Source-linked AI summary
Super sub-Nyquist single-pixel imaging by means of cake-cutting Hadamard basis sort
Wen-Kai Yu
TL;DR
Single-pixel compressed-sensing imaging must balance acquisition time, spatial resolution, and signal-to-noise ratio. The paper addresses this by optimally reordering deterministic Hadamard patterns with cake-cutting and exploiting fast Hadamard structure. It reports low-sampling reconstruction, including 0.78% acquisition for 1024 × 768 resolution and clear imaging through partially obscuring scenes.
Problem
Single-pixel compressed-sensing imaging faces a trade-off among acquisition time, spatial resolution, and signal-to-noise ratio.
Method
The method orders deterministic Hadamard basis patterns by their reconstruction contribution and applies the ordering to a structured Walsh–Hadamard computation.
Results
0.78% sampling enabled 1.97 ms acquisition at 64 × 64 pixels and 377.51 ms acquisition at 1024 × 768 pixels, while 8192 measurements reconstructed 512 × 512 images in partially obscuring cases.
Takeaways & Limitations
The approach supports super sub-Nyquist single-pixel imaging with reduced acquisition time, reduced matrix-storage needs, and reconstruction under noise and partial obscuration.
Takeaways & Limitations
The proposed pattern-ordering regularity is only effective for Walsh-ordered patterns, and extending the work to other orthogonal or deterministic matrices remains future work.
Abstract
from arXiv · showhide
Single-pixel imaging via compressed sensing can reconstruct high-quality images from a few linear random measurements of an object/scene known a priori to be sparse or compressive, by using a point/bucket detector without spatial resolution. Nevertheless, it still faces a harsh trade-off among the acquisition time, the spatial resolution and the signal-to-noise ratio. Here we present a new compressive imaging approach with use of a strategy called cake-cutting which optimally reorders the deterministic Hadamard basis. By this means, the number of measurements can be dramatically reduced by more than two orders of magnitude. Furthermore, by exploiting the structured characteristic of the Hadamard matrix, we can accelerate the computational process and simultaneously reduce the memory consumption of storing the matrix. The proposed method is capable of recovering an image of the object, of pixel size $1024\times1024$, with a sampling ratio of even 0.2%, thereby realizing super sub-Nyquist sampling and significantly reducing the acquisition time. Moreover, through the differential modulation/measurements, we demonstrate this method with a single-photon single-pixel camera under low light condition and retrieve clear images through partially obscuring scenes. This described practical method complements the single-pixel imaging approaches and can be applied to a variety of fields, such as video, night vision goggles and automatic drive.
Results
The method orders Hadamard patterns by their contribution to reconstruction, combines this ordering with compressed sensing and fast Walsh–Hadamard computation, and evaluates reconstruction quality, sequence generation, and noisy imaging performance.
- Principles description: M = O(K · log(N/K)) < N random patterns can reduce measurements for sparse images, while total variation minimization recovers the image from the resulting linear system.The sampling rate is defined as measurements divided by pixels; the reconstruction uses a TVAL3 solver.
- Results: Significant Hadamard patterns yield better reconstruction at the same sampling ratio, motivating selection of patterns with the largest contribution first.Descending order of the relevant intensity contribution outperforms ordering that begins with unessential patterns.
- Fast Hadamard computation: The fast Walsh–Hadamard transform reduces Hadamard multiplication complexity from O(n^2) to O(n log n), while the cake-cutting sequence is applied to the matching Walsh-ordered operator.Using another FWHT ordering without applying the corresponding cake-cutting sort produces an incorrect optimized order.
- Results: The piece-number regularity further reduces cake-cutting sequence-generation time for large Walsh-ordered Hadamard matrices, although it is effective only for Walsh ordering.The method avoids nested grouping used by the Russian Dolls method, but computing connected regions remains time-consuming before applying the regularity.
- Results: At 12.5% sampling, gray-scale and color images were reconstructed across different objects, while partially obscured 512 × 512 images used 8192 measurements.The color reconstruction used red, green, and blue layers and produced 255 tones with little color distortion; the large-scale simulations included additive white Gaussian noise.
Discussion and Conclusion
The method combines cake-cutting Hadamard ordering, structured fast computation, and differential single-photon measurements to reduce sampling and acquisition demands while supporting practical imaging extensions.
- Acquisition speed: 32.55 kHz DMD operation enables 1.97 ms total measurement at 64×64 resolution and 377.51 ms at 1024×768 resolution, both at 0.78% sampling ratio.The measurement includes two adjacent complementary patterns for differential measurements.
- Scope and future work: The method's fast computation depends on Hadamard orthogonality together with compressed sensing, while reconstruction from other orthogonal or deterministic matrices remains future work.The authors identify Hadamard orthogonality as important but not the only factor in fast computation.
- Potential applications: The approach may extend to color and non-visible imaging by using spectral filters and changing the lens and single-pixel detector for the target wavelength.The stated wavelength range supports extension toward infrared and ultraviolet imaging.
- Method and implementation: The proposed method uses cake-cutting Hadamard ordering and structured matrix operations to reduce computational overhead and memory consumption.Predetermined patterns can be loaded in real time without storing them all on the DMD.
- Experimental demonstration: The system was demonstrated with a single-photon single-pixel camera using differential DMD modulation under low-light conditions.The setup uses a photomultiplier tube and can acquire reconstructions through increasing tissue obstacles at a 3.13% sampling ratio.
Methods
The methods define the test target, image-quality measures, normalization procedures, and computational environment used to evaluate reconstructed images.
- Target object: The 1951 USAF resolution chart contains line groups whose widths in the tested red square are 793.70 µm, 707.11 µm, and 629.96 µm for Elements 3–5.These values correspond to Group −1 of the chart.
- Image analysis: Relative error compares the reconstructed image with the original image using the Frobenius norm over all p × q pixels.The reconstructed and original images are denoted by Ũ and Uo.
- Image analysis: The Frobenius norm is defined through the conjugate transpose and singular values of the image-related matrix.The passage identifies ∗ as the conjugate transpose operator and σ_i as a singular value.
- Normalization: Recovered numerical images are normalized to 0–255, while optical-experiment results are normalized directly to 0–1.The distinction reflects whether an original reference image is available.
Additional information
The paper provides supplementary information, declares no competing financial interests, and supplies online reprint and citation information.
- Supplementary material: Supplementary Information accompanies the paper online.The passage provides an online location for the supplementary material.
- Competing interests: The authors declare no competing financial interests.
- Permissions: Reprints and permission information is available online.
- Citation: The paper provides a citation format including the title, authorship, journal placeholder, volume placeholder, DOI placeholder, and year placeholder.