# Super-Resolution and Feature Extraction for Ocean Bathymetric Maps Using Sparse Coding

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Dictionary Learning

**α**obtained by learning the LR dictionary ${\mathit{D}}_{\mathrm{L}}$,

#### 2.2. Sparse-Coding and Reconstruction

Algorithm 1. Reconstruction algorithm for sparse coding super-resolution (ScSR). |

0: Learn HR and LR dictionaries, ${\mathit{D}}_{\mathrm{H}}$ and ${\mathit{D}}_{\mathrm{L}}$ 1: Input: dictionaries, ${\mathit{D}}_{\mathrm{H}}$ and ${\mathit{D}}_{\mathrm{L}}$, edge components of an LR image $\mathit{Y}$2: Split an LR image ${\mathit{Y}}_{\mathbf{0}}$ into high- and low-frequency components, $\mathit{Y}$ and ${\mathit{Y}}_{\mathrm{blur}}$. 3: Extract LR patches ${\mathit{P}}_{\mathrm{L}}$ from the edge components of $\mathit{Y}$. 4: $\widehat{\mathit{\alpha}}\leftarrow \mathit{S}\mathit{p}\mathit{a}\mathit{r}\mathit{s}\mathit{e}\mathit{C}\mathit{o}\mathit{d}\mathit{i}\mathit{n}\mathit{g}\left({\mathit{D}}_{\mathrm{L}},{\mathit{P}}_{\mathrm{L}}\right)$ 5: Generate the HR patch: ${\widehat{\mathit{P}}}_{\mathrm{H}}\leftarrow {\mathit{D}}_{\mathrm{H}}\widehat{\mathit{\alpha}}$. 6: Up-sample the high-frequency component of the LR image, $\mathit{U}\leftarrow \mathit{Q}\mathit{Y}$ 7: Superpose the adjacent patches and add $\mathit{U}$: ${\widehat{\mathit{X}}}_{\mathrm{p}}\leftarrow \mathit{F}{\widehat{\mathit{P}}}_{\mathrm{H}}+\mathit{U}.$ 8: Find ${\mathit{X}}^{*}$ which satisfies the constraint: ${\mathit{X}}^{*}=\mathrm{arg}\underset{\mathit{X}}{\mathrm{min}}\Vert \mathit{S}\mathit{H}\mathit{X}-\mathit{Y}{\Vert}_{2}^{2}$. 9: Up-sample the low-frequency component of the LR image: ${\widehat{\mathit{Y}}}_{\mathrm{blur}}\leftarrow \mathit{Q}{\mathit{Y}}_{\mathrm{blur}}$. 10: Take a summation of reconstructed component ${\mathit{X}}^{*}$ and up-sampled component ${\widehat{\mathit{Y}}}_{\mathrm{blur}}$: $\widehat{\mathit{X}}\leftarrow {\mathit{X}}^{*}+{\widehat{\mathit{Y}}}_{\mathrm{blur}}$. 11: Output: SR image $\widehat{\mathit{X}}$. |

## 3. Data and Implementation

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Flowchart of the outline of ScSR. HR and LR indicate high-resolution and low-resolution, respectively.

**Figure 2.**Detailed flowchart of ScSR. It consists of two processes: dictionary learning and reconstruction by sparse coding.

**Figure 3.**Map of the area used in this study. (

**a**) The Japanese Islands (top left) and the sea around Okinawa; (

**b**) Topographical map in the red box in (

**a**). The area is equally divided into eight squares. The ridge in the area 0_2 is the Iheya–Minor Ridge.

**Figure 5.**Seabed topography in area 0_2. (

**a**) The original image; (

**b**) Low-resolution input image; (

**c**) Sparse coding super-resolution (ScSR: the proposed method); (

**d**) Bicubic interpolation.

**Figure 6.**Residual images between the super-resolution images and original image of the area 0_2. (

**a**) ScSR (RMSE: 1.157 m); (

**b**) Bicubic interpolation (RMSE: 1.713 m). The same colour scale is used in both images.

**Figure 7.**(

**a**) Visualisation of 256 bases from the “0_0” dictionary as embedded by UMAP. (

**b**) Distribution of clusters of bases on the embedded space by UMAP. The colour shading of the symbols on (

**b**) corresponds to the sum of the absolute values of the coefficients of each basis in the reconstruction. “c1” represents “cluster 1”, and the same applies to “c2” and beyond.

**Figure 8.**Distribution of clusters 8 and 14 based on the clustering (Figure 7) in the reconstructed area. The small figure to the top left of the coloured maps shows the representative basis within the cluster, that is, the one with the sum of the absolute values of the coefficients throughout the reconstruction is the largest in each cluster. The left figure shows the original image of the reconstructed area. The geological edifices in the green and yellow circles are the small sea knolls. See text for details.

**Figure 9.**Distribution of the 23 clusters in the reconstructed area. The small figure to the left of each map shows the representative basis of the cluster. The scale of the colour bar is identical to that shown in Figure 8.

**Table 1.**RMSEs for eight regions reconstructed using the dictionaries learned in the other seven other sea areas, the RMSEs for bicubic interpolation, and their ratio. The unit of RMSE is metre in this table.

Reconstruct Area | 0_0 | 0_1 | 0_2 | 0_3 | 1_0 | 1_1 | 1_2 | 1_3 | Mean |
---|---|---|---|---|---|---|---|---|---|

ScSR | 0.803 | 1.183 | 1.156 | 1.853 | 1.193 | 1.259 | 1.414 | 1.723 | 1.323 |

bicubic | 1.066 | 1.458 | 1.713 | 2.501 | 1.794 | 1.789 | 2.293 | 2.524 | 1.892 |

ScSR/bicubic | 0.753 | 0.812 | 0.675 | 0.741 | 0.665 | 0.703 | 0.617 | 0.682 | 0.709 |

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**MDPI and ACS Style**

Yutani, T.; Yono, O.; Kuwatani, T.; Matsuoka, D.; Kaneko, J.; Hidaka, M.; Kasaya, T.; Kido, Y.; Ishikawa, Y.; Ueki, T.;
et al. Super-Resolution and Feature Extraction for Ocean Bathymetric Maps Using Sparse Coding. *Sensors* **2022**, *22*, 3198.
https://doi.org/10.3390/s22093198

**AMA Style**

Yutani T, Yono O, Kuwatani T, Matsuoka D, Kaneko J, Hidaka M, Kasaya T, Kido Y, Ishikawa Y, Ueki T,
et al. Super-Resolution and Feature Extraction for Ocean Bathymetric Maps Using Sparse Coding. *Sensors*. 2022; 22(9):3198.
https://doi.org/10.3390/s22093198

**Chicago/Turabian Style**

Yutani, Taku, Oak Yono, Tatsu Kuwatani, Daisuke Matsuoka, Junji Kaneko, Mitsuko Hidaka, Takafumi Kasaya, Yukari Kido, Yoichi Ishikawa, Toshiaki Ueki,
and et al. 2022. "Super-Resolution and Feature Extraction for Ocean Bathymetric Maps Using Sparse Coding" *Sensors* 22, no. 9: 3198.
https://doi.org/10.3390/s22093198