# Geodesic Length Measurement in Medical Images: Effect of the Discretization by the Camera Chip and Quantitative Assessment of Error Reduction Methods

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. In-Silico Model

- Choose a function (randomly).
- Choose a window width s (in pixels) in the image within the limits of [65 350] (randomly) (the mathematical function will be projected into this window).
- The function will be rotated with a random angle $\alpha $.
- The windowed and rotated mathematical function will be projected onto the resolution grid and thereby discretized.

#### 2.2. Centerline Extraction

#### 2.2.1. Erosion

#### 2.2.2. Voronoi Diagram

#### 2.2.3. Centerline Post-Processing

#### 2.3. Length Measurement

#### 2.3.1. Ground Truth Length

#### 2.3.2. Discrete Length of Centerline

#### 2.3.3. Continuous Length by Polynomial Approximation

#### 2.3.4. Continuous Length by Bézier Curve

#### 2.4. Physical Length Measurement

- Carl Zeiss Meditec AG PENTERO
^{®}900 (Surgical microscope) - Silicone tubes RCT THOMAFLUID
^{®} - Rotational plate (Thorlabs PR01(/M))
- Blood analog (52.4 mL demineralized water, 41.5 mL Glycerin (99.5%) and 6.7 g protein powder)
- ICG (PULSION Medical Systems SE)

#### 2.5. Evaluation

- Continuous ground truth length
- Discrete ground truth length without any centerline reconstruction
- Discrete length with the centerline reconstruction by erosion
- Discrete length with the centerline reconstruction by Voronoi diagram
- Continuous length with the centerline reconstruction by erosion and interpolation by Bézier
- Continuous length with the centerline reconstruction by erosion and interpolation by polynomial approximation
- Continuous length with the centerline reconstruction by Voronoi diagram and interpolation by Bézier
- Continuous length with the centerline reconstruction by Voronoi diagram and interpolation by polynomial approximation

## 3. Results

#### 3.1. In Silico Results

#### 3.2. Experimental Results

## 4. Discussion and Conclusions

## 5. Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ICG | Indocyanine Green |

ROI | Region of interest |

PAL | Phase Alternating Line |

## References

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**Figure 1.**An overview of the methods used in this paper. Section 2.1, Section 2.2 and Section 2.3 deal with the in silico method and Section 2.4 with the experimental method.

**Figure 2.**To the

**left**the Voronoi diagram of a segmentation mask, to the

**right**a zoom into a section of this diagram—blue are the Voronoi edges, red is the centerline extracted from the Voronoi edges.

**Figure 3.**To the

**left**, an incorrect measurement due to a step-like connection with a distance of 2, to the

**right**, the corrected centerline with a measured distance of $\sqrt{2}$.

**Figure 4.**An exemplary image of a silicone tube filled with Indocyanine Green (ICG). It is fixed to a rotational board and the red line indicates 0° of rotation.

**Figure 5.**In silico simulation of two vessel segmentations.

**Left**: Section of a parabola with a stenosis.

**Right**: Bifurcation including sections of two Gaussian bells and a sinusoidal.

**Figure 6.**Relative length error using the erosion method in dependency on the angle for straight silicone tubes. The angle is measured to the horizontal pixel grid structure.

**Table 1.**Mathematical function, their formula and variables [25].

Function Type | Detailed Type | Formula | Variables’ Limit |
---|---|---|---|

Straight lines | $f\left(x\right)=tan\left(\theta \right)\xb7x$ | $\theta =[-{45}^{\circ},{45}^{\circ}]$, $x\phantom{\rule{0.222222em}{0ex}}is\phantom{\rule{0.222222em}{0ex}}arbitrary$ | |

Parabolas | Left arm | $f\left(x\right)=a\phantom{\rule{3.33333pt}{0ex}}\xb7{x}^{2}$ | $a=[1,2]$ |

${x}_{low}=[-\sqrt{2s},-2]$, ${x}_{high}=[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2,0]$ | |||

Right arm | $a=[1,2]$ | ||

${x}_{low}=[0,\sqrt{2s}-2]$, ${x}_{high}=[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2,\sqrt{2s}]$ | |||

Vertex | $a=[1,2]$ | ||

${x}_{low}=[-\sqrt{2s},-1]$, ${x}_{high}=[1,\sqrt{2s}]$ | |||

L.a. inverted | $a=[-1,-2]$ | ||

${x}_{low}=[-\sqrt{2s},-3]$, ${x}_{high}=[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2,1]$ | |||

R.a. inverted | $a=[-1,-2]$ | ||

${x}_{low}=[1,\sqrt{2s}-2]$, ${x}_{high}=[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2,\sqrt{2s}]$ | |||

V. inverted | $a=[-1,-2]$ | ||

${x}_{low}=[-\sqrt{2s},-2]$, ${x}_{high}=[2,\sqrt{2s}]$ | |||

Polynomials | $f\left(x\right)={\sum}_{k=4}^{n}{a}_{k}\xb7{x}^{k}$ | $n\le 15$, $x=[-5,5]$ | |

Sinusoids | Low frequency | $f\left(x\right)=a\xb7sin(f\xb7\pi \xb7x)$ | $a=[0.2s,0.7s]$, $f=[1,2]$ |

${x}_{low}\phantom{\rule{0.0pt}{0ex}}=\phantom{\rule{0.0pt}{0ex}}[0,2\pi ]$, ${x}_{high}\phantom{\rule{0.0pt}{0ex}}=\phantom{\rule{0.0pt}{0ex}}[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}0.5\pi ,{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2\pi ]$ | |||

High frequency | $a=[0.1s,0.1+\frac{s}{2f}]$, $f=[1.5,7]$ | ||

${x}_{low}\phantom{\rule{0.0pt}{0ex}}=\phantom{\rule{0.0pt}{0ex}}[0,2\pi ]$, ${x}_{high}\phantom{\rule{0.0pt}{0ex}}=\phantom{\rule{0.0pt}{0ex}}[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}0.5\pi ,{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2\pi ]$ | |||

Curved Waves | Quadratic | $f\left(x\right)=q\xb7{x}^{2}+$ | $q=[-10,15]$, $a=[0.05s,0.15s]$, $f=[0.5,2]$ |

$a\xb7sin(2\xb7\pi \xb7f\xb7x)$ | ${x}_{low}=[-3,3]$, ${x}_{high}=[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2,{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}6]$ | ||

Cubic | $f\left(x\right)=c\xb7{x}^{3}+q\xb7{x}^{2}+$ | $c=[-15,10]$, $q=[-15,30]$ | |

$a\xb7sin(2\xb7\pi \xb7f\xb7x)$ | $a=[0.1s,0.3s]$, $f=[0.5,2]$ | ||

${x}_{low}=[-3,3]$, ${x}_{high}=[{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}2,{x}_{low}\phantom{\rule{0.0pt}{0ex}}+\phantom{\rule{0.0pt}{0ex}}6]$ | |||

Bell curves | Gaussian | $f\left(x\right)=a\phantom{\rule{3.33333pt}{0ex}}\xb7{e}^{-\frac{{x}^{2}}{2{\sigma}^{2}}}$ | $a=[0.2s,s]$, $\sigma =[0.2,3]$ |

${x}_{low}=[-5,1]$, ${x}_{high}=[1,-5]$ | |||

G. inverted | $a=[-0.2s,-s]$, $\sigma =[0.2,3]$ | ||

${x}_{low}=[-5,1]$, ${x}_{high}=[1,-5]$ | |||

Polynomials | $f\left(x\right)=a\xb716\xb7$ | $a=[0.2s,s]$ | |

$({x}^{4}-2\xb7{x}^{3}+{x}^{2})$ | ${x}_{low}=0$, ${x}_{high}=1$ | ||

P. inverted | $a=[-0.2s,-s]$ | ||

${x}_{low}=0$, ${x}_{high}=1$ | |||

Bifurcations | Combination of all |

Inner Diameter in mm | Wall Thickness in mm |
---|---|

1 | 1 |

2 | 0.5 |

3 | 0.5 |

4 | 0.5 |

5 | 0.5 |

6 | 1 |

7 | 1 |

8 | 1 |

**Table 3.**Mean relative error of the discrete length measurement with no reconstruction compared to the continuous ground truth [25].

Function Type | Relative Error | Quantity in the Set |
---|---|---|

Straight lines | 2.3% | 184 |

Parabolas | 7.2% | 312 |

Polynomials | 6.3% | 168 |

Sinusoids | 6.8% | 132 |

Curved Waves | 7.6% | 102 |

Bell curves | 8.5% | 108 |

Bifurcations | 6.7% | 198 |

Mean of all | 6.3% |

**Table 4.**Mean relative error of the length measurement methods in % after centerline extraction by erosion compared to the continuous ground truth. Significance was proven for the reduction of the error compared to the discrete length for each function type ${}^{****}$ p < 0.0001.

Discrete μ ± σ | Bézier μ ± σ | Polynomial μ ± σ | |
---|---|---|---|

Straight lines | 5.2 ± 3.8 | 2.6 ± 1.8 ${}^{****}$ | 2.1 ± 1.6 ${}^{****}$ |

Parabolas | 7.5 ± 4.0 | 2.4 ± 1.5 ${}^{****}$ | 1.9 ± 1.3 ${}^{****}$ |

Polynomials | 6.5 ± 4.8 | 3.9 ± 6.2 ${}^{****}$ | 7.2 ± 7.5 ${}^{****}$ |

Sinusoids | 8.5 ± 5.8 | 4.2 ± 7.8 ${}^{****}$ | 6.6 ± 12.6 ${}^{****}$ |

Curved waves | 7.6 ± 2.6 | 2.5 ± 1.4 ${}^{****}$ | 2.6 ± 2.2 ${}^{****}$ |

Bell curves | 10.4 ± 3.2 | 3.1 ± 1.7 ${}^{****}$ | 1.9 ± 1.4 ${}^{****}$ |

Bifurcations | 6.1 ± 5.4 | 2.1 ± 5.3 ${}^{****}$ | 2.6 ± 5.1 ${}^{****}$ |

Mean of all | 7.0 ± 4.8 | 2.7 ± 4.5 ${}^{****}$ | 3.2 ± 5.8 ${}^{****}$ |

**Table 5.**Mean relative error of the length measurement methods in % after centerline extraction by Voronoi diagrams compared to the continuous ground truth. Significance was proven for the reduction of the error compared to the discrete length for each function type ${}^{****}$ p < 0.0001.

Discrete μ ± σ | Bézier μ ± σ | Polynomial μ ± σ | |
---|---|---|---|

Straight lines | 5.1 ± 5.8 | 3.3 ± 5.8 ${}^{****}$ | 3.2 ± 6.0 ${}^{****}$ |

Parabolas | 8.6 ± 7.0 | 4.2 ± 8.5 ${}^{****}$ | 4.5 ± 8.7 ${}^{****}$ |

Polynomials | 9.1 ± 8.5 | 7.7 ± 11.0 ${}^{****}$ | 11.2 ± 11.3 ${}^{****}$ |

Sinusoids | 8.6 ± 5.0 | 5.6 ± 7.8 ${}^{****}$ | 8.9 ± 12.8 ${}^{****}$ |

Curved waves | 8.3 ± 4.5 | 4.5 ± 7.0 ${}^{****}$ | 5.7 ± 7.5 ${}^{****}$ |

Bell curves | 8.2 ± 4.2 | 3.4 ± 5.1 ${}^{****}$ | 3.4 ± 5.6 ${}^{****}$ |

Bifurcations | 7.9 ± 11.2 | 4.6 ± 11.6 ${}^{****}$ | 5.9 ± 11.2 ${}^{****}$ |

Mean of all | 7.9 ± 8.1 | 4.7 ± 9.2 ${}^{****}$ | 5.9 ± 10.0 ${}^{****}$ |

Erosion | Voronoi | |
---|---|---|

Bézier curve | 13.08 s | 21.09 s |

Polynomial approximation | 2.33 s | 10.34 s |

**Table 7.**Mean relative error of the length measurement methods in % after the centerline extraction by both methods and the spatial interpolation compared to the ground truth for the experimental data set (mean also over all angles). Significance was proven for the reduction of the error compared to the discrete length for each function type ${}^{****}$ p < 0.0001. Angle resolved relative errors are given in Table 8 and Table 9.

Discrete μ ± σ | Bézier μ ± σ | Polynomial μ ± σ | |
---|---|---|---|

Erosion | 4.7 ± 3.0 | 1.9 ± 1.3 ${}^{****}$ | 1.6 ± 1.6 ${}^{****}$ |

Voronoi | 5.0 ± 3.0 | 2.0 ± 1.3 ${}^{****}$ | 1.5 ± 1.5 ${}^{****}$ |

**Table 8.**Mean relative error of the length measurement methods in % after the centerline extraction by erosion and the spatial interpolation compared to the ground truth for the experimental data set. Significance was proven for the reduction of the error compared to the discrete length as indicated ${}^{*}$ p < 0.05, ${}^{***}$p < 0.001, ${}^{****}$ p < 0.0001.

Discrete | Bézier | Polynomial | |
---|---|---|---|

Angle | μ ± σ | μ ± σ | μ ± σ |

0° | 2.4 ± 1.3 | 1.8 ±1.5 ${}^{*}$ | 1.5 ± 1.5 ${}^{***}$ |

15° | 7.0 ± 2.4 | 2.3 ± 1.0 ${}^{****}$ | 1.7 ± 1.6 ${}^{****}$ |

30° | 6.6 ± 2.3 | 1.9 ± 1.3 ${}^{****}$ | 1.6 ± 1.6 ${}^{****}$ |

45° | 2.2 ± 1.7 | 1.6 ± 1.6 ${}^{*}$ | 1.6 ± 1.7 ${}^{*}$ |

60° | 7.0 ± 2.3 | 1.8 ± 1.3 ${}^{****}$ | 1.4 ± 1.6 ${}^{****}$ |

75° | 5.8 ± 2.6 | 2.1 ± 1.2 ${}^{****}$ | 1.9 ± 1.7 ${}^{****}$ |

90° | 1.9 ± 1.3 | 1.7 ± 1.6 | 1.6 ± 1.7 |

**Table 9.**Mean relative error of the length measurement methods in % after the centerline extraction by Voronoi diagrams and the spatial interpolation compared to the ground truth for the experimental data set. Significance was proven for the reduction of the error compared to the discrete length as indicated ${}^{*}$p < 0.05, ${}^{**}$p < 0.01, ${}^{***}$p < 0.001, ${}^{****}$ p < 0.0001.

Discrete | Bézier | Polynomial | |
---|---|---|---|

Angle | μ ± σ | μ ± σ | μ ± σ |

0° | 2.4 ± 1.6 | 1.8 ±1.3 | 1.5 ± 1.4 ${}^{*}$ |

15° | 7.4 ± 2.3 | 2.6 ± 1.0 ${}^{****}$ | 1.6 ± 1.2 ${}^{****}$ |

30° | 6.8 ± 2.2 | 2.1 ± 1.3 ${}^{****}$ | 1.7 ± 1.6 ${}^{****}$ |

45° | 2.5 ± 1.5 | 1.6 ± 1.5 ${}^{**}$ | 1.5 ± 1.6 ${}^{***}$ |

60° | 7.1 ± 2.2 | 1.9 ± 1.1 ${}^{****}$ | 1.5 ± 1.5 ${}^{****}$ |

75° | 6.4 ± 2.2 | 2.3 ± 1.0 ${}^{****}$ | 1.5 ± 1.6 ${}^{****}$ |

90° | 2.3 ± 1.7 | 1.7 ± 1.5 | 1.5 ± 1.6 ${}^{**}$ |

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**MDPI and ACS Style**

Naber, A.; Berwanger, D.; Nahm, W.
Geodesic Length Measurement in Medical Images: Effect of the Discretization by the Camera Chip and Quantitative Assessment of Error Reduction Methods. *Photonics* **2020**, *7*, 70.
https://doi.org/10.3390/photonics7030070

**AMA Style**

Naber A, Berwanger D, Nahm W.
Geodesic Length Measurement in Medical Images: Effect of the Discretization by the Camera Chip and Quantitative Assessment of Error Reduction Methods. *Photonics*. 2020; 7(3):70.
https://doi.org/10.3390/photonics7030070

**Chicago/Turabian Style**

Naber, Ady, Daniel Berwanger, and Werner Nahm.
2020. "Geodesic Length Measurement in Medical Images: Effect of the Discretization by the Camera Chip and Quantitative Assessment of Error Reduction Methods" *Photonics* 7, no. 3: 70.
https://doi.org/10.3390/photonics7030070