Holographic Latent Mean Squared Error

• The paper introduces holographic latent MSE as an indirect reconstruction error measured after transformations like diffraction, temporal multiplexing, and quantisation.
• It reveals that in display holography, variance reduction via temporal averaging does not eliminate a non-zero asymptotic bias floor, affecting perceived replay intensity quality.
• Different formulations in HMIMO and beamforming highlight that local control errors may misalign with physically relevant replay-space fidelity, prompting reconstruction-aware optimisations.

“Holographic latent mean squared error” is not a standardized term in the cited literature. As an Editor’s term, it denotes a mean-squared reconstruction error that is not most naturally expressed at the local hologram-parameter level, but instead emerges after diffraction, temporal multiplexing, quantisation, or channel formation in a holographic system. In display holography, the clearest formal instance is the non-vanishing replay-intensity MSE floor in time-multiplexed One-Step Phase-Retrieval (OSPR), where variance decreases with the number of subframes but MSE converges to a non-zero bias term (Christopher et al., 2019). Closely related uses appear in reconstruction-aware hologram quantisation, where the relevant error is the propagated replay-space distortion rather than local hologram-space quantisation distance (Christopher et al., 2020). By contrast, in holographic communications the term MSE usually retains its standard estimation or beamforming meaning, as in self-supervised MMSE channel estimation for holographic MIMO and WMMSE reformulations for reconfigurable holographic surfaces (Yu et al., 2023, Sheemar et al., 21 Mar 2025).

1. Terminological scope and taxonomic distinctions

The phrase is best understood as covering several distinct technical objects that share a common pattern: the physically or operationally relevant error is measured in a domain different from the one in which the hologram, beamformer, or estimator is directly manipulated. In display CGH, that domain is typically the replay-plane intensity. In communication settings, it is usually a Bayesian channel-estimation loss or a per-user symbol-recovery loss.

Paper	Holographic setting	MSE notion
(Christopher et al., 2019)	Time-multiplexed OSPR displays	Replay-intensity MSE with non-zero asymptotic bias floor
(Christopher et al., 2020)	CGH quantisation	Propagated replay-intensity MSE induced by hologram quantisation
(Christopher et al., 2020)	Holographic search algorithms	Phase-insensitive replay-plane amplitude-magnitude MSE
(Yu et al., 2023)	Holographic MIMO	Bayesian channel-estimation MMSE
(Sheemar et al., 21 Mar 2025)	Reconfigurable holographic surface beamforming	Per-user MSE/WMMSE for sum-rate maximization

This taxonomy matters because the word “latent” can otherwise be misleading. In the display papers, it refers most naturally to an indirect or propagated reconstruction error. In the HMIMO paper, the low-dimensional structure appears only in PCA-based SNR estimation rather than in a latent-variable generative model of the channel (Yu et al., 2023). In the beamforming paper, the auxiliary receive filters and MSE weights are optimization devices, not latent variables in the machine-learning sense (Sheemar et al., 21 Mar 2025).

2. OSPR as the canonical display-holography formulation

Christopher and Wilkinson study OSPR for real-time holographic displays under time multiplexing, where many low-quality holograms are displayed rapidly and the eye temporally integrates their replay intensities (Christopher et al., 2019). In each subframe, a target replay field $T$ is assigned random phase, back-propagated by an inverse Fourier transform, quantized, and displayed: $$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$

The paper uses the unitary DFT convention and invokes Parseval energy conservation between hologram and replay planes. The relevant image-quality quantity is not the complex field itself but the reconstructed intensity $R_{u,v}\overline{R_{u,v}}$, compared against the target intensity $T_{u,v}\overline{T_{u,v}}$. That distinction is central: the dominant analysis is in the image-plane intensity domain, whereas hologram generation acts on complex amplitudes.

For one subframe, the replay intensity is modeled as

$$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$

with $\epsilon'_{u,v}$ a complex circularly symmetric noise term. For $N$ subframes, the perceived intensity is the temporal average,

$$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\frac{1}{N}\sum_{n=1}^N \epsilon'_{n,u,v}.$$

Under i.i.d. assumptions across frame index and pixels, the Central Limit Theorem yields an approximately bivariate Gaussian averaged error with

$$\mu_\epsilon=\mu_{\epsilon'},\qquad \sigma_\epsilon^2=\frac{\sigma_{\epsilon'}^2}{N}.$$

This is the formal variance law for time multiplexing in OSPR: temporal noise variance decays as $1/N$. The paper’s contribution is to show that this variance law is not the same thing as the behavior of perceptually relevant reconstruction error.

3. Non-vanishing asymptotic MSE and the bias floor

The paper defines replay-intensity MSE as the mean squared difference between reconstructed and target intensities: $$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$0 Its central analytical statement is

$$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$1

Equivalently, if the measured curves are fit empirically as

$$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$2

then

$$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$3

Hence

$$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$4

This is the most precise source-equivalent formulation of a holographic “latent MSE” floor: the random fluctuation component averages away, but a deterministic asymptotic mismatch remains (Christopher et al., 2019). The paper explicitly argues that it is tempting to identify MSE with variance when $$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$5, but that “this does not follow” because the system is non-linear. The concluding explanation is explicit: variance does not converge to zero in tandem with MSE because intensity depends quadratically on amplitude, and replay quality is judged in the intensity domain rather than the complex-amplitude domain.

The same theme appears in the SSIM analysis. Using the standard local-window SSIM decomposition, the paper approximates

$$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$6

which implies

$$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$7

The abstract states that SSIM “converges quadratically to a non-unitary value.” A cautious reading is appropriate here: the printed approximation is rational in $$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$8, whereas the asymptotic conclusion that SSIM saturates below $$R'_{u,v}=T_{u,v}\angle \mathrm{Rand}[0,2\pi],\qquad H=\mathcal{F}^{-1}\{R'\},\qquad H'_n=\mathrm{Quantise}(H).$$9 is directly supported by the displayed equations.

4. Bias sources, scene statistics, and empirical structure

The paper identifies two bias sources in OSPR. The first is conjugate-symmetry bias, which arises on binary devices when replay fields are constrained by conjugate symmetry. The cited relation

$R_{u,v}\overline{R_{u,v}}$0

couples nominally distinct replay pixels and invalidates the i.i.d. assumptions used in the variance derivation. This produces a deterministic mismatch unless the target itself has the required symmetry. The paper notes that this term is expected to be zero for rotationally symmetric target images (Christopher et al., 2019).

The second and more general source is intensity-distribution bias, denoted $R_{u,v}\overline{R_{u,v}}$1. The argument is distributional rather than purely variance-based. Replay-amplitude noise may average down, but intensity is a squared magnitude, so averaging random complex-amplitude realizations does not, in general, produce unbiased intensity. Under the assumption of circularly symmetric Gaussian error, the replay-intensity distribution becomes Rician; the paper’s printed PDF is garbled, but the stated point is that the replay-intensity law is not generically Gaussian and varies with signal-to-noise ratio.

The formal bias expression is written as a double integral over the target and replay intensity distributions: $R_{u,v}\overline{R_{u,v}}$2

A notable consequence is scene-statistics dependence. The reported values show that constant-amplitude images have almost no asymptotic bias, whereas uniform-amplitude images have large bias; Mandrill and Peppers, with more centralized amplitude distributions, have smaller bias than uniform random images. The paper reports the following measured values:

Amplitude distribution	Measured $R_{u,v}\overline{R_{u,v}}$3	Measured $R_{u,v}\overline{R_{u,v}}$4
Uniform	$R_{u,v}\overline{R_{u,v}}$5	$R_{u,v}\overline{R_{u,v}}$6
Constant	$R_{u,v}\overline{R_{u,v}}$7	$R_{u,v}\overline{R_{u,v}}$8
Mandrill	$R_{u,v}\overline{R_{u,v}}$9	$T_{u,v}\overline{T_{u,v}}$0
Peppers	$T_{u,v}\overline{T_{u,v}}$1	$T_{u,v}\overline{T_{u,v}}$2

Under the model $T_{u,v}\overline{T_{u,v}}$3, these imply highly unequal asymptotic floors. For uniform amplitude the floor is approximately $T_{u,v}\overline{T_{u,v}}$4, whereas for constant amplitude it is essentially zero. The Mandrill experiment on a $T_{u,v}\overline{T_{u,v}}$5 image further reports $T_{u,v}\overline{T_{u,v}}$6 and $T_{u,v}\overline{T_{u,v}}$7, averaged over $T_{u,v}\overline{T_{u,v}}$8 independent runs with error bars of two standard deviations. This empirical agreement with the $T_{u,v}\overline{T_{u,v}}$9 form is the paper’s main validation of a persistent bias-driven error floor.

5. Reconstruction-aware quantisation and search-based reductions of indirect MSE

A second major use of the underlying idea appears in “Sympathetic quantisation -- a new approach to hologram quantisation” (Christopher et al., 2020). That paper argues that nearest-neighbour quantisation is suboptimal because hologram pixels are not independent under diffraction: a local hologram-plane perturbation changes every replay pixel. For a one-pixel change,

$$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$0

This establishes a global coupling between quantisation choices and replay-space distortion. The paper therefore treats the physically relevant criterion as replay-plane intensity MSE,

$$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$1

rather than local hologram-space quantisation distance. Its soft sympathetic quantisation (SSQ) uses rotationally symmetric pixel pairs $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$2, narrow-band target-phase randomisation based on a dual von Mises distribution, and pairwise geometric projection onto the unit circle. For the considered single-transform case, it reports that MSE is reduced to under $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$3 of traditional approaches and that SSIM improves by more than $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$4, with FFT cost still dominating runtime at $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$5 and SSQ overhead $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$6.

A different but related response appears in “Improving Holographic Search Algorithms using Sorted Pixel Selection” (Christopher et al., 2020). There the optimized quantity is not intensity MSE but a phase-insensitive replay-plane amplitude-magnitude MSE,

$$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$7

The proposed Sorted Pixel Selection (SPS) modification does not change the objective or the acceptance rule of Direct Search or Simulated Annealing. Instead, it replaces random candidate-pixel selection with a deterministic ordering based on the magnitude of the quantization-induced change after back-projection. Across more than $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$8 tests, the paper reports MSE reductions in the range $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\epsilon'_{u,v},$$9, including $\epsilon'_{u,v}$0 improvement after $\epsilon'_{u,v}$1 iterations on a $\epsilon'_{u,v}$2 Mandrill image with a binary phase SLM and $\epsilon'_{u,v}$3 improvement on a $\epsilon'_{u,v}$4 rotationally symmetric text target after $\epsilon'_{u,v}$5 direct-search iterations.

Taken together, these two papers suggest two complementary operationalizations of indirect or propagated holographic MSE. Sympathetic quantisation changes the quantisation rule so that propagated replay error is less harmful. SPS changes the search schedule so that iterations are spent first on hologram pixels whose quantization displacement is most strongly associated with replay-field MSE. The MSE definitions differ across the papers, but both reject the assumption that local hologram-domain proximity is the right fidelity criterion.

6. MMSE in holographic communications: related, but conceptually distinct

In holographic communication systems, MSE usually enters in standard estimation-theoretic form rather than as a replay-space artifact. “Learning Bayes-Optimal Channel Estimation for Holographic MIMO in Unknown EM Environments” formulates HMIMO channel estimation as minimizing

$\epsilon'_{u,v}$6

whose minimizer is the posterior mean $\epsilon'_{u,v}$7 (Yu et al., 2023). For the additive Gaussian pilot model, the paper derives Tweedie’s formula,

$\epsilon'_{u,v}$8

and learns the score $\epsilon'_{u,v}$9 self-supervisedly from pilots alone using denoising score matching. Its low-dimensional element is not a latent channel model but a PCA-based SNR estimator exploiting rank deficiency in dense holographic arrays. The paper states that the resulting method approaches oracle MMSE NMSE with extremely low complexity and reports, for $N$0, about $N$1 ms runtime on an Intel Core i7-9750H CPU versus about $N$2 ms for oracle MMSE.

“Minimum Mean Squared Error Holographic Beamforming for Sum-Rate Maximization” uses MSE in yet another orthodox sense: symbol-recovery error in a reconfigurable holographic surface downlink (Sheemar et al., 21 Mar 2025). With receive scalar $N$3 and user estimate $N$4, the per-user loss is

$N$5

The paper exploits the WMMSE equivalence and, after writing the holographic beamformer as

$N$6

shows that the weighted sum-MSE is quadratic in each real holographic amplitude $N$7: $N$8 This yields the closed-form scalar update $N$9, followed by projection onto $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\frac{1}{N}\sum_{n=1}^N \epsilon'_{n,u,v}.$$0. The paper gives per-iteration complexity

$$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\frac{1}{N}\sum_{n=1}^N \epsilon'_{n,u,v}.$$1

which is linear in the RHS size $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\frac{1}{N}\sum_{n=1}^N \epsilon'_{n,u,v}.$$2.

These communication papers are relevant because they show that “holographic MSE” is not a single construct. In display holography, the latent aspect is a propagated replay-space distortion or an asymptotic perceptual error floor. In HMIMO and RHS beamforming, MSE retains its customary statistical meaning.

7. Boundaries, misconceptions, and significance

A common misconception is that time multiplexing in OSPR should eventually eliminate error because it eliminates variance. The statistical model in (Christopher et al., 2019) refutes that identification directly: variance follows $$R_{u,v}\overline{R_{u,v}}=T_{u,v}\overline{T_{u,v}}+\frac{1}{N}\sum_{n=1}^N \epsilon'_{n,u,v}.$$3, but MSE contains an additional bias term, so the limiting error is generically non-zero. Another misconception is that any holographic paper invoking MMSE is therefore about the same phenomenon. The HMIMO and RHS beamforming papers show otherwise: they concern Bayesian estimation and WMMSE optimization, not replay-plane bias floors or propagated quantisation error (Yu et al., 2023, Sheemar et al., 21 Mar 2025).

A second boundary concerns the word “latent.” The display-quantisation paper is the most natural source for that interpretation because it explicitly distinguishes direct hologram-domain quantisation error from propagated reconstruction-domain error (Christopher et al., 2020). By contrast, the HMIMO paper explicitly does not use a latent-variable generative model of the channel; its “low-dimensional” structure appears only in PCA-based SNR estimation. This suggests that “latent MSE” is a useful editorial umbrella only when it is tied to an indirect, transform-mediated, or hidden-domain fidelity criterion.

The broader significance is methodological. Across the display papers, the central lesson is that local optimization in hologram space can be misaligned with the quantity that actually determines visible quality. OSPR reveals a non-zero asymptotic bias floor in replay-intensity MSE. Sympathetic quantisation replaces nearest-neighbour projection with a reconstruction-aware rule that exploits diffraction coupling. Sorted Pixel Selection improves search efficiency by prioritizing pixels whose quantization-induced change is most strongly associated with replay-plane error. These developments do not define a single formal theory of “holographic latent mean squared error,” but they do define a coherent research theme: holographic system design must often be judged by an indirect MSE in the physically reconstructed domain rather than by a local error in the variable space where control is applied.