Image Memory Effect in Scattering & Imaging

• Image memory effect is a phenomenon where optical correlations persist under small angular tilts, enabling speckle-correlation imaging in scattering media.
• Selective excitation of transmission eigenchannels can extend the angular memory range, enhancing signal fidelity in both transmission and reflection.
• Generalized memory operators and customized imaging methods allow control over correlation properties with applications in atomic storage, gravitational imaging, and machine learning.

“Image memory effect” denotes a family of phenomena in which an image-like observable retains a reproducible relation to an earlier wavefront, perturbation, or stored excitation. In multiple-scattering optics, the canonical case is the angular memory effect: a small tilt of coherent illumination causes the transmitted speckle pattern to shift laterally rather than decorrelate completely, within a finite angular range. Closely related work shows that this range can be enlarged by selectively exciting transmission eigenchannels, generalized to coupled shift–tilt correlations, or customized to arbitrarily chosen input and output tilt pairs. Other literatures use the same phrase or closely related language for spatially addressable optical image storage, permanent interference changes induced by relativistic transients, black-hole image drift after soft-hair changes, and several distinct memory-like mechanisms in machine learning and image analysis (Yılmaz et al., 2019, 1705.01373, Yılmaz et al., 2020, Clark et al., 2013, Kodwani et al., 2016, Hou et al., 13 Mar 2026, Ouderaa et al., 2019, Shi et al., 2021, Sidorov, 2018).

1. Conventional angular image memory in scattering media

In a diffusive slab, the optical memory effect is the correlation between input and output speckle patterns under a small angular tilt of coherent illumination. If a beam incident on the slab is tilted by a small angle $\theta$, the transmitted wavefront is tilted by the same angle and the far-field speckle pattern shifts laterally rather than changing completely. The phenomenon is described as arising from hidden correlations in the transmission matrix $t$ of a thick diffusive slab: for a slab much wider than it is thick, the transmission matrix is banded in real space and diagonally correlated in spatial-frequency space, and those diagonal correlations produce the angular memory effect (Yılmaz et al., 2019).

The operational importance of the effect is that, within the correlation range, an object’s speckle “fingerprint” remains sufficiently similar under small angular changes to enable wide-field speckle-correlation imaging. The finite correlation range therefore sets the usable scan range and field of view. The correlation width $\theta_0$ is defined from the speckle-intensity correlation function $C(\theta)$ by

$C(\theta_0)=\frac{1}{2}C(0),$

and for random incident wavefronts in the reported experiment the width $\theta_0^{(r)}$ is about $1^\circ$ (Yılmaz et al., 2019).

A complementary formulation characterizes the conventional effect through a field-correlation coefficient,

$$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$

for which the strongest correlation occurs around $\theta_i=\theta_o$. In a diffusive slab of thickness $L$, the conventional angular range is approximately

$t$0

This finite range underlies both the utility and the limitation of conventional memory-effect imaging, scanning through turbid media, and related sensing and communication schemes (Yılmaz et al., 2020).

2. Transmission eigenchannels and enlargement of the angular range

The most direct route to extending the angular memory-effect range in transmission is selective excitation of transmission eigenchannels. These are the eigenvectors $t$1 of the Hermitian matrix $t$2,

$t$3

with $t$4 the corresponding transmission eigenvalue. Physically, $t$5 is the fraction of incident power transmitted through the medium when the input wavefront matches that eigenchannel. High-transmission channels have large $t$6; low-transmission channels have small $t$7 and are strongly attenuated (Yılmaz et al., 2019).

The reported experiment used a ZnO nanoparticle film with thickness $t$8, transport mean free path $t$9, and 532 nm illumination shaped by an SLM. The full field transmission matrix was measured by common-path interferometry in the far field; selected eigenchannel phase fronts were then loaded onto the SLM. The measured angular correlation was obtained by tilting the incident eigenchannel wavefront by $\theta_0$0, recording the transmitted intensity pattern $\theta_0$1, numerically tilting it back by $\theta_0$2, and evaluating the Pearson correlation

$\theta_0$3

The highest-transmission channel yielded

$\theta_0$4

whereas the lowest-transmission channel yielded

$\theta_0$5

Thus the highest-transmission channel provided about a 52% larger angular memory range, while the lowest-transmission channel gave about a 23% smaller one (Yılmaz et al., 2019).

The physical explanation is robustness against tilt perturbations. A tilted eigenchannel input is no longer an eigenvector of $\theta_0$6, so it excites the original eigenchannel together with additional eigenchannels. Those extra contributions behave approximately like a random superposition, causing the output to become a mixture of the original localized eigenchannel response and a more broadly spread random-wavefront response. For a high-transmission channel, the original contribution remains dominant, so decorrelation is slow; for a low-transmission channel, the random background dominates and decorrelation is rapid. A phenomenological model introduces a tilted transmission matrix

$\theta_0$7

and a channel-specific intensity correlation

$\theta_0$8

which is approximated in closed form as a function of $\theta_0$9 and the perturbation strength $C(\theta)$0. The simulations show that $C(\theta)$1 increases with $C(\theta)$2, and that for high-transmission channels the width scales inversely with an effective thickness $C(\theta)$3 including extrapolation lengths (Yılmaz et al., 2019).

The transmission enhancement has a reflection counterpart with the opposite trend. In reflection, the angular memory-effect width is about 7% smaller for the highest-transmission eigenchannel and 6% larger for the lowest-transmission eigenchannel than for random wavefronts. This reversal is explained by replacing $C(\theta)$4 with $C(\theta)$5: a high-transmission channel implies low reflectance, so the reflected field is more sensitive to tilt. A practical implication is that reflection correlations may serve as an indirect signature of coupling into high-transmission channels when transmitted light is inaccessible (Yılmaz et al., 2019).

3. Generalized and customized memory operators

The conventional angular effect is only one slice of a broader correlation structure. The generalized optical memory effect formulates scattering correlations in joint position–angle phase space and shows that the familiar tilt memory effect and the anisotropic shift memory effect are limiting cases of a single framework. In forward-scattering tissue, the generalized correlation contains a shift–tilt cross-term, implying that the optimal scan is achieved by a combined shift and tilt rather than by either operation alone. In the adaptive-optics interpretation reported for forward-scattering slabs, the shift memory effect, the tilt memory effect, and the generalized memory effect correspond to scan ranges $C(\theta)$6, $C(\theta)$7, and $C(\theta)$8, respectively, and the optimal generalized correction plane lies at depth $C(\theta)$9 inside the sample (1705.01373).

A distinct extension replaces passive use of the medium’s natural memory by active design of the correlation itself. The angular memory operator is defined as

$C(\theta_0)=\frac{1}{2}C(0),$0

with chosen input and output tilt angles $C(\theta_0)=\frac{1}{2}C(0),$1 and $C(\theta_0)=\frac{1}{2}C(0),$2. Its eigenvectors satisfy

$C(\theta_0)=\frac{1}{2}C(0),$3

When the transmission matrix is square, the eigenvalue equation implies that the tilted output field is identical to the original output field up to a complex scalar factor, so $C(\theta_0)=\frac{1}{2}C(0),$4 at the chosen $C(\theta_0)=\frac{1}{2}C(0),$5. The customized memory is therefore not restricted to $C(\theta_0)=\frac{1}{2}C(0),$6, to small angles, or to identical tilt directions (Yılmaz et al., 2020).

Experimentally, the method was implemented in a diffusive ZnO nanoparticle layer of about $C(\theta_0)=\frac{1}{2}C(0),$7 thickness. The paper reports a case with $C(\theta_0)=\frac{1}{2}C(0),$8 and $C(\theta_0)=\frac{1}{2}C(0),$9, for which the transmitted pattern remained highly correlated, with $\theta_0^{(r)}$0 in one experiment. The framework also supports multiple designed memories through

$\theta_0^{(r)}$1

which creates a single incident wavefront with two distinct memory pairs. An important limitation is explicit: the customized memory effect does not enlarge the fundamental angular width. Its peak width is essentially the same as that of the conventional memory effect around $\theta_0^{(r)}$2; it moves and duplicates the memory peak rather than broadening it. This distinguishes customization of the correlation location from enlargement of the intrinsic correlation width (Yılmaz et al., 2020).

4. Material image memories and spatially addressable storage

A different usage of image memory effect concerns direct storage of two-dimensional optical information in a material medium. In gradient echo memory (GEM), a weak probe field is mapped into a ground-state spin wave by Raman coupling with a strong control beam. Because the probe image is imprinted onto the transverse structure of the spin wave, transverse intensity modulation can be recovered later. The experiment on “Spatially Addressable Readout and Erasure of an Image in a Gradient Echo Memory” used a $\theta_0^{(r)}$3 cm long $\theta_0^{(r)}$4Rb vapor cell with $\theta_0^{(r)}$5 Torr Kr buffer gas at about $\theta_0^{(r)}$6C, a $\theta_0^{(r)}$7 mm binary mask of the NIST logo, Gaussian $\theta_0^{(r)}$8 probe pulses, Zeeman broadening of the Raman line to about $\theta_0^{(r)}$9 MHz, and typical storage times of $1^\circ$0–$1^\circ$1 (Clark et al., 2013).

The key result is spatially selective access to subregions of the stored image. Three independently addressable read beams, all at the same optical frequency, were imaged into the cell so that different transverse zones could be recalled at different times during a single rephasing cycle. The three zones were read sequentially for about $1^\circ$2 ns each, at $1^\circ$3, $1^\circ$4, and $1^\circ$5. The resulting intensity profile changed from about 0.9 to 0.1 over roughly $1^\circ$6m, matching the optical imaging resolution of the read beams rather than a fundamental GEM limit (Clark et al., 2013).

The same platform also demonstrated spatially addressable erasure. A bright beam near the $1^\circ$7Rb D$1^\circ$8 line at $1^\circ$9 nm, detuned by about $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$0, was focused onto only part of the memory during storage. The beam scattered photons, induced decoherence of the spin wave, and locally deleted the stored image information. For a resolution target with $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$1 line pairs/mm, a selected fringe could be completely removed with negligible impact on nearby fringes. The decoherence rate extracted from fringe visibility was about $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$2, consistent with a best-fit exponential of about $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$3 (Clark et al., 2013).

The principal constraint on persistence and multiplexing capacity is atomic diffusion. Using a measured diffusion coefficient

$$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$4

and a visibility threshold $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$5, the paper estimated a maximum linear channel density

$$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$6

for up to three readouts over $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$7. In this setting, image memory is literal storage, delayed retrieval, and local erasure of a spatial optical pattern encoded in collective atomic coherence (Clark et al., 2013).

5. Permanent image changes in gravitational and black-hole settings

In relativistic settings, image memory can denote a lasting change in an observed interference or image pattern after a transient change in spacetime geometry. For supernova neutrino shells, the effect arises because a shell of relativistic neutrinos changes the surrounding Schwarzschild geometry from mass $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$8 outside the shell to mass $$C_E(\theta_i,\theta_o)\equiv \frac{ \langle \psi|\, t^\dagger X^\dagger(\theta_o)\, t\, X(\theta_i)\,|\psi\rangle }{ \sqrt{ \langle \psi|t^\dagger t|\psi\rangle\; \langle \psi|X^\dagger(\theta_i)t^\dagger t X(\theta_i)|\psi\rangle } },$$9 inside it. In pulsar scintillation, unresolved rays interfere, and a neutrino shell passing near the ray bundle induces slightly different time delays on nearby paths. For two rays separated by $\theta_i=\theta_o$0, the relative delay approaches the permanent asymptotic value

$\theta_i=\theta_o$1

for $\theta_i=\theta_o$2. The observed scintillation pattern therefore retains a permanent record of the supernova; in the paper’s language, a bright spot can become a dark spot over $\theta_i=\theta_o$3 years (Kodwani et al., 2016).

The same work derives interferometric signatures. For a longitudinal arm $\theta_i=\theta_o$4,

$\theta_i=\theta_o$5

while for a transverse arm $\theta_i=\theta_o$6,

$\theta_i=\theta_o$7

The transverse component thus contains a term that grows linearly with time, unlike the usual Christodoulou gravitational memory. The paper emphasizes four distinctions from the Christodoulou effect: the source is a matter shell of neutrinos, the response includes a longitudinal component, the transverse piece contains a linearly growing term, and the mechanism does not require strong spherical-symmetry breaking. Observationally, pulsar scintillation is identified as the most promising channel; by contrast, even a galactic supernova at $\theta_i=\theta_o$8 light-years gives only $\theta_i=\theta_o$9 at $L$0 Hz for LISA- or BBO-like interferometers (Kodwani et al., 2016).

A different gravitational example is the black-hole image memory effect associated with soft hair. For an eternal soft-haired Kerr black hole, the image in the observer’s celestial plane is rotated, dilated, and drifting relative to the bald Kerr image. Rotation and dilation are time-independent, while the drift is a constant-speed motion in a fixed direction. The transformed celestial coordinates are written as

$L$1

with

$L$2

so that $L$3 is a time-independent displacement and $L$4 a constant drift velocity (Hou et al., 13 Mar 2026).

When the black hole emits gravitational or electromagnetic radiation, the soft hair changes, and the image before and after emission drifts along different straight lines; during the emission, the trajectory is curved. That permanent change in drift direction is identified as the black-hole image memory effect. In the small-hair limit, the angular displacement is approximated by

$L$5

For the fiducial intermediate-mass-ratio inspiral with $L$6, $L$7, $L$8, and distance $L$9, the dominant mode is usually $t$00, and the total image-memory shift is about

$t$01

with a last-10-year partial effect of about

$t$02

The paper concludes that detection is extremely difficult with current or near-future VLBI angular resolution, and explicitly notes that the analysis assumes asymptotic flatness and ignores cosmological expansion (Hou et al., 13 Mar 2026).

6. Terminological extensions in machine learning and image analysis

Outside wave physics, “image memory” often denotes computational memory, memorability, or latent memory representations rather than a scattering-induced correlation. These uses are technically distinct, even when the phrase resembles optical memory terminology.

Area	Meaning of “memory”	Reported result
Reversible image-to-image translation	Activation-memory efficiency via approximate invertibility	Additive coupling gives $t$03 spatial complexity in depth (Ouderaa et al., 2019)
Image memorability editing	Human-consistent memorability score of visual content	AttGAN changes memorability by about $t$04 or $t$05 on a 220-image test set (Sidorov, 2018)
Semantic image manipulation	Learned latent texture bank	MIM-Net uses memories $t$06 and a TLU to localize edits (Shi et al., 2021)

In reversible GANs for image-to-image translation, the relevant “memory effect” is memory efficiency. RevGAN uses an approximately invertible core $t$07, so forward activations need not be stored throughout the reversible part of the network. With additive coupling,

$t$08

the inverse is analytically available, and the paper states that intermediate activations do not have to be stored during backpropagation, giving constant spatial complexity $t$09 in terms of layer depth. On Maps, activation memory for unpaired RevGAN remains at $t$10 MiB for depths $t$11 and $t$12, while the CycleGAN baseline grows from $t$13 MiB to $t$14 MiB (Ouderaa et al., 2019).

In image memorability work, the “memory” is human recall rather than persistence of a physical field. Memorability is treated as a continuous score in $t$15, estimated with AMNet and edited with GANs or conventional photo-processing operations. In the reported AttGAN experiment on faces, increasing memorability produced a mean change of about $t$16, decreasing memorability a mean change of about $t$17, and at least 95% of the test images changed in the intended direction under AMNet evaluation. Human validation with a Memory Game on Amazon Mechanical Turk supported the same trend for about 92% of images. Among common editing tools, sharpening was the only operation that showed a stable positive effect overall, with mean change $t$18 (Sidorov, 2018).

In semantic image manipulation with memory, memory denotes a learned latent texture bank rather than a persistent observation. MIM-Net defines a memory set

$t$19

uses word-level attention to retrieve texture information from those slots, and combines it with a Target Localization Unit

$t$20

to confine edits to the region mentioned by the text. The method adds a reconstruction stage, a pseudo ground-truth feature loss, and a randomized memory training loss so that every memory slot becomes decodable into a realistic image. Here the central claim is not persistence under perturbation, but reuse of stored texture representations for semantically guided image synthesis (Shi et al., 2021).

This spread of meanings suggests a useful boundary. In scattering optics, gravitational imaging, and atomic storage, image memory refers to a persistent correlation, a permanent displacement, or a recoverable stored spatial pattern. In machine learning, the same language usually refers instead to activation storage, memorability as an image attribute, or latent-memory retrieval. Conflating these senses obscures the core physical distinction between medium-induced image correlation and algorithmic or perceptual notions of memory.