Conjugate Phase Retrieval Overview

• Conjugate phase retrieval is the recovery of a complex-valued signal up to a global phase and conjugation ambiguity, enabling reconstruction with real sensing vectors.
• It employs techniques like phase lifting and exploits symmetry groups (e.g., S¹ ⋉ {±1}) to reduce algebraic complexity compared to standard phase retrieval.
• Applications span Fourier intensity analysis, MRI calibration, and sparse signal recovery, with robust performance in both finite- and infinite-dimensional settings.

Searching arXiv for recent and foundational papers on conjugate phase retrieval. Conjugate phase retrieval is the problem of recovering a complex-valued signal from phaseless measurements when the admissible ambiguity class is enlarged from global phase alone to global phase together with conjugation. In its most common form, one seeks to identify $x$ only up to

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$

or, in structured Fourier settings, up to global phase and conjugate-reflection symmetries. This formulation arises whenever the measurement operator is intrinsically insensitive to conjugation, as happens for real sensing vectors in $\mathbb C^M$, for one-dimensional Fourier intensity, and in several infinite-dimensional sampling models. Across finite-dimensional frame theory, Fourier phase retrieval, Paley–Wiener and shift-invariant spaces, sparse super-resolution, graph-based models, and MRI calibration, the central organizing principle is that reconstruction is an orbit-recovery problem: one can only hope to recover the orbit of the unknown signal under the symmetry group preserved by the measurements (Evans et al., 2017, Bendory et al., 2022).

1. Concept and formal ambiguity class

Conjugate phase retrieval was introduced as a relaxed version of complex phase retrieval in which conjugation is accepted as an unavoidable ambiguity. For a finite family $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$, the defining property is

$$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$

for some $\theta\in[0,2\pi)$ (Evans et al., 2017). In complex Hilbert-space formulations, the same ambiguity appears as $x=\lambda y$ with $|\lambda|=1$, but conjugate phase retrieval enlarges the target quotient by also identifying conjugates when the measurements do not separate them (Rahimlou et al., 10 Jun 2026).

This enlargement is not merely terminological. For real measurement vectors $\varphi_n\in\mathbb R^M$, one has

$$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$

so real frames can never be phase retrievable on $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$0 in the ordinary sense, but they may still be conjugate phase retrievable (Evans et al., 2017). The same phenomenon is highlighted in the general phase retrieval literature: real measurement vectors in the complex setting fail to distinguish complex conjugates, which directly motivates a weakened injectivity notion modulo phase and conjugation rather than phase alone (Mixon, 2014).

A related but distinct ambiguity arises in Fourier phase retrieval. There the natural symmetry is not entrywise conjugation by itself, but the action of

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$1

where $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$2 acts by multiplication by a global phase and $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$3 acts by conjugation and reflection (Bendory et al., 2022). This is the canonical setting in which “conjugate phase retrieval” is best understood as symmetry-aware recovery under a larger intrinsic ambiguity group.

2. Finite-dimensional theory over $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$4

The foundational finite-dimensional treatment establishes conjugate phase retrieval as a genuine intermediate notion between complex phase retrieval and norm retrieval (Evans et al., 2017). Its key structural equivalence is phase-lift based: for $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$5,

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$6

whereas

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$7

Thus ordinary complex phase retrieval seeks to recover the full rank-one Hermitian matrix $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$8, while conjugate phase retrieval with real measurements seeks only $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$9 (Evans et al., 2017). This reduction explains why the obstruction set is governed by real symmetric rank-$\mathbb C^M$0 geometry rather than the rank-$\mathbb C^M$1 Hermitian geometry of standard complex phase retrieval.

For a real frame $\mathbb C^M$2, the relevant linear map is

$\mathbb C^M$3

acting on real symmetric matrices. The kernel criterion states that $\mathbb C^M$4 is conjugate phase retrievable if and only if

$\mathbb C^M$5

with a sufficient condition

$\mathbb C^M$6

because $\mathbb C^M$7 (Evans et al., 2017). This formulation is the direct analogue of the rank-nullspace criteria used in standard complex phase retrieval (Mixon, 2014), but adapted to the weaker quotient.

The low-dimensional cases admit complete characterizations. In $\mathbb C^M$8,

$\mathbb C^M$9

and a real $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$0 frame is conjugate phase retrievable if and only if the determinant of the associated quadratic measurement matrix is nonzero (Evans et al., 2017). In this dimension, conjugate phase retrieval is equivalent to the complement property. In $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$1,

$\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$2

and a real $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$3 frame is conjugate phase retrievable if and only if the corresponding $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$4 determinant condition holds; unlike the two-dimensional case, the complement property is no longer sufficient (Evans et al., 2017).

For $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$5, a generic real frame with at least

$\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$6

measurements is conjugate phase retrievable in $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$7 (Evans et al., 2017). This generic threshold contrasts with the $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$8 heuristic and conjectural landscape for standard complex phase retrieval surveyed in the broader literature (Mixon, 2014). A plausible implication is that allowing conjugation lowers the effective algebraic complexity of the injectivity problem.

The same $\Phi=\{\varphi_n\}_{n\in I}\subset \mathbb C^M$9 count reappears in an infinite-dimensional-to-finite-dimensional specialization for spline Hermite sampling. For $$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$0, the explicit real frame

$$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$1

with

$$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$2

does conjugate phase retrieval (Chen et al., 2023). This provides concrete deterministic examples of real conjugate phase retrievable frames beyond generic existence.

3. Fourier intensity and algebraic ambiguity structure

One-dimensional Fourier phase retrieval furnishes the clearest algebraic model of conjugation-related ambiguity. In the general phase retrieval model

$$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$3

the relevant symmetry group depends on the measurement operator (Bendory et al., 2022). For complex generic linear measurements it is $$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$4, but for one-dimensional Fourier intensity the invariance group is

$$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$5

with $$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$6 acting by conjugation and reflection (Bendory et al., 2022). The data are

$$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$7

and the Fourier intensity is equivalent to the aperiodic autocorrelation $$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$8 (Bendory et al., 2022).

The ambiguity set is completely characterized by the root-flip theorem quoted there. If $$|\langle x,\varphi_n\rangle|=|\langle y,\varphi_n\rangle|\ \forall n\in I \quad\Longrightarrow\quad x=e^{i\theta}y \ \text{or}\ x=e^{i\theta}\overline{y},$$9 are the roots of $\theta\in[0,2\pi)$0, then $\theta\in[0,2\pi)$1 has the same Fourier intensity as $\theta\in[0,2\pi)$2 if and only if there is a subset $\theta\in[0,2\pi)$3 and angle $\theta\in[0,2\pi)$4 such that

$\theta\in[0,2\pi)$5

Each root may be kept or replaced by its reflected-conjugate reciprocal

$\theta\in[0,2\pi)$6

and the extreme case $\theta\in[0,2\pi)$7 yields a scalar multiple of the conjugate-reflected signal (Bendory et al., 2022). If the roots are distinct and none is paired with its reciprocal-conjugate partner, then there are $\theta\in[0,2\pi)$8 distinct signals modulo the trivial symmetry group $\theta\in[0,2\pi)$9 (Bendory et al., 2022).

This algebraic picture clarifies two points. First, conjugation is an intrinsic ambiguity of Fourier intensity, not an artifact of a particular proof technique. Second, modding out by phase and conjugate reflection does not generally restore uniqueness in one dimension. The paper gives a concrete example of four distinct vectors $x=\lambda y$0 with the same Fourier intensity

$x=\lambda y$1

where $x=\lambda y$2, and notes that these vectors are not related by a trivial ambiguity (Bendory et al., 2022). This establishes that the fiber geometry can remain large even after quotienting by the natural conjugation symmetry.

Sparse Fourier models sharpen this uniqueness picture. For discrete-time sparse signals of length $x=\lambda y$3, almost all $x=\lambda y$4-sparse signals with aperiodic support can be uniquely recovered by solving the sparse autocorrelation inverse problem, but only up to time-shift, conjugate-flip, and global phase (Jaganathan et al., 2013). The conjugate-flipped signal is

$x=\lambda y$5

and a central structural lemma states that if two non-equivalent signals $x=\lambda y$6 have the same autocorrelation, then there exist $x=\lambda y$7 and $x=\lambda y$8 such that

$x=\lambda y$9

which makes conjugate-flip a built-in algebraic mechanism of nonuniqueness (Jaganathan et al., 2013). The same paper develops the Two-stage Sparse Phase Retrieval algorithm and proves that it can provably recover most $|\lambda|=1$0-sparse signals, up to a time-shift, conjugate-flip and global phase, with a noise-robust regime for most $|\lambda|=1$1-sparse signals (Jaganathan et al., 2013).

In sparse multivariate super-resolution, the inevitable ambiguity class is again explicit: $|\lambda|=1$2 so recovery is only unique modulo global phase shift, translation, and conjugated reflection (Beinert et al., 2023). For structured signals

$|\lambda|=1$3

the paper shows that multivariate recovery can be reduced to line-wise one-dimensional problems and resolved, up to the inevitable ambiguities, from $|\lambda|=1$4 phaseless Fourier samples on $|\lambda|=1$5 lines (Beinert et al., 2023). This suggests that additional geometry and adaptive sampling can control, but not eliminate, conjugation-generated ambiguity.

4. Infinite-dimensional sampling theories

Conjugate phase retrieval has been developed extensively in function spaces where the signal class is closed under conjugation and the measurements are phaseless samples or Hermite-type samples.

In Paley–Wiener space $|\lambda|=1$6, the ambiguity is

$|\lambda|=1$7

where

$|\lambda|=1$8

A principal result is that conjugate phase retrieval can be accomplished in $|\lambda|=1$9 by sampling only on the real line by using structured convolutions, and can also be accomplished by sampling both $\varphi_n\in\mathbb R^M$0 and $\varphi_n\in\mathbb R^M$1 only on the real line (Lai et al., 2019). If $\varphi_n\in\mathbb R^M$2 is a sampling sequence for $\varphi_n\in\mathbb R^M$3, then the maps

$\varphi_n\in\mathbb R^M$4

and

$\varphi_n\in\mathbb R^M$5

are one-to-one on $\varphi_n\in\mathbb R^M$6 under the stated conditions (Lai et al., 2019). The same work shows that, generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate (Lai et al., 2019). This is one of the clearest quantitative separations between the standard and conjugate formulations.

A closely related graph-based extension proves that for signals in the Paley-Wiener space, any complex-valued function can be recovered, up to a unimodular constant and conjugation, from structured phaseless samples taken at three times the Nyquist rate (Cheng et al., 30 Jul 2025). In that framework, the sampled values are treated as a graph signal, and connectivity of a signal-dependent triangle graph is sufficient for conjugate recovery from absolute values on vertices and relative magnitudes between neighboring vertices (Cheng et al., 30 Jul 2025). The same paper provides two numerical reconstruction algorithms for Paley–Wiener and general shift-invariant spaces (Cheng et al., 30 Jul 2025).

For compactly supported real generators $\varphi_n\in\mathbb R^M$7, the complex shift-invariant space

$\varphi_n\in\mathbb R^M$8

supports a local-to-global theory of conjugate phase retrievability (Chen et al., 2023). When $\varphi_n\in\mathbb R^M$9 has the spanning property

$$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$0

there exists a finite set

$$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$1

such that any $$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$2 satisfying the global coefficient conditions can be reconstructed, up to unimodular constant and conjugation, from

$$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$3

(Chen et al., 2023). For spline spaces $$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$4, point-sampling alone is insufficient when $$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$5, but Hermite phaseless sampling restores uniqueness: there are sets $$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$6 with

$$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$7

such that $$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$8 can be determined, up to unimodular constant and conjugation, from $$|\langle x,\varphi_n\rangle|=|\langle \overline{x},\varphi_n\rangle|,$$9 on $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$00 and $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$01 on $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$02 (Chen et al., 2023).

The Gaussian-generated shift-invariant space

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$03

admits an especially sharp theory (Chen et al., 2024). Because the generator is real-valued, the space is conjugate invariant and the natural ambiguity is

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$04

If $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$05 is separated with lower Beurling density

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$06

then $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$07 is uniquely determined, up to a unimodular constant and conjugation, from the phaseless Hermite samples

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$08

(Chen et al., 2024). For finitely supported coefficient sequences, the same paper gives an explicit reconstruction procedure based on recovering the exponential coefficients in $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$09 and an auxiliary derivative-related quantity, then reconstructing the coefficients recursively (Chen et al., 2024).

Across these infinite-dimensional models, the common mechanism is that the measurements determine $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$10, $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$11, or finite-dimensional local Gram data, and entire-function or algebraic arguments then show that equality of these phaseless quantities forces equality up to phase or conjugation. This suggests that conjugate phase retrieval is particularly natural in analytically structured function spaces where conjugation preserves the model class.

5. Group-theoretic and graph-theoretic formulations

A group-theoretic version of conjugate phase retrieval appears in permutation representations. For a complex subspace $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$12 with real-valued measurement vectors $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$13, a system does conjugate phase retrieval if

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$14

implies the existence of $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$15 such that either

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$16

with conjugation entrywise (Bartusel et al., 2021). For the affine group

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$17

acting on the zero-sum subspace

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$18

the orbit of a difference vector under a doubly transitive action does conjugate phase retrieval on $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$19 (Bartusel et al., 2021). The geometric reason is that equality of magnitudes across the orbit forces equality of all pairwise distances among the complex coordinates, and Euclidean rigidity then leaves only phase or conjugation (Bartusel et al., 2021).

The same representation supports stronger properties. The canonical irreducible representation $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$20 on $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$21 admits explicit generating vectors whose orbits achieve matrix recovery, the strongest retrieval notion considered there (Bartusel et al., 2021). Since matrix recovery implies phase retrieval, and the paper also develops conjugate phase retrieval and sign retrieval in the same framework, it places conjugate recovery within a hierarchy of information-completeness properties tied to representation theory (Bartusel et al., 2021).

Graph-based conjugate phase retrieval provides a complementary local-geometric model (Cheng et al., 30 Jul 2025). For a complex-valued graph signal $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$22, the measurements are vertex magnitudes $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$23 and edge relative magnitudes $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$24. A signal-dependent graph $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$25 is built from triangles of the underlying graph, and two triangles are adjacent if they share a common non-collinear edge, meaning

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$26

If every vertex belongs to some triangle and $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$27 is connected, then $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$28 is determined, up to a unimodular constant and conjugation, from $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$29 and $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$30 (Cheng et al., 30 Jul 2025). The paper further constructs explicit reference-point and circulant graph models where these conditions are easy to verify, and applies the theory to shift-invariant spaces and STFT-based sampling (Cheng et al., 30 Jul 2025).

These results emphasize that conjugate phase retrieval can be framed either globally, via quotient manifolds and lifted operators, or locally, via propagation of orientation choices across overlaps, non-collinear edges, or sample neighborhoods.

6. Algorithms, applications, and neighboring frameworks

Algorithmic work on conjugate phase retrieval falls into two broad classes: methods that solve the standard phaseless inverse problem but are interpreted modulo the larger ambiguity class, and methods designed to exploit conjugate symmetry explicitly.

In finite-dimensional standard phase retrieval, the PhaseCut formulation rewrites the problem

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$31

as a nonconvex quadratic program over unit-modulus phases

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$32

with the semidefinite relaxation

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$33

(Waldspurger et al., 2012). This paper is not about conjugate phase retrieval per se, but its complex-valued phase-lifting perspective is directly relevant because it treats the recovery variable as a phase vector on the complex torus and supplies a convex framework that can accommodate symmetry-aware formulations (Waldspurger et al., 2012). A plausible implication is that conjugate phase retrieval relaxations may be designed by replacing the ordinary quotient geometry with its conjugate-modified analogue.

Alternating projections also remain useful. In $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$34, the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces, and experiments with an explicit $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$35 conjugate phase retrievable matrix yielded 850 of 1000 reconstructions succeeded within 900 iterations (Lai et al., 2019). The paper notes the usual traps and tunnels behavior, indicating that conjugate injectivity does not by itself remove the nonconvexity of iterative algorithms (Lai et al., 2019).

MRI provides an application where conjugation symmetry is operational rather than purely abstract. In the signal model

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$36

with

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$37

phase-constrained reconstructions require absolute-phase maps $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$38, not merely relative coil sensitivities (Uecker et al., 2015). VCC-ESPIRiT extends ESPIRiT to physical and virtual conjugate coils by defining

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$39

so that

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$40

The crucial identity

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$41

reveals the unknown pixelwise phase up to a $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$42 ambiguity (Uecker et al., 2015). This is not presented as “conjugate phase retrieval” in the abstract mathematical sense, but it is a symmetry-based phase-recovery mechanism in which conjugate channels provide redundant information needed to estimate phase-inclusive sensitivities.

The same application also exposes a limitation. When the image phase contains high-frequency variation, VCC-ESPIRiT produces a second set of maps with eigenvalue near 1, indicating that the data space is not well represented by a single real-valued phase-constrained model (Uecker et al., 2015). This suggests that conjugation-aware recovery may require multi-map or relaxed formulations when the phase model is only approximately valid.

7. Relation to broader retrieval notions and open directions

Conjugate phase retrieval belongs to a wider family of quotient-based inverse problems. Generalized phase retrieval for operator-valued measurements replaces frame vectors by bounded operators $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$43 and seeks recovery from $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$44 up to a unimodular scalar (Rahimlou et al., 10 Jun 2026). The exact definition is

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$45

implies

$$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$46

The paper does not explicitly develop conjugate phase retrieval as a separate notion, but it places the standard phase/sign ambiguity in a unified Hilbert-space setting and proves that g-phase retrieval is characterized by stability of both phase retrieval and norm retrieval under all invertible pre-compositions (Rahimlou et al., 10 Jun 2026). This broader perspective suggests that a generalized conjugate phase retrieval theory would likely require replacing the unimodular quotient by a larger symmetry quotient analogous to the $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$47 structure seen in Fourier intensity.

Several recurring themes define the current state of the subject. First, the relevant ambiguity group is measurement-model dependent. In generic complex linear measurements it is $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$48; with real sensing vectors it becomes phase plus conjugation; in one-dimensional Fourier intensity it is global phase and conjugate reflection (Evans et al., 2017, Bendory et al., 2022). Second, algebraic injectivity and practical identifiability can diverge sharply: even after quotienting by the correct symmetry group, one-dimensional Fourier fibers may remain exponentially large (Bendory et al., 2022). Third, additional side information can break or localize conjugation ambiguities. The algebraic survey notes that a generic signal can be recovered from Fourier magnitude plus one extra time-domain sample, up to either global phase or conjugate reflection depending on the sample location (Bendory et al., 2022). In function-space models, derivative data or relative magnitude data play a similar symmetry-breaking role (Lai et al., 2019, Chen et al., 2023, Chen et al., 2024, Cheng et al., 30 Jul 2025).

A persistent open issue is the lack of a simple injectivity characterization for complex measurement systems even in ordinary phase retrieval. For a generic complex matrix $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$49 with $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$50, every vector $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$51 can be recovered, up to multiplication by a scalar $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$52, from the phaseless measurements $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$53, but a simple complete characterization of complex matrices guaranteeing injectivity modulo $$x=e^{i\theta}y \quad \text{or} \quad x=e^{i\theta}\overline y,$$54 is not known (Bendory et al., 2022). A plausible implication is that symmetry-enlarged variants, including conjugate phase retrieval outside highly structured settings, are likely to remain algebraically subtle.

Taken together, the literature shows that conjugate phase retrieval is not a single theorem but a family of inverse problems organized by symmetry. Its finite-dimensional core is now well developed; its Fourier and sparse incarnations reveal rich algebraic nonuniqueness; its infinite-dimensional sampling theories exhibit sharp density and sample-complexity gains over standard phase retrieval; and its applications, especially in MRI and structured sensing, demonstrate that conjugation symmetry is often a physically meaningful feature of the measurement process rather than a mathematical nuisance (Evans et al., 2017, Bendory et al., 2022, Lai et al., 2019, Chen et al., 2023, Chen et al., 2024, Cheng et al., 30 Jul 2025).