# Spectral Image Reconstruction Using Recovered Basis Vector Coefficients

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## Abstract

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## 1. Introduction

- We investigated the characterization of spectral reflectance by RGB values and demonstrated the superiority of their corresponding basis vector coefficients for the reconstruction of spectral images.
- Accordingly, we developed one data-driven algebraic method for recovering these coefficients and used them as inputs for the employed CNN networks.
- To ensure the convenience of the spectral imaging systems, we validated the algorithm on a large spectral dataset and our real-world dataset with RGB images as input.
- To strike a balance between accuracy and convenience, we also conducted further research to investigate the effect of channels on the reconstruction performance and offered recommendations for optimal channel selection.

## 2. Methodology

#### 2.1. RGB Digital Camera Imaging Principle

#### 2.2. Basis Vector Coefficients

#### 2.3. Spectral Reconstruction

## 3. Experiments on a Public Dataset

#### 3.1. Settings

**Spectral dataset**. The experiments of our proposed method and benchmarks are conducted on one new natural spectral image dataset provided by the NTIRE 2020 challenge [42]. Containing two tracks and consisting of 510 high-quality natural spectral images in total, this dataset is one of the most comprehensive datasets available. The “clean” track aims to recover spectral images from the noise-free RGB images, while the “real world” track requires participants to rebuild the spectral images from noisy JPEG-compression RGB images created by an unknown camera response function [58]. The original spectral images have a uniform spatial resolution of $482\times 512$, and the visible wavelength range of 400–700 nm is sampled at $\Delta \lambda =10\mathrm{nm}$ intervals.

**Evaluation metrics**. In order to adequately perform the quantitative evaluation of our proposed method, several metrics are used, including the root mean squared error (RMSE), mean relative absolute error (MRAE), structural similarity index (SSIM) [59], and peak signal-to-noise ratio (PSNR). Among them, MRAE is the metric recommended by the NTIRE challenges for the ability to avoid over-weighting errors in higher luminance areas of the test images [58]. In addition, compared to the other metrics, that can only evaluate the spectral reflectance reconstruction accuracy of the individual pixels, the SSIM metric is used to assess the similarity of the whole spectral image to the ground truth. For the metrics mentioned above, including RMSE, MRAE, and SSIM, all values fall within the range of 0 to 1. Higher values of SSIM and PSNR indicate superior reconstruction performance, while the converse holds true for RMSE and MRAE.

**Implementation details**. With reference to the employed benchmarks in this work, the metric RMSE is used as the loss function and minimized with the well-known adaptive moment estimation method (Adam) [60] by setting ${\beta}_{1}=0.9$, ${\beta}_{2}=0.999$, and $\u03f5={10}^{-8}$. All neural networks are trained for 35 epochs while the learning rate is initialized as ${10}^{-5}$ and scaled to one-tenth of the previous one every 10 epochs. During the training process, RGB patches of size $256\times 256$ and their corresponding spectral data cubes are fed into the models. Data augmentation is performed by means of random horizontal and vertical flipping. All networks are implemented within the PyTorch framework and trained on an NVIDIA RTX 2080Ti hardware platform.

#### 3.2. Results

#### 3.3. Visualization

## 4. Experiments on the Real-World Dataset

#### 4.1. Settings

**Implementation details**. In order to achieve improved reconstruction results, adjustments were made to various hyperparameters and learning rate strategies of the neural network, considering the differences in the intrinsic properties of the real-world dataset. Specifically, all neural networks were trained for 60 epochs while the learning rate was initialized as $4\times {10}^{-4}$ and the cosine annealing scheme was adopted.

#### 4.2. Results

## 5. Discussion

#### 5.1. Computational Efficiency and Flexibility

#### 5.2. The Effect of Channels

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Results of statistical analysis based on the paired t-test (* p < 0.05). (

**a**–

**d**) the statistical analysis of RMSE, MRAE, SSIM and PSNR. The blue and red boxplots represent the results for the benchmarks with vanilla input-end (denoted as “direct”) and modified input-end (denoted as “coeff”), respectively.

**Figure 3.**Visual comparison of the metric RMSE between the ground truth and the corresponding reconstructed spectral image under the employed benchmarks with vanilla input-end and modified input-end.

**Figure 4.**Visual comparison of the metric MRAE between the ground truth and the corresponding reconstructed spectral image under the employed benchmarks with vanilla input-end and modified input-end.

**Table 1.**Quantitative results of spectral reconstruction on the public dataset. The better result is shown in bold.

Data Type | Metrics | ZYYNet | HSCNN-D | ||

Direct | Coeff | Direct | Coeff | ||

“Real world” track | RMSE | 0.03084 | 0.02371 | 0.04465 | 0.02988 |

MRAE | 0.16508 | 0.11817 | 0.23345 | 0.15860 | |

SSIM | 0.88178 | 0.91654 | 0.86875 | 0.90669 | |

PSNR (dB) | 29.883 | 32.378 | 27.102 | 30.001 | |

“Clean” track | RMSE | 0.03130 | 0.02301 | 0.04596 | 0.02772 |

MRAE | 0.15510 | 0.10936 | 0.22371 | 0.14196 | |

SSIM | 0.88884 | 0.93197 | 0.88509 | 0.93387 | |

PSNR (dB) | 29.705 | 32.736 | 26.943 | 30.935 |

**Table 2.**Quantitative results of spectral reconstruction on the real-world dataset. The better result is shown in bold.

Metrics | ZYYNet | HSCNN-D | ||

Direct | Coeff | Direct | Coeff | |

RMSE | 0.08995 | 0.03487 | 0.09201 | 0.03894 |

MRAE | 0.29733 | 0.14442 | 0.29992 | 0.14397 |

SSIM | 0.78487 | 0.87190 | 0.78376 | 0.86707 |

PSNR (dB) | 20.291 | 28.187 | 20.103 | 27.431 |

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## Share and Cite

**MDPI and ACS Style**

Xu, W.; Wei, L.; Yi, X.; Lin, Y.
Spectral Image Reconstruction Using Recovered Basis Vector Coefficients. *Photonics* **2023**, *10*, 1018.
https://doi.org/10.3390/photonics10091018

**AMA Style**

Xu W, Wei L, Yi X, Lin Y.
Spectral Image Reconstruction Using Recovered Basis Vector Coefficients. *Photonics*. 2023; 10(9):1018.
https://doi.org/10.3390/photonics10091018

**Chicago/Turabian Style**

Xu, Wei, Liangzhuang Wei, Xiangwei Yi, and Yandan Lin.
2023. "Spectral Image Reconstruction Using Recovered Basis Vector Coefficients" *Photonics* 10, no. 9: 1018.
https://doi.org/10.3390/photonics10091018