Blind Deconvolution & Phase Retrieval

• Blind Deconvolution and Phase Retrieval (BDPR) is a class of inverse problems that jointly recovers unknown signals and blur kernels from phaseless data.
• Key methodologies include convex relaxations with lifting techniques and nonconvex optimization strategies such as spectral initialization and Wirtinger-gradient descent.
• Applications span microscopy, ptychography, and turbulence imaging, where leveraging physical priors and hardware constraints is crucial to resolving ambiguities.

Blind Deconvolution and Phase Retrieval (BDPR) denotes a class of inverse problems in which both an unknown object and an unknown blur, kernel, probe, pupil, or illumination must be recovered from phaseless or intensity-only measurements. In the unified notation used for blind phase retrieval, the task is to find $(w,u)$ such that $|\mathcal A(w,u)|^2=f$, which subsumes ptychography, Fourier ptychography, coherent diffractive imaging, FROG, and convolutional phase retrieval (Chang et al., 2022). In this sense, BDPR combines the bilinear ambiguity of blind deconvolution with the phase loss of classical phase retrieval. Representative measurement models include phaseless Fourier measurements of circular convolution, $\tilde y = |F(w\circledast x)|$ (Ahmed et al., 2019), low-resolution phaseless convolution measurements $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$ (Pinilla et al., 2021), and incoherent turbulence imaging with coded aperture, $g=f\otimes h_{\rm i}$ and $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$ (Muneta et al., 2022).

1. Core inverse-problem structure

BDPR problems are characterized by two coupled sources of nonconvexity. First, the forward map is bilinear or multiplicative in two unknowns. Second, the data are phaseless, so only magnitudes or intensities are observed. The survey formulation $|\mathcal A(w,u)|^2=f$ makes this explicit and places a wide range of imaging models under a single algebraic template (Chang et al., 2022). In convolutional settings, the model may be written as phaseless Fourier data of a circular convolution, $y = |F(w\circledast x)|$, whereas in blind ptychography the phaseless data are $\bm a_j = |\mathcal F(\omega\circ \mathcal S_j u)|$ for multiple scan positions $j$ (Ahmed et al., 2019).

The role of additional physical structure varies by modality. In microscopy, origin-symmetric point spread functions imply that the blurred image and the original image share the same Fourier-transform phase (Tofighi et al., 2015). In turbulence imaging with a coded aperture, the aberrated incoherent PSF is generated by a pupil field $|\mathcal A(w,u)|^2=f$0 and the coded aperture acts as a known support mask in the pupil plane (Muneta et al., 2022). In low-resolution phaseless blind deconvolution, only low-frequency Fourier coefficients are retained, so the problem combines blind deconvolution, phase retrieval, and super-resolution in a single model (Pinilla et al., 2021).

Setting	Representative measurement model	Structural ingredient
Convolutional BDPR	$|\mathcal A(w,u)|^2=f$1	Known subspaces for both factors
Low-resolution BDPR	$|\mathcal A(w,u)|^2=f$2	Partial Fourier sampling with $|\mathcal A(w,u)|^2=f$3
Blind ptychography	$|\mathcal A(w,u)|^2=f$4	Multiple scan positions and overlap
Turbulence imaging	$|\mathcal A(w,u)|^2=f$5	Coded aperture support on pupil plane

This unification is not merely terminological. It indicates that many seemingly different imaging problems are governed by the same identifiability and optimization obstacles: bilinear factor ambiguity, lost phase, and strong sensitivity to side information, redundancy, or hardware design.

2. Ambiguities, identifiability, and structural priors

The most basic ambiguity in BDPR is bilinear scaling. In subspace-based convolutional formulations, $|\mathcal A(w,u)|^2=f$6, and the corresponding lifted matrices satisfy $|\mathcal A(w,u)|^2=f$7 (Ahmed et al., 2019). Phase retrieval introduces additional global phase or sign ambiguities; for low-resolution phaseless blind deconvolution, the paper explicitly notes the inherent global phase ambiguity, while the tensorized formulation of simultaneous blind deconvolution and phase retrieval identifies the sign ambiguity $|\mathcal A(w,u)|^2=f$8 (Pinilla et al., 2021).

Beyond these trivial ambiguities, one-dimensional linear convolution admits a richer algebraic ambiguity structure. In the $|\mathcal A(w,u)|^2=f$9-domain, convolution becomes polynomial multiplication, and ambiguities are in one-to-one correspondence with factorization ambiguities generated by swapping zeros between the two factors (Walk et al., 2017). Exact recovery from correlation data is therefore tied to a coprimeness condition: the two $\tilde y = |F(w\circledast x)|$0-transforms must have no common factors. The same paper shows that adding both autocorrelations and cross-correlations turns blind deconvolution into a deterministic convex recovery problem, whereas omitting one autocorrelation destroys uniqueness (Walk et al., 2017).

Several BDPR approaches attack ambiguity through engineered priors or known context rather than through generic regularization alone. In microscopy, the convex set

$\tilde y = |F(w\circledast x)|$1

encodes the fact that symmetric blur preserves Fourier phase (Tofighi et al., 2015). In turbulence imaging, the coded aperture $\tilde y = |F(w\circledast x)|$2 reduces the unknown pupil support to the transmitting coordinates $\tilde y = |F(w\circledast x)|$3, which reduces the number of unknowns and improves conditioning (Muneta et al., 2022). In Fourier phase retrieval with background information, known background pixels are embedded into the reconstruction domain; for $\tilde y = |F(w\circledast x)|$4-dimensional signals, the paper states that background lengths satisfying

$\tilde y = |F(w\circledast x)|$5

are sufficient to ensure uniqueness with probability $\tilde y = |F(w\circledast x)|$6, assuming i.i.d. Gaussian background entries (Yuan et al., 2023).

A common misconception is that BDPR is identifiable once enough measurements are taken. The literature summarized here does not support that claim. Instead, identifiability is repeatedly tied to concrete side conditions: random known subspaces, no common zeros, sufficient scan overlap, known background information, known probe magnitudes, or hardware support constraints (Chang et al., 2018).

3. Convex formulations and exact-recovery theory

One major line of BDPR research converts the original bilinear phaseless inverse problem into a convex program by lifting the unknown factors to positive semidefinite rank-one matrices. For phaseless Fourier measurements of a circular convolution, $\tilde y = |F(w\circledast x)|$7 and $\tilde y = |F(w\circledast x)|$8 are assumed to lie in known subspaces, and the lifted matrices $\tilde y = |F(w\circledast x)|$9 and $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$0 yield bilinear measurements in nonnegative scalar functionals. The nonconvex equalities are then relaxed to first-quadrant hyperbolic inequalities, and the rank constraints are replaced by trace minimization: $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$1 subject to

$$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$2

Under random Gaussian subspaces of dimensions $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$3 and $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$4, exact recovery up to scaling holds with high probability when $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$5, and the method does not require initialization (Ahmed et al., 2019).

A deterministic convex alternative arises when additional autocorrelation information is available. In the one-dimensional aperiodic convolution setting, blind deconvolution with the autocorrelations $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$6 and $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$7, together with the cross-correlations, can be written as a feasibility SDP over the lifted rank-one matrix $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$8. The main theorem states that if the $$\boldsymbol y = |\boldsymbol F_{lo}(\boldsymbol x\circledast \boldsymbol h)|^2+\boldsymbol\eta$$9-transforms of $g=f\otimes h_{\rm i}$0 and $g=f\otimes h_{\rm i}$1 have no common factors and $g=f\otimes h_{\rm i}$2, then the feasible convex program has the unique solution $g=f\otimes h_{\rm i}$3, hence exact recovery up to global phase (Walk et al., 2017).

Convexity also appears at the level of modular constraints rather than whole-program lifting. For microscopy deblurring, the closed convex set of images with prescribed Fourier phase is combined with the epigraph set of total variation,

$g=f\otimes h_{\rm i}$4

so that blind deconvolution alternates between blur updates, phase-consistency projection, TV-epigraph projection, and positivity or support constraints (Tofighi et al., 2015). The paper argues that these projections are non-expansive and can be inserted into many blind deconvolution algorithms.

Convexification is therefore not a single method but a family of strategies. It may act globally, through lifted semidefinite recovery with exact-recovery theorems, or locally, through closed convex feasibility sets embedded inside alternating schemes. This suggests that the boundary between “convex BDPR” and “nonconvex BDPR” is often determined by which part of the original bilinear phase-loss structure is being relaxed.

4. Nonconvex algorithms, initialization, and optimization geometry

A second major line of work solves BDPR directly in factorized variables. In low-resolution phaseless blind deconvolution, BliPhaSu minimizes a non-convex least-squares objective over low-dimensional coefficients $g=f\otimes h_{\rm i}$5 after imposing known subspaces $g=f\otimes h_{\rm i}$6 and $g=f\otimes h_{\rm i}$7 (Pinilla et al., 2021). The algorithm consists of a spectral initialization based on the leading eigenvectors of two moment matrices,

$g=f\otimes h_{\rm i}$8

followed by stochastic Wirtinger-gradient iterations. Under $g=f\otimes h_{\rm i}$9, bounded noise, and proper step sizes, the paper proves linear convergence in expectation to the true pair up to a noise floor (Pinilla et al., 2021).

For single-shot turbulence imaging, the reconstruction is based on the ptychographic iterative engine. The method alternates among updating the object spectrum $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$0, the incoherent transfer function $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$1, and the pupil function $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$2 on the coded-aperture support. The amplitude constraint

$h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$3

and the phase-only pupil constraint

$h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$4

enforce the non-absorbing turbulence model during iteration (Muneta et al., 2022).

In blind ptychographic phase retrieval, generalized ADMM introduces split variables $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$5 and optimizes a Poisson-noise-motivated nonlinear least-squares or maximum-likelihood objective with amplitude constraints on both $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$6 and $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$7. The augmented-Lagrangian subproblems for $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$8 and $h_{\rm i}=|\mathcal F^{-1}[P\times A]|^2$9 admit fast element-wise updates because of the bilinear structure and orthogonality of the Fourier transform, while the $|\mathcal A(w,u)|^2=f$0-update is a proximal step for penalized AGM or pIPM data terms (Chang et al., 2018). The same paper proves convergence of the generalized ADMM to a stationary point under sufficient-overlap and bounded-preconditioner conditions.

Several theoretical works analyze why such nonconvex methods can nevertheless behave regularly. For phase retrieval and blind deconvolution, vanilla gradient descent can remain inside a region of incoherence and contraction without explicit regularization, a phenomenon called implicit regularization (Ma et al., 2017). For the tensorized surrogate of simultaneous blind deconvolution and phase retrieval, the population-risk landscape on the unit sphere has global minimizers at $|\mathcal A(w,u)|^2=f$1, and Riemannian gradient descent converges linearly under local initialization conditions; analogous local linear convergence and noise-floor guarantees are proved under a tensor restricted isometry property for the surrogate sensing model (Liang et al., 13 Sep 2025).

These results do not imply that generic nonconvex BDPR is globally benign. They show, rather, that benign geometry can emerge under very specific models: random subspaces, sufficient overlap, proper initialization, or TRIP-like sensing conditions. The survey literature treats convergence for many practical projection methods as still incomplete (Chang et al., 2022).

5. Experimental modalities and domain-specific formulations

BDPR appears in substantially different physical regimes, and the literature repeatedly adapts the inverse problem to the governing optics. In microscopy, the key assumption is that many point spread functions are approximately origin-symmetric, so the Fourier phase of the object is preserved and can be used as a convex constraint inside a blind deconvolution loop (Tofighi et al., 2015). In turbulence imaging under spatially incoherent light, the object and the turbulence-induced pupil aberration are estimated jointly from a single captured image, with a binary random coded aperture on the pupil plane acting as a support prior (Muneta et al., 2022).

Ptychographic BDPR treats the sample and the probe jointly as unknowns. The survey formulation $|\mathcal A(w,u)|^2=f$2 unifies conventional ptychography with related models such as Fourier ptychography and FROG (Chang et al., 2022). One blind-pycthography line formulates the task directly as constrained nonlinear optimization under Poisson noise and solves it with convergent ADMM (Chang et al., 2018). Another line rewrites far-field blind ptychography as a blind deconvolution problem over autocorrelations,

$|\mathcal A(w,u)|^2=f$3

then combines blind deconvolution, angular synchronization, and regularized Wirtinger descent (Roach, 2023). This algebraic recasting makes the ptychography-to-BDPR connection explicit.

Phase-retrieval imaging can also be blind in a parametric sense. In the regularization view of Paganin-type phase retrieval, the recovered quantity $|\mathcal A(w,u)|^2=f$4 is obtained from the convolutional filter

$|\mathcal A(w,u)|^2=f$5

where $|\mathcal A(w,u)|^2=f$6 depends on physical sample parameters and propagation geometry (Miqueles et al., 2017). The paper treats the unknown $|\mathcal A(w,u)|^2=f$7 as the blur parameter of a Tikhonov-regularized deconvolution problem and determines it by a one-dimensional optimization based on L-curve curvature. This suggests a broader BDPR interpretation in which the “blind” variable is not necessarily a free-form kernel but may be a physically meaningful parameter controlling a convolution model.

Learned priors have introduced a different kind of structure. Deep PBD assumes that the sharp image and blur kernel lie in the ranges of two pretrained generators and minimizes a phaseless measurement-misfit in latent space with alternating gradient descent and latent $|\mathcal A(w,u)|^2=f$8 regularization (Shamshad et al., 2020). The paper reports performance on MNIST, CelebA, and Shoes, under both oversampled Fourier and subsampled Fourier ptychography forward models. This replaces analytical priors such as support, TV, or symmetry with class-specific generator manifolds.

6. Limitations, misconceptions, and open directions

A persistent misconception is that BDPR is simply “blind deblurring plus phase retrieval.” The literature indicates a more heterogeneous picture. Some problems are blind because both factors are unknown vectors in known random subspaces; some are blind because an unknown probe modulates an unknown object; some are blind because a physical blur parameter such as $|\mathcal A(w,u)|^2=f$9 is unknown; and some are made identifiable only by additional autocorrelations, background information, or coded apertures (Miqueles et al., 2017). The commonality lies in joint recovery under phaseless data, not in a single universal forward model.

Another misconception is that convex theory has resolved BDPR in general. The strongest exact-recovery results rely on restrictive side conditions: random Gaussian subspaces with $y = |F(w\circledast x)|$0, deterministic correlation designs with no common factors, or sufficient known background information (Ahmed et al., 2019). The survey literature explicitly notes that uniqueness and convergence theory remain subtle for realistic blind ptychographic and Fourier ptychographic models, especially under noise, probe drift, partial coherence, or scan pathologies (Chang et al., 2022).

Hardware assistance improves conditioning but introduces its own tuning problems. In coded-aperture single-shot blind deconvolution, successful experimental reconstruction occurs for transmittance ratios between $y = |F(w\circledast x)|$1 and $y = |F(w\circledast x)|$2; lower transmittance leads to photon-starved noisy measurements, whereas higher transmittance weakens the support constraint (Muneta et al., 2022). In background-assisted phase retrieval, decreased background length improves practicality, but the theory and algorithms still depend on sufficient known context and on the interaction between measurement noise and background perturbation (Yuan et al., 2023). In blind ptychography, extra phaseless measurements of the probe can suppress periodic artifacts, but obtaining them requires additional calibration (Chang et al., 2018).

The main open directions identified in the survey include stronger convergence theory for blind problems, sharper identifiability results beyond idealized random or subspace assumptions, more noise-robust formulations, scalable algorithms for large 2D and 3D imaging, and better handling of experimental artifacts such as probe drift, background scattering, and partial coherence (Chang et al., 2022). The recent tensor-landscape analysis suggests one possible path: exposing structured low-rank tensor geometry and proving local linear convergence for Riemannian algorithms under tractable surrogate models (Liang et al., 13 Sep 2025). A plausible implication is that future BDPR theory will continue to alternate between physically faithful formulations and analytically simpler surrogates, with hardware design, side information, and optimization geometry treated as inseparable parts of the same inverse problem.