ge 3$. We know that $G_0=C_k$ has a crumby coloring, but we need more.
For $k=5$, we must observe that there exist adjacent vertices $x$ and $y$ in $C_5$, which are not the endpoints of any ear[Vertex $x$ might be the endpoint of ear $E_i$, but in that case $y$ is not the other endpoint.].
Therefore, we color $x$ and $y$ blue and the remaining 3 vertices red in order to establish . The required crumby colorings of $C_k$ are shown in the following table for $k\le 8$.
For larger $k$, we get a crumby coloring by starting with $rrb$ and continue with the crumby coloring of $C_{k-3}$.

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
k & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\mathrm{crumby~coloring~of~}C_k & rrb & rrrb & rrrbb & rrbrrb & rrbrrrb & rrrbrrrb \\
\hline
\end{array}
\]

If $k\ne 5$, then we can rotate the above given crumby colorings such that $v$ gets its prescribed color.
We notice the following for $k=5$.
If the prescribed color of $v$ is red, then we can choose two adjacent vertices $x$ and $y$ of the $5$-face containing $v$ (distinct from $v$), for which there is no (later) ear connecting them.
We color $x$ and $y$ blue and the rest red to establish .
If $v$ is supposed to be blue, then  holds immediately as we rotate the given coloring of $C_5$ to make $v$ blue.

If $d(v)=3$, then we show a crumby coloring of $G_1$, the subgraph spanned by the two bounded faces containing $v$.
One endpoint of the first ear $E_1$ is $v$.
We denote the other endpoint by $u$.
Suppose that the boundary of $G_0$ is $(u, w_1, w_2, \dots, w_k, v)$, where $u$ and $v$ are adjacent, and the internal points of $E_1$ are $(z_1,z_2,\dots,z_{\ell})$ from $u$ to $v$.
Firstly, we give a crumby coloring for $k=\ell=2$.
There are two cases depending on the prescribed color of $v$, see Figure <ref>.

Crumby colorings for $k=\ell=2$

In the remaining cases, we color $u$ and $v$ differently and
assume the prescribed color of $v$ to be red.
If it was blue, then $u$ plays the role of $v$. In the following table, we summarize the initial colorings depending on the values of $k$ and $\ell$.

\[
\begin{array}{|c|c|c|c|}
\hline
~ & \makecell{\ell=3d+1 ~(d \in \mathbb{N}) \\ \mathrm{the~color~of~} \\ (z_1,z_2,\dots,z_{\ell})} & \makecell{\ell=3d+2 ~(d \in \mathbb{N}) \\ \mathrm{the~color~of~} \\ (z_1,z_2,\dots,z_{\ell})} & \makecell{\ell=3d+3 ~(d \in \mathbb{N}) \\ \mathrm{the~color~of~} \\ (z_1,z_2,\dots,z_{\ell})} \\
\hline
\makecell{k=3c+1 ~ (c\in \mathbb{N}) \\ \mathrm{let~the~color~of~} \\ (w_1,w_2,\dots,w_k) \\ \mathrm{be~} (rrb)^{c}r} & (rrb)^{d}r & \makecell{\mathrm{if~}d=0:~ br\\ \mathrm{if~}d\ge1:~ (rrb)^{d-1}rrrbr} & (rrb)^{d}rrb \\
\hline
\makecell{k=3c+2 ~ (c\in \mathbb{N}) \\ \mathrm{let~the~color~of~} \\ (w_1,w_2,\dots,w_k) \\ \mathrm{be~} \underbrace{(rrb)^{c-1}rrr}_{\mathrm{if~} c~\ge~1}br} & (rrb)^{d}r & \makecell{\mathrm{if~}d=0:~ br~(\mathrm{for~}\\ c=0,\mathrm{~see~Figure~}\ref{kl2}) \\ \mathrm{if~}d\ge1:~ (rrb)^{d-1}rrrbr} & (rrb)^{d}rrb \\
\hline
\makecell{k=3c+3 ~ (c\in \mathbb{N}) \\ \mathrm{let~the~color~of~} \\ (w_1,w_2,\dots,w_k) \\ \mathrm{be~} (rrb)^{c}rrb} & (rrb)^{d}r & \makecell{\mathrm{if~}d=0:~ br \\ \mathrm{if~}d\ge1:~ (rrb)^{d-1}rrrbr} &  \makecell{\mathrm{if~}d=0:~ brr \\ \mathrm{if~}d\ge1:~ (rrb)^{d-1}rrrbrr} \\
\hline
\end{array}
\]

The crumby colorings we use if $d(v)=3$, depending on the values of $k$ and $\ell$

It is immediate that these are crumby colorings, and have , thus the coloring process can start.

Adding a new ear: Let us assume that $G_i$ has already been colored and  holds.
We consider the next ear $E_{i+1}=(x,z_1,z_2,\dots,z_{\ell},y)$ of the ear decomposition.
implies that the color of the endpoints of this ear cannot be both blue.
We assume $x$ is red.

Case $\ell=1$, $rb$: If $x$ is not an endpoint of any red $P_3$, then we color $z_1$ red.
If $y$ is a singleton blue vertex, then we color $z_1$ blue. Otherwise, we interchange the color of $x$ and $y$, their previous components remained admissible, and color $z_1$ red.

Case $\ell=1$, $rr$: We color $z_1$ blue.

Case $\ell=2$, $rb$:
In the following table, we summarize the possibilities and give a suitable coloring for the endpoints and the internal points of the ear.

\[
\begin{array}{|c|c|c|c|c|}
\hline
\ell=2,~rb: & \makecell{x \mathrm{~is~an~endpoint} \\ \mathrm{of~a~red~} P_3 \\ \& \\ y \mathrm{~is~a~singleton} \\ \mathrm{~blue~vertex}} & \makecell{x \mathrm{~is~an~endpoint} \\ \mathrm{of~a~red~} P_3 \\ \& \\ y \mathrm{~is~in~a~blue} \\ K_2 \mathrm{~component}} & \makecell{x \mathrm{~is~not~an~endpoint} \\ \mathrm{of~a~red~} P_3 \\ \& \\ y \mathrm{~is~a~singleton} \\ \mathrm{~blue~vertex}} & \makecell{x \mathrm{~is~not~an~endpoint} \\ \mathrm{of~a~red~} P_3 \\ \& \\ y \mathrm{~is~in~a~blue} \\ K_2 \mathrm{~component}} \\
\hline
\makecell{\mathrm{color~of~} \\ (x,z_1,z_2,y)} & brrb & brrr & rrbb & rrbr \\
\hline
\end{array}
\]

Case $\ell=2$, $rr$: If both $x$ and $y$ are red, then at most one of them can be an endpoint of a red $P_3$.
We may assume that there is no red $P_3$ in $G_i$ ending in $x$. We color $z_1$ red and $z_2$ blue.

Case $\ell=3$, $rb$: We color $z_1,z_2,z_3$ to $brr$.

Case $\ell=3$, $rr$: We may assume $x$ is not an endpoint of any red $P_3$.
If there is no (later) ear with endpoints $z_2$ and $z_3$, then we color $z_1,z_2,z_3$ to $rbb$ maintaining .
On the other hand, if there exists an ear with endpoints $z_2$ and $z_3$ together with $m$ internal points (denote them by
$w_1,\dots,w_m$), then we merge the two ears and add them to
$G_i$ in one step.
We give a crumby coloring of the resulting graph in the following table.
Independent of the value $m$, we color $z_1$ and $z_3$ blue in order to avoid conflicts with the rest of the coloring.
We color $z_2$ red as well as $w_1$.
The coloring of $(w_1,w_2,\dots,w_m)$ is only shown for $m\le 6$ in the following table.
For greater $m$, we use the crumby coloring for $m-3$ and add $brr$ at the end.

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\ell=3,~rr: & m=1 & m=2 & m=3 & m=4 & m=5 & m=6 \\
\hline
\makecell{\mathrm{color~of~} \\ (w_1,w_2,\dots,w_m)} & r & rr & rrb & rbrr & rrbrr & rrbrrr \\
\hline
\end{array}
\]

Case $\ell=4$, $rb$: We color $z_1,z_2,z_3,z_4$ to $brrr$.

Case $\ell=4$, $rr$: We color $z_1,z_2,z_3,z_4$ to $brrb$.

Case $\ell=5$, $rb$: Depending on the type of the components of $x$ and $y$ we need to color the points of the next ear a bit differently. In the following table, we summarize the possibilities and give a suitable coloring for the endpoints and the internal points of the ear.

\[
\begin{array}{|c|c|c|c|}
\hline
\ell=5,~rb: & \makecell{x \mathrm{~is~an~endpoint} \\ \mathrm{of~a~red~} P_3 \\ \& \\ y \mathrm{~is~a~singleton} \\ \mathrm{~blue~vertex}} & \makecell{x \mathrm{~is~an~endpoint} \\ \mathrm{of~a~red~} P_3 \\ \& \\ y \mathrm{~is~in~a~blue} \\ K_2 \mathrm{~component}} & \makecell{x \mathrm{~is~not~an~endpoint} \\ \mathrm{of~a~red~} P_3} \\
\hline
\makecell{\mathrm{color~of~} \\ (x,z_1,z_2,z_3,z_4,z_5,y)} & brrbrrb & brrbrrr & rrbrrrb \\
\hline
\end{array}
\]

Case $\ell=5$, $rr$: We color $z_1,z_2,z_3,z_4,z_5$ to $brrrb$.

Case $\ell=6$, $rb$: We color $z_1,z_2,z_3,z_4,z_5,z_6$ to $brrbrr$.

Case $\ell=6$, $rr$: We may assume $x$ is not an endpoint of any red $P_3$. We color
$z_1,z_2,z_3,z_4,z_5,z_6$ to $rbrrrb$.

For $\ell\ge 7$, we create a crumby coloring using the cases for smaller values of $\ell$.
We start by $brr$, and continue with the given coloring of the ear with $\ell-3$ internal points. By starting with $brr$, we trace back to a similar situation for $\ell-3$ internal points in which $z_3$ takes over the role of $x$.
We remark that $z_3$ is in a red $K_2$ component, thus cannot be an endpoint of a red $P_3$.

A general outerplanar graph is not necessarily 2-connected. It is glued together from 2-connected blocks in a tree-like manner.
Some of the edges can form a tree hanging from a vertex of a block, or connecting a number of $2$-connected outerplanar components. In our case, the maximum degree 3 condition gives some extra structural information.
We are convinced that the natural extension of Theorem <ref> to all subcubic outerplanar graphs holds.

Every outerplanar graph with maximum degree $3$ admits a crumby coloring.

Considering this problem, one gets the impression that particular small trees attached to the vertices of a 2-connected outerplanar graph make the difficulty. It turns out, that most trees do not cause any problems at all.
To prove our statement, we also need the following.

Any subcubic tree $T$ admits a crumby coloring such that the color of an arbitrary vertex of degree $2$ is prescribed, unless $T=P_3$.

If $T=P_3$, then the middle vertex cannot be blue in a crumby coloring. Therefore, this is an exception. From now on, we assume that $T$ has at least 4 vertices. Every tree admits a crumby coloring by Theorem <ref>. Let us suppose that $T$ is a minimal example of a tree, which has a vertex $v$ of degree 2 such that in any crumby coloring of $T$, the color of $v$ must be red. We think of $v$ as the root, and denote the two neighbors of $v$ by $x$ and $y$.

If any of the neighbors of $v$ is of degree 2, say $x$, then we can delete the edge $vx$ and consider the two remaining trees $T_v$ rooted at $v$ and $T_x$ rooted at $x$. We get a contradiction by using Theorem <ref> with prescribed color red on $x$ and blue on $v$ in the respective trees.

If $d_T(x)=2$, then we get a contradiction

Since $T$ has at least 4 vertices, we may assume that $d_T(x)=3$. As before, we get a contradiction if the color of $x$ can be red in a crumby coloring of $T_x$, since we can color $v$ blue and use Theorem <ref> on $T_v$. Therefore, let us suppose that $T_x$ is a tree, for which the degree 2 vertex $x$ can only be colored blue in a crumby coloring. Denote the neighbors of $x$ in $T_x$ by $z$ and $w$.

Due to the same reasons as above, the degree of $z$ and $w$ cannot be 2 in $T_x$. It cannot be 1 either, since in that case $T_x$ has a crumby coloring in which that leaf is prescribed red. Consequently $x$ is also red, which is a contradiction. Hence $d_{T_x}(z)=d_{T_x}(w)=3$, and by the minimality of $T$, we know that $T_z$ admits a crumby coloring such that the degree 2 vertex $z$ is blue. Now we may delete the edge $xz$ and precolor the degree 1 vertex $x$ red and find a crumby coloring of a subgraph of $T_x$. However, we can add back the edge $xz$ giving a crumby coloring of $T_x$ with red $x$, a contradiction. The same holds for $T_w$, but there is one exception: if both $T_z=T_w=P_3$. In Figure <ref>, we give a crumby coloring of $T_x$ so that $x$ is red, which concludes the proof.

A crumby coloring of $T_x$ such that $x$ is red

If $G$ is a graph that admits a crumby coloring, and $T$ is an arbitrary tree with a leaf $v$, then let $G_T$ denote a graph which we get by identifying $v$ with any vertex of $G$.
Observe that if an attachment tree $T$ is not $K_2$ or $K_{1,3}$, then it is trivial to get a crumby coloring of $G_T$. The key idea is to assign different colors to $v$ and its only neighbor $x$ inside $T$. Consider a crumby coloring of $G$, therefore the color of $v$ is given, and color $x$ differently. By Theorem <ref> and Theorem <ref> (depending on $d_T(x)$), we can extend this coloring to a crumby coloring of $T-v$ which results in a crumby coloring of $G_T$.

Therefore, it is indifferent with respect to crumby colorings to attach trees, which are not isomorphic to $K_2$ or $K_{1,3}$. In the sequel, we assume that every attachment tree is either $K_2$ or $K_{1,3}$.

Now, we prove a basic instance of Conjecture <ref> relying on Theorem <ref>.

Let $C$ be a cycle with vertices $v_1,\dots,v_k$, plus we might attach arbitrary trees $\{T_i\}$ to vertices $\{v_i\}$ of $C$, where $i\in I$ and $I\subseteq [k]$. The resulting graph $G$ admits a crumby coloring.

We may assume that each attachment tree is isomorphic to $K_2$ or $K_{1,3}$ by Remark <ref>.

Our arguments slightly vary depending on some properties of $G$, thus we explain them separately.

Notice that some vertices of $C$ have attachments and some do not.
In the latter case, the vertex is called empty. First, let us assume that there are no empty vertices at all.

We notice that even $k$ is simple.
We color the vertices of $C$ alternately red and blue.
This gives the prescribed color of a leaf $v_i$ in the tree $T_i$. We color $T_i$ using Theorem <ref> for each $i=1,\dots,k$.
These colorings together form a crumby coloring of $G$.

Assume now that $k$ is odd.
We try to reuse the previous strategy by cutting off two consecutive vertices $v_i$ and $v_{i+1}$ and the trees $T_i$ and $T_{i+1}$ from $G$.
We notice that the remaining graph $H$ admits a crumby coloring by the previous argument. In particular, the first and last vertex on $C-\{v_i,v_{i+1}\}$ receive the same color.

For every $j$ between 1 and $k$, the tree $T_j-v_j$ admits a crumby coloring.

Let us record for every $j$ the color of $u_j$, the neighbor of $v_j$ in $T_j$.
Since $k$ is odd, there is an index $\ell$ such that $u_{\ell}$ and $u_{\ell+1}$ received the same color, say blue.
Now we color $v_{\ell}$ and $v_{\ell+1}$ red and cut the cycle $C$ by removing $\{v_{\ell},v_{\ell+1}\}$. We color $H$ as before such that we color the first and last vertex on $C-\{v_{\ell},v_{\ell+1}\}$ blue.
If $u_{\ell}$ was red, then we interchange colors accordingly.
Altogether, a crumby coloring of $G$ arises.

Unless there are no attachment trees at all (which case is easy), we can find two consecutive vertices of $C$, say $v_1$ and $v_2$ such that there is a tree attached to $v_1$, but $v_2$ has none. We use the following algorithm to color the vertices on $C$ starting by coloring $v_1$ red and $v_2$ blue. Our aim is to color the vertices along $C$ alternately, except in one case, when after a blue vertex we color an empty vertex red.
In that case, the next vertex must be also red.
Observe that if a red vertex is non-empty, then no matter if the tree is $K_2$ or $K_{1,3}$, we can color its vertices maintaining the crumby property. If $v_{i-1}$ is blue, and $v_i$ is an empty red, then $v_{i+1}$ must also be red.
However, it is attainable that $v_{i+1}$ is not an end of a red $P_3$. Only two problems can occur during this algorithm.
Both of them might happen, when we color $v_k$.

If $v_k$ was blue, then $v_1$ might remain a red singleton. However, this cannot be the case by the existence of $T_1$. Otherwise if $v_k$ is red, then we might create a large red component.
If $T_1=K_2$, then the leaf of $T_1$ can be blue.
Hence the red component cannot contain a red $P_4$, since $v_k$ was not an end of a red $P_3$. If $T_1=K_{1,3}$, then the center of $T_1$ must be red, which causes a problem if $v_{k-1}$ is an empty red or $T_k=K_{1,3}$. If we created a red $P_4$, then we recolor $v_1$ to blue and color the remaining vertices in $T_1$ red.

Using the ideas of the previous proofs, we can show Conjecture $\ref{outerplanar}$ for a few other classes.
For instance, if $G$ is glued together from $2$-connected pieces in a tree-like fashion by single edges or paths of any length. Actually the paths might be replaced by any tree, as long as the first vertex outside of a $2$-connected piece has degree $2$. Even if the degree is $3$, our algorithm works except when the tree part between two $2$-connected components is precisely $P_3$, see Figure $\ref{f:fa_p3}$.

In these good cases, we use Theorem $\ref{t:treesplus}$ and Theorem $\ref{t:fa2foku}$ as follows.
We first color a $2$-connected outerplanar subgraph $G_1$.
There is at least one vertex of attachment $x_1$, where a tree $T$ is glued on $G_1$.
Let $y_1$ be the neighbor of $x_1$ in $T$, which we know has degree $1$ or $2$ in $T-x_1$.
We prescribe the color of $y_1$ to be different from that of $x_1$.
We continue this way, until the entire graph is colored.

Problematic situation between two 2-connected outerplanar components $G_1$ and $G_2$

§ SUBDIVISIONS OF THE COMPLETE GRAPH ON 4 VERTICES

Here we consider subdivisions of $K_4$, that has played interesting role in coloring problems [6].
As a strengthening of Hajós' conjecture, Toft [9] posed the problem if every 4-chromatic graph contains a totally odd subdivision of $K_4$.
Thomassen [7] and independently Wang [11] gave an affirmative answer for this.

Bellitto et al. [2] constructed planar graphs refuting Conjecture <ref>.
Characteristically, those counterexamples have $K_4$-minors.
Therefore, we study whether this property has fundamental importance.
We conjecture that every $K_4$-minor-free subcubic graph possesses a crumby coloring.
On the other hand, we show that one topological appearance of $K_4$ is not yet an obstacle.

As the core of the problem, we first consider $\le2$-subdivisions of $K_4$.
That is, every edge contains 0, 1 or 2 subdivision vertices.
It feels straightforward to give a computer-assisted proof, which we did.
We decided to include it as an appendix. However, we opted for a human proof argument.

Let $G$ be a subdivision of $K_4$ such that every edge is divided into at most $3$ parts. The graph $G$ admits a crumby coloring.

Let $V(K_4)=\{A,B,C,D\}$.
Every edge of $K_4$ may remain intact or might be subdivided by either one or two new internal vertices.
Our arguments are organized by the number of intact edges.

If there are no intact edges (genuine subdivision), then color the vertices of $K_4$ red and every subdivision vertex blue.
Since the red vertices are isolated, we must recolor some internal vertices red.
If there are two independent edges of $K_4$ with one internal vertex each, then recolor these internal vertices red.
Otherwise, there exists a vertex of $K_4$, vertex $B$ say, with at least two incident original edges $BC$ and $BD$ with two internal vertices.
There are two possibilities regarding the number of internal points on $AD$ and $AC$ as one can see in Figure <ref>.
If there is only one internal vertex on one of these edges, then change its color to red, and change the color of two internal vertices on $BD$ and $BC$ as shown. In the other case, one can create a crumby coloring just like in the picture on the right. Note that every dashed edge have at least one internal vertex and these are blue. Hence we got a crumby coloring.

Crumby colorings of genuine subdivisions of $K_4$, where vertex $B$ is incident with 2 edges with 2 internal vertices

Next assume there is precisely one intact edge, $AB$ say.
If $CD$ contains exactly one internal vertex $x$, then color $x$ and the vertices of $K_4$ red, and color the other internal vertices blue to get a crumby coloring.
Thus we may assume that $CD$ contains two internal vertices.
If any of the remaining 4 edges of $K_4$ contains two internal vertices, then there is a path on 7 vertices formed along these two particular edges of $K_4$.
We color the vertices of the path $rrbrrbr$ starting from $C$.
It can be extended to a crumby coloring by coloring the vertices of $K_4$ red and the internal vertices blue. In Figure <ref>, we illustrate such a coloring and also cover the only remaining case. Again all dashed edges contain blue internal vertices.

Extendable coloring of the path and a crumby coloring of the remaining case

Suppose there are exactly two intact edges of $K_4$.
If these two edges are independent, then again color the vertices of $K_4$ red and the internal points blue.
This results in a crumby coloring.
Therefore, we may assume that the intact edges are $AB$ and $BD$. If one of the edges incident to $C$ contains two internal vertices, then color the internal vertex adjacent to $C$ red along with $A,B,C,D$. Color the other vertices blue. In Figure <ref>, we give crumby colorings in the case, where the three edges incident to $C$ have 1 subdivision vertex. There are two such cases depending on the number of internal vertices on $AD$.

Two intact edges and all three edges incident to $C$ have exactly one subdivision vertex

Assume there are at least three intact edges.
There might be three of them, which form a path on the vertices of $K_4$ or there are exactly three intact edges either incident with the same vertex of $K_4$ or forming a triangle. In Figure <ref>, we give crumby colorings for the latter two cases by coloring the internal vertices on the dotted edge red, and on the dashed edges blue.

Precisely three intact edges incident to the same vertex or forming a triangle

Let us suppose that there is a path on the vertices of $K_4$, which consists of three intact edges.
We may assume that these are $AB$, $BC$ and $CD$.
Our idea is to color $A$ and $C$ red and $B$ blue (the color of $D$ might vary), and depending on the number of internal vertices on the remaining edges (which again form a path) color the vertices of this path suitably. Let $(i,j,k)$ denote the case, in which the number of internal vertices on $CA$, $AD$ and $DB$ are exactly $i$, $j$ and $k$, in that order.
There are three subcases depending on the value of $i$. In the following tables, we summarize the possibilities and give crumby colorings.

$(j,k)$:   color of the path
(0,0)    $R~bb~R~R~B$
(0,1)    $R~bb~R~R~b~B$
(0,2)    $R~bb~R~R~rb~B$

$(j,k)$:   color of the path
(1,0)    $R~br~R~b~R~B$
(1,1)    $R~br~R~b~R~b~B$
(1,2)    $R~br~R~b~R~rb~B$

$(j,k)$:   color of the path
(2,0)    $R~br~R~bb~R~B$
(2,1)    $R~br~R~bb~R~b~B$
(2,2)    $R~br~R~bb~R~rb~B$

Crumby colorings if there are three undivided edges which form a path and $i=2$

$(j,k)$:   color of the path
(0,0)    $R~b~R~R~B$
(0,1)    $R~b~R~R~b~B$
(0,2)    $R~b~R~R~rb~B$

$(j,k)$:   color of the path
(1,0)    $R~b~R~b~B~R$
(1,1)    $R~b~R~b~B~r~R$
(1,2)    $R~r~B~b~R~bb~R$

$(j,k)$:   color of the path
(2,0)    $R~b~R~rb~R~B$
(2,1)    $R~b~R~rb~R~b~B$
(2,2)    $R~b~R~rb~R~rb~B$

Crumby colorings if there are three undivided edges which form a path and $i=1$

$(j,k)$:   color of the path
(0,0)    $B~R~R~R$
(0,1)    $B~R~R~b~R$
(0,2)    $B~R~R~bb~R$

$(j,k)$:   color of the path
(1,0)    $B~R~b~R~R$
(1,1)    $R~B~r~R~r~B$
(1,2)    $B~R~b~R~rb~R$

$(j,k)$:   color of the path
(2,0)    $B~R~bb~R~R$
(2,1)    $B~R~br~R~b~R$
(2,2)    $B~R~br~R~rb~R$

Crumby colorings if there are three intact edges which form a path and $i=0$

This finishes all cases of the lemma.

Using the solutions for the restricted cases in the previous lemma, we can prove the crumby colorings for all subdivisions of $K_4$.

Let $G$ be a subdivision of $K_4$. The graph $G$ admits a crumby coloring.

Let $G'$ be the following reduction of $G$.
Independently, on each edge of $K_4$,
we replace the $k$ subdivision vertices by $k$ $\mathrm{mod}~3$ vertices.
By Lemma <ref>, there is a crumby coloring of $G'$.

We extend the coloring of $G'$ independently on each edge of $K_4$. If along an edge of $K_4$ both colors appear, then we can find two consecutive vertices $x$ and $y$ with different colors.
We insert the necessary number of blocks of $brr$ between $x$ and $y$ such that it remains a crumby coloring.

Otherwise, the considered edge of $K_4$ is monochromatic.
If every vertex is red, then we insert $rbr$ between any two of them.  Now we continue the extension (if necessary) just like in the previous case, since there exist consecutive vertices with different colors.
A monochromatic blue edge means two blue vertices in $K_4$.
In that case, we insert $rrr$ between them.
Again, we continue the extension (if necessary) just like in the first case, since there exist consecutive vertices with different colors.

In a slightly opposite direction, motivated by our proof idea of Theorem <ref>, we pose the following

Every $K_4$-minor-free graph admits a crumby coloring.

§ ACKNOWLEDGEMENTS

The first author would like to thank Bjarne Toft for 25 years of friendship and encouragement. He influenced our work in Section <ref>. The first author was partially supported by ERC Advanced Grant "GeoScape" and NKFIH Grant K. 131529.
The second author was supported in part by the Hungarian National Research, Development
and Innovation Office, OTKA grant no. SNN 132625 and K 124950.

[1]J. Barát.
Decomposition of cubic graphs related to Wegner’s conjecture.
Disc. Math 342(5) 1520–1527, (2019).

[2]T. Bellitto, T. Klimošová, M. Merker, M. Witkowski, Y. Yuditsky.
Counterexamples to Thomassen's conjecture on decomposition of cubic graphs.

[3] J.A. Bondy, U.S.R. Murty.
Graph Theory. Springer-Verlag, London, XII+663 pages, (2008).

[4] J. Edmonds.
Paths, trees, and flowers,
Can. J. Math. 17 449–467, (1965).

[5]T. Gallai.
Maximale  Systeme  unabhänginger  Kanten,
Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 401–413,

[6]T.R. Jensen and B. Toft. Graph Coloring Problems.
Wiley (1994).

[7] C. Thomassen.
Totally Odd $K_4$-subdivisions in $4$-chromatic Graphs.
Combinatorica 21, 417–443 (2001).

[8] C. Thomassen. The square of a planar cubic graph is 7-colorable.
JCTB 128 192–218, (2017).

[9] B. Toft: Problem 11, in: Recent Advances in Graph Theory, Academia Praha 543–544, (1975).

[10] G. Wegner.  Graphs with given diameter and a coloring problem. Technical  Report, University of Dortmund, (1977).

[11] W. Zang.
Proof of Toft’s Conjecture: Every graph containing no fully odd $K_4$ is 3-colorable.
J. Combin. Optimization 2 117–188, (1998).

§ APPENDIX

Computer-assisted proof of Lemma <ref>


# Learning MRI Artifact Removal With Unpaired Data

Siyuan Liu1 Kim-Han Thung1 Liangqiong Qu1 Weili Lin1 Dinggang Shen1 Pew-Thian
Yap1,✉ and the UNC/UMN Baby Connectome Project Consortium

###### Abstract

Retrospective artifact correction (RAC) improves image quality post
acquisition and enhances image usability. Recent machine learning driven
techniques for RAC are predominantly based on supervised learning and
therefore practical utility can be limited as data with paired artifact-free
and artifact-corrupted images are typically insufficient or even non-existent.
Here we show that unwanted image artifacts can be disentangled and removed
from an image via an RAC neural network learned with unpaired data. This
implies that our method does not require matching artifact-corrupted data to
be either collected via acquisition or generated via simulation. Experimental
results demonstrate that our method is remarkably effective in removing
artifacts and retaining anatomical details in images with different contrasts.

Department of Radiology and Biomedical Research Imaging Center (BRIC),
University of North Carolina at Chapel Hill, NC, U.S.A.

††footnotetext: ✉ Corresponding author: Pew-Thian Yap<EMAIL_ADDRESS>
## Introduction

Structural magnetic resonance imaging (sMRI) captures high spatial-resolution
details of brain anatomy, but is susceptible to artifacts caused for example
by eye and head motions[1], especially when scanning pediatric, elderly,
claustrophobic, and epileptic patients[2]. Artifacts can result in unusable
images and hence cause financial losses for imaging studies[3]. Motion
artifact correction[4] can be used to remove artifacts, improve image quality,
and increase the amount of usable images. This is particularly important in
view of the fact that the accuracy and reliability of subsequent analysis or
diagnosis can be jeopardized by poor image quality.

Methods for correction of motion artifacts can be prospective or
retrospective. Prospective techniques[5, 6, 7, 8, 9, 10] utilize either
optical tracking of target markers placed on the head or continuously
reacquired images from dedicated navigator scans for real-time motion
prediction[8]. However, prospective methods require additional hardware and/or
scanner modifications. Motion markers can cause patient discomfort and optical
tracking requires expensive hardware, needs clear visibility of target
markers, and may be sensitive to facial movements. Moreover, these methods
typically assume quiescent periods with minimal motion and reacquire data when
this condition is not met. This prolongs acquisition time without necessarily
bringing substantial improvements to image quality when there is little
motion.

In contrast, retrospective artifact correction (RAC)[4] can be used to remove
artifacts, improve image quality, and enhance image usability without
requiring additional hardware, as motion estimation and correction are
considered a part of the image reconstruction process. Retrospective
techniques can be acquisition-based or software-based. One representative
technique of acquisition-based methods is PROPELLER[11], which is a self-
navigation technique that utilizes a number of concentric blades rotated at
different angles to cover the whole k-space. The k-space center is repeatedly
sampled and used as self-navigators for the estimation of rotation and
translation information. It has been shown to be effective for 2D motion
correction[12]. However, acquisition-based methods such as PROPELLER can only
estimate in-plane motion parameters, and the through-plane motion might
disrupt signals across slices. Moreover, retrospective techniques often
require additional purposefully designed navigator sequences, more
complicatedly designs, longer acquisition times, and impose additional
constraints on imaging parameters (i.e., TR/TE/TI).

Software-based methods for post-acquisition RAC[4] can be used to remove
artifacts without modification of sequences, mounting of markers, and
constraining acquisition parameters. They are inexpensive post-processing
method that can be readily incorporated across all scanners. Particularly,
deep neural networks (DNNs), such as convolutional neural networks (CNNs),
have demonstrated great potential for simultaneous removal of a variety of
artifacts irrespective of the acquisition scheme[13, 14]. CNNs are typically
trained in a supervised manner, which in RAC requires paired artifact-
corrupted and artifact-free images. Such paired data can be collected by
scanning the same subjects without and with motions, which can be impractical,
costly, and time-consuming. Artifact-corrupted images can also be generated by
adding simulated artifacts to artifact-free images [15, 16, 17, 18]. However,
simulations might not accurately and sufficiently reflect all possible forms
of real artifacts.

In this paper, we consider the artifact removal problem as image translation
from an artifact-corrupted domain to an artifact-free domain. We gain
inspirations from unsupervised image translation techniques, such as UNIT[19],
CycleGAN[20], BicycleGAN[21], and Pix2Pix[22], which employ auto-encoders to
learn invertible mappings between domain pairs using unpaired images. We
introduce an end-to-end disentangled unsupervised cycle-consistent adversarial
network (DUNCAN), which can be trained using unpaired data for flexible and
simultaneous removal of various sMRI artifacts. We employ cycle translations
between artifact-corrupted and artifact-free domains, where each cycle
translation is defined as a forward translation from one domain to its target
domain, followed by a backward translation from the target domain to the
original domain. Both the forward and backward translations are realized with
auto-encoders. Note that each MR image, even deemed good in quality, may
inevitably contain some artifacts. Therefore, we assume that images from
either artifact-corrupted or artifact-free domains are composed of an
anatomical content component, residing in a domain-invariant content space,
and an artifact component, residing in a domain-specific artifact space. The
auto-encoders disentangle these two components in an image via two kinds of
encoders in each domain translation mapping, i.e., a content encoder, which
captures anatomical structures shared across domains, and an artifact encoder,
which captures artifacts specific to a domain. Then, the decoder ensembles the
extracted content and artifact features from both encoders to translate images
to the target domain. To ensure complete disentanglement of content and
artifact components, we propose a multi-scale content consistency (MS-CC) loss
and a content-swapping mechanism supervised by adversarial learning. We also
design a multi-scale reconstruction consistency (MS-RC) loss, including a
pixel reconstruction consistency (PRC) loss, an edge reconstruction
consistency (ERC) loss, and a structure reconstruction consistency (SRC) loss,
to avoid degradation of structural details. In addition, we propose an image
quality consistency (IQC) loss to ensure that no structural details are
removed from artifact-free images. The architecture of DUNCAN is summarized in
Figure 1 and detailed in the Methods section.

Figure 1: Overview of DUNCAN. a, Disentangled cycle translation (DCT) mapping
$\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$ consists of two
sequential domain mappings $\mathcal{M}_{\text{c}\rightarrow\text{f}}$ and
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$. For
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$, an artifact-corrupted image
$x_{\text{c}}$ is first encoded in the domain-invariant content space
$\mathcal{C}$ and the domain-specific artifact space $\mathcal{A}_{\text{c}}$
to obtain the content (CT) information $z_{\text{c}}^{\text{CT}}$ and artifact
(AF) information $z_{\text{c}}^{\text{AF}}$, respectively. Then,
$z_{\text{c}}^{\text{CT}}$ and $z_{\text{c}}^{\text{AF}}$ are decoded to
remove artifacts from $x_{\text{c}}$, and to obtain the intermediate
translated image $x_{\text{c}\rightarrow\text{f}}$ in the artifact-free domain
$\mathcal{I}_{\text{f}}$. For $\mathcal{M}_{\text{f}\rightarrow\text{c}}$,
$x_{\text{c}\rightarrow\text{f}}$ is first encoded in the content space
$\mathcal{C}$ and the artifact space $\mathcal{A}_{\text{f}}$ to obtain the
content information $z_{\text{f}}^{\text{CT}}$ and the artifact information
$z_{\text{f}}^{\text{AF}}$, respectively. Then $z_{\text{f}}^{\text{CT}}$ and
$z_{\text{f}}^{\text{AF}}$ are decoded to add artifacts to
$x_{\text{c}\rightarrow\text{f}}$ to reconstruct image $\hat{x}_{\text{c}}$.
DCT mapping $\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$ is
hence
$\\{x_{\text{c}}\in\mathcal{I}_{\text{c}}\\}\rightarrow\\{z_{\text{c}}^{\text{CT}}\in\mathcal{C},z_{\text{c}}^{\text{AF}}\in\mathcal{A}_{\text{c}}\\}\rightarrow\\{x_{\text{c}\rightarrow\text{f}}\in\mathcal{I}_{\text{f}}\\}\rightarrow\\{z_{\text{c}\rightarrow\text{f}}^{\text{CT}}\in\mathcal{C},z_{\text{c}\rightarrow\text{f}}^{\text{AF}}\in\mathcal{A}_{\text{f}}\\}\rightarrow\\{\hat{x}_{\text{c}}\in\mathcal{I}_{\text{c}}\\}$.
Conversely, DCT mapping
$\mathcal{M}_{\text{f}\rightarrow\text{c}\rightarrow\text{f}}$ is
$\\{x_{\text{f}}\in\mathcal{I}_{\text{f}}\\}\rightarrow\\{z_{\text{f}}^{\text{CT}}\in\mathcal{C},z_{\text{f}}^{\text{AF}}\in\mathcal{A}_{\text{f}}\\}\rightarrow\\{x_{\text{f}\rightarrow\text{c}}\in\mathcal{I}_{\text{c}}\\}\rightarrow\\{z_{\text{f}\rightarrow\text{c}}^{\text{CT}}\in\mathcal{C},z_{\text{f}\rightarrow\text{c}}^{\text{AF}}\in\mathcal{A}_{\text{c}}\\}\rightarrow\\{\hat{x}_{\text{f}}\in\mathcal{I}_{\text{f}}\\}$.
b, DUNCAN takes any two unpaired images, i.e., one image $x_{\text{c}}$ from
artifact-corrupted domain $\mathcal{I}_{\text{c}}$ and one image
$x_{\text{f}}$ from artifact-free domain $\mathcal{I}_{\text{f}}$, as inputs.
The artifact-corrupted and artifact-free domain cycles incorporate the DCT
mappings $\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{f}\rightarrow\text{c}\rightarrow\text{f}}$, respectively.
The content-swapping translation (CST) and identity translation (IT),
respectively, give the content-swapped translated images, i.e.,
$\hat{x}_{\text{f}\leftrightarrow\text{c}}$ and
$\hat{x}_{\text{c}\leftrightarrow\text{f}}$, and identity translated images,
i.e., $\tilde{x}_{\text{c}}$ and $\tilde{x}_{\text{f}}$. c, CST in artifact-
corrupted and artifact-free domains. d, DCT and IT in artifact-corrupted and
artifact-free domains. e, Network architecture of the proposed auto-encoder
($G_{\text{c}}$ or $G_{\text{f}}$). f, Network architecture of the
discriminator. The discriminator employs a fully convolutional network (FCN)
to determine if the generate image is real or fake based on a semantic
map[22].

## Results

Figure 2: Visual comparison of corrected in vivo images. a, T1-weighted images
and, b, T2-weighted images corrected using various methods. From top to bottom
are images with heavy, moderate, and minor artifacts. In a and b, the original
artifact-corrupted images are shown in the first column and the images
corrected using CycleGAN, Pix2Pix, and DUNCAN are shown respectively in the
second to fourth columns. DUNCAN outperforms the other methods in removing
artifacts and in preserving anatomical details. Figure 3: Visual comparison of
corrected in silico images. a, T1-weighted images and, b, T2-weighted images
corrected using various methods. From top to bottom are images with heavy,
moderate, and minor artifacts. In a and b, the ground truth is shown in the
first column, the original artifact-corrupted images in the second column, and
the images corrected using U-Net, CycleGAN, Pix2Pix, and DUNCAN, respectively,
in the third to sixth columns. DUNCAN removes more artifacts and preserves
more anatomical details in agreement with the ground truth. Figure 4:
Quantitative comparison of corrected in silico T1-weighted images. Numerical
evaluation conducted with different levels of artifacts (heavy, moderate, and
minor) and various metrics (MSE, SSIM, MS-SSIM, PSNR, VIF, and UQI). The bars
show the means and the error bars show the standard errors on the means. The
sample sizes of IS_T1_HVY, IS_T1_MOD, and IS_T1_MIN are 200, 300, 300,
respectively. Compared with the other methods, DUNCAN yields lower MSE and
higher SSIM, MS-SSIM, PSNR, VIF, and UQI. Figure 5: Quantitative comparison of
corrected in silico T2-weighted images. Numerical evaluation conducted with
different levels of artifacts (heavy, moderate, and minor) and various metrics
(MSE, SSIM, MS-SSIM, PSNR, VIF, and UQI). The bars show the means and the
error bars show the standard errors on the means. The sample sizes of
IS_T2_HVY, IS_T2_MOD, and IS_T2_MIN are 200, 300, 300, respectively. Compared
with the other methods, DUNCAN yields lower MSE and higher SSIM, MS-SSIM,
PSNR, VIF, and UQI. Figure 6: Segmentation accuracy of in silico images. a,
DSC comparison of artifact-corrupted images with and without correction,
indicating artifact removal improves image usability. b, Applying DUNCAN on
the artifact-free images do not degrade image details, as indicated by the
high DSCs. The bars show the means and the error bars show the standard errors
on the means. The sample size is 10 for each case.

###  0.1 Datasets

We evaluated DUNCAN using  (i) An in vivo dataset of T1- and T2-weighted
images of children scanned from one month to six years of age[23]; and  (ii)
An in silico dataset of T1- and T2-weighted images with simulated artifacts.

We denote the in vivo datasets for T1- and T2-weighted MR images as IV_T1 and
IV_T2, respectively. For each modality, we selected 20 artifact-free and 20
artifact-corrupted volumes for training, and 10 artifact-corrupted volumes for
testing. We extracted 76 to 85 axial slices from each image volume, resulting
in a total of 1620, 1600, and 800 axial slices respectively from the 20
artifact-free, 20 artifact-corrupted, and 10 artifact-corrupted T1-weighted
volumes for IV_T1 and 1520, 1550, and 800 axial slices respectively from the
20 artifact-free, 20 artifact-corrupted, and 10 artifact-corrupted T2-weighted
volumes for IV_T2. Each artifact-corrupted image volume was labeled with one
of three artifact severity levels: minor, moderate, or heavy.

To generate the in silico datasets, we synthesized artifact-corrupted images
from the artifact-free images from IV_T1 and IV_T2 with three levels of
artifacts, i.e., minor, moderate, and heavy. The resulting datasets are
respectively denoted as IS_T1_MIN, IS_T1_MOD, and IS_T1_HVY for T1-weighted
images, and IS_T2_MIN, IS_T2_MOD, and IS_T2_HVY for T2-weighted images. We
simulated the motion artifacts in k-space, reflecting background noise
movements, swallowing-like movements, and random sudden movements. We
generated the background noise movement via a pseudorandomized series (Perlin
noise[24]) with magnitude of 5, the swallowing-like movements via
multiplications with linear phase shifts in motion directions, i.e.,
translations along $z$-axis and rotations along $x$-axis, and the random
sudden movements via sudden changes in the magnitudes of motions in all
directions. For IS_T1_MIN, IS_T1_MOD, and IS_T1_HVY, 1620, 1620, 800, and 800
axial slices were extracted, respectively, from the 20 artifact-free, 20
synthesized artifact-corrupted, 10 artifact-free, and 10 synthesized artifact-
corrupted T1-weighted volumes. For IS_T2_MIN, IS_T2_MOD, and IS_T2_HVY, 1520,
1520, 800, and 800 axial slices were extracted, respectively, from the 20
artifact-free, 20 synthesized artifact-corrupted, 10 artifact-free, and 10
synthesized artifact-corrupted T2-weighted volumes.

###  0.2 Evaluation Metrics

To quantitatively evaluate the performance of DUNCAN on the in silico
datasets, several image quality metrics, including mean square error (MSE),
structural similarity index (SSIM)[25], multi-scale structural similarity
index (MS-SSIM)[26], peak signal-to-noise ratio (PSNR), visual information
fidelity (VIF)[27], and universal quality index (UQI)[28], were utilized to
gauge the quality of the artifact-corrected images. We used the default
settings for all the hyper-parameters of the evaluation metrics. For all
metrics, except MSE, higher values indicate better performance.

###  0.3 Compared Methods

To verify the effectiveness and superiority of DUNCAN, we compared it with
three state-of-the-art methods that are closely related to our task, i.e., one
supervised method – U-Net[13] – and two unsupervised methods – CycleGAN[20]
and Pix2Pix[22] – implemented with Keras0.3. Pix2Pix differs from CycleGAN by
using a least-squares adversarial loss[29] and PatchGAN[22] as the
discriminator. 11footnotetext: https://github.com/eriklindernoren/Keras-GAN/

###  0.4 Performance Evaluation Using In Vivo Datasets

Since no ground truth is available for the in vivo images, only qualitative
comparisons were conducted. The comparison results for different levels of
artifacts are shown for the T1-weighted and T2-weighted datasets in Figures 2a
and 2b, respectively. CycleGAN and Pix2Pix are unable to remove the artifacts
completely for different levels of artifacts in the T1- and T2-weighted
images. In comparison, DUNCAN is able to remove artifacts with varying
severity without introducing new artifacts.

###  0.5 Performance Evaluation Using In Silico Datasets

Visual comparison results for the in silico T1- and T2-weighted datasets are
provided in Figures 3a and 3b, respectively. The error maps, gradient maps,
and gradient error maps for T1- and T2-weighted images are provided in
Supplementary Figures 1–3, respectively. Quantitative comparison results using
various evaluation metrics are summarized in Figures 4 and 5, respectively.
Quantitative comparison results of gradient maps using various evaluation
metrics on in silico T1- and T2-weighted images are included in Supplementary
Figure 4. CycleGAN and Pix2Pix yield similar performance for the various
evaluation metrics, but in terms of visual appearance, Pix2Pix is
significantly better than CycleGAN due to its PatchGAN discriminator and
least-squares adversarial loss. Although U-Net was trained with paired data
and performs better than CycleGAN and Pix2Pix for the various evaluation
metrics, CycleGAN and Pix2Pix generate images that are sharper than U-Net,
both qualitatively and quantitatively, as illustrated in Supplementary Figures
2–4. This is due to the use of adversarial learning in CycleGAN and Pix2Pix.
In comparison, as DUNCAN utilizes both adversarial learning and disentangled
representation learning of artifacts and contents, it yields better
performance in artifact removal and better capability in maintaining
structural information in artifact-corrupted images. Even when corrupted with
heavy artifacts, image details can still be satisfactorily recovered by
DUNCAN.

###  0.6 Tissue Segmentation

To further demonstrate that DUNCAN can improve image usability, we applied BET
[30] and FAST [31] on the testing data in IS_T1 and IS_T2 for brain extraction
and tissue segmentation, respectively. We report in Figure 6 the tissue
segmentation accuracy before and after artifact correction, as measured by the
Dice similarity coefficient (DSC). Tissue segmentation maps from the artifact-
free images were used as references. The results shown in Figure 6a indicate
that DSCs are improved remarkably by DUNCAN correction. To validate the
quality preservation property of DUNCAN for artifact-free images, we also
evaluated the tissue segmentation accuracy of artifact-free images processed
with DUNCAN. The high DSCs shown in Figure 6b indicate that the quality of
artifact-free images is preserved.

###  0.7 Discussion

We have demonstrated that DUNCAN can be applied to MR images for post-
acquisition artifact removal. DUNCAN is therefore useful when raw k-space data
and reconstruction algorithms are not available. DUNCAN is a flexible method
for RAC irrespective of the acquisition technique. For training, the user only
needs to label the images as artifact-corrupted or artifact-free. No
additional images need to be acquired and no knowledge of MR physics is needed
to simulate artifacts. DUNCAN can potentially allow image imperfections such
as noise, streaking, and ghosting to be removed without explicitly generating
them for supervised training. DUNCAN can be incorporated in a quality control
pipeline to improve image usability. We also note that DUNCAN can be used, via
translation of artifact-free to artifact-corrupted images, to generate natural
and realistic artifacts that can be used for supervised or unsupervised
training of machine learning algorithms.

This work was supported in part by National Institutes of Health grants
(EB006733, AG053867, MH117943, MH104324, MH110274) and the efforts of the
UNC/UMN Baby Connectome Project Consortium. The authors thank Dr. Xiaopeng
Zong of the University of North Carolina at Chapel Hill for an initial
discussion on motion artifact simulation and Dr. Yoonmi Hong of the University
of North Carolina at Chapel Hill and Dr. Yong Chen of Case Western Reserve
University for proofreading the paper.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to Pew-Thian Yap
(email: ptyap@med.unc.edu).

The data used in this paper were provided by the investigative team of the
UNC/UMN Baby Connectome Project. The data can be obtained from the National
Institute of Mental Health Data Archive (NDA) (http://nda.nih.gov/) or by
contacting the investigative team[23].

The source code and trained models for this study are publicly available on
Zenodo (https://zenodo.org/record/3742351)[32].

SL designed the framework and network architecture, carried out the
implementation, performed the experiments, and analyzed the data. SL and PTY
wrote the manuscript. SL, KHT, and PTY revised the manuscript. LQ contributed
to the initial formulation of the method before moving to Stanford University.
WL provided the infant data for training and testing. PTY conceived the study
and were in charge of overall direction and planning. DS was involved in the
initial discussion of the problem when he was with the University of North
Carolina at Chapel Hill. All work was done at the University of North Carolina
at Chapel Hill.

In this work, we  (i) consider the artifact removal problem as image
translation from an artifact-corrupted domain to an artifact-free domain;
(ii) propose an end-to-end unsupervised RAC framework based on a disentangled
unsupervised cycle-consistent adversarial network (DUNCAN, see Figure 1),
which employs two auto-encoders to learn a cycle translation mapping that
translates the images forward and backward between the artifact-corrupted and
artifact-free domains;  (iii) adopt two encoders to embed the images in a
domain-invariant content space, which contains anatomical information, and a
domain-specific artifact space, which captures artifact and noise information,
and adopt a decoder to translate the images to a target domain using the
encoded content and artifact features;  (iv) realize content-artifact
disentanglement, hinging on determining the domain-invariant content space
using two strategies: a multi-scale content consistency (MS-CC) loss to keep
content features consistent across domains and a content-swapping mechanism to
ensure the domain-invariance of the content space; and  (v) introduce a
quality preservation mechanism to ensure that no image details are removed.

###  0.8 Disentangled Cycle Translation Mapping

Let $\mathcal{I}_{\text{c}}$ and $\mathcal{I}_{\text{f}}$ be the domains of
artifact-corrupted and artifact-free MR images, respectively. Our method aims
to learn the nonlinear mappings between the two domains, i.e.,
$\mathcal{M}_{\text{c}\rightarrow\text{f}}:\mathcal{I}_{\text{c}}\rightarrow\mathcal{I}_{\text{f}}$
and
$\mathcal{M}_{\text{f}\rightarrow\text{c}}:\mathcal{I}_{\text{f}}\rightarrow\mathcal{I}_{\text{c}}$,
using unpaired training images. In practice, each acquired MR image, even with
good quality, may inevitably contain artifacts. Therefore, we assume that each
MR image is a nonlinear combination of content and artifact components. For
two unpaired images
$(x_{\text{c}}\in\mathcal{I}_{\text{c}},x_{\text{f}}\in\mathcal{I}_{\text{f}})$,
disentangled cycle translation (DCT) mapping
$\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}:\mathcal{I}_{\text{c}}\rightarrow\mathcal{I}_{\text{f}}\rightarrow\mathcal{I}_{\text{c}}$
is accomplished with sequential forward and backward translation mappings
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$ and
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$; conversely, the DCT mapping
$\mathcal{M}_{\text{f}\rightarrow\text{c}\rightarrow\text{f}}:\mathcal{I}_{\text{f}}\rightarrow\mathcal{I}_{\text{c}}\rightarrow\mathcal{I}_{\text{f}}$
is realized with $\mathcal{M}_{\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$, as illustrated in Figure 1a.
Specifically, taking DCT mapping
$\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$ as an example,
for forward translation mapping $\mathcal{M}_{\text{c}\rightarrow\text{f}}$,
we first encode $x_{\text{c}}$ in two latent spaces, i.e., the domain-
invariant content space $\mathcal{C}$ and the domain-specific artifact space
$\mathcal{A}_{\text{c}}$, to obtain two disentangled representations, i.e.,
the artifact (AF) representation $z_{\text{c}}^{\text{AF}}$ and the content
(CT) representation $z_{\text{c}}^{\text{CT}}$, respectively. We then build a
decoder based on the disentangled representations to construct intermediate
image $x_{\text{c}\rightarrow\text{f}}$ in the artifact-free domain
$\mathcal{I}_{\text{f}}$. The forward translation mapping
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$ for $x_{\text{c}}$ can be
summarized as
$\\{x_{\text{c}}\in\mathcal{I}_{\text{c}}\\}\rightarrow\\{z_{\text{c}}^{\text{CT}}\in\mathcal{C},z_{\text{c}}^{\text{AF}}\in\mathcal{A}_{\text{c}}\\}\rightarrow\\{x_{\text{c}\rightarrow\text{f}}\in\mathcal{I}_{\text{f}}\\}$.
We then further conduct a backward translation mapping
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ on
$x_{\text{c}\rightarrow\text{f}}$. We first encode
$x_{\text{c}\rightarrow\text{f}}$ as
$z_{\text{c}\rightarrow\text{f}}^{\text{CT}}$ in content space $\mathcal{C}$
and $z_{\text{c}\rightarrow\text{f}}^{\text{AF}}$ in artifact space
$\mathcal{A}_{\text{f}}$. Note that this artifact space is specific to input
from domain $\mathcal{I}_{\text{f}}$, whereas $\mathcal{A}_{\text{c}}$ is
specific to input from domain $\mathcal{I}_{\text{c}}$, as images from both
domains have different types of artifacts manifesting in different styles.
Feeding $z_{\text{c}\rightarrow\text{f}}^{\text{CT}}$ and
$z_{\text{c}\rightarrow\text{f}}^{\text{AF}}$ as input to a decoder, we obtain
the reconstructed artifact-corrupted image $\hat{x}_{\text{c}}$. The backward
translation mapping $\mathcal{M}_{\text{f}\rightarrow\text{c}}$ for
$x_{\text{c}\rightarrow\text{f}}^{\text{CT}}$ can be summarized as
$\\{x_{\text{c}\rightarrow\text{f}}^{\text{CT}}\in\mathcal{I}_{\text{f}}\\}\rightarrow\\{z_{\text{c}\rightarrow\text{f}}^{\text{CT}}\in\mathcal{C},z_{\text{c}\rightarrow\text{f}}^{\text{AF}}\in\mathcal{A}_{\text{f}}\\}\rightarrow\\{\hat{x}_{\text{c}}\in\mathcal{I}_{\text{c}}\\}$.
Similarly, we also perform DCT mapping for $x_{\text{f}}$ via
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$ to obtain in sequence the
intermediate artifact-corrupted image $x_{\text{f}\rightarrow\text{c}}$ and
artifact-free image $\hat{x}_{\text{f}}$, i.e.,
$\\{x_{\text{f}}\in\mathcal{I}_{\text{f}}\\}\rightarrow\\{z_{\text{f}}^{\text{CT}}\in\mathcal{C},z_{\text{f}}^{\text{AF}}\in\mathcal{A}_{\text{f}}\\}\rightarrow\\{x_{\text{f}\rightarrow\text{c}}\in\mathcal{I}_{\text{c}}\\}\rightarrow\\{z_{\text{f}\rightarrow\text{c}}^{\text{CT}}\in\mathcal{C},z_{\text{f}\rightarrow\text{c}}^{\text{AF}}\in\mathcal{A}_{\text{c}}\\}\rightarrow\\{\hat{x}_{\text{f}}\in\mathcal{I}_{\text{f}}\\}$.

###  0.9 The DUNCAN Architecture

Figure 1b shows an overview of the network architecture of DUNCAN, consisting
of two DCT mappings (artifact-corrupted and artifact-free domain cycles), two
content-swapping and identity translations (in both artifact-corrupted and
artifact-free domains), and four adversarial constraints (for the generated
artifact-corrupted and artifact-free images). As described in the previous
section, the artifact-corrupted and artifact-free domain cycles aim to perform
DCT mappings $\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$
and $\mathcal{M}_{\text{f}\rightarrow\text{c}\rightarrow\text{f}}$,
respectively, using two domain translation mappings, i.e.,
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$. Each domain translation mapping
is realized by two encoders to disentangle an image into content and artifact
features, and one decoder to reconstruct the target-domain image using the
disentangled features, as illustrated in Figure 1c. More specifically, the
mapping $\mathcal{M}_{\text{f}\rightarrow\text{c}}$ is realized by content
encoder $E_{\text{f}}^{\text{CT}}$, artifact encoder
$E_{\text{f}}^{\text{AF}}$, and decoder $D_{\text{f}}$, whereas the mapping
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$ is realized by content encoder
$E_{\text{c}}^{\text{CT}}$, artifact encoder $E_{\text{c}}^{\text{AF}}$, and
decoder $D_{\text{c}}$. With any two unpaired images
$x_{\text{c}}\in\mathcal{I}_{\text{c}}$ and
$x_{\text{f}}\in\mathcal{I}_{\text{f}}$ as inputs, the encoders and decoders
are learned to respectively reconstruct images $\hat{x}_{\text{c}}$ and
$\hat{x}_{\text{f}}$ via DCT mappings
$\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{f}\rightarrow\text{c}\rightarrow\text{f}}$.

Using the domain-invariant property of the content space, we propose content-
swapping translation (CST) for complete representation disentanglement, as
illustrated in Figure 1c. The idea behind this mechanism is that when the
content and artifact information are completely disentangled, swapping the
domain-invariant content information between domain translations should not
lead to changes in translation outcomes. Specifically, we replace content
information from $E_{\text{c}}^{\text{CT}}(x_{\text{c}})$ in domain
translation $\mathcal{M}_{\text{c}\rightarrow\text{f}}$ with content
information from $E_{\text{f}}^{\text{CT}}(x_{\text{f}})$ to construct
content-swapped translated image
$\hat{x}_{\text{c}\leftrightarrow\text{f}}\in\mathcal{I}_{f}$ via decoder
$D_{\text{c}}$. Similarly, we replace content information from
$E_{\text{f}}^{\text{CT}}(x_{\text{f}})$ in domain translation
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ with content information from
$E_{\text{c}}^{\text{CT}}(x_{\text{c}})$ to generate content-swapped
translated image
$\hat{x}_{\text{f}\leftrightarrow\text{c}}\in\mathcal{I}_{\text{c}}$ via
decoder $D_{\text{f}}$. The translated images
$\hat{x}_{\text{c}\leftrightarrow\text{f}}$ and
$\hat{x}_{\text{f}\leftrightarrow\text{c}}$, respectively, are constrained by
$x_{\text{f}}$ and $x_{\text{c}}$ via discriminators
$D_{\text{c}\leftrightarrow\text{f}}^{\textrm{ADV}}$ and
$D_{\text{f}\leftrightarrow\text{c}}^{\textrm{ADV}}$.

Furthermore, we constrain the forward translation mappings with identity
translation (IT) mappings, as illustrated in Figure 1d, to maintain image
quality when no alteration is expected. Specifically, when translation
mappings $\mathcal{M}_{\text{c}\rightarrow\text{f}}$ and
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ are applied to $x_{\text{f}}$ and
$x_{\text{c}}$, respectively, the mappings are constrained to result in
identity reconstructions $\tilde{x}_{\text{f}}\in\mathcal{I}_{\text{f}}$ and
$\tilde{x}_{\text{c}}\in\mathcal{I}_{\text{c}}$. A set of consistency losses
are used to ensure that the identity translated images are consistent with the
corresponding input images.

###  0.10 Auto-Encoder Architecture

We devise two auto-encoders $G_{\text{c}}$ and $G_{\text{f}}$ to respectively
generate artifact-free and artifact-corrupted images. Each auto-encoder, with
architecture illustrated in Figure 1e, consists of  (i) a content encoder,
$E_{\text{c}}^{\text{CT}}$ or $E_{\text{f}}^{\text{CT}}$, and an artifact
encoder, $E_{\text{c}}^{\text{AF}}$ or $E_{\text{f}}^{\text{AF}}$, that
respectively map an input image to the domain-invariant latent space
$\mathcal{C}$ and the domain-specific latent space, $\mathcal{A}_{\text{c}}$
or $\mathcal{A}_{\text{f}}$, to respectively extract the content and artifact
information of the image, and  (ii) a decoder, $\mathcal{D}_{\text{c}}$ or
$\mathcal{D}_{\text{f}}$, that generates from the extracted features an image
in the target domain of the auto-encoder.  We describe next the details for
each component of the auto-encoder.

Content Encoder The content encoder takes the original image as input, and
extracts content features through 4 residual blocks. Each residual block
consists of 4$\times$4 convolution, leaky ReLU, and instance normalization
(IN)[33] layers. We use an IN layer instead of a batch normalization layer[33]
to accelerate model convergence and maintain independence between features.
All normalized feature maps are activated by leaky ReLU with negative slope
0.2.

Artifact Encoder We first down-sample the input image using 2$\times$2
average-pooling in the artifact encoder. Then, we extract features from the
down-sampled image using 3 residual blocks without IN layers since IN removes
the feature means and variances, which contain important artifact information.

Decoder The decoder takes the extracted content and artifact features as input
and generates, using a set of up-sampling layers and residual blocks, a
content image and an artifact image, which are concatenated and then fused
through a residual block and an 1$\times$1 convolution layer to generate the
translated image.

###  0.11 Adversarial Learning

We employ generative adversarial networks (GANs) to better learn the
translation mappings between the artifact-free and artifact-corrupted image
domains. A GAN is comprised of a generator network and a discriminator
network. In our case, the auto-encoder acts as the generator network by
translating an input image to a target-domain image. The discriminator network
is a classifier that distinguishes between real and fake images. As training
progresses, the generator is getting better at fooling the discriminator, and
the discriminator is getting better at distinguishing real and fake images. We
employ two discriminators $D_{\text{c}}^{\textrm{ADV}}$ and
$D_{\text{f}\leftrightarrow\text{c}}^{\textrm{ADV}}$ in the artifact-corrupted
domain and another two discriminators $D_{\text{f}}^{\textrm{ADV}}$ and
$D_{\text{c}\leftrightarrow\text{f}}^{\textrm{ADV}}$ in the artifact-free
domain. We use PatchGAN[22], shown in Figure 1f, as the discriminators. The
numbers of filters are 64, 128, 256, 512 for the convolution layers and the
number of output channel is 1. Leaky ReLU activation is implemented with a
negative slope coefficient of 0.2.

###  0.12 Disentangled Representation Learning

We took several measures to ensure that our encoders can properly disentangle
content and artifact information from an input image. First, as content space
$\mathcal{C}$ is domain-invariant, i.e., shared between the artifact-free and
artifact-corrupted domains, the content information of an image and its
generated counterpart in the target domain should be consistent. For example,
the content information of $x_{\text{c}}$ and
$x_{\text{c}\rightarrow\text{f}}$ should be consistent, and so should
$x_{\text{f}}$ and $x_{\text{f}\rightarrow\text{c}}$. To this end, we propose
a multi-scale content consistency (MS-CC) loss based on the low- and high-
level features of the content encoders to respectively encourage the
consistency of structural and semantic content information. Second,
discriminating between real and content-swapped generated images via
discriminators $D_{\text{f}\leftrightarrow\text{c}}^{\text{ADV}}$ and
$D_{\text{c}\leftrightarrow\text{f}}^{\text{ADV}}$ also ensures better
disentanglement by the encoders.

###  0.13 Image Quality Consistency (IQC)

To ensure that the image quality of an input image is similar to the
translated image, we propose a pixel-wise image quality consistency (IQC) loss
to encourage the auto-encoders in the translation mappings
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{c}\rightarrow\text{f}}$ to perform as identity translation
mappings
$\mathcal{M}_{\text{c}\rightarrow\text{c}}:\mathcal{I}_{\text{c}}\rightarrow\mathcal{I}_{\text{c}}$
and
$\mathcal{M}_{\text{f}\rightarrow\text{f}}:\mathcal{I}_{\text{f}}\rightarrow\mathcal{I}_{\text{f}}$
when respectively given artifact-corrupted and artifact-free images. The IQC
loss encourages the artifact-free image generator to not remove image details
when given a good quality image. Similarly, the IQC loss encourages the
artifact-corrupted image generator to not introduce any additional artifacts
when given an artifact-corrupted image.

###  0.14 Loss Functions

We leverage two types of loss functions, i.e., consistency losses and
adversarial losses to facilitate model training, as illustrated in Figures 1c
and 1d.

Consistency Losses We utilize three consistency losses: multi-scale content
consistency (MS-CC) loss $\mathcal{L}_{\textrm{MS-CC}}$, which measures
content consistency between the input and output of the forward translation of
each DCT (i.e., $\mathcal{M}_{\text{c}\rightarrow\text{f}}$ in
$\mathcal{M}_{\text{c}\rightarrow\text{f}\rightarrow\text{c}}$ and
$\mathcal{M}_{\text{f}\rightarrow\text{c}}$ in
$\mathcal{M}_{\text{f}\rightarrow\text{c}\rightarrow\text{f}}$), multi-scale
reconstruction consistency (MS-RC) loss $\mathcal{L}_{\textrm{MS-RC}}$, which
computes the reconstruction consistency between an image and its reconstructed
counterpart in the same domain, and image quality consistency (IQC) loss,
which computes the image quality consistency between an image and its identity
translated counterpart in the same domain.

The MS-CC loss measures low- and high-level content feature differences and is
formulated as

$\mathcal{L}_{\textrm{MS-
CC}}=\sum_{i}[\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}\|\phi^{i}_{\text{c}}(x_{\text{c}})-\phi^{i}_{\text{f}}(x_{\text{c}\rightarrow\text{f}})\|_{1}+\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}\|\phi^{i}_{\text{f}}(x_{\text{f}})-\phi^{i}_{\text{c}}(x_{\text{f}\rightarrow\text{c}})\|_{1}],$
(1)

where $x_{\text{c}}$ and $x_{\text{f}}$ denote the artifact-corrupted and
artifact-free images, respectively, and
$x_{\text{c}\rightarrow\text{f}}=D_{\text{c}}(z_{\text{c}}^{\text{CT}},z_{\text{c}}^{\text{AF}})$
and
$x_{\text{f}\rightarrow\text{c}}=D_{\text{f}}(z_{\text{f}}^{\text{CT}},z_{\text{f}}^{\text{AF}})$
denote the corresponding artifact-free and artifact-corrupted images generated
by the decoders, respectively. $\phi^{i}_{\text{c}}(\cdot)$ and
$\phi^{i}_{\text{f}}(\cdot)$ denote the outputs of the $i$-th residual block
of the content encoders $E_{\text{c}}^{\text{CT}}$ and
$E_{\text{f}}^{\text{CT}}$, respectively.
$z_{\text{c}}^{\text{CT}}=E_{\text{c}}^{\text{CT}}(x_{\text{c}})\in\mathcal{S}$
and
$z_{\text{f}}^{\text{CT}}=E_{\text{f}}^{\text{CT}}(x_{\text{f}})\in\mathcal{S}$
denote the content information extracted respectively from $x_{\text{c}}$ and
$x_{\text{f}}$, whereas
$z_{\text{c}}^{\text{AF}}=E_{\text{c}}^{\text{AF}}(x_{\text{c}})\in\mathcal{A}_{\text{c}}$
and
$z_{\text{f}}^{\text{AF}}=E_{\text{f}}^{\text{AF}}(x_{\text{f}})\in\mathcal{A}_{\text{f}}$
denote the artifact information extracted respectively from $x_{\text{c}}$ and
$x_{\text{f}}$.

We compute the MS-RC loss by combining three reconstruction consistency
losses, i.e., the pixel reconstruction consistency (PRC) loss
$\mathcal{L}_{\textrm{PRC}}$, the edge reconstruction consistency (ERC) loss
$\mathcal{L}_{\textrm{ERC}}$, and the structure reconstruction consistency
(SRC) loss $\mathcal{L}_{\textrm{SRC}}$, as defined below:

$\displaystyle\mathcal{L}_{\textrm{MS-RC}}$
$\displaystyle=\mathcal{L}_{\textrm{PRC}}+\mathcal{L}_{\textrm{ERC}}+\mathcal{L}_{\textrm{SRC}},$
(2) $\displaystyle\mathcal{L}_{\textrm{PRC}}$
$\displaystyle=\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}\|x_{\text{c}}-\hat{x}_{\text{c}}\|_{1}+\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}\|x_{\text{f}}-\hat{x}_{\text{f}}\|_{1},$
(3) $\displaystyle\mathcal{L}_{\textrm{ERC}}$
$\displaystyle=\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}\|\psi_{\text{L}}(x_{\text{c}})-\psi_{\text{L}}(\hat{x}_{\text{c}})\|_{1}+\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}\|\psi_{\text{L}}(x_{\text{f}})-\psi_{\text{L}}(\hat{x}_{\text{f}})\|_{1},$
(4) $\displaystyle\mathcal{L}_{\textrm{SRC}}$
$\displaystyle=\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}\|\psi_{\text{H}}(x_{\text{c}})-\psi_{\text{H}}(\hat{x}_{\text{c}})\|_{1}+\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}\|\psi_{\text{H}}(x_{\text{f}})-\psi_{\text{H}}(\hat{x}_{\text{f}})\|_{1},$
(5)

where
$\hat{x}_{\text{c}}=D_{\text{f}}(E_{\text{f}}^{\text{CT}}(x_{\text{c}\rightarrow\text{f}}),E_{\text{f}}^{\text{AF}}(x_{\text{c}\rightarrow\text{f}}))$
and
$\hat{x}_{\text{f}}=D_{\text{c}}(E_{\text{c}}^{\text{CT}}(x_{\text{f}\rightarrow\text{c}}),E_{\text{c}}^{\text{AF}}(x_{\text{f}\rightarrow\text{c}}))$
are, respectively, the reconstructed images in the artifact-corrupted and
artifact-free domains, and $\psi_{\text{L}}(\cdot)$ and
$\psi_{\text{H}}(\cdot)$ are, respectively, the low-level structural
information that reflects edges and high-level semantic information that
reflects contents measured by a pre-trained network (i.e., VGG19[34] trained
on ImageNet).

To preserve image quality after translation when no alteration is expected, an
image quality consistency (IQC) loss is devised to measure the pixel-wise
difference between the input and identity mapped images as

$\mathcal{L}_{{}_{\textrm{IQC}}}=\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}\|x_{\text{c}}-\tilde{x}_{\text{c}}\|_{1}+\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}\|x_{\text{f}}-\tilde{x}_{\text{f}}\|_{1},$
(6)

where
$\tilde{x}_{\text{c}}=D_{\text{f}}(E_{\text{f}}^{\text{CT}}(x_{\text{c}}),E_{\text{f}}^{\text{AF}}(x_{\text{c}}))$
and
$\tilde{x}_{\text{f}}=D_{\text{c}}(E_{\text{c}}^{\text{CT}}(x_{\text{f}}),E_{\text{c}}^{\text{AF}}(x_{\text{f}}))$
are the identity mapped images of $x_{\text{c}}$ and $x_{\text{f}}$,
respectively.

Adversarial Losses We apply two types of adversarial losses, i.e., single-
domain adversarial (SD-ADV) loss and cross-domain adversarial (CD-ADV) loss,
to enhance the judgment accuracy of the discriminators. All adversarial losses
are designed with the mean square error function. The SD-ADV loss is
calculated in a specific domain, i.e., $\mathcal{I}_{\text{c}}$ or
$\mathcal{I}_{\text{f}}$, as

$\begin{split}\mathcal{L}_{{}_{\textrm{SD-
ADV}}}=&\frac{1}{2}\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}(D^{\textrm{ADV}}_{\text{c}}(x_{\text{c}})-I)^{2}+\frac{1}{2}\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}(D^{\textrm{ADV}}_{\text{c}}(x_{\text{f}\rightarrow\text{c}}))^{2}\\\
+&\frac{1}{2}\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}(D^{\text{ADV}}_{\text{f}}(x_{\text{f}})-I)^{2}+\frac{1}{2}\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}(D^{\text{ADV}}_{\text{f}}(x_{\text{c}\rightarrow\text{f}}))^{2},\end{split}$
(7)

where $D^{\text{ADV}}_{\text{c}}$ and $D^{\text{ADV}}_{\text{f}}$ are the
discriminators used to distinguish between real and fake images respectively
in domains $\mathcal{I}_{\text{c}}$ and $\mathcal{I}_{\text{f}}$, $I$ is a
matrix of ones with size $N_{1}\times N_{2}$ matching the output of the
discriminator. The cross-domain adversarial (CD-ADV) loss is defined as

$\begin{split}\mathcal{L}_{{}_{\textrm{CD-
ADV}}}=&\frac{1}{2}\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}}}(D^{\text{ADV}}_{\text{f}\leftrightarrow\text{c}}(x_{\text{c}})-I)^{2}+\frac{1}{2}\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}},x_{\text{c}}\sim\mathcal{I}_{\text{c}}}(D^{\text{ADV}}_{\text{f}\leftrightarrow\text{c}}(\hat{x}_{\text{f}\leftrightarrow\text{c}}))^{2}\\\
+&\frac{1}{2}\mathbb{E}_{x_{\text{f}}\sim\mathcal{I}_{\text{f}}}(D^{\text{ADV}}_{\text{c}\leftrightarrow\text{f}}(x_{\text{f}})-I)^{2}+\frac{1}{2}\mathbb{E}_{x_{\text{c}}\sim\mathcal{I}_{\text{c}},x_{\text{f}}\sim\mathcal{I}_{\text{f}}}(D^{\text{ADV}}_{\text{c}\leftrightarrow\text{f}}(\hat{x}_{\text{c}\leftrightarrow\text{f}}))^{2},\end{split}$
(8)

where
$\hat{x}_{\text{c}\leftrightarrow\text{f}}=D_{\text{c}}(z_{\text{f}}^{\text{CT}},z_{\text{c}}^{\text{AF}})$
and
$\hat{x}_{\text{f}\leftrightarrow\text{c}}=D_{\text{f}}(z_{\text{c}}^{\text{CT}},z_{\text{f}}^{\text{AF}})$
are the images reconstructed by cross-domain content information, i.e.,
$z_{\text{f}}^{\text{CT}}$ or $z_{\text{c}}^{\text{CT}}$, and current-domain
artifact information, i.e., $z_{\text{c}}^{\text{AF}}$ or
$z_{\text{f}}^{\text{AF}}$.

Total Loss In summary, the total loss function of DUNCAN is

$\mathcal{L}_{\textrm{total}}=\mathcal{L}_{{}_{\textrm{SD-
ADV}}}+\mathcal{L}_{{}_{\textrm{CD-ADV}}}+\lambda_{{}_{\textrm{MS-
CC}}}\mathcal{L}_{{}_{\textrm{MS-
CC}}}+\lambda_{{}_{\textrm{PRC}}}\mathcal{L}_{{}_{\textrm{PRC}}}+\lambda_{{}_{\textrm{ERC}}}\mathcal{L}_{{}_{\textrm{ERC}}}+\lambda_{{}_{\textrm{SRC}}}\mathcal{L}_{{}_{\textrm{SRC}}}+\lambda_{{}_{\textrm{IQC}}}\mathcal{L}_{{}_{\textrm{IQC}}},$
(9)

where $\lambda_{{}_{\textrm{MS-CC}}}$, $\lambda_{{}_{\textrm{PRC}}}$,
$\lambda_{{}_{\textrm{ERC}}}$, $\lambda_{{}_{\textrm{SRC}}}$, and
$\lambda_{{}_{\textrm{IQC}}}$ are the loss weights used for controlling the
contributions of the terms in term in Equation (9).

###  0.15 Implementation Details

DUNCAN was implemented using Keras with Tensorflow backend. Evaluation was
based on a machine with a CPU (Intel i7-8700K) and a GPU (NVIDIA GeForce GTX
1080Ti 11GB RAM). The Adam optimizer with $1\times 10^{-4}$ learning rate was
utilized for minimizing the loss function. For in vivo T1- and T2-weighted
datasets, i.e., IV_T1 and IV_T2, we used $\lambda_{{}_{\textrm{MS-CC}}}=5$,
$\lambda_{{}_{\textrm{PCC}}}=10$, $\lambda_{{}_{\textrm{ERC}}}=5$,
$\lambda_{{}_{\textrm{SRC}}}=5$, and $\lambda_{{}_{\textrm{IQC}}}=1$ for MS-
CC, PRC, ERC, SRC, and IQC losses, respectively. For in silico T1- and
T2-weighted datasets, i.e., IS_T1 and IS_T2, we used $\lambda_{{}_{\textrm{MS-
SCC}}}=10$, $\lambda_{{}_{\textrm{PCC}}}=20$,
$\lambda_{{}_{\textrm{ERC}}}=10$, $\lambda_{{}_{\textrm{SRC}}}=10$, and
$\lambda_{{}_{\textrm{IQC}}}=5$ for MS-CC, PRC, ERC, SRC, and IQC losses,
respectively. For both the in vivo and in silico datasets, every three
adjacent slices in each volume were inserted into RGB channels of a color
image, which was then normalized to have a range between -1 and 1 and cropped
to 208$\times$256 from the geometric center. During training, one artifact-
corrupted image and one artifact-free image were randomly selected each time
as inputs.

###  0.16 References

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