Differential Fourier-Domain Regularized Inversion (D-FDRI)

• D-FDRI is a reconstruction framework for single-pixel imaging that uses a differential operator combined with Fourier-domain regularization to suppress biases and recover high-quality images.
• The method leverages a precomputed reconstruction matrix, decoupling computational cost from resolution, which enables real-time imaging even under extreme compression.
• It achieves superior performance with improved PSNR and SSIM compared to traditional compressive sensing methods, making it effective for applications like dual-channel and polarization-sensitive imaging.

Differential Fourier-Domain Regularized Inversion (D-FDRI) is a closed-form reconstruction framework for highly compressive, real-time single-pixel imaging (SPI). By integrating a differential operator to suppress measurement bias with sophisticated Fourier-domain regularization, D-FDRI enables faithful image recovery at extreme compression ratios and high resolution from binary, non-adaptive patterns. The technique leverages a single, precomputed reconstruction matrix, thereby decoupling computational cost from image resolution at runtime and effectively addressing the speed–quality bottleneck inherent to traditional SPI and compressive sensing solvers (Stojek et al., 2022, Pastuszczak et al., 2022, Czajkowski et al., 2018).

1. Mathematical Formulation and Inverse Problem

D-FDRI treats SPI as an underdetermined linear inverse problem. The physical measurement corresponds to

$$\mathbf{y}_0 = \mathbf{M} (\mathbf{x}+\mathbf{n}_s) + \mathbf{n}_d,$$

where $\mathbf{x} \in \mathbb{R}^n$ is the image of interest, $\mathbf{M} \in \{0,1\}^{k \times n}$ is the sampling matrix comprising binary DMD patterns, $\mathbf{n}_s$ is a signal-dependent noise (e.g., background light), and $\mathbf{n}_d$ is detector noise. To minimize systematics from unknown DC offset and to encode both positive and negative weights, the measurement vector is transformed using a finite-difference operator,

$$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$

defining the effective forward operator $\mathbf{A} := \mathbf{D} \mathbf{M}$ and noise $$\eta := \mathbf{D} \mathbf{M} \mathbf{n}_s + \mathbf{D} \mathbf{n}_d$$. The reconstruction target is then to solve

$\mathbf{y} = \mathbf{A} \mathbf{x} + \eta.$

The differential operator $\mathbf{D}$ can be first or higher order and is constructed such that $\mathbf{x} \in \mathbb{R}^n$0 is invariant to additive or even affine offsets, providing robustness against baseline drift (Stojek et al., 2022, Pastuszczak et al., 2022).

2. Differential Measurement and Regularization in the Fourier Domain

A defining feature of D-FDRI is its hybrid data-fidelity and regularization cost function, imposed in the frequency domain:

$\mathbf{x} \in \mathbb{R}^n$1

where $\mathbf{x} \in \mathbb{R}^n$2 is the (unitary) 2D discrete Fourier transform and $\mathbf{x} \in \mathbb{R}^n$3 is a diagonal spectral filter. The regularizer penalizes both high spatial frequencies and, via adjustable weighting, large gradients (as in TV-like priors). Specifically, for each spatial frequency $\mathbf{x} \in \mathbb{R}^n$4,

$\mathbf{x} \in \mathbb{R}^n$5

where $\mathbf{x} \in \mathbb{R}^n$6 trades off between finite-difference and pure-frequency penalties and $\mathbf{x} \in \mathbb{R}^n$7 is a small constant to avoid singularities. The typical value $\mathbf{x} \in \mathbb{R}^n$8 balances edge preservation and noise suppression, and $\mathbf{x} \in \mathbb{R}^n$9 ensures numerical stability (Stojek et al., 2022, Pastuszczak et al., 2022, Czajkowski et al., 2018).

3. Closed-form Solution and Computational Strategy

D-FDRI bypasses iterative solvers by leveraging a "one-shot" closed-form estimator. The unique minimum of the cost function is achieved by precomputing the reconstruction matrix

$\mathbf{M} \in \{0,1\}^{k \times n}$0

where $\mathbf{M} \in \{0,1\}^{k \times n}$1 denotes the Moore–Penrose pseudoinverse. The reconstruction proceeds as follows:

• Offline (once per mask set):

1. Compute $\mathbf{M} \in \{0,1\}^{k \times n}$2
2. Form $\mathbf{M} \in \{0,1\}^{k \times n}$3
3. Compute $\mathbf{M} \in \{0,1\}^{k \times n}$4 (via SVD or regularized QR)
4. Assemble $\mathbf{M} \in \{0,1\}^{k \times n}$5

• Online (per frame):

1. Acquire measurements $\mathbf{M} \in \{0,1\}^{k \times n}$6
2. Compute differential $\mathbf{M} \in \{0,1\}^{k \times n}$7
3. Recover image $\mathbf{M} \in \{0,1\}^{k \times n}$8
4. Apply non-negativity (e.g., $\mathbf{M} \in \{0,1\}^{k \times n}$9)

This structure yields per-frame complexity $\mathbf{n}_s$0 (for $\mathbf{n}_s$1 measurements and $\mathbf{n}_s$2 pixels), enabling real-time operation at frame rates up to 17 Hz for $\mathbf{n}_s$3 resolution and $\mathbf{n}_s$4 s for $\mathbf{n}_s$5 at extreme compression (Stojek et al., 2022, Pastuszczak et al., 2022, Czajkowski et al., 2018).

4. Sampling Patterns and System Design

D-FDRI is agnostic to the choice of sampling patterns, supporting random binary, Hadamard, DCT-based, or Morlet-wavelet–derived masks. The method achieves higher entropy in detection traces by randomized thresholding of DCT patterns or explicit partitioning (for DC-probing) and can accommodate non-adaptive, binary DMD constraints efficiently. The differential nature of the scheme suppresses DC bias and enhances A/D tolerance, facilitating the use of low bit-depth (8–16 bit) AC-coupled digitizers (Pastuszczak et al., 2022, Stojek et al., 2022).

Optical implementations typically utilize high-speed DMDs (e.g., Vialux V-7001 with DLP7000 at 22.7 kHz) and support multi-channel detection (polarization, VIS–IR dual-band) without modification to the core algorithm. Pattern design accommodates requirements for DC recovery via auxiliary matrices (e.g., $\mathbf{n}_s$6 for grouping DMD pixels), and differential operators of order 1 (gradient) or 2 (Laplacian) can be selected for additional bias invariance (Pastuszczak et al., 2022).

5. Performance, Noise Robustness, and Comparative Analysis

D-FDRI achieves high-quality reconstructions under strong compression. For $\mathbf{n}_s$7 (e.g., $\mathbf{n}_s$8, $\mathbf{n}_s$9):

• Sparse images: PSNR up to 30 dB, with 3–5 dB improvement over DCT+FDRI and Walsh-Hadamard FDRI.
• Dense images: PSNR $\mathbf{n}_d$0 dB.
• SSIM is also favorably improved, especially in empty image regions.
• Frame rates: Up to 17 Hz for $\mathbf{n}_d$1 with CR = 2%; up to 7 Hz for $\mathbf{n}_d$2 at CR = 0.4%.
• Noise robustness: Maintains superior performance for relative noise up to $\mathbf{n}_d$3; tolerates low A/D bit depth with negligible PSNR loss compared to non-differential schemes.
• Real-time reconstructability: $\mathbf{n}_d$4 cost allows per-frame inversion in $\mathbf{n}_d$5 s CPU time even at megapixel resolutions (Stojek et al., 2022, Pastuszczak et al., 2022, Czajkowski et al., 2018).

In comparison to iterative compressive sensing (e.g., TV-regularized NESTA), D-FDRI closely matches or exceeds PSNR (within 0.1–0.3 dB) at a fraction of the runtime (e.g., $\mathbf{n}_d$6 vs. $\mathbf{n}_d$7 at $\mathbf{n}_d$8) (Pastuszczak et al., 2022).

6. Limitations, Practical Trade-offs, and Extensions

D-FDRI’s reliance on precomputation introduces an $\mathbf{n}_d$9 storage requirement for the reconstruction matrix $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$0, which becomes significant when $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$1. The method employs a globally fixed regularizer $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$2, precluding frame-specific or spatially adaptive regularization. Under aggressive compression, the method is ideally suited to sparse or limited-FOV scenes; dense, high-frequency image content tends to be overly smoothed. Extensions and enhancements include:

• Map-based sparsity correction: A subsequent stage (MD-FDRI) can suppress “empty” sectors in highly sparse data (Stojek et al., 2022).
• Learned regularization: Data-driven approaches to select $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$3, $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$4, or $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$5 (e.g., via Stein’s unbiased risk estimate).
• Hybrid solvers: Post hoc iterative refinement (such as proximal methods) may be used.
• Adaptive/online updating: Possibility of incrementally updating $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$6 to accommodate hardware drift or slowly varying scenes.
• Integration with dual-channel or neural network correction: Dual-channel detection or neural post-correction can address artifacts or enhance robustness (Stojek et al., 2022).

7. Implementation Considerations and Applications

The D-FDRI framework has been experimentally validated across multiple platforms and spectral regimes. Typical configurations feature DMDs operated at $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$7 kHz, effective pixel groupings (e.g., $$\mathbf{y} = \mathbf{D} \mathbf{y}_0 = \mathbf{D} \mathbf{M} (\mathbf{x} + \mathbf{n}_s) + \mathbf{D} \mathbf{n}_d,$$8 mirrors per pixel), and bucket detectors suitable for VIS–NIR imaging. Data acquisition is performed using streaming DAQ (e.g., PicoScope) at 8–16-bit resolution with AC coupling. Online reconstruction is implemented in single-precision for speed; precomputation can be performed in double-precision to maximize precision in matrix inversion and FFT-based construction.

Applications include real-time, high-resolution SPI, polarization-sensitive imaging, and dual-channel VIS–IR acquisition. D-FDRI robustly supports arbitrary binary pattern sets and remains compatible with hardware-imposed constraints. The technique is especially attractive where scene sparsity, rapid field-of-view shifts, or hardware non-idealities (e.g., DC offset, low bit depth) preclude slower, adaptive or conventional compressive sensing solvers (Pastuszczak et al., 2022, Stojek et al., 2022).