Coded Diffraction Patterns: Models & Reconstruction

• Coded Diffraction Patterns (CDP) are a technique that applies known modulations to an object to generate structured, intensity-only diffraction measurements for phase retrieval.
• They replace single ambiguous Fourier measurements with multiple quadratic observations, allowing for advanced recovery methods like PhaseLift and alternating-projection algorithms.
• CDP frameworks extend to various propagation models, including Fresnel and one-bit schemes, enhancing resolution and robustness in imaging and crystallography.

Coded diffraction patterns (CDPs) are phaseless diffraction measurements acquired after a known modulation is applied to an object before Fourier or Fresnel propagation. In the canonical discrete formulation, for an unknown $x\in\mathbb C^n$, DFT rows $f_k^*$, and diagonal masks $D_\ell$, the data take the form

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$

so each mask produces a distinct intensity-only diffraction pattern of a modulated object. This measurement design replaces a single highly ambiguous Fourier-magnitude observation by a structured family of quadratic measurements, and it underlies convex lifting, alternating-projection, spectral, learning-based, and photonic optimization approaches to phase retrieval (Candes et al., 2013).

1. Canonical model and principal variants

In the basic CDP model, one begins with a complex-valued object

$x=\{x[t]\}_{t=0}^{n-1}\in\mathbb C^n$

and modulates it by a known waveform $d[t]$, or equivalently by a diagonal matrix $D=\operatorname{diag}(d[0],\dots,d[n-1])$. A single coded diffraction pattern is then the Fourier magnitude of the modulated object,

$$y_k=\left|\sum_{t=0}^{n-1} x[t]\bar d[t] e^{-i2\pi \omega_k t}\right|^2 = |f_k^*D^*x|^2,$$

and the multi-mask model used for theory and reconstruction is

$$y_{\ell,k}=\left|\sum_{t=0}^{n-1}x[t]\bar d_\ell[t]e^{-i2\pi kt/n}\right|^2,\qquad 0\le k\le n-1,\ 1\le \ell\le L.$$

The total number of scalar measurements is $m=nL$. A single uncoded diffraction pattern is recovered as the special case $f_k^*$0 (Candes et al., 2013).

A broader formulation replaces the raw intensities by distorted or nonlinear measurements,

$f_k^*$1

where $f_k^*$2 is a possibly unknown scalar nonlinearity. In the one-bit CDP model, masks are used in pairs, and only the sign of the difference between two coded patterns is retained: $f_k^*$3 This formulation preserves order statistics rather than intensity values and is central to robustness, super-resolution, and blind-deconvolution results derived for CDPs under nonlinear distortions and diffraction limits (Mroueh, 2014).

The term CDP also covers propagation models beyond the far-field DFT. In coherent diffractive imaging, randomly coded masks can be placed in a near-field Fresnel geometry, yielding measurements of the form

$f_k^*$4

with $f_k^*$5 the Fresnel propagator. By contrast, some learning-based formulations retain the same physical acquisition but optimize against Fourier amplitudes,

$f_k^*$6

because amplitude-based losses were found easier for recovery, while explicitly noting that the physical measurement is $f_k^*$7 (Seaberg et al., 2015, Cai et al., 2020).

2. Identifiability and uniqueness

The motivation for CDP is the non-injectivity of single-pattern Fourier phase retrieval. Intensity-only Fourier data arise naturally in X-ray crystallography and coherent diffraction imaging, but with a single diffraction pattern the inverse problem is nonconvex and highly ambiguous. In the PhaseLift formulation of Candès, Li, and Soltanolkotabi, random masks make the feasible set

$f_k^*$8

collapse to the unique point $f_k^*$9 with high probability once the number of coded diffraction patterns obeys

$D_\ell$0

yielding exact recovery up to a global phase and total measurement complexity $D_\ell$1 (Candes et al., 2013).

That polylogarithmic requirement was sharpened almost immediately. Gross, Krahmer, and Kueng proved that $D_\ell$2 independent diffraction patterns suffice for PhaseLift recovery in the same random-mask Fourier setting, improving the $D_\ell$3 bound. Their analysis relies on near-isotropicity of the CDP measurement ensemble, robust injectivity on the tangent space $D_\ell$4, and an improved golfing-scheme construction of dual certificates (Gross et al., 2014).

Uniqueness can also be expressed in nonconvex fixed-point language. For two oversampled coded diffraction patterns generated by independent random phase masks, Fannjiang and Strohmer showed that the corresponding Fourier-domain Difference Map family—encompassing Fourier-domain Hybrid-Projection-Reflection and Fourier-domain Douglas–Rachford—has a unique fixed point in the object domain, up to global phase, for generic complex objects in dimension $D_\ell$5. In a related alternating-projection analysis, one oversampled coded diffraction pattern plus a real or nonnegative object constraint is sufficient for uniqueness almost surely, while two oversampled coded diffraction patterns suffice for generic complex objects without object-domain constraints (Fannjiang, 2014, Chen et al., 2015).

The same logic extends to tomography. Under the Born and projection approximations for a discrete $D_\ell$6 object, standard computed tomography with full projected field measurements requires $D_\ell$7 projections for uniqueness, while tomographic phase retrieval from coded projected diffraction patterns requires $D_\ell$8 projection directions for unique determination up to a global phase factor. The extra projection removes a residual degeneracy that survives after random-mask phase retrieval is reduced to projection consistency (Fannjiang, 2021).

3. Reconstruction frameworks

The foundational convex framework is PhaseLift. By lifting $D_\ell$9 to

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$0

one rewrites CDP measurements as linear functionals,

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$1

The nonconvex rank constraint is then relaxed through trace minimization,

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$2

or through regularized least squares

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$3

which was implemented via an interior-type subgradient optimization method of Auslender and Teboulle (2006) in TFOCS for the original CDP numerics (Candes et al., 2013).

Convex relaxations were also used in the first experimental CDI demonstration with randomly coded masks. There, PhaseCut supplied an initial estimate and a variation on Fienup’s input-output algorithm refined it. The crucial point was that the algorithm no longer required the usual object-space phase constraints or support constraints; instead, it exploited consistency across multiple known masks and their associated diffraction measurements (Seaberg et al., 2015).

Projection-based and fixed-point methods form a second major family. In Fourier-domain Difference Map methods, one alternates between the projector onto the object-consistent Fourier subspace,

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$4

and the projector onto the magnitude constraint,

$y_{\ell,k} = |f_k^* D_\ell^* x|^2,$5

through a three-parameter map [ \mathcal D = I + \beta\Delta,\qquad \Delta=P_O\big((1+\gamma_2)P_M-\gamma_2I\big)-P_M\big