# A Novel Robust Topological Denoising Method Based on Homotopy Theory for Virtual Colonoscopy

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## Abstract

**:**

## 1. Introduction

- (1)
- Mathematical rigor. Our proposed method is based on rigorous mathematical theory in homotopy, which guarantees the computation of the non-trivial loops.
- (2)
- Novel framework. The proposed algorithm is novel and has been first applied to colon surface denoising in virtual colonoscopy.
- (3)
- Robust computation. Compared to the State-of-the-Art topological denoising method, our method is more robust, based on the experimental results.

## 2. Materials and Methods

#### 2.1. Theoretic Background

**Definition**

**1**(Homotopy)

**.**

**Definition**

**2**(Fundamental Group)

**.**

**Definition**

**3**(Algebraic Intersection Number)

**.**

**Definition**

**4**(Canonical Basis)

**.**

**Theorem**

**1**(Surface Fundamental Group Canonical Representation)

**.**

**Definition**

**5**(Exponential Map)

**.**

**Definition**

**6**(Cut Locus)

**.**

**Theorem**

**2**(Main)

**.**

**Proof.**

#### 2.2. Overview

#### 2.3. Algorithms

**Definition**

**7**(Simplex)

**.**

**Definition**

**8**(Simplicial Complex)

**.**

- 1.
- Every face of a simplex from $\mathcal{K}$ is also in $\mathcal{K}$.
- 2.
- The non-empty intersection of any two simplices ${\sigma}_{1},{\sigma}_{2}\in \mathcal{K}$ is a face of both ${\sigma}_{1}$ and ${\sigma}_{2}$.

#### 2.3.1. Cut Graph

Algorithm 1 Algorithm for Cut Graph |

Require: A closed triangle mesh MEnsure: $\mathcal{C}$ is a cut graph of M1: Compute the dual mesh $\overline{M}$ of the input mesh M; 2: Compute a spanning tree $\overline{T}$ of $\overline{M}$; 3: The cut graph is given by $\mathcal{C}:=\{e\in M|\overline{e}\notin \overline{T}\}$; 4: Prune all the leaves of $\mathcal{C}$ recursively. |

#### 2.3.2. Shortest Loop

Algorithm 2 Algorithm for Shortest Loop |

Require: A closed triangle mesh MEnsure: The shortest non-trivial loop $\gamma $ of M1: Compute the cut graph $\mathcal{C}$ of M, using Algorithm 1; 2: Slice M along $\mathcal{C}$, to obtain a simply connected mesh $\tilde{M}$; 3: for all vertex ${v}_{i}\in \mathcal{C}$ with $\mathrm{deg}\left({v}_{i}\right)=k$ do4: Find all $\{{v}_{i}^{1},{v}_{i}^{2},\dots ,{v}_{i}^{k}\}\subset \partial \tilde{M}$; 5: for all pair $({v}_{i}^{j},{v}_{i}^{k})$ on the boundary $\partial \tilde{M}$ do6: Compute the shortest path ${\tilde{\gamma}}_{i}^{jk}$, using Dijkstra’s algorithm; 7: Find the loop ${\gamma}_{i}^{jk}\subset M$ corresponding to ${\tilde{\gamma}}_{i}^{jk}\subset \tilde{M}$; 8: end for9: end for10: Sort all the shortest loops ${\gamma}_{i}^{jk}$ in ascending order by their total lengths; 11: Return the shortest loop $\gamma $. |

#### 2.3.3. Topological Surgery

Algorithm 3 Algorithm for Topological Surgery |

Require: A closed triangle mesh M, a non-trivial loop $\gamma $Ensure: A mesh N with one handle removed from M1: Slice M along $\gamma $ to obtain a mesh ${M}_{\gamma}$, $\partial {M}_{\gamma}={\gamma}_{0}-{\gamma}_{1}$; 2: for all boundary component ${\gamma}_{k}$, $k=0,1$ do3: ${w}_{k}\leftarrow 0$; 4: for all vertex ${v}_{i}\in {\gamma}_{k}$ do5: ${w}_{k}\leftarrow {w}_{k}+{v}_{i}$; 6: end for7: ${w}_{k}\leftarrow {w}_{k}/\left|{\gamma}_{k}\right|$; 8: for all edge ${e}_{j}\in {\gamma}_{k}$ do9: ${e}_{j}$ and ${w}_{k}$ form a triangle face ${f}_{j}^{k}$; 10: ${M}_{\gamma}\leftarrow {M}_{\gamma}\cup {f}_{j}^{k}$; 11: end for12: end for13: $N\leftarrow {M}_{\gamma}$. |

Algorithm 4 Topological Denoising Algorithm |

Require: A high-genus closed triangle mesh MEnsure: The mesh N with all handles removed from M1: Compute the Euler number $\left(M\right)$ and the genus g of M; 2: Set ${M}_{1}\leftarrow M$; 3: for $k=1,2,\dots ,g$ do4: Compute the shortest loop ${\gamma}_{k}$ on ${M}_{k}$, using Algorithm 2; 5: Perform topological surgery on ${M}_{k}$ along ${\gamma}_{k}$, to obtain ${M}_{k+1}$, using Algorithm 3; 6: end for7: Set $N\leftarrow {M}_{g+1}$. |

## 3. Results

#### 3.1. Visual Evaluation on General Test Models

#### 3.2. Topological Denoising for Two Colon Datasets

#### 3.3. Colon Flattening in Virtual Colonoscopy

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Poincaré Duality: (

**right**) a triangulated mesh M, indicated by the black triangles, corresponds to a dual mesh $\overline{M}$, as shown in the green cells; (

**left**) a 2D simplex $\Delta $ of M containing vertex $\sigma $.

**Figure 4.**A cut graph of a genus-2 surface, using Algorithm 1: (

**a**) the computed initial cut graph represented by the red tree structure; (

**b**) the pruned cut graph is obtained by removing all leaf nodes with only one link in the initial cut graph in (

**a**).

**Figure 6.**Results of our method for visual illustration on general test models: double torus, genus-3 model, amphora, genus-6 model, deco-cube, knotty, LoveMe model, and TwoKids model.

**Figure 7.**Results of our method on a colon wall surface: (

**left**) a colon wall mesh with topological noises denoted by the green loops; (

**middle**,

**right**) zoom-in views of the two regions marked with red boxes in the left figure.

**Figure 8.**Result of flattening the 3D colon wall surface to a 2D plane after topological denoising using the proposed algorithm: (

**top**) the original 3D colon surface before denoising, with green highlighted loops, and the corresponding colon surface after denoising; (

**bottom**) the 2D flattened colon mesh.

**Figure 9.**Result of applying the proposed topological denoising method to colon data with a clinically detected polyp: (

**a**,

**b**) two different views of the 3D colon surface, with a yellow arrow pointing to the location of the polyp highlighted by the red points; (

**c**,

**d**) endoscopic views of the polyp on the colon wall; (

**e**) a 2D visualization of the polyp on the flattened colon surface.

Mesh | #Faces | #Genus |
---|---|---|

Genus-6 model | 2 K | 6 |

Double torus | 7 K | 2 |

Knotty | 10 K | 2 |

Genus-3 model | 12 K | 3 |

Amphora | 20 K | 2 |

LoveMe | 50 K | 3 |

TwoKids | 80 K | 3 |

Deco-cube | 120 K | 5 |

**Table 2.**Comparison, on the first colon dataset with 10 colon meshes, of persistent homology method (denoted by PHM) to our method.

Success? | All Loops Found? | #Vertices of All Loops | ||||
---|---|---|---|---|---|---|

Mesh (#Faces, #Genus) | PHM | Our | PHM | Our | PHM | Our |

Colon1 (127 K, 67) | No | Yes | No | Yes | NA | 311 |

Colon2 (44 K, 13) | Yes | Yes | Yes | Yes | 65 | 61 |

Colon3 (158 K, 31) | No | Yes | No | Yes | NA | 141 |

Colon4 (152 K, 8) | Yes | Yes | Yes | Yes | 56 | 28 |

Colon5 (140 K, 38) | Yes | Yes | Yes | Yes | 660 | 170 |

Colon6 (133 K, 30) | Yes | Yes | Yes | Yes | 177 | 141 |

Colon7 (167 K, 14) | Yes | Yes | Yes | Yes | 161 | 66 |

Colon8 (176 K, 13) | Yes | Yes | Yes | Yes | 131 | 67 |

Colon9 (236 K, 7) | Yes | Yes | Yes | Yes | 24 | 23 |

Colon10 (147 K, 19) | Yes | Yes | Yes | Yes | 118 | 74 |

**Table 3.**Comparison, on the second colon dataset with 20 colon meshes, of persistent homology method (denoted by PHM) to our method.

Success? | All Loops Found? | #Vertices of All Loops | ||||
---|---|---|---|---|---|---|

Mesh (#Faces, #Genus) | PHM | Our | PHM | Our | PHM | Our |

Colon1 (1091 K, 4) | Yes | Yes | Yes | Yes | 77 | 24 |

Colon2 (1679 K, 6) | Yes | Yes | Yes | Yes | 115 | 65 |

Colon3 (1679 K, 6) | Yes | Yes | Yes | Yes | 118 | 65 |

Colon4 (1528 K, 26) | Yes | Yes | Yes | Yes | 1104 | 492 |

Colon5 (1181 K, 17) | Yes | Yes | Yes | Yes | 231 | 119 |

Colon6 (1350 K, 12) | Yes | Yes | Yes | Yes | 321 | 124 |

Colon7 (1185 K, 3) | Yes | Yes | Yes | Yes | 60 | 36 |

Colon8 (1144 K, 9) | Yes | Yes | Yes | Yes | 158 | 107 |

Colon9 (1389 K, 17) | Yes | Yes | Yes | Yes | 675 | 436 |

Colon10 (1259 K, 2) | Yes | Yes | Yes | Yes | 11 | 11 |

Colon11 (1204 K, 4) | Yes | Yes | Yes | Yes | 116 | 58 |

Colon12 (1692 K, 2) | Yes | Yes | Yes | Yes | 122 | 36 |

Colon13 (1236 K, 11) | Yes | Yes | Yes | Yes | 206 | 129 |

Colon14 (1300 K, 11) | Yes | Yes | Yes | Yes | 298 | 250 |

Colon15 (1428 K, 17) | Yes | Yes | Yes | Yes | 326 | 213 |

Colon16 (1631 K, 22) | No | Yes | No | Yes | NA | 410 |

Colon17 (1531 K, 8) | Yes | Yes | Yes | Yes | 409 | 277 |

Colon18 (1774 K, 11) | Yes | Yes | Yes | Yes | 210 | 83 |

Colon19 (1482 K, 18) | Yes | Yes | Yes | Yes | 483 | 454 |

Colon20 (1202 K, 19) | No | Yes | No | Yes | NA | 177 |

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## Share and Cite

**MDPI and ACS Style**

Ma, M.; Chen, W.; Lei, N.; Gu, X.
A Novel Robust Topological Denoising Method Based on Homotopy Theory for Virtual Colonoscopy. *Axioms* **2023**, *12*, 942.
https://doi.org/10.3390/axioms12100942

**AMA Style**

Ma M, Chen W, Lei N, Gu X.
A Novel Robust Topological Denoising Method Based on Homotopy Theory for Virtual Colonoscopy. *Axioms*. 2023; 12(10):942.
https://doi.org/10.3390/axioms12100942

**Chicago/Turabian Style**

Ma, Ming, Wei Chen, Na Lei, and Xianfeng Gu.
2023. "A Novel Robust Topological Denoising Method Based on Homotopy Theory for Virtual Colonoscopy" *Axioms* 12, no. 10: 942.
https://doi.org/10.3390/axioms12100942