Source-linked AI summary
Hadamard single-pixel imaging versus Fourier single-pixel imaging
Zibang Zhang, Xueying Wang, Guoan Zheng, Jingang Zhong
TL;DR
Single-pixel imaging offers useful non-visible and low-light capabilities but historically required many measurements and suffered limited reconstruction quality. This paper compares HSI and FSI through theoretical analysis and experiments, finding that FSI is more efficient while HSI is more noise-robust. The results guide technique selection according to application priorities.
Problem
Single-pixel imaging historically faced low reconstruction quality and long acquisition times, motivating comparison of HSI and FSI for improved imaging applicability.
Method
The paper theoretically and experimentally compares HSI and FSI across their principles, imaging efficiency, noise robustness, and implementation characteristics.
Results
FSI is more efficient than HSI, while HSI has better noise robustness; under undersampling below 10%, FSI produces clearer and sharper reconstructions.
Takeaways & Limitations
Choose FSI when imaging efficiency matters and HSI when image quality or accuracy is the priority.
Takeaways & Limitations
Hadamard transform is limited to square input images with side lengths that are powers of 2.
Abstract
from arXiv · showhide
Single-pixel imaging is an innovative imaging scheme and has received increasing attentions in recent years. It is applicable to imaging at non-visible wavelengths and imaging under low light conditions. However, single-pixel imaging has once encountered problems of low reconstruction quality and long data-acquisition time. This situation has been changed thanks to the developments of Hadamard single-pixel imaging (HSI) and Fourier single-pixel imaging (FSI). Both techniques are able to achieve high-quality and efficient imaging, remarkably improving the applicability of single-pixel imaging scheme. In this paper, we compare the performances of HSI and FSI with theoretical analysis and experiments. The results show that FSI is more efficient than HSI while HSI is more noise-robust than FSI. Our work may provide a guideline for researchers to choose suitable single-pixel imaging technique for their applications.
1. Introduction
Single-pixel imaging avoids pixelated detectors, enabling potentially lower-cost imaging at non-visible wavelengths and under low-light conditions. The paper compares deterministic HSI and FSI as responses to earlier reconstruction-quality and acquisition-time limitations.
- Single-pixel imaging can operate at non-visible wavelengths and under low-light conditions without pixelated light detectors.Large-area single-pixel detectors can be easier to fabricate and more sensitive than pixelated detectors.
- Early single-pixel imaging used random, overcomplete non-orthogonal illumination patterns, requiring many measurements and long acquisition times.Reconstruction quality also remained difficult to match with conventional two-dimensional detector imaging.
- Deterministic-model techniques emerged to address the earlier limitations of single-pixel imaging.
- HSI and FSI use complete orthogonal basis patterns for illumination, with Hadamard patterns in HSI and Fourier patterns in FSI.The basis-pattern approach supports spatial-information acquisition and perfect reconstruction in principle.
- The paper theoretically and experimentally compares HSI and FSI in terms of principles, imaging efficiency, and noise robustness.The comparison identifies advantages and disadvantages of both techniques.
2. Comparison of theory
HSI and FSI reconstruct images from deterministic transform-domain measurements, but their pattern-generation requirements create different efficiency, implementation, quantization, and sampling trade-offs. Fourier representations concentrate natural-image energy more effectively and support rapid low-frequency sampling, while Hadamard patterns remain advantageous for binary devices and quantization robustness.
- HSI principle: HSI acquires Hadamard coefficients through inner-product measurements and reconstructs the image with an inverse Hadamard transform.Differential HSI uses paired complementary patterns to obtain each real-valued coefficient and suppress noise.
- Applicability: Hadamard transforms require square input images whose side length is a power of 2.
- Measurement schemes: 4-step FSI uses symmetric differential measurements, whereas 3-step FSI uses an asymmetric differential form with weaker noise suppression.The paper explicitly attributes the difference in noise performance to the symmetry of the measurement schemes.
- Basis pattern generation: Hadamard patterns are binary and naturally compatible with high-speed DMD illumination, while Fourier patterns are grayscale and cannot exploit DMD binary-mode speed.The cited example reports approximately 20,000 binary patterns per second versus approximately 250 8-bit grayscale patterns per second for a state-of-the-art DMD.
- Basis pattern generation: Binary Fourier patterns avoid grayscale-device speed limitations but introduce dithering-related quantization errors and reduced spatial resolution.Hadamard patterns are discrete and can be generated without quantization errors, whereas Fourier patterns can also be generated by physical optical means.
- Overall trade-offs: HSI can benefit more from high-speed binary DMDs, whereas FSI offers more flexible and varied illumination-pattern generation.
- Efficiency: FSI provides stronger energy concentration for natural images, enabling efficient sampling of low-frequency Fourier components and rapid reconstruction-quality convergence.The paper identifies circular low-frequency sampling as the most effective approach for diffraction-limited systems in principle.
3. Comparison of experiment
Numerical simulations and experiments compare HSI and FSI across undersampling, noise, photography, fast imaging, and microscopy. FSI generally reconstructs more clearly with few measurements, while HSI becomes more robust or higher quality with many measurements and uniform patterns.
- Numerical simulations: For natural images, circular sampling is best for FSI, whereas zig-zag sampling is best for HSI.The comparison covers 632 USC-SIPI images using PSNR, SSIM, and power ratio.
- Numerical simulations: FSI reconstructs natural images better than HSI under undersampling because Fourier transforms concentrate image energy more effectively.The simulations evaluate multiple images and sampling strategies, including the USC-SIPI database.
- Noise robustness: HSI is more robust to measurement noise than FSI, revealing a tradeoff between energy concentration and noise robustness.White Gaussian noise was added to measurements to produce different SNRs; PSNR, SSIM, RMSE, and reconstructed images were compared.
- Single-pixel photography: In photography, FSI produces clearer and sharper reconstructions than HSI when the sampling ratio is below 10%, while full-sampled results are comparable.The reconstructed images have 256×256 pixels, and the experimental trend agrees with the numerical simulations.
- Fast single-pixel imaging: In fast imaging, FSI performs better with few measurements, while HSI gives cleaner uniform-pattern reconstructions and becomes comparable or better at high measurement counts.At 4,000 measurements, 3-step binary FSI distinguishes all five pencils; at 98,304 measurements, FSI background noise becomes noticeable from quantization errors.
4. Conclusion
The paper’s comparison finds that FSI is more efficient than HSI, while HSI offers better noise robustness. The preferred technique therefore depends on whether efficiency or image quality and accuracy matter more.
- FSI is more efficient than HSI because Fourier transform concentrates image energy better than Hadamard transform.
- 3-step FSI reduces measurements for differential measurement applications.
- HSI has better noise robustness than FSI and is suitable for DMD-based single-pixel imaging systems.
- Applications prioritizing efficiency should select FSI, whereas those prioritizing image quality or accuracy should select HSI.