# A Multiresolution Grid Structure Applied to Seafloor Shape Modeling

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## Abstract

**:**

## 1. Introduction

#### 1.1. Introduction to the Multibeam Swath Bathymetry

#### 1.2. Existing Methods

## 2. Materials and Methods

#### 2.1. Initial Assumptions

#### 2.2. IHO Standards

- special—areas where under-keel clearance is critical (a = 0.25 m, b = 0.0075),
- 1a—areas shallower than 100 m where under-keel clearance is less critical but features of concern to surface shipping may exist (a = 0.5 m, b = 0.013),
- 1b—areas shallower than 100 m where under-keel clearance is not considered to be an issue for the type of surface shipping expected to transit the area (a = 0.5 m, b = 0.013),
- 2—areas generally deeper than 100 m where a general description of the sea floor is considered adequate (a = 1 m, b = 0.023).

#### 2.3. Characteristics of the Analyzed Method

- a significant reduction of data volume stored in a DTM (in comparison to the traditional, uniform grid structure);
- high reconstruction accuracy (maximal error in any model node does not exceed a maximal error provided by the operator—the parental error ($PE$));
- a possibility for a fast transformation between regular and multiresolution girds in both ways, often found in the bathymetric data processing;
- a structure that ensures a precise description of both small and irregular objects found on the sea floor, e.g., wrecks, rocks, shear areas, and irregular land forms, as well as large plain areas;
- efficient storage of information about surface contours, both outer and inner (blanks and holes in the surface);
- block-based processing, in order to easily manage large areas of DTM.

#### 2.4. Method Description

Algorithm 1 Create multiresolution depth grid. | |

function EncodeGrid ($in$, $out$) | ▹ IN: source grid $in$, |

▹ OUT: destination structure $out$ | |

load $in$ | |

while size $\left(in\right)>4$ do | ▹$size\left(\right)$ returns the dimension of the block |

$error=$ max{abs(mean $\left(in\right))-$max $\left(in\right),$ abs (mean $\left(in\right))-$min $\left(in\right)\}$ | |

if $error\le PE$ then | |

$level=$ log2 (size $\left(in\right))$ | |

$mean\_val=$ mean $\left(in\right)$ | |

create new cell ${c}_{i}$ in $out$ | |

save $level$ in ${c}_{i}$ | |

save $mean\_val$ in ${c}_{i}$ | |

else | |

decompose $(in,4)$ | ▹ decompose $in$ into 4 square submatrices |

▹ and repeat recursively for every submatrix | |

end if | |

end while | |

save $out$ | ▹ save all stored data in destination structure $out$ |

end function |

Algorithm 2 Reconstruct depth grid from multiresolution representation. | |

function ReconstructGrid ($in$, $out$) | ▹ IN: source structure $in$, |

▹ OUT: reconstructed grid $out$ | |

load $in$ | |

initialize $out$ | ▹ create empty matrix $out$ |

for ${c}_{i}$ in $in$ do | ▹ for each cell ${c}_{i}$ in input structure $in$ |

get $level$ | ▹$level$ stores 0 for block 1 × 1, 1 for block 2 × 2 etc. |

get $mean\_val$ | |

append $(mean\_val\times $ones $\left({2}^{level}\right),out)$ | ▹ append a submatrix |

▹ of $mean\_val$ to the matrix $out$ | |

end for | |

for $level=0:(maximum\_level-1)$ do | |

for each submatrix ${s}_{i}$ in $out$ do | ▹ submatrix ${s}_{i}$ contains data after |

▹ sub-division of a larger matrix | |

concatenate $({s}_{i},{s}_{i+1},{s}_{i+2},{s}_{i+3})$ | ▹ concatenate all 4 neighbouring |

▹ submatrices and create one submatrix | |

▹ of $level=level+1$ and remove used matrices | |

end for | |

Store last remaining element of $out$ | |

end for | |

Trim $out$ to original size | |

end function |

## 3. Results and Discussion

#### 3.1. Test Data

#### 3.2. Test Protocol

#### 3.2.1. Gate Area

#### 3.2.2. Anchorage Area

#### 3.2.3. Swinging Area

#### 3.2.4. Wrecks Area

#### 3.3. Analysis of the Results

- for $PE$ = 1 cm, the compression ratio is low, approximately 50% for surfaces that do not vary much and 10–20% for surfaces that have more variation;
- for $PE$ = 5 cm, the compression ratio is equal to 60–75% for surfaces that do not vary much and 90–95% for surfaces that have more variation;
- for $PE$ = 10 cm and $PE$ = 25 cm, the compression ratio is higher than 90%.

## 4. Comparison with Other Methods

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Wessex Archaeology. Explore the Seafloor. 2010. Available online: http://ets.wessexarch.co.uk/recs/how-we-study-the-seafloor/geophysical-survey/ (accessed on 19 February 2018).
- Łubczonek, J.; Stateczny, A. Concept of Neural Model of the Sea Bottom Surface; Rutkowski, L., Kacprzyk, J., Eds.; Neural Networks and Soft Computing; Physica-Verlag HD: Heidelberg, Germany, 2003; pp. 861–866. [Google Scholar]
- Maleika, W. Development of a Method for the Estimation of Multibeam Echosounder Measurement Accuracy. Prz. Elektrotech.
**2012**, 88, 205–208. [Google Scholar] - Gosciewski, D. Selection of interpolation parameters depending on the location of measurement points. GISci. Remote Sens.
**2013**, 50, 515–526. [Google Scholar] - Wlodarczyk-Sielicka, M.; Stateczny, A. Clustering Bathymetric Data for Electronic Navigational Charts. J. Navig.
**2016**, 69, 1143–1153. [Google Scholar] [CrossRef] - Maleika, W.; Palczynski, M.; Frejlichowski, D. Interpolation Methods and the Accuracy of Bathymetric Seabed Models Based on Multibeam Echosounder Data; Pan, J.S., Chen, S.M., Nguyen, N.T., Eds.; Intelligent Information and Database Systems; Springer: Berlin/Heidelberg, Germany, 2012; pp. 466–475. [Google Scholar]
- Samet, H. Foundations of Multidimensional and Metric Data Structures (The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling); Morgan Kaufmann Publishers Inc.: San Francisco, CA, USA, 2005. [Google Scholar]
- Heckbert, P.S.; Garland, M. Survey of Polygonal Surface Simplification Algorithms. 1997. Available online: http://www.dtic.mil/dtic/tr/fulltext/u2/a461098.pdf (accessed on 19 February 2018).
- Cignoni, P.; Montani, C.; Scopigno, R. A comparison of mesh simplification algorithms. Comput. Graph.
**1998**, 22, 37–54. [Google Scholar] [CrossRef] - Luebke, D.P. A Developer’s Survey of Polygonal Simplification Algorithms. IEEE Comput. Graph. Appl.
**2001**, 21, 24–35. [Google Scholar] [CrossRef] - Gerstner, T. Multiresolution Compression and Visualization of Global Topographic Data. GeoInformatica
**2003**, 7, 7–32. [Google Scholar] [CrossRef] - Ben-Moshe, B.; Serruya, L.; Shamir, A. Image Compression Terrain Simplification. In Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG2007), Ottawa, ON, Canada, 20–22 August 2007; pp. 125–128. [Google Scholar]
- Pajarola, R. Overview of Quadtree-based Terrain Triangulation and Visualization; Volume 2, Issue 1 of Technical Report; Department of Information & Computer Science, University of California: Irvine, CA, USA, 2002. [Google Scholar]
- Lau, T.Y.; Li, Y.; Xie, Z.; Franklin, W.R. Sea Floor Bathymetry Trackline Surface Fitting Without Visible Artifacts Using ODETLAP. In Proceedings of the GIS’09 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, Seattle, WA, USA, 4–6 November 2009; ACM: New York, NY, USA, 2009; pp. 508–511. [Google Scholar]
- Xie, Z.; Franklin, W.R.; Cutler, B.; Andrade, M.A.; Inanc, M.; Tracy, D.M. Surface compression using over-determined Laplacian approximation. In Proceedings of the Advanced Signal Processing Algorithms, Architectures, and Implementations XVII, San Diego, CA, USA, 26–30 August 2007. [Google Scholar]
- Pajarola, R.; Gobbetti, E. Survey of Semi-regular Multiresolution Models for Interactive Terrain Rendering. Vis. Comput.
**2007**, 23, 583–605. [Google Scholar] [CrossRef] [Green Version] - Danovaro, E.; Floriani, L.D.; Magillo, P.; Puppo, E.; Sobrero, D. Level-of-detail for data analysis and exploration: A historical overview and some new perspectives. Comput. Graph.
**2006**, 30, 334–344. [Google Scholar] [CrossRef] - Mahdavi-Amiri, A.; Harrison, E.; Samavati, F. Hierarchical grid conversion. Comput. Aided Des.
**2016**, 79, 12–26. [Google Scholar] [CrossRef] - Weiss, K.; De Floriani, L. Simplex and Diamond Hierarchies: Models and Applications. Comput. Graph. Forum
**2011**, 30, 2127–2155. [Google Scholar] - Mahdavi-Amiri, A.; Samavati, F. Atlas of connectivity maps. Comput. Graph.
**2014**, 39, 1–11. [Google Scholar] [CrossRef] - Sadeghi, J.; Samavati, F.F. Smooth reverse subdivision. Comput. Graph.
**2009**, 33, 217–225. [Google Scholar] [CrossRef] - International Hydrographic Organization. IHO Standards for Hydrographic Surveys, Special Publication No. 44, 5th Edition. 2008. Available online: http://www.iho.int/iho_pubs/standard/S-44_5E.pdf (accessed on 27 July 2016).
- Forczmanski, P.; Maleika, W. Predicting the Number of DCT Coefficients in the Process of Seabed Data Compression. In Proceedings of the CAIP 2015 16th International Conference Computer Analysis of Images and Patterns, Valletta, Malta, 2–4 September 2015; pp. 77–87. [Google Scholar]
- Maleika, W.; Forczmanski, P. Wavelets in Adaptive Compression of Data Describing Sea-Bottom. In Proceedings of the ACS 2002 9th International Conference Advanced Computer Systems, Miedzyzdroje, Poland, 23–25 October 2002; pp. 381–387. [Google Scholar]
- Maleika, W.; Czapiewski, P. Evaluation of KLT method for controlled lossy compression of high-resolution seabed’s DTM. Earth Sci. Inform.
**2015**, 8, 595–607. [Google Scholar] [CrossRef] - Forczmanski, P.; Maleika, W. Near-Lossless PCA-Based Compression of Seabed Surface with Prediction. In Proceedings of the ICIAR 2015 12th International Conference Image Analysis and Recognition, Niagara Falls, ON, Canada, 22–24 July 2015; pp. 119–128. [Google Scholar]

**Figure 1.**Scheme of an MBES survey (based on [1]).

**Figure 7.**The visualization of the Swinging area represented with multiresolution depth grids for $PE$ = 1, 5, 10, and 20 cm.

Surface | Grid Resolution | Grid Area | No. Points | Binary Filesize |
---|---|---|---|---|

(meters) | (meters) | (Thousand Nodes) | (MBytes) | |

Achorage | 1 | 3008 × 1696 | 5101 | 30.38 |

Swinging | 0.75 | 2464 × 1760 | 4336 | 19.61 |

Gate | 0.5 | 1888 × 1632 | 3081 | 13.59 |

Wrecks | 0.01 | 1856 × 672 | 1247 | 4.00 |

Sub-Block Size | $\mathit{PE}$ | |||
---|---|---|---|---|

1 cm | 5 cm | 10 cm | 25 cm | |

1 × 1 | 686,516 | 183,424 | 57,428 | 4232 |

2 × 2 | 24,130 | 106,667 | 77,918 | 22,385 |

4 × 4 | 2959 | 8396 | 18,070 | 16,766 |

8 × 8 | 261 | 696 | 1595 | 4423 |

16 × 16 | 12 | 155 | 155 | 417 |

32 × 32 | 0 | 21 | 49 | 96 |

$CR$ | 5.5% | 60.4% | 79.5% | 93.6% |

Sub-Block Size | $\mathit{PE}$ | |||
---|---|---|---|---|

1 cm | 5 cm | 10 cm | 25 cm | |

1 × 1 | 6435 | 6431 | 6431 | 6431 |

2 × 2 | 15,884 | 2925 | 2925 | 2925 |

4 × 4 | 56,311 | 1447 | 1439 | 1439 |

8 × 8 | 26,529 | 1003 | 697 | 697 |

16 × 16 | 2454 | 1079 | 344 | 320 |

32 × 32 | 24 | 2871 | 3074 | 3080 |

$CR$ | 70.8% | 95.7% | 96.0% | 96.0% |

Sub-Block Size | $\mathit{PE}$ | |||
---|---|---|---|---|

1 cm | 5 cm | 10 cm | 25 cm | |

1 × 1 | 812,458 | 136,166 | 49,178 | 4838 |

2 × 2 | 101,880 | 96,733 | 46,388 | 18,473 |

4 × 4 | 4225 | 32,924 | 26,055 | 10,113 |

8 × 8 | 92 | 3242 | 7249 | 5700 |

16 × 16 | 0 | 137 | 623 | 1820 |

32 × 32 | 0 | 1 | 18 | 217 |

$CR$ | 20.1% | 76.6% | 88.7% | 96.4% |

Sub-Block Size | $\mathit{PE}$ | |||
---|---|---|---|---|

1 cm | 5 cm | 10 cm | 25 cm | |

1 × 1 | 109,570 | 8010 | 2798 | 2030 |

2 × 2 | 39,088 | 9526 | 3777 | 1305 |

4 × 4 | 4205 | 7391 | 2998 | 1084 |

8 × 8 | 118 | 1440 | 1755 | 636 |

16 × 16 | 0 | 285 | 347 | 484 |

32 × 32 | 0 | 11 | 68 | 148 |

$CR$ | 49.5% | 91.2% | 96.1% | 98.1% |

**Table 6.**CR values [%] for different benchmark surfaces and other previously developed near-lossless compression methods ($PE$ = 5 cm). DCT: discrete cosine transform; PCA: principal component analysis.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Maleika, W.; Koziarski, M.; Forczmański, P.
A Multiresolution Grid Structure Applied to Seafloor Shape Modeling. *ISPRS Int. J. Geo-Inf.* **2018**, *7*, 119.
https://doi.org/10.3390/ijgi7030119

**AMA Style**

Maleika W, Koziarski M, Forczmański P.
A Multiresolution Grid Structure Applied to Seafloor Shape Modeling. *ISPRS International Journal of Geo-Information*. 2018; 7(3):119.
https://doi.org/10.3390/ijgi7030119

**Chicago/Turabian Style**

Maleika, Wojciech, Michał Koziarski, and Paweł Forczmański.
2018. "A Multiresolution Grid Structure Applied to Seafloor Shape Modeling" *ISPRS International Journal of Geo-Information* 7, no. 3: 119.
https://doi.org/10.3390/ijgi7030119