Deep Learning-based Measurement Matrix for Phase Retrieval

• The paper introduces a learnable Fourier measurement operator within an unrolled deep architecture that jointly optimizes sensing and recovery for phase retrieval.
• It employs iterative subgradient descent and learned proximal mappings to enforce data fidelity and sparsity, thereby improving reconstruction accuracy.
• Experiments show that DLMMPR consistently outperforms baseline methods in PSNR and SSIM, especially under high noise conditions.

The Deep Learning-based Measurement Matrix for Phase Retrieval (DLMMPR) framework fundamentally redefines phase retrieval by jointly optimizing the measurement operator and recovery modules in an end-to-end deep learning architecture. By parameterizing the multiplexed Fourier measurement process as learnable tensors and coupling this with unrolled subgradient descent and learned proximal mappings, DLMMPR achieves robust signal recovery under diverse noise conditions and outperforms prevailing state-of-the-art methodologies such as DeepMMSE and PrComplex in both quantitative and qualitative dimensions (Liu et al., 16 Nov 2025).

1. Measurement Matrix Parameterization

DLMMPR departs from traditional fixed coded-diffraction pattern (CDP) designs, which utilize a stack of masked Fourier transforms, by embedding the entire multi-mask Fourier measurement operator as a learnable parameter within the deep network. The conventional CDP operator is given by

$$y = |\mathbf{A}x| + w,\quad \mathbf{A} = [(F D_1)^\top;\, (F D_2)^\top;\, (F D_3)^\top;\, (F D_4)^\top]$$

where $F$ is the discrete Fourier matrix, $D_j$ are diagonal matrices with entries sampled on the complex unit circle, and $w$ denotes noise. DLMMPR replaces the fixed $F$ with a trainable matrix $T$, such that at the $k$-th stage:

$$W^{k} = [(T^k D_1)^\top;\, (T^k D_2)^\top;\, (T^k D_3)^\top;\, (T^k D_4)^\top]$$

Both $W^{k}$ and its adjoint $(W^{k})^\top$ are treated as weight tensors, initialized to $F$, and updated via backpropagation with all other network parameters. This learnable parameterization facilitates joint optimization of the sensing process and iterative recovery procedure.

2. Unrolled Recovery Structure

DLMMPR utilizes K=7 unrolled ISTA-like iterations, each comprising a learnable subgradient descent (SGD) module and a learned proximal-projection mapping (PPM). At each stage $k$:

• Subgradient-Descent (SGD) Module: Enforces data fidelity using the non-differentiable cost

$f(x) = \frac{1}{2}\|\,|W x| - y\,\|_2^2$

with subgradient

$$\partial f(x) \ni W^\top[W x - y \odot (W x / |W x|)]$$

and update

$$r^k = x^{k-1} - t^k (W^{k-1})^\top\left[ W^{k-1}x^{k-1} - y \odot \frac{W^{k-1}x^{k-1}}{|W^{k-1}x^{k-1}|} \right]$$

• Proximal-Projection Mapping (PPM) Module: Enforces sparsity in a learned transform domain $\Psi(\cdot)$, parameterized by a CNN block $\mathcal{F}^k$ and its left-inverse $\mathcal{G}^k = (\mathcal{F}^k)^{-1}$.

Soft-thresholding is applied to sparse codes:

$$z^k = \mathcal{F}^k(r^k), \quad \hat{z}^k = \mathrm{soft}(z^k,\, \eta^k), \quad x^k = r^k + \mathcal{G}^k(\hat{z}^k)$$

Thus, each stage combines measurement consistency (SGD) and learned sparsification (PPM) in a hybrid iterative framework.

3. End-to-End Training and Optimization

Supervised training employs batches of image patches $\{x_i\}$ and simulated measurements $y_i = |W_\star x_i| + w_i$, where $W_\star$ are the initial CDP masks. The objective is to minimize the mean squared-error:

$$\mathcal{L} = \frac{1}{N_s N} \sum_{i=1}^{N_s}\| x_i^K - x_i \|_2^2$$

Gradients with respect to all parameters $$\{W^k,\,(W^k)^\top,\,t^k,\,\eta^k,\,\mathcal{F}^k,\,\mathcal{G}^k\}$$ are computed via backpropagation and updated using the Adam optimizer. DLMMPR simultaneously fits recovered signals to ground truth, enforcing both measurement-model consistency and accurate reconstruction through learned sparsity in transform domains.

4. Network Architecture and Training Protocol

• Unrolled Stages: Seven (K=7), no weight sharing across stages.
• PPM Modules: Each stage’s transform $\Psi$ comprises:
  1. $3 \times 3$ convolution,
  2. Convolutional Block Attention Module (CBAM),
  3. ConvU (three $3 \times 3$ convolution+ReLU layers),
  4. Fourier Residual Block (ResFBlock).

The inverse chain $\mathcal{G}^k$ mirrors these sub-modules in reverse, wrapped in a residual skip connection.

• Initialization: $W^k$ and $(W^k)^\top$ from Fourier operator $F$, Xavier initialization for CNN weights, thresholds $\eta^1 = 0.5$, step sizes $t^1 = 0.01$.

• Training: PyTorch-based, Adam optimizer ($\beta_1 = 0.9$, $\beta_2 = 0.999$), initial $lr=1\times10^{-3}$, batch size 10, 150 epochs. Input patches are $128 \times 128$, normalized to $[0,1]$.
• Datasets: 6,000 training patches from BSD400 and PASCAL-VOC. Testing on prDeep “Set12”: UNT-6 and NT-6. Poisson noise levels $\alpha\in\{9,27,81\}$.

5. Quantitative Evaluation and Empirical Performance

DLMMPR's performance is benchmarked against DeepMMSE and PrComplex (Liu et al., 16 Nov 2025). The quantitative results for Poisson noise levels $\alpha$ on UNT-6 and NT-6 test datasets are summarized:

Noise ($\alpha$)	DeepMMSE	PrComplex	DLMMPR
9	40.26/0.98, 39.45/0.98	41.11/0.98, 39.78/0.99	41.89/0.98, 40.41/0.99
27	33.03/0.94, 32.32/0.93	34.64/0.93, 33.49/0.94	35.23/0.95, 33.51/0.95
81	26.64/0.81, 25.41/0.78	28.27/0.84, 26.55/0.82	29.70/0.86, 27.44/0.83

Across all regimes, DLMMPR achieves the highest PSNR and SSIM, with typical gains exceeding +1 dB, especially at high noise levels. Visual assessment affirms preservation of sharper edges and finer textures, and superior suppression of speckle noise compared to competing methods.

6. Theoretical Insights, Convergence, and Limitations

DLMMPR is informed by LISTA/ISTA convergence theory and the weight-coupling principle, also leveraged in prior work on unfolded algorithms for phase retrieval (Naimipour et al., 2020). Although explicit proofs are absent, the architecture leverages theoretical rationale from ISTA-Net (Theorem 1) to justify the $\ell_2$-norm approximation in the proximal step. Empirical evidence shows rapid convergence (network saturation at $\sim$150 epochs and 150 unrolled iterations), with stable performance across noise regimes.

DLMMPR's computational overhead scales with image size due to its multi-stage network structure; current instantiation assumes $J=4$ masks. Anticipated future work includes lightweight models and distributed optimization for large-scale imaging, and generalization of learned matrix paradigms to broader inverse problems.

7. Relationship to Prior Unfolded and Deep Measurement Matrix Approaches

Prior frameworks, such as Unfolded Phase Retrieval (UPR) (Naimipour et al., 2020), employ deep unfolding to jointly learn sensing matrices $\mathbf{A}$ and solver parameters via interpretable iterative networks. DLMMPR advances this by generalizing the measurement operator to parameterized Fourier-like transforms and embedding learned proximal mappings via CNN modules. Both approaches advocate joint, end-to-end training that enhances measurement efficiency, convergence rate, and recovery accuracy, substantiating the benefit of data-driven measurement matrix optimization in phase retrieval.

A plausible implication is that learned measurement operators, when coupled with unrolled and interpretable recovery networks, can systematically improve imaging performance even beyond phase retrieval, motivating further methodological exploration in computational imaging and signal processing.