Single-Pixel Coherent Diffraction Imaging

• Single-Pixel Coherent Diffraction Imaging (SPI-CDI) is an imaging technique that reconstructs both amplitude and phase from sequential binary mask illuminations using a single-pixel detector.
• It significantly reduces sensor dynamic range demands by measuring only the DC component, enabling applications across visible, IR, THz, and X-ray regimes.
• Advanced phase retrieval algorithms and homodyne digital ghost holography variants facilitate high-resolution, quantitative imaging in fields such as life sciences and materials metrology.

Single-pixel coherent diffraction imaging (SPI-CDI) is an imaging modality that enables full-field complex (amplitude and phase) object reconstruction using only a single-pixel detector in the far field, facilitated by sequential structured illumination. It circumvents the need for high-dynamic-range two-dimensional detector arrays intrinsic to conventional coherent diffraction imaging (CDI) by measuring only the DC (zero spatial frequency) component of modulated far-field diffraction patterns under binary mask illumination, and subsequently reconstructing the entire complex field via computational phase retrieval algorithms (Li et al., 2020). The technique operates across a wide spectral range—encompassing visible, infrared (IR), terahertz (THz), and potentially X-ray regimes—thereby broadening the application space of complex-field imaging.

1. Principles and Forward Model

In SPI-CDI, the object $O(x, y) \in \mathbb{C}$ is illuminated by a sequence of $M$ known binary modulation patterns $P_k(x, y)$, where each mask is typically a Bernoulli random array of 0s and 1s. For transmission-mode implementations under the Fraunhofer approximation, the modulated field after the object, $\psi_k(x, y) = P_k(x, y) O(x, y)$, propagates to the far field where a single-pixel detector records only the central (DC) intensity per mask. The $k$-th measurement is

$$I_k = |F[P_k \odot O](0, 0)|^2 = \left| \sum_{i=1}^N \sum_{j=1}^N P_{k,ij} O_{ij} \right|^2 = |\langle P_k, O \rangle|^2,$$

where $F[\cdot](0, 0)$ denotes the value of the 2D Fourier transform at the zero spatial frequency, $\odot$ stands for entry-wise multiplication, and $\langle\cdot, \cdot\rangle$ is the inner product over the $N \times N$ sampled object and mask arrays (Li et al., 2020). This reduces a high-dimensional diffraction measurement acquisition to a set of DC-only, single-pixel measurements across multiple patterns.

2. Binary Mask Design and Dynamic Range Considerations

SPI-CDI employs binary patterns, commonly generated by a digital micromirror device (DMD), to achieve statistical diversity and phase retrieval capability. Pattern sizes of $M$0 or $M$1 (i.e., $M$2 or $M$3) and a sampling ratio $M$4 are typical; for instance, $M$5 gives $M$6 patterns per image (Li et al., 2020). In contrast to standard CDI, which requires the acquisition of full 2D diffraction patterns spanning $M$7 to $M$8 orders of magnitude in intensity (center-to-edge), SPI-CDI collects only DC components whose intensity spans are reduced to $M$9, thus decreasing sensor dynamic range requirements by $P_k(x, y)$0–$P_k(x, y)$1 orders of magnitude. This attribute enables extension into wavelength regimes (e.g., IR, THz) where array detectors with high dynamic range are either unavailable or prohibitively expensive.

3. Phase Retrieval and Reconstruction Algorithm

SPI-CDI reconstructs $P_k(x, y)$2 by minimizing

$P_k(x, y)$3

a nonconvex intensity-matching loss across all measurements. Efficient solution is achieved via a complex-field alternating projection algorithm, a variant of Gerchberg–Saxton phase retrieval, as follows (Li et al., 2020): 1. Compute $P_k(x, y)$4; 2. Fourier transform: $P_k(x, y)$5; 3. Set $P_k(x, y)$6 and leave other frequencies unchanged; 4. Inverse transform: $P_k(x, y)$7; 5. Update $P_k(x, y)$8, with step size $P_k(x, y)$9.

No object support or nonnegativity constraint is imposed; the sequence of binary patterns fully lifts the phase ambiguity. Typically, 200–400 passes through all $\psi_k(x, y) = P_k(x, y) O(x, y)$0 patterns suffice for convergence.

4. Experimental Implementation and Quantitative Performance

A typical SPI-CDI setup includes a collimated laser source (e.g., 488 nm or 980 nm), a DMD to modulate the incident field with binary masks, the object under test, and a single-pixel detector (avalanche photodiode preceded by a $\psi_k(x, y) = P_k(x, y) O(x, y)$1 $\psi_k(x, y) = P_k(x, y) O(x, y)$2m pinhole) for DC-only detection in the far field (object-to-detector distances of $\psi_k(x, y) = P_k(x, y) O(x, y)$3–$\psi_k(x, y) = P_k(x, y) O(x, y)$4 m are used) (Li et al., 2020). Representative samples include calibrated etched glass phase targets and biological specimens. Phase calibration is achieved by acquiring a reference measurement with a plane glass in place of the sample and subtracting the resulting phase map.

Quantitative metrics achieved include:

• Phase standard deviation $\psi_k(x, y) = P_k(x, y) O(x, y)$5 rad; phase depth resolution after unwrapping $\psi_k(x, y) = P_k(x, y) O(x, y)$6 nm.
• Lateral resolution set by the pattern and optics (e.g., $\psi_k(x, y) = P_k(x, y) O(x, y)$7 $\psi_k(x, y) = P_k(x, y) O(x, y)$8m pixel size for red-blood-cell imaging).
• Robustness to noise: $\psi_k(x, y) = P_k(x, y) O(x, y)$9 dB PSNR at measurement SNR as low as $k$0 dB.
• Acquisition times: $k$1 s for $k$2, $k$3 s for $k$4 at $k$5 kHz pattern rates.
• Computational costs: $k$6, with total runtime in the tens of seconds on commodity GPUs.

5. Extension: Homodyne and Single-Pixel Digital "Ghost" Holography

Traditional SPI-CDI measures intensity-only DC projections, requiring iterative phase-retrieval. By contrast, single-pixel digital “ghost” holography (DGH) integrates homodyne detection via a Mach–Zehnder interferometer and phase-shifting to access the cross-term $k$7 per pattern directly (Clemente et al., 2013). Here, the complex field between a reference and the object-modulated arm is encoded through phase-shifted measurements and recovered by correlating the measured homodyne signals $k$8 with computed intensity patterns. The cross-correlation

$k$9

is proportional to the Fourier transform of $$I_k = |F[P_k \odot O](0, 0)|^2 = \left| \sum_{i=1}^N \sum_{j=1}^N P_{k,ij} O_{ij} \right|^2 = |\langle P_k, O \rangle|^2,$$0, the object’s complex transmission. In this scheme, iterative phase retrieval is obviated; the Fourier amplitude and phase are recovered directly, increasing SNR and potentially reducing acquisition and computational overhead (Clemente et al., 2013).

6. Advantages, Limitations, and Future Prospects

Advantages:

• Eliminates the need for high-dynamic-range or large-format 2D array detectors; robust operation across visible, NIR, THz, and X-ray regimes (Li et al., 2020).
• No mechanical scanning or explicit object support required.
• Quantitative complex-field recovery (amplitude and phase), enabling downstream digital refocusing, DIC, and three-dimensional profiling.
• Homodyne (DGH) variants provide nanometric depth sensitivity, three-dimensional focusing by digital propagation, and SNR scaling linearly with pattern number (Clemente et al., 2013).

Limitations:

• Acquisition time increases with $$I_k = |F[P_k \odot O](0, 0)|^2 = \left| \sum_{i=1}^N \sum_{j=1}^N P_{k,ij} O_{ij} \right|^2 = |\langle P_k, O \rangle|^2,$$1, necessitating high-speed spatial modulators for practical imaging rates.
• Computational effort can be significant for large $$I_k = |F[P_k \odot O](0, 0)|^2 = \left| \sum_{i=1}^N \sum_{j=1}^N P_{k,ij} O_{ij} \right|^2 = |\langle P_k, O \rangle|^2,$$2 or high pattern counts.
• Spatial resolution limited by mask size and optical setup (e.g., speckle size in DGH).
• Requires phase stability (in interferometric adaptations), and pattern diversity for robust phase retrieval.
• No support or nonnegativity constraints required; however, compressive acquisition and postprocessing can further reduce $$I_k = |F[P_k \odot O](0, 0)|^2 = \left| \sum_{i=1}^N \sum_{j=1}^N P_{k,ij} O_{ij} \right|^2 = |\langle P_k, O \rangle|^2,$$3 (Li et al., 2020).

Prospective Developments:

• Multi-pixel detector arrays (e.g., SPAD arrays) could sample multiple spatial-frequency components per shot, reducing total pattern count linearly (Li et al., 2020).
• Compressive sensing and prior-based algorithms (e.g., sparsity, total-variation regularization) for improved efficiency and noise resilience.
• Adaptations to reflection-mode and passive-illumination for surface metrology, remote sensing, and 3D holographic refocusing (Li et al., 2020).

7. Applications

SPI-CDI has been demonstrated and prospectively targeted for:

• Quantitative phase imaging in life sciences (cell morphology, tissue histology, digital DIC).
• Thin-film and nanostructure depth metrology.
• Non-destructive testing and concealed-object detection in the THz regime.
• High-precision surface profiling in semiconductor and photonics manufacturing (Li et al., 2020).

A plausible implication is that integration of DGH-based homodyne measurement strategies within SPI-CDI could further expand its reach to wavelengths or experimental geometries where array detection is impractical and phase stability is achievable (Clemente et al., 2013).