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<EMAIL_ADDRESS><EMAIL_ADDRESS>Tanaka

[a]Department of Materials Science and Engineering, Kyoto University, Sakyo,
Kyoto 606-8501, Japan [b]Research and Services Division of Materials Data and
Integrated System, National Institute for Materials Science, Tsukuba, Ibaraki
305-0047, Japan [c]Center for Elements Strategy Initiative for Structural
Materials, Kyoto University, Sakyo, Kyoto 606-8501, Japan [d]Nanostructures
Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan

#  Algorithms for magnetic symmetry operation search and identification of
magnetic space group from magnetic crystal structure

Isao

###### Abstract

A crystal symmetry search is crucial for computational crystallography and
materials science. Although algorithms and implementations for the crystal
symmetry search have been developed, their extension to magnetic space groups
(MSGs) remains limited. In this paper, algorithms for determining magnetic
symmetry operations of magnetic crystal structures, identifying magnetic
space-group types of given MSGs, searching for transformations to a BNS
setting, and symmetrizing the magnetic crystal structures using the MSGs are
presented. The determination of magnetic symmetry operations is numerically
stable and is implemented with minimal modifications from the existing crystal
symmetry search. Magnetic space-group types and transformations to the BNS
setting are identified by a two-step approach combining space-group type
identification and the use of affine normalizers. Point coordinates and
magnetic moments of the magnetic crystal structures are symmetrized by
projection operators for the MSGs. An implementation is distributed with a
permissive free software license in spglib v2.0.2:
https://github.com/spglib/spglib.

In this paper, algorithms for determining magnetic symmetry operations of
magnetic crystal structures, identifying magnetic space-group types from a
given magnetic space group (MSG), searching for transformations to a BNS
setting, and symmetrizing the magnetic crystal structures on the basis of the
determined MSGs are presented.

##  1 Introduction

A crystal symmetry search and the standardization of crystal structures play
crucial roles in computational materials science. For example, symmetry
operations are required in irreducible representations of electronic states
[GAO2021107760], band paths [HINUMA2017140], phonon calculations [phonopy,
phono3py], a random structure search [FREDERICKS2021107810], and crystal
structure description [ganose_jain_2019]. Moreover, the standardization of
crystal structures is indispensable for comparing crystal structures in
different settings and analyzing magnetic crystal structures in high-
throughput first-principles calculations [Horton2019].

Owing to the development of a computer-friendly description of space groups
[Hall:a19707, shmueli2010symmetry] and algorithms [Opgenorth:js0065, Grosse-
Kunstleve:au0146, Grosse-Kunstleve:au0265, Eick2006], we can automatically
perform the crystal symmetry search nowadays. For example, spglib implements
the symmetry-search algorithm and an iterative method to robustly determine
crystal symmetries [spglibv1], which one of the authors has developed and
maintained.

On the other hand, algorithms and implementations for magnetic space groups
(MSGs) [litvin2016magnetic] remains limited. MSGs are essential when we
consider time-reversal operations or magnetic crystal structures. To the best
of our knowledge, existing implementations only partly provide MSG
functionalities. aflow-sym [Hicks:ae5042] proposed and implemented a robust
space-group analysis algorithm; however, it does not seem to support MSGs yet.
identify magnetic group [doi:10.1146/annurev-matsci-070214-021008] in the
Bilbao Crystallographic Server [aroyo2011crystallography] and CrysFML2008
[RODRIGUEZCARVAJAL199355, Gonzalez-Platas:tu5004] can identify MSGs from
magnetic symmetry operations; however, the determination of magnetic symmetry
operations from magnetic crystal structures is not supported. findsym
[https://doi.org/10.1107/S0021889804031528, FINDSYM] supports the
determination of magnetic symmetry operations and the identification of MSGs;
however, the source code is not freely available.

Here, we present algorithms for determining magnetic symmetry operations of
given magnetic crystal structures, identifying magnetic space-group types of
given MSGs, searching for transformations to a BNS setting, and symmetrizing
the magnetic crystal structures on the basis of the determined MSGs. Note that
the implementation of these algorithms is virtually unattainable without
recent development in crystallography: litvin2014magnetic provided extensive
tables for the 1651 MSGs. iso-mag [ISOMAG] provides tables of MSGs in both
human and computer readable formats. Magnetic Hall symbols [Gonzalez-
Platas:tu5004] and unified (UNI) MSG symbols [Campbell:ib5106] have been
developed to represent MSGs or magnetic space-group types unambiguously, which
are based on BNS symbols [belov1957neronova, Bradley2009-ze]. In this paper,
we use the magnetic Hall symbols and the MSG datasets tabulated in Gonzalez-
Platas:tu5004. The implementation is distributed under the BSD 3-clause
license in spglib v2.0.2.

This paper is organized as follows. In Sec. 2, we recall the mathematical
structures of MSGs and present definitions and terminology for describing
MSGs. In Sec. 3, we provide an algorithm for determining magnetic symmetry
operations of a given magnetic crystal structure on the basis of equivalence
relationships between sites in the magnetic crystal structure. In Sec. 4, we
provide an algorithm to identify a magnetic space-group type of the determined
MSG and to search for a transformation from the determined MSG to one in the
BNS setting. In Sec. 5, we provide an algorithm to symmetrize point
coordinates and magnetic moments of the magnetic crystal structure from the
determined MSG.

##  2 Definitions

Before we discuss algorithms for MSGs and magnetic crystal structures, we
describe definitions and terminology for MSGs. In Sec. 2.1, we define MSGs and
derived space groups, which are essential in identifying a magnetic space-
group type and searching for a transformation between MSGs. In Sec. 2.2, we
define equivalence relationships between MSGs. In Sec. 2.3, we mention BNS
symbols and their settings, which specify representatives of MSGs, and we use
them to standardize given MSGs. Finally, in Sec. 2.4, we give examples of
actions of magnetic symmetry operations for magnetic moments.

###  2.1 MSG and its construct type

We consider a _time-reversal operation_ $1^{\prime}$ and call an index-two
group generated from $1^{\prime}$ as a _time-reversal group_
$\\{1,1^{\prime}\\}\,(\cong\mathbb{Z}_{2})$, where $1$ represents an identity
operation. Let $\mathcal{M}$ be a subgroup of a direct product of three-
dimensional Euclidean group $\mathrm{E}(3)$ and $\\{1,1^{\prime}\\}$. An
element $(\bm{W},\bm{w})\theta$ of $\mathcal{M}$ is called a _magnetic
symmetry operation_ , where we call $\bm{W}$ as a matrix part, $\bm{w}$ as a
translation part, and $\theta\in\\{1,1^{\prime}\\}$ as a time-reversal part of
the magnetic symmetry operation. In particular, $(\bm{W},\bm{w})1^{\prime}$ is
called an _antisymmetry_ operation. A translation subgroup of $\mathcal{M}$ is
defined as

$\displaystyle\mathcal{T}(\mathcal{M})=\left\\{(\bm{E},\bm{t})\mathrel{}\middle|\mathrel{}(\bm{E},\bm{t})1\in\mathcal{M}\right\\},$
(1)

where $\bm{E}$ represents the identity matrix. The subgroup $\mathcal{M}$ is
called a _magnetic space group_ (MSG) when $\mathcal{T}(\mathcal{M})$ is
generated from three independent translations. We write a _magnetic point
group_ of $\mathcal{M}$ as

$\displaystyle\mathcal{P}(\mathcal{M})$
$\displaystyle=\left\\{\bm{W}\theta\mathrel{}\middle|\mathrel{}\exists\bm{w}\in\mathbb{R}^{3},(\bm{W},\bm{w})\theta\in\mathcal{M}\right\\}.$
(2)

We consider two derived space groups from $\mathcal{M}$. A _family space
group_ (FSG) of $\mathcal{M}$ is a space group obtained by ignoring time-
reversal parts in magnetic symmetry operations:

$\displaystyle\mathcal{F}(\mathcal{M})=\left\\{(\bm{W},\bm{w})\mathrel{}\middle|\mathrel{}\exists\theta\in\\{1,1^{\prime}\\},(\bm{W},\bm{w})\theta\in\mathcal{M}\right\\}.$
(3)

A _maximal space subgroup_ (XSG) of $\mathcal{M}$ is a space group obtained by
removing antisymmetry operations:

$\displaystyle\mathcal{D}(\mathcal{M})=\left\\{(\bm{W},\bm{w})\mathrel{}\middle|\mathrel{}(\bm{W},\bm{w})1\in\mathcal{M}\right\\}.$
(4)

The MSGs are classified into the following four _construct types_
[Bradley2009-ze, Campbell:ib5106]:

  * •

(Type I) $\mathcal{M}=\mathcal{F}(\mathcal{M})1=\mathcal{D}(\mathcal{M})1$:
The MSG $\mathcal{M}$ does not have antisymmetry operations.

  * •

(Type II)
$\mathcal{M}=\mathcal{F}(\mathcal{M})1\,\sqcup\,\mathcal{F}(\mathcal{M})1^{\prime},\mathcal{F}(\mathcal{M})=\mathcal{D}(\mathcal{M})$:
The MSG $\mathcal{M}$ has antisymmetry operations and corresponding ordinary
symmetry operations.

  * •

(Type III)
$\mathcal{M}=\mathcal{D}(\mathcal{M})1\sqcup(\mathcal{F}(\mathcal{M})\backslash\mathcal{D}(\mathcal{M}))1^{\prime}$
and $\mathcal{D}(\mathcal{M})$ is an index-two _translationengleiche_ subgroup
of $\mathcal{F}(\mathcal{M})$111 The notation
$\mathcal{F}(\mathcal{M})\backslash\mathcal{D}(\mathcal{M})$ indicates a
complement set,
$\mathcal{F}(\mathcal{M})\backslash\mathcal{D}(\mathcal{M})=\left\\{g\theta\in\mathcal{F}(\mathcal{M})\mathrel{}\middle|\mathrel{}g\theta\notin\mathcal{D}(\mathcal{M})\right\\}$.
. Thus, translation subgroups of $\mathcal{F}(\mathcal{M})$ and
$\mathcal{D}(\mathcal{M})$ are identical.

  * •

(Type IV)
$\mathcal{M}=\mathcal{D}(\mathcal{M})1\sqcup(\mathcal{F}(\mathcal{M})\backslash\mathcal{D}(\mathcal{M}))1^{\prime}$
and $\mathcal{D}(\mathcal{M})$ is an index-two _klassengleiche_ subgroup of
$\mathcal{F}(\mathcal{M})$. Thus, point groups of $\mathcal{F}(\mathcal{M})$
and $\mathcal{D}(\mathcal{M})$ are identical.

For a type-III MSG example, Fig. 1(a) shows an antiferromagnetic (AFM) rutile
structure whose MSG is
$\mathcal{M}_{\mathrm{rutile}}=P4_{2}^{\prime}/mn^{\prime}m$ (BNS number
136.498) in the BNS symbol. The FSG and XSG of $\mathcal{M}_{\mathrm{rutile}}$
are $P4_{2}/mnm$ (No. 136) and $Pnnm$ (No. 58), respectively.

For a type-IV MSG example, Fig. 1(b) shows an AFM bcc structure whose MSG is
$\mathcal{M}_{\mathrm{bcc}}=P_{I}m\overline{3}m$ (BNS number 221.97) in the
BNS symbol. The FSG and XSG of $\mathcal{M}_{\mathrm{bcc}}$ are
$Im\overline{3}m$ (No. 229) and $Pm\overline{3}m$ (No. 221), respectively.

###  2.2 Magnetic space-group type

We consider a transformation $(\bm{P},\bm{p})$ between two coordinate systems
specified with basis vectors $\bm{A}=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3})$ with
origin $\bm{O}$ and basis vectors $\bm{AP}$ with origin $\bm{O}+\bm{Ap}$. A
transformation $(\bm{P},\bm{p})$ with $\det\bm{P}>0$ is called orientation-
preserving. We assume that a magnetic symmetry operation
$(\bm{W},\bm{w})\theta$ is transformed into
$(\bm{W}^{\prime},\bm{w}^{\prime})\theta^{\prime}$ by $(\bm{P},\bm{p})$ as

$\displaystyle(\bm{W}^{\prime},\bm{w}^{\prime})$
$\displaystyle=(\bm{P},\bm{p})^{-1}(\bm{W},\bm{w})(\bm{P},\bm{p})$ (5)
$\displaystyle\theta^{\prime}$ $\displaystyle=\theta.$ (6)

We refer to the criteria to choose a representative of each space-group type
as a _setting_. The _standard ITA setting_ is one of the conventional
descriptions for each space-group type used in the International Tables for
Crystallography Vol. A [ITA2016]: unique axis b setting, cell choice 1 for
monoclinic groups, hexagonal axes for rhombohedral groups, and origin choice 2
for centrosymmetric groups. Similarly to space groups, each equivalent class
of MSGs up to orientation-preserving transformations is called a _magnetic
space-group type_.

###  2.3 BNS setting

The BNS symbol represents each magnetic space-group type [belov1957neronova].
We refer to a setting of the BNS symbol as a _BNS setting_ : For types-I, -II,
and -III MSGs, it uses the same setting as the standard ITA setting of the
FSG. For type-IV MSG, it uses that of the XSG. In Sec. 4, we consider
standardizing a given magnetic crystal structure by applying a transformation
to an MSG in the BNS setting.

###  2.4 Action of magnetic symmetry operations

In general, we can arbitrarily choose how magnetic symmetry operations act on
objects as long as they satisfy the definition of actions. For a magnetic
moment $\bm{m}$, a symmetry operation $(\bm{W},\bm{w})$ acts on $\bm{m}$ as an
axial vector, and the time-reversal operation $1^{\prime}$ reverses the sign
of $\bm{m}$. When we choose the Cartesian coordinates for $\bm{m}$, the matrix
part of $(\bm{W},\bm{w})$ is expressed as $\bm{A}\bm{W}\bm{A}^{-1}$ in
Cartesian coordinates with basis vectors
$\bm{A}=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3})$. Therefore, the magnetic symmetry
operations act on $\bm{m}$ as

$\displaystyle(\bm{W},\bm{w})\theta\bm{m}$
$\displaystyle=\begin{cases}(\mathrm{det}\bm{W})\bm{A}\bm{W}\bm{A}^{-1}\bm{m}&(\theta=1)\\\
-(\mathrm{det}\bm{W})\bm{A}\bm{W}\bm{A}^{-1}\bm{m}&(\theta=1^{\prime})\\\
\end{cases}.$ (7)

##  3 Magnetic symmetry operation search

We provide a procedure to search for magnetic symmetry operations from a given
magnetic crystal structure represented by basis vectors, point coordinates,
atomic types, and magnetic moments within a unit cell. Formally, our input for
the magnetic symmetry operation search is the following four objects: (1)
basis vectors of its lattice $\bm{A}=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3})$, (2)
an array of point coordinates of sites in its unit cell
$\bm{X}=(\bm{x}_{1},\cdots,\bm{x}_{N})$, (3) an array of atomic types of sites
in its unit cell $\bm{T}=(t_{1},\cdots,t_{N})$, and (4) an array of magnetic
moments of sites in its unit cell $\bm{M}=(\bm{m}_{1},\cdots,\bm{m}_{N})$,
where $N$ is the number of sites in the unit cell.

We search for a magnetic symmetry operation
$g\theta\in\mathrm{E}(3)\times\\{1,1^{\prime}\\}$ that preserves the magnetic
crystal structure $(\bm{A},\bm{X},\bm{T},\bm{M})$. Therefore, the symmetry
operation $g$ should map point coordinates $\bm{x}_{i}$ into
$\bm{x}_{\sigma_{g}(i)}$ up to translations, where $\sigma_{g}$ is a
permutation of $N$ sites induced by $g$. Also, $g\theta$ should equate a
magnetic moment $\bm{m}_{i}$ with a mapped one $\bm{m}_{\sigma_{g}(i)}$. Such
a magnetic symmetry operation forms an MSG of the magnetic crystal structure
as a stabilizer of $\mathrm{E}(3)\times\\{1,1^{\prime}\\}$,

$\displaystyle\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$
$\displaystyle=\mathrm{Stab}_{\mathrm{E}(3)\times\\{1,1^{\prime}\\}}\,(\bm{A},\bm{X},\bm{T},\bm{M})$
$\displaystyle=\left\\{g\theta\in\mathrm{E}(3)\times\\{1,1^{\prime}\\}\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\exists\sigma_{g}\in\mathfrak{S}_{N},\forall
i,\\\ g\bm{x}_{i}\equiv\bm{x}_{\sigma_{g}(i)}\,(\mathrm{mod}\,1)\\\
t_{i}=t_{\sigma_{g}(i)},\\\
g\theta\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}\end{array}\right\\},$ (12)

where $\mathfrak{S}_{N}$ is a symmetric group of degree $N$222 We recall that
the condition of $\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$ in Eq. (3) can be
read as there exists a permutation $\sigma_{g}$ such that point coordinates,
atomic types, and magnetic moments are preserved for every site $i$. .

Because the domain of the symmetry operation $g$ in Eq. (3) is not restricted,
we cannot search thoroughly for $g$ at this point. To narrow down the
candidates for $g$, we consider a crystal structure $(\bm{A},\bm{X},\bm{T})$
obtained by ignoring the magnetic moments of $(\bm{A},\bm{X},\bm{T},\bm{M})$.
A space group of $(\bm{A},\bm{X},\bm{T})$ is written as a stabilizer of
$\mathrm{E}(3)$ that preserves $(\bm{A},\bm{X},\bm{T})$:

$\displaystyle\mathcal{S}(\bm{A},\bm{X},\bm{T})$
$\displaystyle=\mathrm{Stab}_{\mathrm{E}(3)}\,(\bm{A},\bm{X},\bm{T})$
$\displaystyle=\left\\{g\in\mathrm{E}(3)\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\exists\sigma_{g}\in\mathfrak{S}_{N},\forall
i,\\\ g\bm{x}_{i}\equiv\bm{x}_{\sigma_{g}(i)}\,(\mathrm{mod}\,1)\\\
t_{i}=t_{\sigma_{g}(i)}\end{array}\right\\}.$ (16)

As shown in Fig. 2, $\mathcal{S}(\bm{A},\bm{X},\bm{T})$ may not be a subgroup
of $\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$ in general because the former
ignores magnetic moments. Because time-reversal operations do not change point
coordinates and atomic types, we can restrict the domain of symmetry
operations $g$ to $\mathcal{S}(\bm{A},\bm{X},\bm{T})$,

$\displaystyle\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$
$\displaystyle=\left\\{g\theta\in\mathcal{S}(\bm{A},\bm{X},\bm{T})\times\\{1,1^{\prime}\\}\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\exists\sigma_{g}\in\mathfrak{S}_{N},\forall
i,\\\ g\bm{x}_{i}\equiv\bm{x}_{\sigma_{g}(i)}\,(\mathrm{mod}\,1)\\\
t_{i}=t_{\sigma_{g}(i)},\\\
g\theta\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}\end{array}\right\\}.$ (21)

The symmetry operations for $(\bm{A},\bm{X},\bm{T})$ can be obtained from
existing crystal symmetry search algorithms such as
https://doi.org/10.1107/S0021889804031528, spglibv1, and Hicks:ae5042.

Based on the formulation of the MSG in Eq. (3), we can search for magnetic
symmetry operations using the following procedure:

  1. 1.

We compute $\mathcal{S}(\bm{A},\bm{X},\bm{T})$ by the existing crystal
symmetry search algorithms.

  2. 2.

If all magnetic moments are zero, both $g1$ and $g1^{\prime}$ belong to
$\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$ for all
$g\in\mathcal{S}(\bm{A},\bm{X},\bm{T})$ and we skip the remaining steps (in
this case, the MSG is type II). Otherwise, we choose a site $i^{\ast}$ with a
non-zero magnetic moment $\bm{m}_{i^{\ast}}(\neq\bm{0})$.

  3. 3.

For each symmetry operation $g\in\mathcal{S}(\bm{A},\bm{X},\bm{T})$, we search
for the time-reversal part as follows:

    1. (a)

We compute a permutation $\sigma_{g}$ and solve

$\displaystyle g\theta\bm{m}_{i^{\ast}}=\bm{m}_{\sigma_{g}(i^{\ast})}$ (22)

for $\theta\in\\{1,1^{\prime}\\}$. We denote the solution of Eq. (22) as
$\theta^{\ast}$ if it exists. If the solution does not exist, we skip the
symmetry operation $g$.

    2. (b)

We check if the condition $g\theta^{\ast}\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}$
holds for other sites. If the condition holds for all sites, $g\theta^{\ast}$
belongs to $\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$.

Note that the comparison of point coordinates and magnetic moments should be
performed within tolerances in practice [Grosse-Kunstleve:sh5006]. We use an
absolute tolerance parameter $\epsilon$ for point coordinates [spglibv1] and
another absolute tolerance $\epsilon_{\mathrm{mag}}$ for magnetic moments.
Then, the comparisons in this section are replaced with the following
inequalities:

$\displaystyle g\bm{x}_{i}\equiv\bm{x}_{\sigma_{g}(i)}\,(\mathrm{mod}\,1)$
$\displaystyle\rightarrow\left\lVert\bm{A}\left[g\bm{x}_{i}-\bm{x}_{\sigma_{g}(i)}\right]_{\mathrm{mod}\,1}\right\rVert_{2}<\epsilon$
(23) $\displaystyle g\theta\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}$
$\displaystyle\rightarrow\left\lVert
g\theta\bm{m}_{i}-\bm{m}_{\sigma_{g}(i)}\right\rVert_{2}<\epsilon_{\mathrm{mag}}.$
(24)

Here, $\left[\cdot\right]_{\mathrm{mod}\,1}$ takes a remainder with modulo one
between $[-0.5,0.5]$.

##  4 Identification of magnetic space-group type and transformation to BNS
setting

For the detected MSG $\mathcal{M}=\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$ in
the previous section, we provide an algorithm to identify its magnetic space-
group type and search for a transformation from $\mathcal{M}$ to a magnetic
space-group representative $\mathcal{M}_{\mathrm{BNS}}$ in the BNS setting.
The algorithms presented in this section are applied to a list of magnetic
symmetry operations in the matrix form either obtained through the magnetic
symmetry operation search in Sec. 3 or provided from outside of the software
package as predetermined operations.

For all the 1651 magnetic space-group types, a magnetic space-group
representative $\mathcal{M}_{\mathrm{BNS}}$ in the BNS setting has already
been tabulated [Gonzalez-Platas:tu5004, Campbell:ib5106]. Thus, we search for
$\mathcal{M}_{\mathrm{BNS}}$ with the same magnetic space-group type as
$\mathcal{M}$ and an orientation-preserving transformation $(\bm{P},\bm{p})$
while satisfying

$\displaystyle(\bm{P},\bm{p})^{-1}\mathcal{M}(\bm{P},\bm{p})=\mathcal{M}_{\mathrm{BNS}}.$
(25)

In Sec. 4.1, we identify a construct type of $\mathcal{M}$ to choose a
candidate $\mathcal{M}_{\mathrm{BNS}}$, which is one of the magnetic space-
group representatives in the BNS setting. In Sec. 4.2, we try to obtain
$(\bm{P},\bm{p})$ from affine normalizers of
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ or
$\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$.

###  4.1 Identification of construct type of MSG

The construct type of $\mathcal{M}$ can be determined from orders of the
magnetic point group and point groups of FSG and XSG. We write a point group
of space group $\mathcal{S}$ as

$\displaystyle\mathcal{P}(\mathcal{S})$
$\displaystyle=\left\\{\bm{W}\mathrel{}\middle|\mathrel{}\exists\bm{w}\in\mathbb{R}^{3},(\bm{W},\bm{w})\in\mathcal{S}\right\\}.$
(26)

When
$|\mathcal{P}(\mathcal{F}(\mathcal{M}))|/|\mathcal{P}(\mathcal{D}(\mathcal{M}))|=1$,
$\mathcal{M}$ is type I or II. Then, when
$|\mathcal{P}(\mathcal{M})|/|\mathcal{P}(\mathcal{F}(\mathcal{M}))|=1$,
$\mathcal{M}$ is type I. When
$|\mathcal{P}(\mathcal{M})|/|\mathcal{P}(\mathcal{F}(\mathcal{M}))|=2$,
$\mathcal{M}$ is type II.

When
$|\mathcal{P}(\mathcal{F}(\mathcal{M}))|/|\mathcal{P}(\mathcal{D}(\mathcal{M}))|=2$,
$\mathcal{M}$ is type III or IV. For type-III or type-IV MSG, we consider a
coset decomposition of $\mathcal{M}$ by $\mathcal{D}(\mathcal{M})$:

$\displaystyle\mathcal{M}=\mathcal{D}(\mathcal{M})1\sqcup\mathcal{D}(\mathcal{M})(\bm{W}_{0},\bm{w}_{0})1^{\prime}.$
(27)

If the coset representative $(\bm{W}_{0},\bm{w}_{0})1^{\prime}$ can be taken
as an anti-translation, $\mathcal{D}(\mathcal{M})$ is a klassengleiche
subgroup of $\mathcal{F}(\mathcal{M})$ and $\mathcal{M}$ is type IV. If not,
$\mathcal{D}(\mathcal{M})$ is a translationengleiche subgroup of
$\mathcal{F}(\mathcal{M})$ and $\mathcal{M}$ is type III.

###  4.2 Transformation of MSG to BNS setting

For each magnetic space-group representative $\mathcal{M}_{\mathrm{BNS}}$ with
the same construct type as $\mathcal{M}$, we consider searching for
$(\bm{P},\bm{p})$ from two consecutive transformations
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ and
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$ with

$\displaystyle(\bm{P},\bm{p})=(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$
(28)

as described below. If such a transformation $(\bm{P},\bm{p})$ is found,
$\mathcal{M}$ belongs to the same magnetic space-group type with
$\mathcal{M}_{\mathrm{BNS}}$.

We rewrite Eq. (25) to an equivalent one in terms of derived space groups
because we would like to use an existing transformation search algorithm to
obtain a transformation between space groups with the same space-group type
proposed by Grosse-Kunstleve:au0146. As shown in Appendix A, the condition of
Eq. (25) is equivalent to satisfying the following two conditions:

$\displaystyle(\bm{P},\bm{p})^{-1}\mathcal{F}(\mathcal{M})(\bm{P},\bm{p})$
$\displaystyle=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ (29a)
$\displaystyle(\bm{P},\bm{p})^{-1}\mathcal{D}(\mathcal{M})(\bm{P},\bm{p})$
$\displaystyle=\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}).$ (29b)

Note that a transformation satisfying Eq. (29a) does not necessarily hold Eq.
(29b) in general, and vice versa.

The present algorithm, based on the new conditions, is outlined as follows.
First, we obtain a temporal transformation
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ to match FSGs or XSGs of
$\mathcal{M}$ and $\mathcal{M}_{\mathrm{BNS}}$ by the existing transformation
search algorithm. Then, we search for a correction transformation
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$ to match FSGs and XSGs
simultaneously. We divide the transformation search into cases by the
construct type of $\mathcal{M}$ in more detail.

####  4.2.1 When $\mathcal{M}$ is type I or II

When $\mathcal{M}$ is type I or II, $\mathcal{M}_{\mathrm{BNS}}$ with the same
construct type with $\mathcal{M}$ uses the standard ITA setting of
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$. Thus, we need to obtain an
orientation-preserving transformation
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ such that
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})^{-1}\mathcal{F}(\mathcal{M})(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$.
The temporal transformation $(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$
can be obtained by the existing transformation search algorithm. We write an
MSG transformed by $(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ as

$\displaystyle\mathcal{M}_{\mathrm{temp}}=(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})^{-1}\mathcal{M}(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}}).$
(30)

By construction, $\mathcal{F}(\mathcal{M}_{\mathrm{temp}})$ and
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ are identical as sets,
$\mathcal{F}(\mathcal{M}_{\mathrm{temp}})=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$.

In this case, the XSGs are also identical to one another,
$\mathcal{D}(\mathcal{M}_{\mathrm{temp}})=\mathcal{F}(\mathcal{M}_{\mathrm{temp}})=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})=\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$.
Thus, we do not need to search for a correction transformation because
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ also satisfies
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})^{-1}\mathcal{D}(\mathcal{M})(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})=\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$.

####  4.2.2 When $\mathcal{M}$ is type III

When $\mathcal{M}$ is type III, $\mathcal{M}_{\mathrm{BNS}}$ with type III
uses the standard ITA setting of $\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$.
Thus, we need to obtain an orientation-preserving transformation
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ such that
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})^{-1}\mathcal{F}(\mathcal{M})(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$.
Then, the FSG of the transformed MSG in Eq. (30),
$\mathcal{F}(\mathcal{M}_{\mathrm{temp}})$, is the space-group representative
in the standard ITA setting.

A correction transformation $(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$
should satisfy the following conditions to simultaneously hold Eqs. (29a) and
(29b),

$\displaystyle(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})^{-1}\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$
$\displaystyle=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ (31a)
$\displaystyle(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})^{-1}\mathcal{D}(\mathcal{M}_{\mathrm{temp}})(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$
$\displaystyle=\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}).$ (31b)

The condition of Eq. (31a) indicates that
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$ belongs to an affine
normalizer of $\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ [koch2016normalizers].
The situation is shown in Fig. 3(a), where we write the affine normalizer of a
space group $\mathcal{S}$ as

$\displaystyle\mathcal{N}_{\mathrm{A}(3)}(\mathcal{S})=\left\\{(\bm{Q},\bm{q})\in\mathrm{A}(3)\mathrel{}\middle|\mathrel{}(\bm{Q},\bm{q})^{-1}\mathcal{S}(\bm{Q},\bm{q})=\mathcal{S}\right\\}$
(32)

and the three-dimensional affine group as $\mathrm{A}(3)$. If a correction
transformation
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})\in\mathcal{N}_{\mathrm{A}(3)}(\mathcal{F}(\mathcal{M}_{\mathrm{BNS}}))$
satisfies Eq. (31b), the combined transformation in Eq. (28) transforms
$\mathcal{M}$ to $\mathcal{M}_{\mathrm{BNS}}$.

Finally, we describe how to prepare transformations in the affine normalizer
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{F}(\mathcal{M}_{\mathrm{BNS}}))$ in
practice. Because $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is a normal
subgroup of $\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$, an operation in
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ does not give another conjugated
subgroup of $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$. Also, although the
affine normalizer may have continuous translations, the continuous
translations do not give another conjugated subgroup of
$\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$. Thus, it is sufficient to consider
coset representatives of
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{F}(\mathcal{M}_{\mathrm{BNS}}))/\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$
other than continuous translations. We divide the affine normalizer
computation into cases whether the number of the coset representatives other
than continuous translations is finite or not.

When $\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ is triclinic or monoclinic, the
number of the coset representatives other than continuous translations is
infinite and we cannot check transformations thoroughly. However, there are no
such conjugate space groups with
$\mathcal{D}(\mathcal{M}_{\mathrm{temp}})\neq\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$
because $\mathcal{P}(\mathcal{F}(\mathcal{M}_{\mathrm{BNS}}))$ does not have a
pair of proper conjugate subgroups in its affine normalizer. Therefore, we do
not need to compute the affine normalizer in this case.

When $\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ belongs to other crystal
systems, the number of the coset representatives other than continuous
translations is finite. To simplify the present algorithm and implementation,
instead of using a list of affine normalizers in koch2016normalizers, we
enumerate matrix parts and origin shifts of orientation-preserving
transformations in the coset representatives other than continuous
translations as follows. For matrix parts, we enumerate integer matrices
$\bm{P}_{\mathrm{corr}}$ whose elements are -1, 0, or 1, and their
determinants are equal to one. For origin shifts, we enumerate vectors
$\bm{p}_{\mathrm{corr}}$ by restricting their vector components to one of
$\left\\{0,\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}\right\\}$.
These will be sufficient because they cover all orientation-preserving coset
representatives of
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{F}(\mathcal{M}_{\mathrm{BNS}}))/\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$
up to translations [koch2016normalizers]. Since
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$ can be tabulated for each
space-group representative in the standard ITA setting, we can precompute them
before performing the transformation search in practice.

####  4.2.3 When $\mathcal{M}$ is type IV

When $\mathcal{M}$ is type IV, $\mathcal{M}_{\mathrm{BNS}}$ with type IV uses
the standard ITA setting of $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$. Thus,
we need to obtain an orientation-preserving transformation
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})$ such that
$(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})^{-1}\mathcal{D}(\mathcal{M})(\bm{P}_{\mathrm{temp}},\bm{p}_{\mathrm{temp}})=\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$.
Then, the XSG of the transformed MSG in Eq. (30),
$\mathcal{D}(\mathcal{M}_{\mathrm{temp}})$, is the space-group representative
in the standard ITA setting.

Similarly to type-III MSGs, we need to search for an orientation-preserving
transformation
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})\in\mathcal{N}_{\mathrm{A}(3)}(\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}))$
such that

$\displaystyle(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})^{-1}\mathcal{F}(\mathcal{M}_{\mathrm{temp}})(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})=\mathcal{F}(\mathcal{M}_{\mathrm{BNS}}).$
(33)

The situation is shown in Fig. 3(b) 333 The FSG
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ is a subgroup of
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}))$:
because $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is a normal subgroup of
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$, every operation in
$\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ stabilizes
$\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ and belongs to
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}))$.
Similarly, $\mathcal{F}(\mathcal{M}_{\mathrm{temp}})$ is a subgroup of
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}))$. .

When $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is neither triclinic nor
monoclinic, the brute-force tabulation in Sec. 4.2.2 also works for
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}))$. For
triclinic and monoclinic cases, a factor group
$\mathcal{N}_{\mathrm{A}(3)}(\mathcal{D}(\mathcal{M}_{\mathrm{BNS}}))/\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$
is not finite, and we cannot prove the completeness in the same manner. Thus,
we show that the enumerated $(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$
covers all conjugated type-IV MSGs by explicitly listing
$(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})$ and the conjugated MSGs in
Appendix B.

####  4.2.4 Examples of conjugated MSGs

We present examples of conjugated MSGs for type III and type IV. For a type-
III MSG example, consider coset representatives of
$\mathcal{M}_{\mathrm{BNS}}$ for $P22^{\prime}2_{1}^{\prime}$ in the BNS
setting (BNS number 17.10) as follows:

$\displaystyle x,y,z,1;x,-y,-z,1;$
$\displaystyle-x,-y,z+1/2,1^{\prime};-x,y,-z+1/2,1^{\prime}.$

There is another MSG $\mathcal{M}_{\mathrm{temp}}$ with the same magnetic
space-group type as $\mathcal{M}_{\mathrm{BNS}}$ and identical FSG to
$\mathcal{M}_{\mathrm{BNS}}$:

$\displaystyle x,y,z,1;-x,y,-z+1/2,1;$
$\displaystyle-x,-y,z+1/2,1^{\prime};x,-y,-z,1^{\prime}.$

Although $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ and
$\mathcal{D}(\mathcal{M}_{\mathrm{temp}})$ belong to the same space-group type
(No. 3), these XSGs are different. The following transformation maps
$\mathcal{M}_{\mathrm{temp}}$ to $\mathcal{M}_{\mathrm{BNS}}$ while satisfying
Eq. (31b):

$\displaystyle(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})=\left(\begin{pmatrix}0&-1&0\\\
-1&0&0\\\ 0&0&-1\\\ \end{pmatrix},\begin{pmatrix}0\\\ 0\\\ \frac{1}{4}\\\
\end{pmatrix}\right).$

For a type-IV MSG example, consider coset representatives of
$\mathcal{M}_{\mathrm{BNS}}$ for $C_{c}c$ in the BNS setting (BNS number 9.40)
as follows:

$\displaystyle x,y,z,1;x,-y,z+1/2,1;$ $\displaystyle
x+1/2,y+1/2,z,1;x+1/2,-y+1/2,z+1/2,1;$ $\displaystyle
x+1/2,-y+1/2,z,1^{\prime};x,-y,z,1^{\prime};$ $\displaystyle
x,y,z+1/2,1^{\prime};x+1/2,y+1/2,z+1/2,1^{\prime}.$

There is another MSG $\mathcal{M}_{\mathrm{temp}}$ with the same magnetic
space-group type as $\mathcal{M}_{\mathrm{BNS}}$ and identical XSG to
$\mathcal{M}_{\mathrm{BNS}}$:

$\displaystyle x,y,z,1;x+1/2,-y+1/2,z+1/2,1;$ $\displaystyle
x+1/2,y+1/2,z,1;x,-y,z+1/2,1;$ $\displaystyle
x+1/2,-y,z,1^{\prime};x,-y+1/2,z,1^{\prime};$ $\displaystyle
x+1/2,y,z+1/2,1^{\prime};x,y+1/2,z+1/2,1^{\prime}.$

Although $\mathcal{F}(\mathcal{M}_{\mathrm{BNS}})$ and
$\mathcal{F}(\mathcal{M}_{\mathrm{temp}})$ belong to the same space-group type
(No. 8), these FSGs are different. The following transformation maps
$\mathcal{M}_{\mathrm{temp}}$ to $\mathcal{M}_{\mathrm{BNS}}$ while satisfying
Eq. (33):

$\displaystyle(\bm{P}_{\mathrm{corr}},\bm{p}_{\mathrm{corr}})=\left(\begin{pmatrix}-1&0&-1\\\
0&-1&0\\\ 0&0&1\\\ \end{pmatrix},\begin{pmatrix}0\\\ \frac{1}{4}\\\ 0\\\
\end{pmatrix}\right).$

##  5 Symmetrization of magnetic crystal structure

We symmetrize the magnetic crystal structure $(\bm{A},\bm{X},\bm{T},\bm{M})$
by magnetic symmetry operations of the determined MSG
$\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$. For convenience, we consider its
coset decomposition with a finite index as follows. Let $\mathcal{T}_{\bm{A}}$
be a translation group formed by basis vectors $\bm{A}$, which may not be
primitive basis vectors. We write a coset decomposition of
$\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$ by $\mathcal{T}_{\bm{A}}$ as

$\displaystyle\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$
$\displaystyle=\bigsqcup_{\kappa}(\bm{W}_{\kappa},\bm{w}_{\kappa})\theta_{\kappa}\mathcal{T}_{\bm{A}}.$
(34)

We write the set of coset representatives as

$\displaystyle\overline{\mathcal{M}}=\\{(\bm{W}_{\kappa},\bm{w}_{\kappa})\theta_{\kappa}\\}_{\kappa}.$
(35)

A centering operation $(\bm{E},\bm{w})1$, where
$\bm{w}\not\equiv\bm{0}\,(\mathrm{mod}\,1)$, may belong to
$\overline{\mathcal{M}}$.

A procedure to symmetrize the array of point coordinates $\bm{X}$ by
$\overline{\mathcal{M}}$ is essentially the same as those used by Grosse-
Kunstleve:au0265 and spglibv1. For the $\kappa$th magnetic symmetry operation
$(\bm{W}_{\kappa},\bm{w}_{\kappa})\theta_{\kappa}$, we denote that its inverse
maps the $\sigma_{\kappa}^{-1}(i)$th point coordinates to the $i$th point
coordinates. Then,
$(\bm{W}_{\kappa},\bm{w}_{\kappa})\bm{x}_{\sigma_{\kappa}^{-1}(i)}$ should be
close to $\bm{x}_{i}$ up to lattice translations in $\mathcal{T}_{\bm{A}}$.
With this observation, each of the point coordinates $\bm{x}_{i}$ can be
symmetrized to $\tilde{\bm{x}}_{i}$ by a projection operator:

$\displaystyle\tilde{\bm{x}}_{i}=\bm{x}_{i}+\frac{1}{|\overline{\mathcal{M}}|}\sum_{\kappa}\left[(\bm{W}_{\kappa},\bm{w}_{\kappa})\bm{x}_{\sigma_{\kappa}^{-1}(i)}-\bm{x}_{i}\right]_{\mathrm{mod}\,1}.$
(36)

The modulo is required because the original and mapped point coordinates in
the unit cell may be displaced by lattice translations.

A procedure to symmetrize the array of magnetic moments $\bm{M}$ is similar to
that to symmetrize the array of point coordinates. Each magnetic moment
$\bm{m}_{i}$ can be symmetrized to $\tilde{\bm{m}}_{i}$ by the following
projection operator:

$\displaystyle\tilde{\bm{m}}_{i}=\frac{1}{|\overline{\mathcal{M}}|}\sum_{\kappa}(\bm{W}_{\kappa},\bm{w}_{\kappa})\theta_{\kappa}\bm{m}_{\sigma_{\kappa}^{-1}(i)}.$
(37)

##  6 Conclusion

We have presented the algorithms for determining magnetic symmetry operations
for a given magnetic crystal structure, identifying a magnetic space-group
type for a given MSG, searching for a transformation to the BNS setting, and
symmetrizing the magnetic crystal structure on the basis of the determined
MSG. Matrix and translation parts of magnetic symmetry operations are
determined from the crystal structure by ignoring magnetic moments. A
transformation between the determined MSG and a BNS-setting MSG is obtained by
considering affine normalizers: that of the FSG for type-III MSGs and that of
the XSG for type-IV MSGs. In particular, we provide exhaustive tables of
conjugated MSGs with triclinic or monoclinic type-IV MSGs in the BNS setting
and corresponding transformations. Projection operators of the determined MSG
symmetrize point coordinates and magnetic moments of the magnetic crystal
structure. These algorithms are designed comprehensively and implemented in
spglib under the BSD 3-clause license. The present algorithms and their
implementations are expected to contribute to computational crystallography
and materials science, including high-throughput first-principles calculations
and crystal structure predictions.

##  Appendix A Condition that two MSGs are identical

For two MSGs $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, we write the FSG and XSG
of $\mathcal{M}_{i}\,(i=1,2)$ as $\mathcal{F}_{i}$ and $\mathcal{D}_{i}$,
respectively. When $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ have the same
construct types, they are identical if and only if their FSGs and XSGs are
also identical, that is, $\mathcal{F}_{1}=\mathcal{F}_{2}$ and
$\mathcal{D}_{1}=\mathcal{D}_{2}$. Although it is trivial, we give proof of
this fact for completeness.

When $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ are identical, their FSGs and
XSGs are also identical by definition. We check the converse for each
construct type. For type-I MSGs,
$\mathcal{M}_{1}=\mathcal{F}_{1}1=\mathcal{F}_{2}1=\mathcal{M}_{2}$. For type-
II MSGs,
$\mathcal{M}_{1}=\mathcal{F}_{1}1\sqcup\mathcal{F}_{1}1^{\prime}=\mathcal{F}_{2}1\sqcup\mathcal{F}_{2}1^{\prime}=\mathcal{M}_{2}$.
For type-III or type-IV MSGs,
$\mathcal{M}_{1}=\mathcal{D}_{1}1\sqcup\left(\mathcal{F}_{1}\backslash\mathcal{D}_{1}\right)1^{\prime}=\mathcal{D}_{2}1\sqcup\left(\mathcal{F}_{2}\backslash\mathcal{D}_{2}\right)1^{\prime}=\mathcal{M}_{2}$.
Thus, if FSGs and XSGs are identical, the two MSGs are also identical.

##  Appendix B Correction transformations for triclinic or monoclinic type-IV
MSGs

We give all anti-translations in conjugated MSGs and corresponding
transformations $(\bm{P},\bm{p})$ for a type-IV MSG
$\mathcal{M}_{\mathrm{BNS}}$ in the BNS setting, where
$\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is triclinic or monoclinic. Because
an anti-translation $(\bm{E},\bm{w})1^{\prime}$ in a type-IV MSG is index-two
up to translations, it is sufficient to consider the following seven anti-
translations:

$\displaystyle
1^{\prime}_{a}=\left(\bm{E},\left(\frac{1}{2},0,0\right)\right)1^{\prime}$
(38) $\displaystyle
1^{\prime}_{b}=\left(\bm{E},\left(0,\frac{1}{2},0\right)\right)1^{\prime}$
(39) $\displaystyle
1^{\prime}_{c}=\left(\bm{E},\left(0,0,\frac{1}{2}\right)\right)1^{\prime}$
(40) $\displaystyle
1^{\prime}_{bc}=\left(\bm{E},\left(0,\frac{1}{2},\frac{1}{2}\right)\right)1^{\prime}$
(41) $\displaystyle
1^{\prime}_{ac}=\left(\bm{E},\left(\frac{1}{2},0,\frac{1}{2}\right)\right)1^{\prime}$
(42) $\displaystyle
1^{\prime}_{ab}=\left(\bm{E},\left(\frac{1}{2},\frac{1}{2},0\right)\right)1^{\prime}$
(43) $\displaystyle
1^{\prime}_{abc}=\left(\bm{E},\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right)1^{\prime}.$
(44)

When $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is a triclinic or monoclinic
$P$-centering space group (Nos. 1, 2, 3, 4, 6, 7, 10, 11, 13, and 14), Tables
1, 2, and 4 show transformations for $\mathcal{M}_{\mathrm{BNS}}$, which are
obtained by the brute force described in Sec. 4.2.2, and anti-translations in
the transformed MSGs. Because each table contains the seven anti-translations,
we confirm that these transformations are sufficient to search for conjugated
type-IV MSGs.

For other cases, $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is a monoclinic
$C$-centering space group (Nos. 5, 8, 9, 12, and 15). Tables 3, 5, and 6 show
transformations for $\mathcal{M}_{\mathrm{BNS}}$ and anti-translations in the
transformed MSGs. Note that the anti-translation $1_{ab}^{\prime}$ should not
be contained in the conjugated MSGs because $\mathcal{M}_{\mathrm{BNS}}$ is
not type II and $\mathcal{D}(\mathcal{M}_{\mathrm{BNS}})$ is $C$-centering.
Then, each table contains the six anti-translations other than
$1_{ab}^{\prime}$. Thus, we also confirm that these transformations are
sufficient.

We would like to thank Juan Rodríguez-Carvajal for providing magnetic space-
group datasets tabulated in Gonzalez-Platas:tu5004.

[references]

Figure 1:  Examples of antiferromagnetic (AFM) crystal structures with (a) type-III and (b) type-IV MSGs. The red arrows represent collinear spins with the same magnitudes.  Figure 2:  Group–subgroup relationship of MSGs and related space groups. The nodes represent space groups or MSGs. Each edge indicates that a lower group is a subgroup of an upper group in the diagram. Although it is not exploited in this study, the XSG $\mathcal{D}(\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M}))$ is a subgroup of the FSG $\mathcal{F}(\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M}))$ because the latter simply ignores time-reversal parts of $\mathcal{M}(\bm{A},\bm{X},\bm{T},\bm{M})$.  Figure 3: Group–subgroup relationship of conjugated MSGs and affine normalizers. Table 1: Transformations between type-IV MSG $\mathcal{M}_{\mathrm{BNS}}$ and conjugated MSGs, where their XSGs are identical to space groups with types Nos. 1 and 2 in the ITA standard setting. BNS number | Transformation $(\bm{P},\bm{p})$ | Anti-translations in $(\bm{P},\bm{p})^{-1}\mathcal{M}_{\mathrm{BNS}}(\bm{P},\bm{p})$
---|---|---
1.3, 2.7 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{c}^{\prime}$
| $(-\bm{c},\bm{a}+\bm{b}+\bm{c},\bm{a}+\bm{c};0,0,0)$ | $1_{a}^{\prime}$
| $(\bm{a}+\bm{b}+\bm{c};\bm{c},\bm{a}+\bm{c};0,0,0)$ | $1_{b}^{\prime}$
| $(\bm{a}+\bm{b}+\bm{c},-\bm{a},\bm{a}+\bm{c};0,0,0)$ | $1_{bc}^{\prime}$
| $(\bm{a}+\bm{b},-\bm{a}-\bm{c},\bm{a}+\bm{b}+\bm{c};0,0,0)$ | $1_{ac}^{\prime}$
| $(\bm{a}+\bm{b}+\bm{c},\bm{a}+\bm{b},-\bm{a}-\bm{c};0,0,0)$ | $1_{ab}^{\prime}$
| $(\bm{a}+\bm{b}+\bm{c},\bm{a}+\bm{c},-\bm{b}-\bm{c};0,0,0)$ | $1_{abc}^{\prime}$
Table 2: Transformations between type-IV MSG $\mathcal{M}_{\mathrm{BNS}}$ and conjugated MSGs, where their XSGs are identical to space groups with types Nos. 3, 4, 6, 10, and 11 in the ITA standard setting. BNS number | Transformation $(\bm{P},\bm{p})$ | Anti-translations in $(\bm{P},\bm{p})^{-1}\mathcal{M}_{\mathrm{BNS}}(\bm{P},\bm{p})$
---|---|---
3.4, 4.10, 6.21, 10.47, 11.55 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{a}^{\prime}$
| $(\bm{a}+\bm{c},\bm{b},-\bm{a};0,0,0)$ | $1_{c}^{\prime}$
| $(\bm{a}+\bm{c},\bm{b},\bm{c};0,0,0)$ | $1_{ac}^{\prime}$
3.5, 4.11, 6.22, 10.48, 11.56 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{b}^{\prime}$
3.6, 4.12, 6.23, 10.49, 11.57 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{ab}^{\prime}$
| $(\bm{a}+\bm{c},\bm{b},-\bm{a};0,0,0)$ | $1_{bc}^{\prime}$
| $(\bm{a}+\bm{c},\bm{b},\bm{c};0,0,0)$ | $1_{abc}^{\prime}$
Table 3: Transformations between type-IV MSG $\mathcal{M}_{\mathrm{BNS}}$ and conjugated MSGs, where their XSGs are identical to space groups with types Nos. 5, 8, and 12 in the ITA standard setting. BNS number | Transformation $(\bm{P},\bm{p})$ | Anti-translations in $(\bm{P},\bm{p})^{-1}\mathcal{M}_{\mathrm{BNS}}(\bm{P},\bm{p})$
---|---|---
5.16, 8.35, 12.63 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ |  $1_{c}^{\prime}$, $1_{abc}^{\prime}$
| $(\bm{a},\bm{b},-\bm{a}+\bm{c};0,0,0)$ |  $1_{ac}^{\prime}$, $1_{bc}^{\prime}$
5.17, 8.36, 12.64 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ |  $1_{a}^{\prime}$, $1_{b}^{\prime}$
Table 4:  Transformations between type-IV MSG $\mathcal{M}_{\mathrm{BNS}}$ and conjugated MSGs, where their XSGs are identical to space groups with types Nos. 7, 13, and 14 in the ITA standard setting. Note that BNS numbers 7.30 and 7.31 are not listed in ascending order.  BNS number | Transformation $(\bm{P},\bm{p})$ | Anti-translations in $(\bm{P},\bm{p})^{-1}\mathcal{M}_{\mathrm{BNS}}(\bm{P},\bm{p})$
---|---|---
7.27, 13.70, 14.80 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{a}^{\prime}$
| $(\bm{a}+\bm{c},\bm{b},\bm{c};0,0,0)$ | $1_{ac}^{\prime}$
7.28, 13.71, 14.81 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{b}^{\prime}$
7.29, 13.72, 14.82 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{c}^{\prime}$
7.31, 13.73, 14.83 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{bc}^{\prime}$
7.30, 13.74, 14.84 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ | $1_{ab}^{\prime}$
| $(\bm{a}+\bm{c},\bm{b},\bm{c};0,0,0)$ | $1_{abc}^{\prime}$
Table 5: Transformations between type-IV MSG $\mathcal{M}_{\mathrm{BNS}}$ and conjugated MSGs, where their XSGs are identical to a space group with type No. 9 in the ITA standard setting. BNS number | Transformation $(\bm{P},\bm{p})$ | Anti-translations in $(\bm{P},\bm{p})^{-1}\mathcal{M}_{\mathrm{BNS}}(\bm{P},\bm{p})$
---|---|---
9.40 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ |  $1_{c}^{\prime}$, $1_{abc}^{\prime}$
| $\left(\bm{a},\bm{b},-\bm{a}+\bm{c};0,\frac{1}{4},0\right)$ |  $1_{ac}^{\prime}$, $1_{bc}^{\prime}$
9.41 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ |  $1_{a}^{\prime}$, $1_{b}^{\prime}$
Table 6: Transformations between type-IV MSG $\mathcal{M}_{\mathrm{BNS}}$ and conjugated MSGs, where their XSGs are identical to a space group with type No. 15 in the ITA standard setting. BNS number | Transformation $(\bm{P},\bm{p})$ | Anti-translations in $(\bm{P},\bm{p})^{-1}\mathcal{M}_{\mathrm{BNS}}(\bm{P},\bm{p})$
---|---|---
15.90 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ |  $1_{c}^{\prime}$, $1_{abc}^{\prime}$
| $\left(\bm{a},\bm{b},-\bm{a}+\bm{c};\frac{1}{4},\frac{1}{4},0\right)$ |  $1_{ac}^{\prime}$, $1_{bc}^{\prime}$
15.91 | $(\bm{a},\bm{b},\bm{c};0,0,0)$ |  $1_{a}^{\prime}$, $1_{b}^{\prime}$


figure table

# Personalized Interiors at Scale: Leveraging AI for Efficient and
Customizable Design Solutions

Kaiwen Zhou  Heilongjiang Institute of Technology  Tianyu Wang  Heilongjiang
Institute of Technology

###### Abstract

In this paper, we introduce an innovative application of artificial
intelligence in the realm of interior design through the integration of Stable
Diffusion and Dreambooth models. This paper explores the potential of these
advanced generative models to streamline and democratize the process of room
interior generation, offering a significant departure from conventional,
labor-intensive techniques. Our approach leverages the capabilities of Stable
Diffusion for generating high-quality images and Dreambooth for rapid
customization with minimal training data, addressing the need for efficiency
and personalization in the design industry. We detail a comprehensive
methodology that combines these models, providing a robust framework for the
creation of tailored room interiors that reflect individual tastes and
functional requirements. We presents an extensive evaluation of our method,
supported by experimental results that demonstrate its effectiveness and a
series of case studies that illustrate its practical application in interior
design projects. Our study contributes to the ongoing discourse on the role of
AI in creative fields, highlighting the benefits of leveraging generative
models to enhance creativity and reshape the future of interior design.

###### keywords:

Design, Stable Diffusion, Dreambooth

##  1 Introduction

In the digital era [1, 2, 3, 4], the intersection of technology and creativity
has given rise to unprecedented opportunities for innovation in the field of
design [5, 6, 7, 8, 9, 10, 11, 12]. Traditional methods of room interior
generation [13, 14], once labor-intensive and time-consuming, are now being
revolutionized by artificial intelligence (AI), leading to a new paradigm of
design synthesis [15, 16, 17]. The emergence of generative models like Stable
Diffusion [18] and Dreambooth [19] marks a significant shift in how designers
conceptualize and materialize their visions.

Stable Diffusion, a diffusion model renowned for generating versatile and
realistic images from conditioning data, has proven its capability in the
domain of image generation and editing [20, 21, 22, 23, 24, 25, 26], and holds
significant importance in fields such as 3D generation [27, 28, 29, 30, 31,
32]. It distinguishes itself from conventional generative adversarial networks
(GANs) [33, 34, 35] by adeptly avoiding mode collapse and imprecision, thereby
achieving a high level of visual output quality. This technology’s proficiency
in learning from extensive datasets and replicating intricate patterns and
fine details positions it as an invaluable tool for designers.

Dreambooth advances customization by enabling users to fine-tune the
generative process with minimal training data. This approach facilitates rapid
adaptation to specific user preferences and requirements, fostering a more
personalized design experience.

The fusion of these technologies addresses a critical need in the design
industry for efficient and scalable solutions that keep pace with the dynamic
preferences of modern consumers. The capacity to generate and modify room
interiors with text-based prompts expedites the design process and
democratizes it, inviting a broader spectrum of individuals to participate in
the creation of their living and working spaces.

In this paper, we introduce an innovative method that leverages the power of
Stable Diffusion Models and Dreambooth to generate room interiors that are
aesthetically pleasing and tailored to users’ unique tastes and functional
needs. We explore the potential of these models to transform the approach
interior designers take to their craft, offering a glimpse into a future where
creativity and technology converge to redefine the boundaries of design.

The subsequent sections of this paper will delve into the theoretical
underpinnings of our approach, the methodology employed, and the implications
of our findings for the field of interior design. We will also discuss
potential challenges and future directions for research in this burgeoning
area of study. We then show the structure of this Paper:

Theoretical Foundation: The paper begins with an introduction to the
theoretical basis of Stable Diffusion and Dreambooth, including the
mathematical principles of diffusion models and fine-tuning techniques.

Methodology: We then describe in detail the method combining Stable Diffusion
and Dreambooth for generating interior designs, encompassing data preparation,
model training, and fine-tuning processes.

Experimental Results: A series of experimental results will be presented,
demonstrating the effectiveness of our method and comparing it with existing
technologies.

Case Studies: Several case studies will illustrate how our method can be
applied to real interior design projects and how it aids designers in
realizing their creative visions.

Discussion and Challenges: We will discuss the challenges encountered in
implementing and applying these technologies, including data quality, model
generalizability, and user interaction.

Future Directions: Finally, we will explore future research directions,
including model optimization, cross-domain applications, and integration with
emerging technologies.

Through this in-depth study, we aim to bring new perspectives to the field of
interior design and promote the application of AI in the creative industry.

##  2 Preliminaries

Figure 1: We show the pipeline of stable diffusion models.

###  2.1 Stable diffusion

Stable Diffusion Models have emerged as a pivotal advancement in the realm of
Artificial Intelligence Generated Content (AIGC), providing a robust and
versatile framework for the generation of high-quality images. These models
harness the power of diffusion processes to create intricate and realistic
visual outputs.

####  2.1.1 Conceptual Overview

As shown in Fig. 1, Stable Diffusion Models are generative models that operate
on the principle of diffusion, a probabilistic process that gradually
transforms a set of data into a more disordered state. In the context of image
generation, this process is reversed, where the model learns to generate new
images by reverting from a disordered state back to a coherent image.

####  2.1.2 Key Features

The allure of Stable Diffusion Models lies in their unique set of features
that set them apart from traditional generative adversarial networks (GANs):

  * •

High-Quality Image Generation: These models are capable of producing images
with remarkable clarity and detail.

  * •

Versatility: They can generate a wide range of images from diverse datasets
without overfitting.

  * •

Mode Collapse Resistance: Unlike some GANs, Stable Diffusion Models are less
prone to mode collapse, ensuring a more diverse set of generated images.

  * •

Scalability: The models can be scaled to generate images at various
resolutions, catering to different application needs.

####  2.1.3 Technical Foundation

The technical foundation of Stable Diffusion Models is rooted in the diffusion
process, which can be mathematically described as follows:

$p_{\theta}(x_{t:T})=p_{\theta}(x_{T})\prod_{t=1}^{T-1}\frac{p_{\theta}(x_{t-1}|x_{t})}{p_{\theta}(x_{t}|x_{t-1})}$

Here, $x_{t:T}$ denotes a sequence of image states from time $t$ to $T$, and
$p_{\theta}$ represents the probability distribution parameterized by
$\theta$. The model learns to reverse this process, allowing it to generate
new images from noise.

####  2.1.4 Applications

The applications of Stable Diffusion Models are extensive and cross-
disciplinary:

  * •

Art and Design: Creating unique artwork and designs that can be used in
various creative industries.

  * •

Virtual Reality and Gaming: Generating realistic environments and characters
for immersive experiences.

  * •

Fashion and Apparel: Designing new clothing items and visualizing fashion
collections.

  * •

Architectural Visualization: Producing detailed visual renderings of
architectural projects.

The introduction of Stable Diffusion Models marks a significant milestone in
the journey towards fully realizing the potential of AI in the creative
process, promising a future where AI is not just a tool, but a collaborative
partner in the realm of content creation.

Figure 2: We show the pipeline of Dreambooth.

###  2.2 Dreambooth

Dreambooth is an innovative approach within the field of Artificial
Intelligence Generated Content (AIGC) that enables personalized image
generation with remarkable efficiency. As shown in Fig. 2, this technique
stands out for its ability to fine-tune generative models, such as Stable
Diffusion, using a minimal set of training images that represent the desired
style or features.

####  2.2.1 Key Characteristics of Dreambooth

Dreambooth’s capabilities can be distilled into several key characteristics
that make it an attractive option for content creators and designers alike:

  * •

Minimal Training Data: Unlike traditional generative adversarial networks
(GANs) that require extensive datasets, Dreambooth can be trained with as few
as 3-4 representative images.

  * •

Rapid Customization: The fine-tuning process with Dreambooth is expedited,
allowing for swift adaptation to new visual styles or personalized
preferences.

  * •

High Fidelity Output: The images generated through Dreambooth maintain a high
level of detail and realism, aligning closely with the input examples.

  * •

User-Centric Design: The process is user-friendly, empowering users to guide
the generative process through simple inputs and adjustments.

####  2.2.2 Technical Overview

The technical foundation of Dreambooth involves a series of steps that
transform a general generative model into a specialized one that can produce
images consistent with a particular set of features. The process can be
summarized as follows:

  1. 1.

Data Preparation: A curated set of images that embody the target style or
features is prepared.

  2. 2.

Model Selection: A base generative model, such as Stable Diffusion, is
selected to undergo fine-tuning.

  3. 3.

Fine-Tuning: The model is trained on the prepared dataset, with the objective
of adjusting the parameters to capture the nuances of the input images.

  4. 4.

Generation: Post fine-tuning, the model can generate new images based on
textual prompts or additional conditioning inputs that align with the learned
style.

####  2.2.3 Mathematical Framework

The mathematical framework underpinning Dreambooth involves optimizing the
model’s parameters $\theta$ to minimize a loss function $\mathcal{L}$ that
measures the discrepancy between the generated images $G(z;\theta)$ and the
target images $x_{i}$ from the training set:

$\theta^{*}=\arg\min_{\theta}\mathcal{L}(G(z;\theta),\\{x_{i}\\})$

Here, $G$ represents the generative model, $z$ is the latent space vector, and
$\mathcal{L}$ typically encompasses a mean squared error or a perceptual loss
function that ensures the generated images closely match the style and quality
of the training images.

####  2.2.4 Applications and Potential

Dreambooth’s applications extend beyond mere aesthetic generation, offering
potential in various domains such as:

  * •

Interior Design: Creating digital mock-ups of room interiors based on user
preferences.

  * •

Entertainment: Producing artwork for movies, games, and other media where
specific visual styles are required.

  * •

Fashion Design: Generating fashion items or runway images that align with
emerging trends.

  * •

Architectural Visualization: Visualizing architectural designs with customized
aesthetic touches.

The potential of Dreambooth lies in its ability to democratize content
creation by reducing the barriers of entry in terms of data requirements and
technical expertise. As research progresses, we anticipate even more
sophisticated applications of this technology in the realm of AIGC.

##  3 How does it work?

The research methodology is structured to explore the efficacy of leveraging
Stable Diffusion and Dreambooth models for the generation of room interior
designs. The comprehensive approach involves a series of interconnected
stages, each designed to maximize the creative and functional capabilities of
the AI-driven design process.

###  3.1 Extensive Dataset Compilation

We initiate our methodology with the compilation of an extensive and diverse
dataset encompassing a wide array of room interior images. These images are
sourced from various domains, including residential, commercial, and virtual
reality environments, to ensure representation of different architectural
styles and design elements. The dataset is curated to reflect current design
trends as well as classic elements, providing a rich tapestry of visual data
for the models to learn from.

###  3.2 Data Preprocessing and Augmentation

Each image in the dataset undergoes a rigorous preprocessing phase to
standardize dimensions, color profiles, and resolutions. We employ image
augmentation techniques, such as rotation, scaling, and flipping, to
artificially expand the dataset and introduce variability. This step is
crucial for enhancing the model’s ability to generalize across different
interior settings.

###  3.3 Model Architecture and Customization

The Stable Diffusion model is selected for its demonstrated proficiency in
generating high-quality images from textual descriptions. We customize the
model’s architecture to accommodate the specific nuances of room interior
design, ensuring that it can interpret and manifest complex design concepts
with precision.

###  3.4 Dreambooth Fine-Tuning Protocol

Utilizing Dreambooth, we implement a fine-tuning protocol that requires only a
handful of images from the target room. This innovative approach allows the
model to quickly adapt to the unique features of the space, including layout,
color schemes, and decorative elements. The fine-tuned model is then capable
of generating images that are consistent with the room’s aesthetic.

###  3.5 Text-to-Image Synthesis

As shown in Fig. 3 and 4, we facilitate text-to-image synthesis by allowing
users to input textual prompts describing desired modifications or additions
to the room interior. The fine-tuned model processes these prompts and
generates a series of images that embody the user’s vision. This interactive
design session is iterative, with users providing feedback that guides
subsequent generations.

Figure 3: Dreambooth style 1. We show some generation results.

###  3.6 Multi-Criteria Evaluation System

The generated images are subjected to a multi-criteria evaluation system that
assesses various aspects such as realism, diversity, creativity, and alignment
with user prompts. This system incorporates both quantitative metrics, like
image fidelity and structural accuracy, and qualitative assessments by a panel
of design experts.

Figure 4: Dreambooth style 2. We show some generation results.

###  3.7 Ablation Study and Component Analysis

To deepen our understanding of the model’s performance, we conduct an ablation
study that isolates the impact of individual components. By selectively
disabling or modifying elements of the model, we analyze their contribution to
the overall outcome, providing insights that inform targeted enhancements.

###  3.8 Integration with User-Centered Design Tools

We develop a user-centered design tool that integrates the generative models
into the workflow of professional designers. This tool allows for real-time
interaction with the model, enabling designers to make adjustments and
generate new design iterations promptly.

To evaluate the practical application of our methodology, we conduct a user
experience study with professional designers and amateur enthusiasts. This
study captures feedback on the usability, functionality, and creative
potential of the AI-assisted design process.

Finally, we implement a longitudinal performance assessment to monitor the
model’s performance over time. This assessment tracks the evolution of the
model’s generative capabilities and its ability to satisfy user needs as it is
exposed to more design prompts and feedback.

Through this detailed and structured methodology, we aim to establish a robust
framework for AI-assisted room interior generation. The goal is to push the
boundaries of what is possible with current technology, creating a synergy
between human creativity and artificial intelligence that enhances the design
process across various domains.

##  4 Discussion

The integration of AI-driven generative models like Stable Diffusion and
Dreambooth into the field of interior design, while immensely promising,
introduces a variety of challenges that need to be carefully navigated. A
critical concern is the quality and diversity of the training data, as the
models’ outputs are inherently biased towards the data they are trained on.
This can lead to a lack of representation and potential cultural insensitivity
in design. Ensuring that these tools are accessible and user-friendly is also
paramount; the development of intuitive interfaces that allow designers to
interact seamlessly with AI systems is an ongoing challenge.

Moreover, the ethical implications of AI in design cannot be overlooked.
Questions of intellectual property, model transparency, and the potential for
AI to supplant human creativity in design must be addressed. It is essential
to foster an environment where AI is seen as a collaborative tool that
enhances human creativity rather than as a substitute.

Another challenge is the generalizability of AI models to diverse design
contexts and the ability to integrate these models into existing design
workflows without disrupting established practices. The computational cost of
training and deploying these models also presents a barrier, particularly for
smaller studios or individual designers with limited resources.

Furthermore, there is a pressing need for education and training to equip
designers with the necessary skills to work alongside AI. As these
technologies evolve, continuous learning and development will be crucial to
fully leverage their potential.

In summary, while the advent of AI in interior design heralds a new era of
innovation and efficiency, it also necessitates a proactive approach to
overcome the associated challenges. It requires a concerted effort from
designers, technologists, ethicists, and industry leaders to steer this
technology towards a future that is inclusive, ethical, and truly enhances the
creative process.

##  5 Conclusion

In conclusion, we has presented a pioneering exploration into the integration
of AI technologies, specifically Stable Diffusion and Dreambooth, within the
creative space of interior design. Through a meticulous examination of these
models’ capabilities, we have demonstrated their profound impact on the design
process, enabling the generation of customized room interiors with
unprecedented efficiency and precision. The experimental results and case
studies provided have not only validated the effectiveness of our method but
have also showcased the transformative potential of AI in making the design
process more accessible and responsive to individual user needs. The
challenges discussed, while significant, underscore the areas ripe for future
research and development. Addressing issues of data quality, model
generalizability, and user interaction will be key to further refining these
AI tools and expanding their application in the design industry. We also
recognize the ethical considerations and the importance of intellectual
property rights in the context of AI-generated content. As we look to the
future, the convergence of creativity and technology holds vast possibilities.
The research presented in this paper is a stepping stone towards a horizon
where AI is seamlessly intertwined with human creativity, augmenting the
designer’s craft and pushing the boundaries of what is achievable in interior
design. We envision a future where generative models like Stable Diffusion and
Dreambooth are commonplace tools in the designer’s arsenal, facilitating the
creation of spaces that are not only functional but also deeply personal and
aesthetically resonant.

## References

  * [1] R. Schroeder, “Towards a theory of digital media,” Information, Communication & Society 21(3), 323–339 (2018).
  * [2] P. N. Howard and M. M. Hussain, “The role of digital media,” J. Democracy 22, 35 (2011).
  * [3] E. G. Coleman, “Ethnographic approaches to digital media,” Annual review of anthropology 39, 487–505 (2010).
  * [4] N. Memon and P. W. Wong, “Protecting digital media content,” Communications of the ACM 41(7), 35–43 (1998).
  * [5] M. R. Lee and T. T. Chen, “Digital creativity: Research themes and framework,” Computers in human behavior 42, 12–19 (2015).
  * [6] R. S. Ulrich, “Effects of interior design on wellness: theory and recent scientific research.,” in Journal of Health Care Interior Design: Proceedings from the… Symposium on Health Care Interior Design. Symposium on Health Care Interior Design, 3, 97–109 (1991).
  * [7] J. F. Pile, A history of interior design, Laurence King Publishing (2005).
  * [8] L. K. Havenhand, “A view from the margin: Interior design,” Design Issues 20(4), 32–42 (2004).
  * [9] M. Kang and D. A. Guerin, “The characteristics of interior designers who practice environmentally sustainable interior design,” Environment and Behavior 41(2), 170–184 (2009).
  * [10] D. Garlan, R. Allen, and J. Ockerbloom, “Exploiting style in architectural design environments,” ACM SIGSOFT software engineering notes 19(5), 175–188 (1994).
  * [11] V. Machairas, A. Tsangrassoulis, and K. Axarli, “Algorithms for optimization of building design: A review,” Renewable and sustainable energy reviews 31, 101–112 (2014).
  * [12] N. K. Booth, Basic elements of landscape architectural design, Waveland press (1989).
  * [13] W. Wu, X.-M. Fu, R. Tang, et al., “Data-driven interior plan generation for residential buildings,” ACM Transactions on Graphics (TOG) 38(6), 1–12 (2019).
  * [14] T. Tutenel, R. Bidarra, R. M. Smelik, et al., “Rule-based layout solving and its application to procedural interior generation,” in CASA workshop on 3D advanced media in gaming and simulation, (2009).
  * [15] S. Amershi, D. Weld, M. Vorvoreanu, et al., “Guidelines for human-ai interaction,” in Proceedings of the 2019 chi conference on human factors in computing systems, 1–13 (2019).
  * [16] J. Auernhammer, “Human-centered ai: The role of human-centered design research in the development of ai,” (2020).
  * [17] R. Verganti, L. Vendraminelli, and M. Iansiti, “Innovation and design in the age of artificial intelligence,” Journal of product innovation management 37(3), 212–227 (2020).
  * [18] R. Rombach, A. Blattmann, D. Lorenz, et al., “High-resolution image synthesis with latent diffusion models,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 10684–10695 (2022).
  * [19] N. Ruiz, Y. Li, V. Jampani, et al., “Dreambooth: Fine tuning text-to-image diffusion models for subject-driven generation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 22500–22510 (2023).
  * [20] T. Brooks, A. Holynski, and A. A. Efros, “Instructpix2pix: Learning to follow image editing instructions,” in CVPR, (2023).
  * [21] A. Hertz, R. Mokady, J. Tenenbaum, et al., “Prompt-to-prompt image editing with cross attention control,” in ICLR, (2023).
  * [22] P. Li, Q. Huang, Y. Ding, et al., “Layerdiffusion: Layered controlled image editing with diffusion models,” in SIGGRAPH Asia 2023 Technical Communications, 1–4 (2023).
  * [23] C. Meng, Y. Song, J. Song, et al., “Sdedit: Image synthesis and editing with stochastic differential equations,” in ICLR, (2022).
  * [24] N. Tumanyan, M. Geyer, S. Bagon, et al., “Plug-and-play diffusion features for text-driven image-to-image translation,” in CVPR, (2023).
  * [25] B. Kawar, S. Zada, O. Lang, et al., “Imagic: Text-based real image editing with diffusion models,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 6007–6017 (2023).
  * [26] P. Li, B. Li, and Z. Li, “Sketch-to-Architecture: Generative AI-aided Architectural Design,” in Pacific Graphics Short Papers and Posters 2023, (2023).
  * [27] P. Li, C. Tang, Q. Huang, et al., “Art3d: 3d gaussian splatting for text-guided artistic scenes generation,” arXiv:2405.10508 (2024).
  * [28] B. Poole, A. Jain, J. T. Barron, et al., “DreamFusion: Text-to-3d using 2d diffusion,” in Int. Conf. Learn. Represent., (2023).
  * [29] C.-H. Lin, J. Gao, L. Tang, et al., “Magic3d: High-resolution text-to-3d content creation,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 300–309 (2023).
  * [30] P. Li and B. Li, “Generating daylight-driven architectural design via diffusion models,” arXiv preprint arXiv:2404.13353 (2024).
  * [31] E. R. Chan, K. Nagano, M. A. Chan, et al., “Generative novel view synthesis with 3d-aware diffusion models,” arXiv preprint arXiv:2304.02602 (2023).
  * [32] J. Tang, T. Wang, B. Zhang, et al., “Make-it-3d: High-fidelity 3d creation from a single image with diffusion prior,” arXiv preprint arXiv:2303.14184 (2023).
  * [33] I. Goodfellow, J. Pouget-Abadie, M. Mirza, et al., “Generative adversarial networks,” Communications of the ACM 63(11), 139–144 (2020).
  * [34] L. Metz, B. Poole, D. Pfau, et al., “Unrolled generative adversarial networks,” arXiv preprint arXiv:1611.02163 (2016).
  * [35] I. Goodfellow, J. Pouget-Abadie, M. Mirza, et al., “Generative adversarial nets,” Advances in neural information processing systems 27 (2014).