††footnotetext: ∗ Corresponding author.

# Characterization of (weak) phase retrieval dual frames

Fahimeh Arabyani-Neyshaburi1<EMAIL_ADDRESS><EMAIL_ADDRESS>Ali Akbar Arefijamaal2 arefijamaal<EMAIL_ADDRESS>Rajab Ali
Kamyabi-Gol3,∗<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recovering a signal up to a unimodular constant from the magnitudes of linear
measurements has been popular and well studied in recent years. However,
numerous unsolved problems regarding phase retrieval still exist. Given a
phase retrieval frame, may the family of phase retrieval dual frames be
classified? And is such a family dense in the set of dual frames? Can we
present the equivalent conditions for a family of vectors to do weak phase
retrieval in complex Hilbert space case? What is the connection between phase,
weak phase and norm retrieval? In this context, we aim to deal with these open
problems concerning phase retrieval dual frames, weak phase retrieval frames,
and specially investigate equivalent conditions for identifying these
features. We provide some characterizations of alternate dual frames of a
phase retrieval frame which yield phase retrieval in finite dimensional
Hilbert spaces. Moreover, for some classes of frames, we show that the family
of phase retrieval dual frames is open and dense in the set of dual frames.
Then, we study weak phase retrieval problem. Among other things, we obtain
some equivalent conditions on a family of vectors to do phase retrieval in
terms of weak phase retrieval.

1 Department of Mathematical sciences, Ferdowsi University of Mashhad,
Mashhad, Iran.

2 Department of Mathematics and Computer Sciences, Hakim Sabzevari University,
Sabzevar, Iran.

3 Department of Mathematical sciences, Faculty of Math, Ferdowsi University of
Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS),
Mashhad, Iran. Email:

Mathematics Subject Classification 2020. Primary 42C15; Secondary 41A58.

Keywords. Phase retrieval, weak phase retrieval, dual frames.

##  1 Introduction and Preliminaries

Signal reconstruction without using phase is a longstanding conjecture,
specially with regard to speech recognition systems which was first introduced
from mathematical point of view by R. Balan, P. Casazza and D. Edidin in [8],
and has been a very popular topic in recent years due to so many applications
of $X$-ray, electron microscopy, optics, image processing and much more [5,
14, 20, 21]. To present the problem in a more precise approach, we first
briefly state some basic definitions and preliminaries in relevant areas. Then
we propound a number of problems to address phase retrieval frames in some new
aspects.

Throughout this paper, we suppose that $\mathcal{H}$ denotes a separable
Hilbert space, $\mathcal{H}_{n}$ denote an $n$-dimensional real or complex
Hilbert space and we use $\mathbb{R}^{n}$ and $\mathbb{C}^{n}$ whenever it is
necessary to differentiate between the two. Also, we consider the notations;
$I_{m}=\\{1,2,...,m\\}$ and $\\{\delta_{i}\\}_{i\in I_{n}}$ as the standard
orthonormal basis of $\mathcal{H}_{n}$.

A family of vectors $\Phi:=\\{\phi_{i}\\}_{i\in I}$ in $\mathcal{H}$ is called
a frame if there exist the constants $0<A_{\Phi}\leq B_{\Phi}<\infty$ such
that

$\displaystyle A_{\Phi}\|f\|^{2}\leq\sum_{i\in I}|f,\phi_{i}\rangle|^{2}\leq
B_{\Phi}\|f\|^{2},\qquad(f\in\mathcal{H}).$ (1.1)

The constants $A_{\Phi}$ and $B_{\Phi}$ are called frame bounds. The sequence
$\\{\phi_{i}\\}_{i\in I}$ is said to be a Bessel sequence whenever the right
hand side of (1.1) holds. A frame $\\{\phi_{i}\\}_{i\in I}$ is called
$A$-tight frame if $A=A_{\Phi}=B_{\Phi}$, and in the case of
$A_{\Phi}=B_{\Phi}=1$ it is called a Parseval frame. Given a frame
$\Phi=\\{\phi_{i}\\}_{i\in I}$, its Grammian matrix formed by the inner
product of the frame vectors is as
$G_{\Phi}=[\langle\phi_{i},\phi_{j}\rangle]_{i,j}$. The frame operator is
defined by $S_{\Phi}f=\sum_{i\in I}\langle f,\phi_{i}\rangle\phi_{i}$. It is a
bounded, invertible, and self-adjoint operator [13]. Also, the synthesis
operator $T_{\Phi}:l^{2}\rightarrow\mathcal{H}$ is defined by
$T_{\Phi}\\{c_{i}\\}=\sum_{i\in I}c_{i}\phi_{i}$. The frame operator can be
written as $S_{\Phi}=T_{\phi}T_{\phi}^{*}$ where
$T_{\Phi}^{*}:\mathcal{H}\rightarrow l^{2}$, the adjoint of $T$, given by
$T_{\Phi}^{*}f=\\{\langle f,\phi_{i}\rangle\\}_{i\in I}$ is called the
analysis operator. The family $\\{S_{\Phi}^{-1}f_{i}\\}_{i\in I}$ is also a
frame for $\mathcal{H}$, called the canonical dual frame. In general, a frame
$\\{\psi_{i}\\}_{i\in I}\subseteq\mathcal{H}$ is called an alternate dual or
simply a dual for $\\{\phi_{i}\\}_{i\in I}$ if $f=\sum_{i\in I}\langle
f,\psi_{i}\rangle\phi_{i},$ for $f\in\mathcal{H}$. All frames have at least a
dual, the canonical dual, and redundant frames have an infinite number of
alternate dual frames. We denote the excess of a frame $\Phi$ by $E(\Phi)$. It
is known that every dual frame is of the form
$\\{S_{\Phi}^{-1}\phi_{i}+u_{i}\\}_{i\in I}$, where $\\{u_{i}\\}_{i\in I}$ is
a Bessel sequence that satisfies $\sum_{i\in I}\langle
f,u_{i}\rangle\phi_{i}=0,$ for all $f\in\mathcal{H}$. Also recall that, two
frames $\Phi$ and $\Psi$ are equivalent if there exists an invertible operator
$U$ on $\mathcal{H}$ so that $\Psi=U\Phi$. See [13, 16] for more detailed
information on frame theory and [3, 4, 7, 17, 18, 19] for the importance of
duality principle.

Consider a frame $\\{\phi_{i}\\}_{i\in I_{m}}$ in a Hilbert space
$\mathcal{H}_{n}$. A finite set of indices $\sigma\subset I_{m}$ satisfies the
minimal redundancy condition (MRC) whenever $\\{\phi_{i}\\}_{i\in\sigma^{c}}$
remains to be a frame for $\mathcal{H}_{n}$. Furthermore, we say $\Phi$
satisfies MRC for $r$-erasures if every subset $\sigma\subset I_{m}$,
$|\sigma|=r$ satisfies MRC for $\Phi$. The spark of a matrix is the size of
the smallest linearly dependent subset of the columns and the spark of a
family $\\{\phi_{i}\\}_{i\in I_{m}}$ is defined as the spark of its synthesis
matrix $\Phi$. Also, the family $\\{\phi_{i}\\}_{i\in I_{m}}$, $m\geq n$ is
said full spark if it has the spark $n+1$. It is shown that if $m\geq n$, then
the set of full spark frames is open and dense in the set of all frames [2].

Suppose now the nonlinear mapping

$\displaystyle\mathbb{M}_{\Phi}:\mathcal{H}\rightarrow
l^{2}(I),\qquad\mathbb{M}_{\Phi}(f)=\\{|\langle
f,\phi_{i}\rangle|^{2}\\}_{i\in I}$ (1.2)

obtained by taking the absolute value element wise of the analysis operator.
Let us denote by $\mathbb{H}=\mathcal{H}/\sim$ considered by identifying two
vectors which are different in a phase factor, i.e., $f\sim g$ whenever there
exists a scalar $\theta$ with $|\theta|=1$ so that $g=\theta f$. Obviously in
a real Hilbert space we have $\mathbb{H}=\mathcal{H}/\\{1,-1\\}$ and in the
complex case $\mathbb{H}=\mathcal{H}/\mathbb{T}$, where $\mathbb{T}$ is the
complex unit circle. So, the mapping $\mathbb{M}_{\Phi}$ can be extended to
$\mathbb{H}$ as $\mathbb{M}_{\Phi}(\hat{f})=\\{|\langle
f,\phi_{i}\rangle|^{2}\\}_{i\in I}$, $f\in\hat{f}=\\{g\in\mathcal{H}:g\sim
f\\}$. The injectivity of the nonlinear mapping $\mathbb{M}_{\Phi}$ leads to
the reconstruction of every signal in $\mathcal{H}$ up to a constant phase
factor from the modula of its frame coefficients. In [8], the authors
investigated the injectivity of $\mathbb{M}_{\Phi}$ in finite dimensional real
Hilbert spaces and moreover, it was proven that $4n-2$ measurements suffice
for the injectivity in $n$-dimensional complex Hilbert spaces. Indeed, the
injectivity of the mapping $\mathbb{M}_{\Phi}$ is equivalent to the following
definition:

######  Definition 1.1.

A family of vectors $\Phi=\\{\phi_{i}\\}_{i\in I}$ in $\mathcal{H}$ does phase
retrieval if whenever $f,g\in\mathcal{H}$ satisfy

$\displaystyle|\langle f,\phi_{i}\rangle|=|\langle
g,\phi_{i}\rangle|,\qquad(i\in I)$ (1.3)

then there exists a scalar $\theta$ with $|\theta|=1$ so that $f=\theta g$.

If for every $f,g\in\mathcal{H}$, which satisfy $(\ref{121})$ we get
$\|f\|=\|g\|$, then it said $\Phi$ to do norm retrieval. Clearly, if $\Phi$
does phase (norm) retrieval, then so does $\alpha_{i}f_{i}$ for every
$0<\alpha_{i}$, $i\in I$. Also, tight frames do norm retrieval. Moreover,
phase retrieval implies norm retrieval, but the converse fails. For example
every orthonormal basis does norm retrieval, but fails at phase retrieval.

The following result states that for two equivalent frames $\Phi$ and $\psi$,
the injectivity of $\mathbb{M}_{\Phi}$ and $\mathbb{M}^{\Psi}$ are the same.

######  Theorem 1.2.

[8] A family $\Phi=\\{\phi_{i}\\}_{i\in I}$ in $\mathcal{H}$ does phase
retrieval if and only if $\\{U\phi_{i}\\}_{i\in I}$ does phase retrieval for
every invertible operator $U$ on $\mathcal{H}$.

Applying the above theorem, shows that a frame does phase retrieval if and
only if its canonical dual does phase retrieval.

######  Definition 1.3.

[8] A family of vectors $\Phi=\\{\phi_{i}\\}_{i\in I}$ in $\mathcal{H}$ has
the complement property if for every $\sigma\subset I$ either
$\overline{span}\\{\phi_{i}\\}_{i\in\sigma}=\mathcal{H}$ or
$\overline{span}\\{\phi_{i}\\}_{i\in\sigma^{c}}=\mathcal{H}$.

As we stated, a fundamental classification of frames which do phase retrieval
was presented for finite dimensional real case in [8] and then for infinite
dimensional case in [11] as follows:

######  Theorem 1.4.

A family $\Phi=\\{\phi_{i}\\}_{i\in I}$ in a real Hilbert space $\mathcal{H}$
does phase retrieval if and only if it has the complement property.

As an immediate result of the above theorem, every phase retrieval frame in
real Hilbert space $\mathcal{H}$ is satisfied in MRC for $(n-1)$-erasures.

######  Proposition 1.5.

[8] If $\Phi=\\{\phi_{i}\\}_{i\in I_{m}}$ does phase retrieval in
$\mathbb{R}^{n}$, then $m\geq 2n-1$. If $m\geq 2n-1$ and $\Phi$ is full spark
then $\mathcal{\phi}$ does phase retrieval. Moreover, $\\{\phi_{i}\\}_{i\in
I_{2n-1}}$ does phase retrieval if and only if $\Phi$ is full spark.

Two vectors $x=\\{x_{i}\\}_{i\in I_{n}}$ and $y=\\{y_{i}\\}_{i\in I_{n}}$ in
$\mathcal{H}_{n}$ weakly have the same phase if there is a $|\alpha|=1$ so
that $phase(x_{i})=\alpha phase(y_{i})$, for all $i\in I_{n}$ which $x_{i}\neq
0\neq y_{i}$.

######  Definition 1.6.

A family $\Phi=\\{\phi_{i}\\}_{i\in I_{m}}$ in $\mathcal{H}_{n}$ does weak
phase retrieval if for any $x,y\in\mathcal{H}_{n}$ with $|\langle
x,\phi_{i}\rangle|=|\langle y,\phi_{i}\rangle|$ for all $i\in I_{m}$, then $x$
and $y$ weakly have the same phase.

It is shown that if $\Phi=\\{\phi_{i}\\}_{i\in I_{m}}$ does weak phase
retrieval in $\mathbb{R}^{n}$, then $m\geq 2n-2$ [10]. Also, clearly phase
retrieval implies weak phase retrieval property, although the converse does
not hold in general. As a simple example $\\{(1,1),(1,-1)\\}$ does weak phase
retrieval for $\mathbb{R}^{2}$, see[1], but clearly does not phase retrieval.

Now, we state the concept of a generic set; A subset
$\Omega\subseteq\mathbb{R}^{n}$ is called generic whenever there exists a
nonzero polynomial $p(x_{1},...,x_{n})$ so that

$\Omega^{c}\subseteq\\{(x_{1},...,x_{n})\in\mathbb{R}^{n}:p(x_{1},...,x_{n})=0\\}.$

It is known that generic sets are open, dense and full measure. Furthermore, a
generic set in $\mathbb{C}^{n}$ is defined as a generic set in
$\mathbb{R}^{2n}$.

Applying the linear operator introduced in [6] gives an equivalent condition
for injectivity of $\mathbb{M}_{\Phi}$, defined by $(\ref{Phase map})$. More
precisely, let $\\{\phi_{i}\\}_{i\in I_{m}}$ be a family of vectors in
$\mathbb{C}^{n}$ and $H_{n\times n}$ denotes the space of all $n\times n$
Hermitian matrices. Consider the operator $\Lambda_{\Phi}:H_{n\times
n}\rightarrow\mathbb{R}^{m}$ by $\Lambda_{\Phi}(A)=\\{\langle
A,\phi_{i}\otimes\phi_{i}\rangle\\}_{i\in I_{m}}$, in which
$\phi_{i}\otimes\phi_{i}$ is the rank one projection onto
$span\\{\phi_{i}\\}$. It is easily proven that $\Lambda_{\Phi}(f\otimes
f)=\mathbb{M}_{\Phi}(f)$ and this equality has a key role for characterizing
the injectivity of $\mathbb{M}_{\Phi}$, [6, 12], See also [15]. So, we get the
following equivalent conditions for a frame to do phase retrieval.

######  Corollary 1.7.

Let $\Phi=\\{\mathcal{\phi}_{i}\\}_{i\in I_{m}}$ be a frame in
$\mathcal{H}_{n}$, then the followings are equivalent;

  * (i)

$\Phi$ does phase retrieval.

  * (ii)

$\mathbb{M}_{\Phi}$ is injective.

  * (iii)

$\Lambda_{\Phi}|_{B_{1}}$ is injective, where $B_{1}$ denotes rank one
matrices.

  * (iv)

There exists no rank 2 matrix in the null space of $\Lambda_{\Phi}$.

###### Proof.

$(i)\Leftrightarrow(ii)$ is obvious by the definition. For
$(ii)\Leftrightarrow(iv)$ we just note that, due to the completeness of $\Phi$
in $\mathcal{H}_{n}$, the rank one matrices $B_{1}=\\{f\otimes f:\quad
f\in\mathcal{H}_{n}\\}$ can not be in the null space of $\Lambda_{\Phi}$.
Indeed, $\Lambda_{\Phi}(f\otimes f)=\mathbb{M}_{\Phi}(f)=0$ implies that
$f\perp\phi_{i}$, for all $i\in I_{m}$ and so $f=0$. So, as a result of Lemma
5.5 of [6], the injectivity of $\mathbb{M}_{\Phi}$ is equivalent to the
statement that; there is no rank 2 matrix in the null space of
$\Lambda_{\Phi}$. On the other hand, the injectivity of $\mathbb{M}_{\Phi}$
along with the fact that the mapping $\gamma:\mathcal{H}_{n}\rightarrow
B_{1}$, $\gamma(f)=f\otimes f$ is invertible, deduce that the linear operator
$\Lambda_{\Phi}|_{B_{1}}$ is also injective. Moreover, for a left inverse
$\mathbb{L}_{\Phi}$ of $\mathbb{M}_{\Phi}$, the mapping
$\gamma\mathbb{L}_{\Phi}$ is a left inverse for $\Lambda_{\Phi}|_{B_{1}}$.
Conversely, for a left inverse $\Gamma_{\Phi}$ of $\Lambda_{\Phi}|_{B_{1}}$,
the mapping $\gamma^{-1}\Gamma_{\Phi}$ is a left inverse for
$\mathbb{M}_{\Phi}$, this completes the proof of $(ii)\Leftrightarrow(iii)$. ∎

This paper is organized as follows: Section 2 is devoted to characterizing
phase retrieval dual frames and full spark dual frames. In this section, for
some classes of frames, we show that the family of all ( full spark) phase
retrieval dual frames is open and dense in the set of all dual frames.
Moreover, we present several examples in this regard. In Section 3, we survey
weak phase retrieval problem and investigate some equivalent conditions for
identifying phase and weak phase retrieval frames. We also, obtain a
sufficient conditions on a family of weak phase retrieval frames to constitute
a frame and its canonical dual yields weak phase retrieval, as well.

##  2 Phase Retrieval Dual Frames

In this section, we address the problem that, given a phase retrieval frame
$\Phi$, characterize phase retrieval dual frames of $\Phi$ in finite
dimensional Hilbert spaces. For some classes of frames we show that phase
retrieval dual frames of a given frame are dense in the set of all dual
frames. For a frame $\Phi$ we denote the set of all its dual frames of $\Phi$
by $D_{\Phi}$ and the subset of phase retrieval dual frames is denoted by
$PD_{\Phi}$. We first investigate the relationship between phase retrieval
duals of equivalent frames, which is a very useful tool for the main results
of this section.

######  Lemma 2.1.

Suppose that $\Phi=\\{\phi_{i}\\}_{i\in I_{m}}$ is a frame for
$\mathcal{H}_{n}$ and $\mathcal{T}$ is an invertible operator on
$\mathcal{H}_{n}$. Then,

  * (i)

$D_{\mathcal{T}\Phi}=(\mathcal{T}^{*})^{-1}D_{\Phi}$.

  * (ii)

$PD_{\mathcal{T}\Phi}=(\mathcal{T}^{*})^{-1}PD_{\Phi}$.

###### Proof.

It is known that $\mathcal{T}\Phi$ is a frame with
$S_{\mathcal{T}\Phi}=\mathcal{T}S_{\Phi}\mathcal{T}^{*}$ [13], and so a simple
computation assures that $D_{\mathcal{T}\Phi}=\\{(\mathcal{T}^{*})^{-1}G:G\in
D_{\Phi}\\}=(\mathcal{T}^{*})^{-1}D_{\Phi}$. Moreover, $(ii)$ is given as an
immediate result of $(i)$ along with Theorem 1.2. ∎

######  Remark 2.2.

If $\Phi=\\{\phi_{i}\\}_{i\in I_{m}}$ is a frame for $\mathcal{H}_{n}$ so that
$E(\Phi)=k$, which $E(\Phi)$ denotes the excess of $\Phi$, then there exist
$i_{1},...,i_{k}$ so that $\Phi\setminus\\{\phi_{i_{j}}\\}_{j=1}^{k}$
constitutes a Riesz basis for $\mathcal{H}_{n}$. Therefore, applying the above
notations and without loss of generality, we may consider
$\Phi=\\{\phi_{i}\\}_{i\in I_{n}}\cup\\{\phi_{i}\\}_{i=n+1}^{m}$, where
$\\{\phi_{i}\\}_{i\in I_{n}}$ is a Riesz basis for $\mathcal{H}_{n}$. In this
point of view, $\Phi$ is indeed equivalent to a form as
$\\{\delta_{i}\\}_{i\in I_{n}}\cup\\{\tilde{\phi}_{i}\\}_{i=n+1}^{m}$ with
redundant elements $\\{\tilde{\phi}_{i}\\}_{i=n+1}^{m}$.

In the next theorem we identify the set of dual frames of a frame
$\\{\phi_{i}\\}_{i\in I_{2n-1}}$ in $n$-dimensional real space and show that
$PD_{\phi}$ is an open and dense subset in $D_{\phi}$.

######  Theorem 2.3.

Let $\phi=\\{\phi_{i}\\}_{i\in I_{2n-1}}$ be a phase retrieval frame in
$\mathbb{R}^{n}$. Then $PD_{\phi}$ is an open and dense subset in $D_{\phi}$.

###### Proof.

First let $\\{\phi_{i}\\}_{i\in I_{n}}$ be the standard orthonormal basis of
$\mathbb{R}^{n}$ and $\phi_{j}=\sum_{i\in I_{n}}a_{i}^{j}\phi_{i}$,
$j=n+1,...,2n-1$, for some non-zero coefficients $\\{a_{i}^{j}\\}_{i\in
I_{n}}$. Due to the fact that

$\displaystyle D_{\Phi}=\left\\{\\{S_{\Phi}^{-1}\phi_{i}+u_{i}\\}_{i\in
I_{2n-1}}:\sum_{i\in I_{2n-1}}\langle f,\phi_{i}\rangle u_{i}=0,\textit{for
all }f\in\mathbb{R}^{n}\right\\}$

by putting $f=\phi_{j}$, $j=1,...,2n-1$ we observe that $(2n-1)\times n$
matrix $[u_{1}|u_{2}|...|u_{2n-1}]^{T}$ is in the null space of
$G_{\Phi}^{T}$, the transposed Gram matrix of $\Phi$. The fact that,
$dim(nullG_{\Phi})=n-1$; assures that just $n-1$ vectors of $\\{u_{i}\\}_{i\in
I_{2n-1}}$ can be independent. More precisely,

$\displaystyle u_{1}+\sum_{i=n+1}^{2n-1}\langle\phi_{1},\phi_{i}\rangle
u_{i}=0,\quad...\quad u_{n}+\sum_{i=n+1}^{2n-1}\langle\phi_{n},\phi_{i}\rangle
u_{i}=0.$

So, by choosing $\\{u_{j}\\}_{j=n+1}^{2n-1}$ we get

$u_{i}=-\sum_{j=n+1}^{2n-1}a_{i}^{j}u_{j},\quad(i\in I_{n}),$ (2.1)

in which $a_{i}^{j}=\langle\phi_{i},\phi_{j}\rangle$, for $n+1\leq j\leq
2n-1$. Hence, every dual frame is uniquely constructed by the following vector

$\displaystyle\mathcal{U}=\left[u_{n+1},...,u_{2n-1}\right]\in\mathbb{R}^{n(n-1)}.$
(2.2)

Considering $D_{\Phi}$ as a metric space by $d(G,H)=\sum_{i\in
I_{2n-1}}\|g_{i}-h_{i}\|$, we define the mapping
$\xi:D_{\Phi}\rightarrow\mathbb{R}^{n(n-1)}$ by $\xi(G)=\mathcal{U}_{g}$,
where $\mathcal{U}_{g}\in\mathbb{R}^{n(n-1)}$ is the unique sequence
associated to $G\in D_{\Phi}$ as in $(\ref{uniqueg})$. Then, clearly $\xi$ is
well-defined and injective. Also, take $\mathcal{A}\in\mathbb{R}^{n(n-1)}$,
then put $u_{n+1}=\\{\mathcal{A}_{1},...,\mathcal{A}_{n}\\}$,…,
$u_{2n-1}=\\{\mathcal{A}_{n^{2}-2n+1},...,\mathcal{A}_{n(n-1)}\\}$ and
construct $u_{i}$, $i\in I_{n}$ by $(\ref{constuctu})$. Thus we get
$\\{S_{\Phi}^{-1}\phi_{i}+u_{i}\\}_{i\in I_{2n-1}}\in D_{\Phi}$, i.e., $\xi$
is a bijective map. Moreover, $\xi$ is a Lipschitz function. Let
$G=\\{S_{\Phi}^{-1}\phi_{i}+u_{i}\\}_{i\in I_{2n-1}}$ and
$H=\\{S_{\Phi}^{-1}\phi_{i}+v_{i}\\}_{i\in I_{2n-1}}$ be dual frames of $\Phi$
with the associated $\mathcal{U}_{g}$ and $\mathcal{U}_{h}$ obtained as in
$(\ref{uniqueg})$, respectively. Then,

$\displaystyle\|\xi(G)-\xi(H)\|=\|\mathcal{U}_{g}-\mathcal{U}_{h}\|=\sum_{i=n+1}^{2n-1}\|u_{i}-v_{i}\|\leq\sum_{i\in
I_{2n-1}}\|u_{i}-v_{i}\|=d(G,H).$

What is more, $\xi$ is a bi-Lipschitz function, but we will not need this
fact. Now, we note that the only cases in which a dual frame
$\\{g_{i}\\}_{i\in I_{2n-1}}$ fails to do phase retrieval is associated to
$det[g_{i_{1}}|\quad...\quad|g_{i_{n}}]=0$ for some index set
$\\{i_{j}\\}_{j=1}^{n}\subset I_{2n-1}$, by Theorem 1.4. Multiplying these
equations yields an $n(n-1)$-variable polynomial in terms of $u_{n+1}$, …,
$u_{2n-1}$, denoted by $P_{\mathcal{\phi}}$. Therefore,

$\displaystyle
PD_{\Phi}=\xi^{-1}\\{\mathcal{U}\in\mathbb{R}^{n(n-1)}:P_{\mathcal{\phi}}(\mathcal{U})\neq
0\\},$ (2.3)

that means $PD_{\Phi}$ is open in $D_{\Phi}$. To show $PD_{\Phi}$ is,
moreover, dense in $D_{\phi}$, let $\epsilon>0$ and $G=\\{g_{i}\\}_{i\in
I_{2n-1}}=\\{S_{\Phi}^{-1}\Phi_{i}+u_{i}\\}_{i\in I_{2n-1}}$ be a dual frame
in $D_{\Phi}\setminus PD_{\Phi}$. Then $G$ is dependent just in vectors
$\\{u_{i}\\}_{i=n+1}^{2n-1}$ such that
$P_{\mathcal{\phi}}(\\{u_{i}\\}_{i=n+1}^{2n-1})=0$. Take a sequence
$\\{\\{v_{i}^{k}\\}_{i=n+1}^{2n-1}\\}_{k\in\mathbb{N}}$ out of the roots of
the polynomial $P_{\mathcal{\phi}}$ in $\mathbb{R}^{n(n-1)}$ so that
$\lim_{k\rightarrow\infty}\\{v_{i}^{k}\\}_{i=n+1}^{2n-1}=\\{u_{i}\\}_{i=n+1}^{2n-1}$.
Also, put $v_{i}^{k}=-\sum_{j=n+1}^{2n-1}a_{i}^{j}v_{j}^{k}$, for all $1\leq
i\leq n$ and $k\in\mathbb{N}$. Then, $\\{v_{i}^{k}\\}_{i\in I_{2n-1}}$
satisfies $(\ref{constuctu})$ and so the sequence
$\\{h_{i}^{k}\\}_{k\in\mathbb{N},i\in I_{2n-1}}$ associated to
$\\{\\{v_{i}^{k}\\}_{i=n+1}^{2n-1}\\}_{k\in\mathbb{N}}$ defined by
$h_{i}^{k}=S_{\Phi}^{-1}\Phi_{i}+v_{i}^{k}$, $i\in I_{2n-1}$ constitutes a
dual frame of $\Phi$, for all $k\in\mathbb{N}$. Furthermore, there exists
$k_{0}\in\mathbb{N}$ so that for $k\geq k_{0}$

$\displaystyle
d\left(\\{v_{i}^{k}\\}_{i=n+1}^{2n-1}\\},\\{u_{i}\\}_{i=n+1}^{2n-1}\\}\right)=\sum_{i=n+1}^{2n-1}\|v_{i}^{k}-u_{i}\|<\epsilon.$

Hence, we can write

$\displaystyle d(\\{h_{i}^{k}\\}_{k\in\mathbb{N},i\in
I_{2n-1}},\\{g_{i}\\}_{i\in I_{2n-1}})$ $\displaystyle=$
$\displaystyle\sum_{i\in I_{2n-1}}\|v_{i}^{k}-u_{i}\|$ $\displaystyle=$
$\displaystyle\sum_{i\in
I_{n}}\|v_{i}^{k}-u_{i}\|+\sum_{i=n+1}^{2n-1}\|v_{i}^{k}-u_{i}\|$
$\displaystyle=$ $\displaystyle\sum_{i\in
I_{n}}\|-\sum_{j=n+1}^{2n-1}a_{i}^{j}(v_{j}^{k}-u_{j})\|+\sum_{i=n+1}^{2n-1}\|v_{i}^{k}-u_{i}\|$
$\displaystyle\leq$ $\displaystyle\sum_{i\in
I_{n}}\sum_{j=n+1}^{2n-1}|a_{i}^{j}|\|v_{j}^{k}-u_{j}\|+\sum_{i=n+1}^{2n-1}\|v_{i}^{k}-u_{i}\|$
$\displaystyle\leq$
$\displaystyle(\alpha+1)\sum_{j=n+1}^{2n-1}\|v_{j}^{k}-u_{j}\|<\epsilon(\alpha+1),$

for $k\geq k_{0}$, i.e., $PD_{\Phi}$ is dense in $D_{\Phi}$.

Now, applying Lemma 2.1 and Remark 2.2 gives the result in general case. In
fact, every frame $\Psi=\\{\psi_{i}\\}_{i\in I_{2n-1}}$ is equivalent to a
frame in the form of $\Phi=\\{\delta_{i}\\}_{i\in
I_{n}}\cup\\{\phi_{i}\\}_{i=n+1}^{2n-1}$, as we discussed in Remark 2.2. So,
there exists an invertible operator $\mathcal{T}$ on $\mathbb{R}^{n}$ so that
$\Psi=\mathcal{T}\Phi$, consequently by Lemma 2.1 we get

$\displaystyle\overline{PD_{\Psi}}=\overline{PD_{\mathcal{T}\Phi}}=\overline{(\mathcal{T}^{*})^{-1}PD_{\Phi}}\supseteq(\mathcal{T}^{*})^{-1}\overline{PD_{\Phi}}=(\mathcal{T}^{*})^{-1}D_{\Phi}=D_{\Psi},$

i.e., $\overline{PD_{\Psi}}=D_{\Psi}$. And using $(\ref{open})$

$\displaystyle
PD_{\Psi}=(\mathcal{T}^{*})^{-1}PD_{\Phi}=(\mathcal{T}^{*})^{-1}\xi^{-1}\\{\mathcal{U}\in\mathbb{R}^{n(n-1)}:P_{\mathcal{\phi}}(\mathcal{U})\neq
0\\},$

i.e., $PD_{\Psi}$ is open in $D_{\Psi}$. This completes the proof. ∎

An analogous approach to the proof of Theorem 2.3 deduces that for any full
spark frame $\Phi$ in $\mathbb{R}^{n}$ with $E(\Phi)=1$, the set of all full
spark dual frames is embedded into a generic set in $\mathbb{R}^{n}$.

######  Corollary 2.4.

Let $\Phi=\\{\phi_{i}\\}_{i=1}^{n+1}$ be a full spark frame for
$\mathbb{R}^{n}$. Then the set of all full spark dual frames of $\Phi$ is an
open and dense subset of $D_{\Phi}$ and is embedded into a generic set in
$\mathbb{R}^{n}$.

###### Proof.

Applying Remark 2.2, a full spark frame $\Phi=\\{\phi_{i}\\}_{i=1}^{n+1}$ of
$\mathbb{R}^{n}$ is equivalent to
$\tilde{\Phi}:=\\{\delta_{i}\\}_{i=1}^{n}\cup\\{\sum_{i=1}^{n}\alpha_{i}\delta_{i}\\}$
for non-zero scalars $\alpha_{i}$, $i\in I_{n}$, and in this case the frame
operator is as follows:

$S_{\tilde{\Phi}}=\left[\begin{array}[]{ccc}1+\alpha_{1}^{2}\quad\alpha_{1}\alpha_{2}\quad...\quad\alpha_{1}\alpha_{n}\\\
\\\ \alpha_{1}\alpha_{2}\quad
1+\alpha_{2}^{2}\quad...\quad\alpha_{2}\alpha_{n}\\\ .\\\ .\\\
\alpha_{1}\alpha_{n}\quad\alpha_{2}\alpha_{n}\quad...\quad 1+\alpha_{n}^{2}\\\
\end{array}\right]$

Also, every dual frame of $\tilde{\Phi}$ is in the form of
$\\{S_{\tilde{\Phi}}^{-1}\tilde{\phi}_{i}+u_{i}\\}_{i=1}^{n+1}$ so that
$T_{u}T^{*}_{\tilde{\Phi}}=0$. By taking $u_{n+1}=[x_{1},...,x_{n}]^{T}$, we
get $u_{i}=-\alpha_{i}u_{n+1}$ for all $i\in I_{n}$. Hence, $D_{\tilde{\Phi}}$
is the set of all $\\{g_{i}\\}_{i=1}^{n+1}$ so that

$\displaystyle
g_{i}^{k}=\begin{cases}\begin{array}[]{ccc}-\alpha\alpha_{i}\alpha_{k}-\alpha_{i}x_{k}&\;{k\neq
i},\\\ \alpha(1+\sum_{j\neq i}\alpha_{j}^{2})-\alpha_{i}x_{k}&\;{k=i},\\\
\end{array}\end{cases}$

where
$\alpha=\dfrac{1}{detS_{\tilde{\Phi}}}=\dfrac{1}{1+\sum_{i=1}^{n}\alpha_{i}^{2}}$,
$g_{i}^{k}$ denotes $k^{th}$ coordinate of $g_{i}$, $i\in I_{n}$, and

$g_{n+1}=\\{\alpha\alpha_{i}+x_{i}\\}_{i=1}^{n}.$

This shows that, every dual frame is associated to $n$-variable
$\\{x_{i}\\}_{i\in I_{n}}$ and a dual frame is full spark frame except the
cases that the sequence $u_{n+1}=\\{x_{i}\\}_{i\in I_{n}}$ belongs to the
roots of the polynomial constructed by
$det[g_{i_{1}}|\quad...\quad|g_{i_{n}}]=0$, for some index set
$\\{i_{j}\\}_{j=1}^{n}\subset I_{n+1}$. Multiplying these $n+1$ polynomials
yield an $n$-variable polynomial $P_{\tilde{\Phi}}(x_{1},...,x_{n})$. So,
every full spark dual frame of $\tilde{\Phi}$ is obtained by a generic choice
of $u_{n+1}$ out of the roots of the polynomial $P_{\tilde{\Phi}}$. Hence, by
a similar approach to the proof of Theorem 2.3, and considering the bijection
map $\xi:D_{\tilde{\Phi}}\rightarrow\mathbb{R}^{n}$ defined as
$\xi(G)=u_{n+1}$, where
$G=\\{S_{\tilde{\Phi}}^{-1}\tilde{\phi}_{i}+u_{i}\\}_{i=1}^{n+1}$, the
complement of all full spark dual frames of $\Phi$ in $D_{\Phi}$ is equivalent
to the set of roots of $P_{\tilde{\Phi}}$, and so, the set of full spark dual
frames of $\Phi$ is embedded into a generic set in $\mathbb{R}^{n}$ by $\xi$.
In general case, if $\Phi=\mathcal{T}\tilde{\Phi}$, for an invertible operator
$\mathcal{T}$ on $\mathbb{R}^{n}$ by using Lemma 2.1, the set of full spark
dual frames of $\Phi$, $FD_{\Phi}$, is the same
$(\mathcal{T}^{*})^{-1}FD_{\tilde{\Phi}}$, and this completes the proof. ∎

###  2.1 Examples

######  Example 2.5.

Suppose that $\phi=\\{\phi_{i}\\}_{i=1}^{3}$ is a full spark frame for
$\mathbb{R}^{2}$. Since $E(\mathcal{\phi})=1$ and equivalent frames do the
same phase retrieval we assume that $\phi$ is equivalent to
$\\{\delta_{1},\delta_{2},\alpha_{1}\delta_{1}+\alpha_{2}\delta_{2}\\}$, in
which both $\alpha_{1}$ and $\alpha_{2}$ are non-zero. In this case,

$D_{\phi}=\left\\{\left[\begin{array}[]{ccc}\alpha(1+\alpha_{2}^{2})-\alpha_{1}x\\\
\\\ -\alpha\alpha_{1}\alpha_{2}-\alpha_{1}y\\\
\end{array}\right],\left[\begin{array}[]{ccc}-\alpha\alpha_{1}\alpha_{2}-\alpha_{2}x\\\
\\\ \alpha(1+\alpha_{1}^{2})-\alpha_{2}y\\\
\end{array}\right],\left[\begin{array}[]{ccc}\alpha\alpha_{1}+x\\\ \\\
\alpha\alpha_{2}+y\end{array}\right];x,y\in\mathbb{R}\right\\}$

where $\alpha=\dfrac{1}{1+\alpha_{1}^{2}+\alpha_{2}^{2}}$. It is worthy of
note that, all dual frames in this case are full spark and so phase retrieval
except dual frames obtained by $(x,y)\in\mathbb{R}^{2}$ on three distinct
lines with slopes of $0$, $\infty$ and $-\alpha_{1}/\alpha_{2}$ such as
$x=-\alpha\alpha_{1}$, $y=-\alpha\alpha_{2}$ and
$\alpha_{1}x+\alpha_{2}y=\alpha$. Considering

$(x,y)\in\mathbb{R}^{2}\setminus\\{(x,y):(x+\alpha\alpha_{1})(y+\alpha\alpha_{2})(\alpha_{1}x+\alpha_{2}y-\alpha)=0\\}$

we get a generic choice of $\\{u_{i}\\}_{i\in I_{4}}$ in $\mathbb{R}^{2}$. The
fact that all dual frames of $\phi$ are translations of this family by
$\\{S_{\phi}^{-1}\phi_{i}\\}_{i\in I_{4}}$ shows that full spark dual frames
are dense in $D_{\Phi}$ and full measure. As a special case, considering
$\alpha_{1}=\alpha_{2}=1$ we get the phase retrieval frame
$\phi=\\{\delta_{1},\delta_{2},\delta_{1}+\delta_{2}\\}$ and we can see that
dual frames of $\Phi$ do phase retrieval for all $(x,y)\in\mathbb{R}^{2}$
except on the lines $x=\dfrac{-1}{3}$, $y=\dfrac{-1}{3}$ and
$y=\dfrac{1}{3}-x$. Put $x=0$, $y=2/3$ we get a phase retrieval dual $\Psi$
and $x=1,y=\dfrac{-2}{3}$, satisfying in $y=\dfrac{1}{3}-x$, gives a non-phase
retrieval dual frame $G$. See Figure 1.

######  Example 2.6.

Let $\phi=\\{\phi_{i}\\}_{i=1}^{4}$ be a full spark frame for
$\mathbb{R}^{3}$. In this case $\phi$ is equivalent to the frame

$\phi:=\\{\delta_{1},\delta_{2},\delta_{3},\alpha_{1}\delta_{1}+\alpha_{2}\delta_{2}+\alpha_{3}\delta_{3}\quad\alpha_{1},\alpha_{2},\alpha_{3}\neq
0\\}.$

The set of all dual frames of $\phi$ is in the form of
$D_{\phi}=\\{g_{1},g_{2},g_{3},g_{4}\\}$ where

$g_{1}=\left[\begin{array}[]{ccc}\alpha(1+\alpha_{2}^{2}+\alpha_{3}^{2})-\alpha_{1}x\\\
\\\ -\alpha\alpha_{1}\alpha_{2}-\alpha_{1}y\\\ \\\
-\alpha\alpha_{1}\alpha_{3}-\alpha_{1}z\\\
\end{array}\right],g_{2}=\left[\begin{array}[]{ccc}-\alpha\alpha_{1}\alpha_{2}-\alpha_{2}x\\\
\\\ \alpha(1+\alpha_{1}^{2}+\alpha_{3}^{2})-\alpha_{2}y\\\ \\\
-\alpha\alpha_{2}\alpha_{3}-\alpha_{2}z\\\ \end{array}\right],$
$g_{3}=\left[\begin{array}[]{ccc}-\alpha\alpha_{1}\alpha_{3}-\alpha_{3}x\\\
\\\ -\alpha\alpha_{2}\alpha_{3}-\alpha_{3}y\\\ \\\
\alpha(1+\alpha_{1}^{2}+\alpha_{3}^{2})-\alpha_{3}z\\\
\end{array}\right],g_{4}=\left[\begin{array}[]{ccc}\alpha\alpha_{1}+x\\\ \\\
\alpha\alpha_{2}+y\\\ \\\ \alpha\alpha_{3}+z\end{array}\right]$

where $\alpha=\dfrac{1}{1+\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}$ and
$x,y,z\in\mathbb{R}$ are arbitrary. All dual frames of $\Phi$ are full spark
except the cases $det[g_{i}|g_{j}|g_{k}]=0$, for $i,j,k\in I_{4}$, in which
$(x,y,z)$ is belong to the four distinct planes $x=-\alpha\alpha_{1}$,
$y=-\alpha\alpha_{2}$, $z=-\alpha\alpha_{3}$ and
$\alpha_{1}x+\alpha_{2}y+\alpha_{3}z=\alpha$. In a similar way, by choosing

$(x,y,z)\in\mathbb{R}^{3}\setminus\\{(x,y,z):(x+\alpha\alpha_{1})(y+\alpha\alpha_{2})(z+\alpha\alpha_{3})(\alpha_{1}x+\alpha_{2}y+\alpha_{3}z-\alpha)=0\\}$

we can say that full spark dual frames are translations of the canonical dual
by a generic choice of the family $\\{u_{i}\\}_{i\in I_{4}}$.

######  Example 2.7.

Consider
$\phi=\\{\delta_{1},\delta_{2},\delta_{3},\sum_{i=1}^{3}\delta_{i},\delta_{1}-\delta_{2}+\delta_{3}\\}$
as a full spark frame for $\mathbb{R}^{3}$. In this case, all dual frames are
constructed by a generic choice of $\left[\begin{array}[]{ccc}u_{4}\\\
u_{5}\\\ \end{array}\right]\in\mathbb{R}^{6}$. In fact, by putting
$u_{4}=[x_{1}\quad y_{1}\quad z_{1}]^{T}$ and $u_{5}=[x_{2}\quad y_{2}\quad
z_{2}]^{T}$, the set of all dual frames of $\phi$, are given by

$g_{1}=\left[\begin{array}[]{ccc}\dfrac{3}{5}-x_{1}-x_{2}\\\ \\\
-y_{1}-y_{2}\\\ \\\ \dfrac{-2}{5}-z_{1}-z_{2}\\\
\end{array}\right],g_{2}=\left[\begin{array}[]{ccc}x_{2}-x_{1}\\\ \\\
\dfrac{1}{3}+y_{2}-y_{1}\\\ \\\ -z_{2}-z_{1}\\\ \end{array}\right],$
$g_{3}=\left[\begin{array}[]{ccc}\dfrac{-2}{5}-x_{1}-x_{2}\\\ \\\
-y_{1}-y_{2}\\\ \\\ \dfrac{3}{5}-z_{1}-z_{2}\\\
\end{array}\right],g_{4}=\left[\begin{array}[]{ccc}\dfrac{1}{5}+x_{1}\\\ \\\
\dfrac{1}{3}+y_{1}\\\ \\\ \dfrac{1}{5}+z_{1}\\\
\end{array}\right],g_{5}=\left[\begin{array}[]{ccc}\dfrac{1}{5}+x_{1}\\\ \\\
\dfrac{-1}{3}+y_{2}\\\ \\\ \dfrac{1}{5}+z_{2}\\\ \end{array}\right]$

where $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}$ are obtained arbitrarily from
$\mathbb{R}$. And all cases in which a dual frame fails to do phase retrieval
is associated to the roots of a $6$-variable polynomial in $\mathbb{R}^{6}$
given by multiplying of the polynomials as
$det[g_{i_{1}}|g_{i_{2}}|g_{i_{3}}]=0,$ for all index set
$\\{i_{j}\\}_{j=1}^{3}\subset I_{5}$. As one case, $det[g_{2}|g_{4}|g_{5}]=0$
deduces that

$z_{1}+x_{2}-\dfrac{3x_{1}}{5}-\dfrac{3z_{2}}{5}+3x_{2}z_{1}-3x_{1}z_{2}=0,$

and the roots of this equation, which are associated to a family of non-phase
retrieval dual frames, constitute a surface in $\mathbb{R}^{3}$.

##  3 Weak Phase Retrieval

The main result of this section is to obtain some equivalent conditions on a
family of vectors to do phase retrieval in terms of weak phase retrieval. This
also derives a relationship between, phase, weak phase and norm retrieval.
First we present some properties of a family of vectors to do weak phase
retrieval.

######  Proposition 3.1.

Assume that $\Phi=\\{\mathcal{\phi}_{i}\\}_{i\in I_{m}}$ is a frame in
$\mathcal{H}_{n}$. Then $\Phi$ does weak phase retrieval if and only if
$P\Phi$ does weak phase retrieval, for every $2$-dimensional orthogonal
projection $P$ on $\mathcal{H}_{n}$.

###### Proof.

In case $\Phi$ does weak phase retrieval, it is simple to see that $P\Phi$
does weak phase retrieval, for every orthogonal projection $P$ on
$\mathcal{H}_{n}$, as well, see also [1]. Conversely, let
$x,y\in\mathcal{H}_{n}$ and $|\langle x,\phi_{i}\rangle|=|\langle
y,\phi_{i}\rangle|$. Then by the assumption that $P\Phi$ does weak retrieval
for the projection $P$ of $\mathcal{H}_{n}$ onto the closed subspace
$span\\{x,y\\}$, implies that $x$ and $y$ weakly have the same phase. ∎

It is shown that [1] a weak phase retrieval family in $\mathbb{R}^{n}$ spans
the space that means such a family constitutes a frame for $\mathbb{R}^{n}$.
In the complex case $\mathbb{C}^{n}$ there is no result available. In the
following, we present sufficient condition in this regard which is also useful
for the main result of this section.

######  Proposition 3.2.

Let $\\{\phi_{i}\\}_{i\in I_{m}}$ be a family of vectors in $\mathbb{C}^{n}$
so that $U\Phi$ does weak phase retrieval for every unitary operator $U$ on
$\mathbb{C}^{n}$. Then $\\{\phi_{i}\\}_{i\in I_{m}}$ is a frame for
$\mathbb{C}^{n}$.

###### Proof.

Assume that there exists an element $x\in\\{\phi_{i}\\}_{i\in I_{m}}^{\perp}$,
and get an ONB for $\mathbb{C}^{n}$ as $\\{e_{i}\\}_{i\in I_{n}}$ with
$e_{1}=\dfrac{x}{\|x\|}$. Also, take $0\neq y\in span\\{\phi_{i}\\}_{i\in
I_{m}}$ so there exists $\\{\beta_{i}\\}_{i=2}^{n}\subset\mathbb{C}$, with
some non-zero elements such that $y=\sum_{i=2}^{n}\beta_{i}e_{i}$. Define

$\mu:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n};\quad\mu(e_{i})=\delta_{i}$

where $\\{\delta_{i}\\}_{i\in I_{n}}$ is the standard ONB of $\mathbb{C}^{n}$.
Clearly $\mu$ is a unitary operator and

$\mu(x+y)=\mu(x)+\mu(y)=\|x\|\delta_{1}+\sum_{i=2}^{n}\beta_{i}\delta_{i},\quad\mu(-x+y)=-\|x\|\delta_{1}+\sum_{i=2}^{n}\beta_{i}\delta_{i}.$

On the other hand,

$\displaystyle|\langle\mu(x+y),\mu\phi_{i}\rangle|$ $\displaystyle=$
$\displaystyle|\langle x+y,\phi_{i}\rangle|$ $\displaystyle=$
$\displaystyle|\langle-x+y,\phi_{i}\rangle|$ $\displaystyle=$
$\displaystyle|\langle\mu(-x+y),\mu\phi_{i}\rangle|,$

for all $i\in I_{m}$. The assumption that $\mu\Phi$ does weak phase retrieval
deduces that $\mu(x+y)$ and $\mu(-x+y)$ weakly have the same phase that is
impossible, unless $x=0$. Therefore, $\Phi$ is a frame for $\mathbb{C}^{n}$. ∎

######  Theorem 3.3.

Let $\Phi=\\{\mathcal{\phi}_{i}\\}_{i\in I_{m}}$ be a family of vectors in
$\mathcal{H}_{n}$. Then the followings are equivalent;

  * (i)

$\Phi$ does phase retrieval.

  * (ii)

$U\Phi$ does phase retrieval for every invertible operator $U$ on
$\mathcal{H}_{n}$.

  * (iii)

$U\Phi$ does norm retrieval for every invertible operator $U$ on
$\mathcal{H}_{n}$.

  * (iv)

$U\Phi$ does weak phase retrieval for every invertible operator $U$ on
$\mathcal{H}_{n}$.

###### Proof.

The equivalency of $(i)$, $(ii)$, and $(iii)$ was proven in [9]. Also, we
clearly have $(i)\Rightarrow(ii)\Rightarrow(iv)$. So, it is sufficient to show
that $(iv)\Rightarrow(i)$. Assuming that $U\Phi$ does weak phase retrieval for
all invertible operators on $\mathcal{H}_{n}$, specially we imply that
$\mathcal{\phi}$ does weak phase retrieval. Moreover, $\Phi$ is a frame for
$\mathcal{H}_{n}$, i.e., $\Phi$ spans $\mathcal{H}_{n}$ by Proposition 3.2. In
the sequel, we are going to show that $\Phi$ yields phase retrieval. On the
contrary, let there exist non-zero elements $x$ and $y$ in $\mathcal{H}_{n}$
so that

$|\langle x,\phi_{i}\rangle|=|\langle y,\phi_{i}\rangle|,\quad(i\in I_{m})$
(3.1)

but $y\neq cx$ for any $|c|=1$. Since $\Phi$ does weak phase retrieval, there
exists some $|\theta|=1$ so that $\theta phase(x_{j})=phase(y_{j})$ for all
$j\in I_{n}$. The assumption that $y\neq\theta x$ implies $|x_{j}|\neq|y_{j}|$
for some $j\in I_{n}$. Since $\Phi$ is a frame, the equality in $(\ref{eq41})$
is non-zero for some $i$ then $y=cx$ immediately implies that $x$ and $y$ are
linearly independent and we can consider a basis for $\mathcal{H}_{n}$
containing $x$ and $y$ as $\\{x,y,e_{3},...,e_{n}\\}$. We face the following
cases

Case $1$. If $x$ and $y$ are disjointly supported, then there exist no non-
zero common coordinates. Without loss of generality we let $x_{1},y_{2}\neq
0$, and so $x_{2}=y_{1}=0$. Define
$U:\mathcal{H}_{n}\rightarrow\mathcal{H}_{n}$ so that $Ux=x-y$, $Uy=x+y$ and
$Ue_{k}=e_{k}$, $3\leq k\leq n$. Then, $U$ is a bounded invertible operator on
$\mathcal{H}_{n}$ and

$|\langle Ux,(U^{-1})^{*}\phi_{i}\rangle|=|\langle
Uy,(U^{-1})^{*}\phi_{i}\rangle|,\quad(i\in I_{m}),$

by $(\ref{eq41})$ and the invertibility of $U$. However, $Ux$ and $Uy$ have
not weakly the same phase due to $phase(Ux)_{1}=phase(Uy)_{1}$ and
$phase(Ux)_{2}=e^{i\pi}phase(Uy)_{2}$. This contradicts the assumption that
$(U^{-1})^{*}\Phi$ does weak phase retrieval.

Case $2$. Let $x$ and $y$ have one non-zero coordinate in common in $i$th
index. If $y_{l}=0=x_{l}$ for all $l\neq i$ then by $(\ref{eq41})$ and using
the assumption $|x_{i}|=|y_{i}|$, i.e., $y=\theta x$ that is a contradiction.
Thus, there exists $l\neq i$ so that $y_{l}\neq 0$ or $x_{l}\neq 0$. Without
loss of generality we let $y_{l}\neq 0$ and
$\dfrac{|x_{l}|}{|y_{l}|}<\dfrac{|x_{i}|}{|y_{i}|}$. In fact, if in all common
non-zero elements $\dfrac{|x_{i}|}{|y_{i}|}=c$ then $y=\dfrac{\theta}{c}x$,
which contradicts by the assumption.

So, we get $\epsilon>0$ so that
$\dfrac{|x_{l}|}{|y_{l}|}<\epsilon<\dfrac{|x_{i}|}{|y_{i}|}$. Define
$U:\mathcal{H}_{n}\rightarrow\mathcal{H}_{n}$ so that $Ux=\theta x-\epsilon
y$, $Uy=\theta x+\epsilon y$ and $Ue_{k}=e_{k}$, $3\leq k\leq n$. Moreover,

$\displaystyle(Ux)_{j}=(|x_{j}|-\epsilon|y_{j}|)\alpha_{j}\theta,\quad(Uy)_{j}=(|x_{j}|+\epsilon|y_{j}|)\alpha_{j}\theta.\quad(j\in
I_{n})$

where $\alpha_{j}=phase(x_{j})$. Hence, we obtain
$phase(Ux)_{i}=phase(Uy)_{i}$, however $phase(Ux)_{l}=e^{i\pi}phase(Uy)_{l}$.
That leads to a contradiction similar to the previous case, as required.

∎

The following result gives a sufficient condition on a family of frames so
that their canonical dual also yields weak phase retrieval.

######  Proposition 3.4.

Let $\Phi=\\{\mathcal{\phi}_{i}\\}_{i\in I_{m}}$ be a frame in
$\mathcal{H}_{n}$ with diagonal frame operator. Then $\Phi$ does weak phase
retrieval if and only if its canonical dual does so.

###### Proof.

Suppose that $\alpha_{1},...,\alpha_{n}$ be the diagonal elements of
$S_{\Phi}$, respectively. Obviously, $\alpha_{i}>0$, for all $i\in I_{n}$. Let
$\Phi$ does weak phase retrieval, take $x,y\in\mathcal{H}_{n}$ such that
$|\langle x,S_{\Phi}^{-1}\phi_{i}\rangle|=|\langle
y,S_{\Phi}^{-1}\phi_{i}\rangle|$. Then, we get
$S_{\Phi}^{-1}x=(x_{1}/\alpha_{1},...,x_{n}/\alpha_{n})$ and
$S_{\Phi}^{-1}y=(y_{1}/\alpha_{1},...,y_{n}/\alpha_{n})$ weakly have the same
phase. Consequently, $x$ and $y$ weakly have the same phase, as well. The
converse is implied by a similar explanation. ∎

Funding. The authors have not disclosed any funding.

Data Availability Statement. Data sharing not applicable to this article as no
datasets were generated during the preparation of this paper.

Conflict-of-interest. This work does not have any conflicts of interest.

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11institutetext: Astronomical Observatory Institute, Faculty of Physics, A.
Mickiewicz University, Słoneczna 36, 60-286 Poznań, Poland 22institutetext:
Lowell Observatory, 14000 W Mars Hill Rd, 86001 Flagstaff, AZ, USA
33institutetext: Astronomical Observatory of Odessa I.I.Mechnikov National
University, Odessa, Ukraine 44institutetext: Korea Astronomy & Space Science
Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
55institutetext: National Youth Space Center, Goheung, Jeollanam-do, 59567,
Korea 66institutetext: Deutsches Zentrum för Luft- und Raumfahrt (DLR),
Institute of Planetary Research, Berlin, Germany 77institutetext: Departamento
de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de
Alicante, Alicante, Spain 88institutetext: Institut de Ciencies del Cosmos
(ICCUB), Universitat de Barcelona, Barcelona, Spain 99institutetext: Platanus
Observatory (IAU code K80), Lusówko, Poland 1010institutetext:
”G.V.Schiaparelli” Astronomical Observatory, Varese, Italy 1111institutetext:
Osservatorio di San Marcello Pistoiese, GAMP Gruppo Astrofili Montagna
Pistoiese, Italy 1212institutetext: Great Shefford Observatory, Berkshire,
United Kingdom 1313institutetext: British Astronomical Association, Burlington
House, Piccadilly, London, United Kingdom 1414institutetext: Las Cumbres
Observatory, Goleta, CA 93117, USA

# Photometry and model of near-Earth asteroid 2021 DW1
from one apparition

T. Kwiatkowski 11 P. Koleńczuk 11 A. Kryszczyńska 11 D. Oszkiewicz 11 K.
Kamiński 11 M. K. Kamińska 11 V. Troianskyi 1133 B. Skiff 22 N. Moskowitz 22
V. Kashuba 33 M.-J. Kim 44 T. Kim 55 S. Mottola 66 T. Santana-Ros T. Kluwak
778899 L. Buzzi 1010 P. Bacci 1111 P. Birtwhistle 1212 R. Miles 1313 J.
Chatelain 1414

(Received xx xx xxxx / Accepted xx xx xxxx)

###### Abstract

Aims. Very Small Asteroids (objects with diameters smaller than about 150 m)
can be spun-up by the YORP effect to rotation periods as short as tens of
seconds. This effect has been observed for many of them. It is also
hypothesized, that in the same process their spin axes are asymptotically
drawn to the position perpendicular to the orbital plane. So far this effect
has been observed only for one VSA and needs further verification. For that,
spin axes of several other VSAs should be determined by observing their
brightness variations at many different positions on the sky.

Methods. On 4 March 2021 at 9 UTC a 30-m in diameter near-Earth asteroid 2021
DW1 passed the Earth at a distance of 570 000 km, reaching the maximum
brightness of $V=14.6\,\mathrm{mag}$. We observed it photometrically from 2
March, when it was visible at $V=16.5\,\mathrm{mag}$, until 7 March
($V=18.2\,\mathrm{mag}$). During that time 2021 DW1 swept a $170\degr$ long
arc in the northern sky, spanning solar phase angles in the range from
$36\degr$ to $86\degr$. This made it an excellent target for physical
characterisation, including spin axis and shape derivation.

Results. Convex inversion of the asteroid lightcurves gives a sidereal period
of rotation $P_{\mathrm{sid}}=0.013760\pm 0.000001\,\mathrm{h}$, and two
solutions for the spin axis ecliptic coordinates: (A) $\lambda_{1}=57\degr\pm
10\degr,\beta_{1}=29\degr\pm 10\degr$, and (B) $\lambda_{2}=67\degr\pm
10\degr,\beta_{2}=-40\degr\pm 10\degr$. The magnitude-phase curve can be
fitted with a standard H, G function with $H=24.8\pm 0.5\,\mathrm{mag}$ and an
assumed $G=0.24$. The asteroid colour indices are $g-i=0.79\pm
0.01\,\mathrm{mag}$, and $i-z=0.01\pm 0.02\,\mathrm{mag}$ which indicates an S
taxonomic class, with an average geometric albedo $p_{V}=0.23\pm 0.02$. The
asteroid effective diameter, derived from $H$ and $p_{V}$, is
$D_{\mathrm{eff}}=30\pm 10\,\mathrm{m}$.

Conclusions. It was found that the inclination of the spin axis of 2021 DW1 is
not perpendicular to the orbital plane (obliquity $\epsilon=54\degr\pm
10\degr$ or $\epsilon=123\degr\pm 10\degr$). More spin axes of VSAs should be
determined to check, if 2021 DW1 is an exception or a typical case.

###### Key Words.:

asteroids – photometric observations – period, spin axis, shape modelling

††offprints: T. Kwiatkowski, e-mail: tkastr at vesta.astro.amu.edu.pl

##  1 Introduction

Very Small Asteroids (VSAs) are objects with diameters $D<150$ m. They often
rotate with periods shorter than 2 h enabling us to study their internal
structure by comparing the centrifugal force with the material forces holding
them together (Holsapple, 2007). Because of their small sizes, VSAs are
sensitive to the YORP effect (Rubincam, 2000), which is a torque induced on
the rotating body by the thermal radiation emitted by its surface complemented
by a torque produced by scattered sunlight. YORP can either spin up or slow
down the asteroid rotation as well as change the obliquity of its spin axis
$\epsilon$, which is an angle between the normal to the asteroid orbital plane
and its rotation axis. While the fast rotation has been observed for many
VSAs, their spin axes were not determined except for one object111To be exact,
in the DAMIT database (Ďurech et al., 2010) there are two other VSAs (2008 TC3
and 2012 TC4), for which spin axes have been determined, but both of them are
non-principal axis rotators and show the effect of tumbling: (54509) YORP
(which name is the same as the name of the effect itself). (54509) was the
first asteroid for which the effect of YORP has been observed (Lowry et al.,
2007; Taylor et al., 2007). The obliquity of the (54509) spin axis is
$\epsilon=173\degr$ which means it is nearly perpendicular to the asteroid
orbital plane. Such orientation of the spin axis was found as an end state of
the YORP evolution in the simulations performed by Čapek & Vokrouhlický (2004)
for objects with finite surface thermal conductivity. If their prediction is
true, then for the VSAs, which experienced a strong YORP effect for a long
time (and the fastest rotating VSAs are such objects) we should observe spin
axis obliquities close to $0\degr$ or $180\degr$. Recently Golubov et al.
(2021) have shown, that for very small objects of highly irregular shape, the
transverse heat conduction (TYORP) can add new asymptotic states for the
obliquity. For this reason it would be interesting to verify those predictions
with observations of VSAs. To do that, we should observe their lightcurves at
many different positions on the sky to be able to determine their spin axes.
For NEAs this condition is met either by the Earth co-orbital asteroids – and
(54509) is an example of such objects – or by objects which during their close
encounter with the Earth can be observed along a long arc on the sky.

The main goal of this paper is to present the observations and modelling of
the near-Earth asteroid 2021 DW1, which allowed us to derive its spin axis.
Apart from that, we were able to determine many other physical properties of
this VSA, such as its rotation period, shape, size, and taxonomy class, adding
new data to this still poorly characterised group of objects.

Figure 1: A trail made by 2021 DW1 on the sky during our observing campaign,
plotted in the RA, Dec equatorial coordinates. The beginning of the arc refers
to 2 Mar, 4 UTC, and the end to 7 Mar, 7 UTC. The length of the trail is about
$170\degr$. Note that two observations made on 7 Mar were finally not used for
lightcurve inversion due to high photometric noise.

##  2 Observations and data reduction

On 16 February 2021 a near-Earth asteroid was discovered by the Pan-STARRS 1
survey at Haleakala, Hawaii. The discovery was reported in MPEC 2021-D73
circular222https://www.minorplanetcenter.net/mpec/K21/K21D73.html and the
object was designated 2021 DW1. Its first estimate of the absolute magnitude
gave a value of $H=25$ mag, which translated roughly to an effective diameter
of 40 m, indicating it is a VSA with a rotation period possibly shorter than
$2.2\,\mathrm{h}$. What differentiated that object from other, similar NEAs,
was its ephemeris which showed that it could be observed in favourable
circumstances ($V<18\,\mathrm{mag}$) along a lengthy arc on the sky. Its
beginning was obscured by the Milky Way, but from 2 March till 7 March the
length of the trail was $170\degr$, with the solar phase angle spanning the
interval from $36\degr$ to $86\degr$ (Fig. 1). Such observing geometry made it
possible to not only determine the rotation period and colour indices, but
also to attempt derivation of the magnitude-phase angle relation, spin axis
coordinates and shape.

We organized an observing campaign for 2021 DW1 contacting a number of
collaborating observatories. A call for observations was also posted on the
Minor Planet Mailing List (MPML). As a result we set up a network of ten
observatories located on the Northern hemisphere, from South Korea in the
East, to Arizona in The West. The spread in longitude helped to secure
observations from different parts of the asteroid trail. There were several
factors which made observations difficult: the weather, bright Moon, fast
movement of asteroid in the sky, and human mistakes. We also encountered
strange problems with pointing some robotic telescopes. They were probably
caused by the systems using outdated ephemerides of 2021 DW1. This once again
proved a golden rule for this kind of target of opportunity observations: to
try to schedule observations, even if any two of them were to be done
simultaneously.

The asteroid aspect data and the observing log are presented in Table 1. We
limit this table only to those observations, that were used in final analysis
(their results are shown in Fig. 2). Details of all observing systems are
shown in Table 2.

Most of the photometric data were reduced with the Starlink package333The
Starlink software is currently supported by the East Asian Observatory.(Currie
et al., 2014). Raw CCD frames were corrected for bias and flat-field (and for
the dark current, if necessary). The aperture photometry was then performed on
the frames, on which the PSF (Point Spread Function) of the asteroid and the
comparison stars was almost circular. On the nights when fast sky motion of
the object (even during short, 5 s exposures) caused significant PSF trailing,
we requested observers to employ the non-sidereal telescope tracking on the
asteroid. This way, all signal from 2021 DW1 was concentrated in a circular
aperture, while the images of stars were trailed. To perform differential
photometry on such CCD frames, we used ”pill-shaped apertures” developed by
Fraser et al. (2016), and used for NEAs by Koleńczuk (2020). In this
technique, an elongated aperture is created as a rectangle with two
semicircular end-caps. It is defined by three parameters: the trail length,
the position angle, and the radius.

The data from the Great Shefford Observatory were reduced in a standard way
with a help of Astrometrica and MPO Canopus commercial packages. Observations
from the Platanus Observatory were done with an exposure time of 1 s. This
produced a circular PSF for a fast moving asteroid, but because of a small
telescope aperture (0.28-m), the SNR was low. However, during a one-hour run,
more than two thousand frames were collected. After reducing the data, every
five points were averaged into one data point, and the composite lightcurve,
even if still quite noisy, produced a well-defined Fourier series fit (see the
C005 plot on Fig. 2).

Table 1: Observing log.

Date | Obs. time | Observatory | r | $\Delta$ | $\alpha$ | $\lambda$ | $\beta$ | $V$ | Mov | Exp | Fltr | Code
---|---|---|---|---|---|---|---|---|---|---|---|---
(UTC) | (UTC) | (see Table 2) | (au) | (au) | $(\degr)$ | $(\degr)$ | $(\degr)$ | [mag] | ”/min | [s] |  |
2021-03-02 | 02:56 – 03:36 | Lowell (1) | 0.9959 | 0.0078 | 51.6 | 114.6 | -25.0 | 16.4 | 25 | 10 | VR | LC01
2021-03-02 | 11:55 – 15:11 | DOAO (2) | 0.9956 | 0.0068 | 48.4 | 116.7 | -19.9 | 16.0 | 35 | 10 | I | LC02
2021-03-02 | 18:46 – 19:35 | Schiap. (3) | 0.9955 | 0.0062 | 46.2 | 118.2 | -16.2 | 15.7 | 43 | 5 | C | LC03
2021-03-03 | 01:58 – 02:22 | Lowell (1) | 0.9953 | 0.0055 | 43.2 | 120.3 | -10.7 | 15.4 | 55 | 4 | VR | LC05
2021-03-03 | 09:01 – 09:59 | Winer (5) | 0.9951 | 0.0049 | 39.9 | 122.9 | -03.4 | 15.0 | 69 | 5 | C | LC06
2021-03-03 | 20:00 – 21:49 | Mayaki (6) | 0.9948 | 0.0041 | 36.5 | 128.6 | +12.2 | 14.5 | 98 | 3 | C | LC07
2021-03-04 | 01:07 – 02:07 | Platanus (10) | 0.9946 | 0.0039 | 37.0 | 131.6 | +19.9 | 14.4 | 112 | $1^{*}$ | C | C005
2021-03-04 | 02:00 – 02:12 | McDonald (7) | 0.9946 | 0.0039 | 37.1 | 131.9 | +20.6 | 14.4 | 112 | 5 | w | LC08
2021-03-04 | 19:22 – 20:26 | DOAO (2) | 0.9942 | 0.0041 | 52.4 | 150.8 | +51.3 | 15.0 | 98 | 5 | I | LC09
2021-03-05 | 02:08 – 02:21 | CalarAlto (8) | 0.9940 | 0.0044 | 59.0 | 161.0 | +59.1 | 15.3 | 88 | 2.5 | C | LC10
2021-03-05 | 03:30 – 03:59 | Lowell (1) | 0.9940 | 0.0045 | 60.5 | 164.0 | +60.8 | 15.5 | 84 | 4 | VR | A001
2021-03-05 | 06:00 – 06:44 | Lowell (1) | 0.9939 | 0.0047 | 63.1 | 169.6 | +63.3 | 15.8 | 77 | 4 | VR | A008
2021-03-05 | 06:45 – 06:51 | Lowell (1) | 0.9939 | 0.0047 | 63.5 | 170.4 | +63.6 | 15.8 | 76 | 4 | g | CI01
2021-03-05 | 06:52 – 06:58 | Lowell (1) | 0.9939 | 0.0047 | 63.6 | 170.6 | +63.7 | 15.8 | 76 | 4 | i | CI02
2021-03-05 | 06:59 – 07:05 | Lowell (1) | 0.9939 | 0.0047 | 63.7 | 170.8 | +63.8 | 15.8 | 76 | 4 | z | CI03
2021-03-05 | 11:40 – 12.17 | Winer (5) | 0.9938 | 0.0052 | 68.0 | 182.4 | +67.1 | 15.9 | 59 | 5 | C | A016
2021-03-06 | 00:37 – 02:32 | Shefford (9) | 0.9935 | 0.0064 | 76.7 | 214.1 | +70.3 | 16.7 | 38 | 4 | C | LC12
2021-03-06 | 08:34 – 10:46 | Winer (5) | 0.9934 | 0.0073 | 80.1 | 227.8 | +69.7 | 17.1 | 28 | 5 | C | B579
2021-03-07 | 08:45 – 08:52 | McDonald (7) | 0.9929 | 0.0100 | 86.6 | 250.1 | +66.2 | 18.2 | 17 | 4 | w | LC13

Note: the table presents the subset of the observing log limited to the
lightcurves which were used in the analysis. The third column shows the
shortened observatory name (full names are presented in Table 2). The next
five columns present the aspect data for the middle of the observing time: $r$
and $\Delta$ are the distances of the asteroid from the Sun and the Earth,
respectively, $\alpha$ is the solar phase angle, while $\lambda$ and $\beta$
are the geocentric, ecliptic (J2000) longitude and latitude. In the next
column, an average brightness $V$ of the asteroid, as predicted by the
Horizons ephemeris, is given. Starting from the tenth column, the table gives
the asteroid movement on the sky (Mov), the exposure time (Exp), and the
filter used in the observations (here ”C” stands for a ”clear” filter). The
last column provides the code to locate the lightcurve in Fig. 2.
(*) For C005 lightcurve more than 2000 exposures were obtained, with a very
short exposure time of 1 s. They were then averaged (every five points into
one) which gave satisfactory result.

Table 2: Observatories, telescopes and detectors used in observations

Observatory | Telescope | Detector
---|---|---
(1) Anderson-Mesa, Lowell Obs. (IAU 688), Arizona | 1.1-m Hall | e2v CCD231
(2) Deokheong Optical Astronomy Obs. (IAU P66), South Korea | 1.0-m RC | PI SOPHIA-2048B CCD
(3) Schiaparelli Obs. (IAU 204), Italy | 0.8-m | SBIG STX-16803 CCD
(4) San Marcello Pistoiese Obs. (IAU 104), Italy | 0.6-m | CCD
(5) Winer Obs. (IAU 648), Arizona | 0.7-m RBT/PST2 | Andor iXon 888 CCD
(6) Odessa-Mayaki Obs. (IAU 583), Ukraine | 0.8-m OMT-800 | FLI ML09000 CCD
(7) McDonald Obs. LCO (IAU V37), Texas | 1.0-m 1m0-08 | FLI ML4720 CCD
(8) Calar Alto Obs. (IAU 493), Spain | 1.23-m | e2v CCD231-84
(9) Great Shefford Obs. (IAU J95), Great Britain | 0.4-m | Apogee Alta U47+ CCD
(10) Planatus Observatory (IAU K80), Poland | 0.28-m | ZWO ASI290MM CMOS

Note: the table includes information about all observatories which provided
data on 2021 DW1. Some of them were used only for cross-validation and are not
presented in this paper.

To search for the rotation period of 2021 DW1 we used a standard Fourier
series analysis. This was done with a PerFit program (Kwiatkowski et al.,
2010a), which in its current version works in the following way: after
correcting all times for the light-time, each differential photometry time
series is divided into parts obtained with the same comparison star. Next, a
Fourier series of a given order is fit to all of them with one synodic period
assumed. In the process, both the Fourier series coefficients and magnitude
shifts (with respect to the first time series) are determined by linear least
squares. This process is repeated for each trial period from a specified
interval. In most cases the 4th to 6th order Fourier series is used. The
accuracy of the fit is measured with the $\chi^{2}$ computed from the
residuals.

The period which gives the smallest $\chi^{2}$ is then selected as a potential
synodic period of rotation. The least square fit takes into account the
accuracy of the measured asteroid differential magnitudes (expressed as
standard deviations). The accuracy of the derived period is estimated by a
Monte Carlo method, in which all data are perturbed 30 times with the
appropriate sigma values.

Figure 2: Selected composite lightcurves of 2021 DW1

Fig. 2, cont.

Using PerFit for the lightcurves of 2021 DW1 it appeared there are two
solutions for the period: $P_{1}=0.00688\,\mathrm{h}$ (this gave mono-modal
composite lightcurves) and $P_{2}=0.01376\,\mathrm{h}$ (bi-modal composite
lightcurves), with $P_{2}=2\cdot P_{1}$. Results for
$P_{2}=0.01376\,\mathrm{h}$ (which is our preferred solution) are presented in
Fig. 2. In the plots we do not draw error bars not to obscure images. The
overall accuracy can be easily judged from the scatter of the data points
about the Fourier fit.

##  3 Sidereal period, spin axis and shape

Since the works of Kaasalainen & Torppa (2001) and Kaasalainen et al. (2001) a
standard method for determination of the asteroid sidereal rotation period,
pole and shape is through the lightcurve inversion. For that, lightcurves
observed at different viewing and illuminating geometries are needed. While
each case is different, a good rule of thumb for NEAs is to have data sampling
an arc on the sky longer than $120\degr$ (Josef Ďurech, personal
communication). While in case of 2021 DW1 we observed its trail extending up
to $170\degr$, the last two observations (from 7 March) were very noisy (the
asteroid brightness dropped to $V=18\,\mathrm{mag}$) and were not used in the
analysis. This shortened the arc to $163\degr$, from which we selected
thirteen composite lighcurves well positioned along the trail (see Fig. 2).
Each of them was obtained with the $P_{2}$ synodical period. Their
brightnesses were converted from magnitudes to fluxes and rescaled so that the
average flux during the rotation was set to unity. We did not use any
lightcurves calibrated to the standard magnitudes, as such callibrations are
usually less accurate than the differential magnitudes. The data were then
used for the lightcurve inversion, for which we used a C-language
implementation of the code, written by Josef Ďurech. The convex shapes used in
computations were approximated by spherical harmonics of the 6th degree and
order. The light scattering law was approximated by the Lommel-Seeliger-
Lambert function, with c=0.1.

Fig. 4 presents a periodogram for the sidereal period, with the best result
obtained for $P_{\mathrm{sid}}=0.013760\pm 0.000001\,\mathrm{h}$. There were
two solutions for the ecliptic coordinates of the spin axis, A:
$\lambda_{1}=57\degr\pm 10\degr,\beta_{1}=29\degr\pm 10\degr,$ and B:
$\lambda_{2}=67\degr\pm 10\degr,\beta_{2}=-40\degr\pm 10\degr$, both with the
same sidereal period. The obliquities resulting from those pole orientations
are: $\epsilon_{1}=54\degr\pm 10\degr$ and $\epsilon_{2}=123\degr\pm 10\degr$.
They are far away from the asymptotic states ($\epsilon=0\degr,180\degr$)
predicted by the theory.

As it often happens, the positions are symmetric with respect to the ecliptic
plane (within the specified uncertainties). Convex shapes resulting from these
pole positions are presented in Fig. 5, and a fit of the modelled lightcurves
to the data, for pole A, in Fig. 6. The fits obtained for solution B are very
similar. For both models the percentage of dark facets was smaller than 0.5.
Note that in the final solution we did not use LC09, which is very close in
time to LC10, and did not give any new information to the optimization
algorithm.

Scanning the whole celestial sphere for other solutions we encountered cases,
where the fit of the model lightcurves to the observed ones was satisfactory,
but the convex shapes were not physical. For example, the body was
significantly elongated in the c-axis direction (which was assumed to be the
axis of rotation), the shape was unrealistically flat (pan-cake like), or it
had more than 5% of dark facets on the surface. The latter would imply surface
albedo variegation, which is difficult to accept for such small objects.

The same happened when we started with the $P_{1}$ synodical period. Folding
all data with $P_{1}$ we obtained a set of mono-modal composite lightcurves,
well fit by the Fourier series. However, when we used such data for lightcurve
inversion, we obtained unrealistic solutions. Based on this, we reject $P_{1}$
as a possible period of rotation (with the assumption of convex shape and the
lack of albedo variegation on the surface).

Figure 4: Results of the search for the sidereal period of rotation for the
$P_{2}$ solution. The search interval has been defined by the range of change
of the synodic period. On the vertical axis there is a value of
$\chi^{2}_{\nu}$ ($\chi^{2}$ per degree of freedom). A horizontal line
indicates a p-value of 0.01. The only statistically significant minimum (which
reaches below p=0.01) is $P_{\mathrm{sid}}=0.013760\pm 0.000001\,\mathrm{h}$.

Figure 5: Projections of convex shapes obtained for two pole solutions. The
upper row is for pole A, the lower one for pole B. In the left column we can
see a projection onto the x-z plane, in the middle column: onto the y-z plane,
while at the right there are projections on the x-y plane (top views, along
the rotation axis).

##  4 Magnitude-phase curve

Figure 6: A superposition of the model A lightcurves (continuous line) on the
data points obtained from observations. All plots are drawn in the same
coordinates: the x-axis is the rotation phase (in the range from 0 to 1), the
y-axis refers to the flux scaled to unity for the average brightness. Note
that our model convex have problems with recreating sharp brightness maxima in
LC03, LC05, and LC06 lightcurves. They are most likely caused by concavities
on the asteroid surface.

There are several problems which make it difficult to derive the magnitude-
phase angle function (hereafter: phase function) for NEAs. Firstly, they are
seldom observed close to opposition, so the non-linear part of the phase
function cannot be properly determined. Secondly, during their passage close
to Earth, the aspect angle (the angle formed by the object’s spin axis and the
direction towards the observer) can change significantly, influencing the
magnitude-phase angle relationship. Thirdly, during astrometric observations
of NEAs (which are used routinely by the Minor Planet
Center444https://www.minorplanetcenter.net/ (hereafter MPC) for determination
of the phase function), asteroid magnitudes from different phases of rotation
are reported and the phase curve is distorted even more by the rotational
brightness variations.

In the case of 2021 DW1 we observed its full lightcurves and were able to
determine its rotationally averaged magnitudes. This removed the third
obstacle mentioned above. Since our observations were done in different
filters (and often were ”unfiltered”), for the purpose of photometric
calibration we used the PanSTARRS catalogue (Tonry et al., 2012) standards of
solar colours. We selected the best lightcurves reported in Table 1, and
calibrated them in the SDSS r band. For each lightcurve we used the Fourier
fit to compute its mean magnitude. We also used the data from 7 March
(lightcurve LC013), which were quite noisy for the convex shape modelling, but
gave a good rotationally averaged magnitude.

In the process we had to discard some results, where calibration was
inaccurate. We also noticed that our data were strongly affected by aspect
changes during the time, when the asteroid was changing its ecliptic latitude
from $\beta=-25\degr$ to $\beta=+20\degr$. For this reason we used only
lightcurves observed from 4 Mar, 2 UTC to 7 Mar, 8:45 UTC. After scaling the
magnitudes to unit distances from Earth and Sun, we tried to fit them with a
standard H, G phase function. Our standard program for doing that, with an
advanced nonlinear minimization, failed, so we used a simpler approach
changing the H and G values in some intervals, fitting the H, G function to
the data, and selecting the result with the smallest residuals. Unfortunately,
the best fit was obtained for $G=1.1$ which has no physical meaning because
the typical values of G for asteroids fall between 0.0 and 0.5. The absolute
magnitudes $H_{\mathrm{r}}$ obtained for $G=0.0$ and $G=0.5$, where
$H_{\mathrm{r}}=24.00$ and $H_{\mathrm{r}}=24.98$, respectively. To translate
them to $H$ values in the V band (which is the standard way of reporting
absolute magnitudes), one can increase them by 0.21 mag. This value can be
obtained using the transformation from $r$ to $V$, provided by Tonry et al.
(2012).

The absolute magnitude for 2021 DW1, derived by the MPC from 284 less
accurate, astrometric observations, is $H_{\mathrm{MPC}}=25.02$. It is
obtained assuming $G=0.15$. To get a final value for the absolute magnitude of
2021 DW1 we used the fact that (as we describe in the next section) it is an
S-type asteroid, for which $G=0.24\pm 0.11$ (Warner et al., 2009). With this
assumption we got $H_{\mathrm{r}}=24.57$ which, after conversion to the V
band, is $H_{\mathrm{V}}=24.8$ (Fig. 7).

Since in our computation of $H$ systematic effects dominate over the
statistical ones, we are not able to derive a standard ”one $\sigma$” error
for it. Instead, we estimate the accuracy of the obtained absolute magnitude
by a maximum uncertainty of $\Delta H=0.5$ mag.

Figure 7: Phase curve of 2021 DW1 and the H, G function fit. During fitting,
the $G$ parameter was kept constant at 0.24 (a value typical for S-type
asteroids). Note that the magnitudes used are in the SDSS r band. The obtained
$H_{\mathrm{r}}=24.57$, after conversion to the Johnson V band, gives
$H_{\mathrm{V}}=24.8$

##  5 Colour indices and taxonomy

On 5 March we observed the asteroid in three Sloan g, i, and z filters (see
the rows in Table 1 with the codes CI01-3). For each of them, a serie of 4 s
exposures where obtained with the telescope tracking on the asteroid.

In the case of the observations in i and g filters, we had no problems with
data reduction, while the exposures in the infrared z-filter left interference
fringes on the CCD frames. The reduction of fringing is possible by
subtracting the so-called fringe map, which is obtained by averaging the
images of the blank fields (fields with a few visible stars). We could not
make blank field observations until 30 April. Unfortunately, it turned out
that the fringe pattern changed too much from 5 March till 30 April to use the
obtained fringe map for the reduction.

The non-sidereal telescope tracking ensured an almost constant position of the
asteroid image in relation to the fringe structure, thanks to which it was
possible to avoid the influence of fringing on instrumental measurements of
the asteroid. The problem was with the comparison stars, which moved quickly
in relation to the fringe structure. To minimize this effect, we selected just
one PanSTARRS comparison star, for which the fringing had little effect on its
brightness (standard deviation of brightness measurements
$\sigma=0.02\,\mathrm{mag}$). We used this star to calibrate the z filter
magnitude of the asteroid.

Next, using PerFit, we phased the g, i, and z lightcurves with the reference
VR lightcurve, obtaining one composite lightcurve. For this purpose, we used
part of the observations made in the VR filter on the same telescope just
before the exposures in the g filter, and for control, 40 minutes after the
exposures in the z filter. In both cases the aspect data were practically the
same. The obtained values of $VR-g$, $VR-i$ and $VR-z$ magnitude shifts gave
the following colour indices: $g-i=0.79\pm 0.01\,\mathrm{mag}$, $i-z=0.01\pm
0.02\,\mathrm{mag}$, $z-g=0.01\pm 0.02\,\mathrm{mag}$.

To determine the taxonomic type of 2021 DW1, we converted the colour indices
to the reflection coefficients $R_{i}$ and $R_{z}$, normalized to the
reflection values in the g band, using the formula given in DeMeo & Carry
(2013):

$R_{f}=10^{-0.4[(f-g)-(f_{\odot}-g_{\odot})]}$ (1)

where $g$ and $g_{\odot}$ are the g magnitudes of the asteroid and Sun, and
$f$ and $f_{\odot}$ are the magnitudes in some other band. In Eq. 1 we assumed
the solar colours $(i-g)_{\odot}=-0.55\pm 0.03\,\mathrm{mag}$ and
$(z-g)_{\odot}=-0.61\pm 0.04\,\mathrm{mag}$ (Holmberg et al., 2006), used our
colour indices, and obtained the following reflectivities:
$R_{\mathrm{g}}\equiv 1$, $R_{\mathrm{i}}=1.25\pm 0.03$, and
$R_{\mathrm{z}}=1.19\pm 0.05$. Next, we compared them with spectra of
different taxonomic classes, as given by DeMeo & Carry (2013). A good match
was found for the S, Sq and K spectra (Fig. 8), with the Sq class beeing the
best.

Since K-class asteroids are typical mainly for the Eos family, and are rare in
the asteroid population, the confirmation of this possibility would require a
good quality spectrum of 2021 DW1. In our campaign we planned to get such a
spectrum, but a fast movement of the object on the sky made it impossible even
for the robotic spectrograph mounted on the 2.0-m LCO telescope. Since K-class
objects are very rare among asteroids, our current conclusion is that 2021 DW1
is a typical S-class NEA.

Figure 8: Plots of reflectivities $R_{\mathrm{g}}$, $R_{\mathrm{r}}$,
$R_{\mathrm{i}}$, and $R_{\mathrm{z}}$ of three asteroid taxonomic classes
(black circles), which are most similar to the reflectivities of 2021 DW1 (red
circles). All reflectivities are normalized to the g band, and the grey areas
denote a dispersion of spectra in a given asteroid class. Data for the
taxonomic classes were taken from DeMeo & Carry (2013).

##  6 Physical characteristic of 2021 DW1

Having obtained the absolute magnitude $H_{\mathrm{V}}=24.8\pm 0.5$, and an
estimate of the geometric albedo $p_{\mathrm{V}}=0.23\pm 0.02$, we are now
able to derive the asteroid effective diameter $D_{\mathrm{eff}}$555Diameter
of a sphere which has the same brightness as the rotationally averaged
magnitude of the asteroid – assuming a zero solar phase angle.. For that, we
used a standard equation $D_{\mathrm{eff}}=1329\times
10^{(-H_{\mathrm{V}}/5)}\times p_{V}^{(-1/2)}$ from Fowler and Chillemi
(1992), and obtained $D_{\mathrm{eff}}=30\,\mathrm{m}$. To get its uncertainty
we first converted the statistical error $\sigma_{\mathrm{pV}}$ into a maximum
one: $\Delta p_{\mathrm{V}}=3\times\sigma_{\mathrm{pV}}$. Then we used both
$\Delta H=0.5$ and $\Delta p_{\mathrm{V}}=0.06$ to compute the maximum
uncertainty $\Delta D=10\,\mathrm{m}$. With that we finally get:
$D_{\mathrm{eff}}=30\pm 10\,\mathrm{m}$.

One of the methods for studying physical properties of asteroids is to plot
their rotation periods versus effective diameters. On Fig. 9 we present
results, taken from the last issue of the Light Curve Data Base
(LCDB)666https://minplanobs.org/mpinfo/index.php (Warner et al., 2009). The
plot includes not only VSAs, but also larger objects – both NEAs and Main Belt
Asteroids – up to 10 kilometres in diameter. What is clearly visible in this
figure is that most asteroids with $D_{\mathrm{eff}}>1\,\mathrm{km}$ (in the
lower-right corner of the plot) have periods $P>2.2\,\mathrm{h}$, while most
of VSAs display much faster rotation. This is possible because VSAs are held
together by tensile strength rather than gravity. However, there is a lower
limit to their periods, set by centrifugal forces. Holsapple (2007) derived an
approximate formula, which makes it possible to draw, on the $\log
D_{\mathrm{eff}}$ – $\log P$ plot, a line of minimum allowable periods (which
we will call critical periods, $P_{\mathrm{c}}$) for the asteroid in the
strength regime. We used a slightly modified equation for $P_{\mathrm{c}}$,
given by Kwiatkowski et al. (2010b), to compute such lines on Fig. 9 for two
tensile strength coefficients ($\kappa=10^{5}$ and
$\kappa=10^{6}\,\mathrm{N}\,\mathrm{m}^{3/2}$). A justification of our choice
of $\kappa$ can be found in Kwiatkowski et al. (2010b). We also assume a
typical angle of friction $\phi=40\degr$ (Richardson et al., 2005), a density
$\rho=2500\,\mathrm{kgm^{-3}}$, and the triaxial ellipsoid approximation of
the 2021 DW1 shape given by $c/a=0.48,b/a=0.88$. The last two parameters are
derived by measuring the extension of the asteroid shape along the x, y, z
axes, and computing the average from Model A and B. Note that the position of
the $P_{\mathrm{c}}$ line is most sensitive to $\kappa$ and $\rho$.

On Fig. 9 we also mark a position of 2021 DW1 with its uncertainty. Since we
ruled out the $P_{1}$ solution for the rotation period, all uncertainty now
lies along the diameter axis. As can be seen, 2021 DW1 is far from the lines
denoting the rotational fission. In the future, depending on the YORP-TYORP
cycles, it will move vertically up (or down), or may be trapped in some
equilibrium point, where its period and spin axis will not change.

##  7 Conclusions

We have derived the spin axis coordinates for 2021 DW1. In the ecliptic
reference frame, the two solutions are: (A) $\lambda_{1}=57\degr\pm
10\degr,\beta_{1}=29\degr\pm 10\degr$, and (B) $\lambda_{2}=67\degr\pm
10\degr,\beta_{2}=-40\degr\pm 10\degr$, with obliquities
$\epsilon_{1}=54\degr\pm 10\degr$ and $\epsilon_{2}=123\degr\pm 10\degr$,
respectively. It shows that the spin axis of 2021 DW1 is far from the
asymptotic states of $\epsilon=0\degr,180\degr$ predicted by simulations and
theory (Čapek & Vokrouhlický, 2004; Golubov et al., 2021) for asteroids with
high thermal conductivity. Interestingly, both $\epsilon_{1}$ and
$\epsilon_{2}$ have the same values as the obliquities, at which the period
change component of YORP vanishes, as shown by Eq. 13 in Golubov et al.
(2021). The same effect has been observed in numerical simulations by Čapek &
Vokrouhlický (2004) (their Figs. 6-8). However, this is only a temporary
situation because the obliquity change component of YORP at those two
positions is still significant. After some time, as the obliquity continues to
change, the rotation period can be altered again.

We also obtained other physical parameters of 2021 DW1: a sidereal period
$P_{\mathrm{sid}}=0.01376\pm 0.00001\,\mathrm{h}$, the parameters of its
magnitude-phase function $H=24.8\pm 0.5\,\mathrm{mag}$ and $G=0.24$. The
asteroid colour indices are $g-i=0.79\pm 0.01\,\mathrm{mag}$, and $i-z=0.01\pm
0.02\,\mathrm{mag}$ which indicates an S taxonomic class, with an average
geometric albedo $p_{V}=0.23\pm 0.02$. The asteroid effective diameter,
derived from $H$ and $p_{V}$, is $D_{\mathrm{eff}}=30\pm 10\,\mathrm{m}$.

Unfortunately, no new observations of this object will be possible in the near
future. According to the JPL Horizons service, the only close approach of 2021
DW1 to Earth in the next 100 years will take place on 20 February 2047\. At
this time, however, the asteroid will reach the maximum brightness of
$V=21\,\mathrm{mag}$ and, with such rapid rotation, would require the biggest
telescopes for observations.

Figure 9: Position of 2021 DW1 on the $\log D_{\mathrm{eff}}-\log P$ plot. The
open circles were taken from the LCDB, while sloped lines indicate minimum
allowable periods (at a given diameter), computed with two different tensile
strength coefficients ($\kappa=10^{5}$ and
$\kappa=10^{6}\,\mathrm{N}\,\mathrm{m}^{3/2}$), and a density of
$\rho=2500\,\mathrm{kgm^{-3}}$. A red square indicates the position of 2021
DW1 with the error bars being a measure of the maximum uncertainty.

##  8 Acknowledgements

T. Kwiatkowski, A. Kryszczyńska, and D. Oszkiewicz were supported by a grant
No.2017/25/B/ST9/00740 from the National Science Centre, Poland. T. Kim was
supported by the Korea Astronomy and Space Science Institute under the R&D
program (Project No. 2020-1-600-05) supervised by the Ministry of Science and
ICT (MSIT), and the National Research Foundation of Korea (NRF) Grant No.
2020R1A2C3011091, funded by MSIT. The work by T. Santana-Ros was carried out
through a grant APOSTD/2019/046 by Generalitat Valenciana (Spain). He was also
supported by the MINECO (Spanish Ministry of Economy) through a grant
RTI2018-095076-B-C21 (MINECO/FEDER, UE). During observations the NASA JPL
Horizons service was used extensively. We would like to thank the reviewer,
David Vokrouhlický, for his useful comments which improved the article.

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