# Chromatic-Aberration-Corrected Hyperspectral Single-Pixel Imaging

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Model and Theoretical Derivation

#### 2.1. Principle of Hyperspectral Differential Ghost Imaging

**I**(x) with the bucket signal

**S**:

**R**(x) is the (intensity) reflection function of the object, and ${\mathrm{A}}_{\mathrm{S}}$ is the plane on which the image is modulated. The retrieved image is thus:

**R**(x) is defined as $\delta R(x)=R(x)-\overline{R}$, where $\overline{R}={\displaystyle \int \u2329I(x)\u232a}R(x){\mathrm{d}}^{2}x/{\displaystyle \int \u2329I(x)\u232a}{\mathrm{d}}^{2}x$ is the average transmission function of the object. We define a new differential bucket signal ${S}_{1}$ as:

**S**can be measured from the following operative form

_{1}**S**with

**S**in Equation (2), namely,

_{1}**X**(x),

**Y**(x) and

**Z**(x):

**X**(x),

**Y**(x), and

**Z**(x) into the three channels of the resultant RGB image, the entire reconstruction process can be completed.

#### 2.2. Principle of Chromatic Aberration Correction

_{1}and another negative meniscus L

_{2}, can reduce the spherical aberration and longitudinal chromatic aberration by splitting one positive lens into two so that the curvature is spread over four surfaces instead of two, while the overall power of the lens is preserved. Subsequently, placing lenses L

_{3}and L

_{4}to face L

_{1}and L

_{2}symmetrically would notably diminish the vertical axis aberration, such as distortion. The vital step to correct the chromatic aberration is through replacing the negative meniscus lenses L

_{2}and L

_{3}with hyperchromatic lenses L

_{21}, L

_{22}and L

_{31}, L

_{32}, respectively. A hyperchromatic lens is a cemented doublet lens with the same refractive index at the given wavelength but different dispersion, such that changing the radius of the curvature of the cemented surface does not alter the focal length of the primary wavelength but adjusts the chromatic aberration [21]. By arranging the above lenses in proper positions, the final double Gaussian lens can correct chromatic aberration.

## 3. Simulation

## 4. Experimental Details

^{3}, which was then imaged by a TV lens onto a common type of SLM—a digital micromirror device (DMD) (ViALUX V-9501, 1920 × 1080) preprogrammed with a series of Hadamard matrices. After modulation, the beam was focused into a spectrometer (Ocean Optics FLAME-S-UV-VIS, 340–1100 nm). We employed 4096 speckle patterns generated under the cake-cutting basis Hadamard basis algorithm [22] for both the control and TV lens measurements.

## 5. Algorithm Optimization

^{i}) or (2

^{i}, 1)(i = 1, 2, 3……). After each correction, its position moves from (1, 2

^{i}) to (1, 2

^{i−1}+ 1) or from (2

^{i}, 1) to (2

^{i−1}+ 1, 1) but never beyond the upper-left quarter of the image. We define the edge region as two line segments from (1, 2) to (1, N/2) and from (2, 1) to (N/2, 1) to simplify the computation. Each time, we search for the first zero point from left to right and from top to bottom. The whole process is finished when the first zero point no longer appears in the edge region. The SPI images of different spectral bands before and after correction are shown in Figure 6c. To quantitatively evaluate the image quality, we use the contrast-to-noise ratio (CNR) instead of the conventional signal-to-noise ratio because we do not have an original object image for comparison. The CNR is defined as [23]:

## 6. Results and Discussion

_{C}and D

_{T}of the single lens and TV lens are, respectively, 30 mm and 16 mm. The smallest unit that was clearly resolved by the single lens system was the 4 mm width of the black gaps between the colored blocks in A1 of Figure 8a, whereas the TV lens system could clearly resolve the white reflective strips in B2 and C2 of Figure 8b, which have a width of less than 1 mm. The distance between the object and the lens is more than 45 cm. Thus, the theoretical and actual angular resolutions can be derived as follows.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic of chromatic aberration along the optical axis. A: object beam. Wavelengths of F, D, and C are 486 nm, 588 nm, and 656 nm, respectively.

**Figure 4.**Simulation results of a single-lens system and TV lens system. Layout of imaging parts in (

**a**) hyperspectral SPI (HSPI) and (

**b**) chromatic-aberration-corrected hyperspectral single-pixel imaging (CHSPI). (

**c**) Lateral chromatic aberration curve: field of view vs. lateral chromatic aberration. (

**d**) The chromatic focal shift curve: wavelength vs. chromatic focal shift in the range from 486 nm of F-light to 656 nm of C-light.

**Figure 6.**Proposed adaptive iterative correction algorithm. (

**a**) The noise ripples (zero points) on the upper-left edge. (

**b**) The adaptive iterative correction algorithm used to correct the erroneous black noise ripples. (

**c**) Comparison of SPI images in different spectral bands before and after correction.

**Figure 7.**Reconstructed hyperspectral images over 400–780 nm in the control (

**a**) and TV lens (

**b**) groups, with their composite-colored images in the last figure. (

**c**,

**d**): Enlargements of the respective images in the red boxes of (

**a**,

**b**) reveal that sharp edges are only seen in the yellow bands of the single-lens system, while they are distinct over the vast majority of band range of the TV system.

**Figure 8.**Reconstructed colored images of the control group (

**a**) and the TV lens group (

**b**). The white boxes show the main differences: purple or yellow haloes appear at the edges of the color blocks in (

**a**), while in (

**b**), even details such as the white edges in boxes B2 and C2 are clearly revealed.

**Figure 9.**Comparison of the performance of chromatic aberration correction. The intensity profiles across the red lines in the reconstructed images of (

**a**) the control group and (

**c**) the TV lens group vs. pixel number for different wave bands are shown in (

**b**,

**d**). Three groups of pixels ①, ②, and ③ are selected and in (

**b**,

**d**) are connected across the spectral bands with a blue line, while the vertical red dotted lines denote the abscissa of the chosen points with the first band (440 nm) as the reference origin.

**Figure 10.**Pixel column number of the ①, ②, and ③ positions vs. wavelength over 400–780 nm for the control group (

**a**) and the TV lens group (

**b**). A flat line indicates no deviation in the focal point position, i.e. no lateral chromatic aberration occurs.

Scheme Name | ① | ② | ③ |
---|---|---|---|

HSPI scheme | 4.8889 | 7.1137 | 1.8659 |

CHSPI scheme | 0.1473 | 0.2459 | 0.2025 |

Improvement (HSPI/CHSPI) | 33.1901 | 28.9292 | 9.2143 |

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**MDPI and ACS Style**

Liu, Y.; Yang, Z.-H.; Yu, Y.-J.; Wu, L.-A.; Song, M.-Y.; Zhao, Z.-H.
Chromatic-Aberration-Corrected Hyperspectral Single-Pixel Imaging. *Photonics* **2023**, *10*, 7.
https://doi.org/10.3390/photonics10010007

**AMA Style**

Liu Y, Yang Z-H, Yu Y-J, Wu L-A, Song M-Y, Zhao Z-H.
Chromatic-Aberration-Corrected Hyperspectral Single-Pixel Imaging. *Photonics*. 2023; 10(1):7.
https://doi.org/10.3390/photonics10010007

**Chicago/Turabian Style**

Liu, Ying, Zhao-Hua Yang, Yuan-Jin Yu, Ling-An Wu, Ming-Yue Song, and Zhi-Hao Zhao.
2023. "Chromatic-Aberration-Corrected Hyperspectral Single-Pixel Imaging" *Photonics* 10, no. 1: 7.
https://doi.org/10.3390/photonics10010007