# Fast PET Scan Tumor Segmentation Using Superpixels, Principal Component Analysis and K-Means Clustering

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## Abstract

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## 1. Introduction

## 2. Implementation

#### 2.1. Pre-Processing

_{enh}is contrast enhanced image.

#### 2.2. Feature Extraction

- (1)
- We computed the average size of the superpixel as shown in Equation (2).$$M=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{n}_{i}$$
_{i}is the number of pixels in ith superpixel. Then, M is an average number of pixels per superpixel. - (2)
- Then, the size of each superpixel is made same as that of the average one by padding some pixel value to the smaller size superpixel and removing some intensity value from the large size superpixels. Instead of appending random intensity values to smaller sized superpixels, we pad by repeating the last pixels value of the superpixel itself. Finally, the superpixel matrix is generated as shown in Equation (3)$$S=\left[\begin{array}{cc}\begin{array}{ccc}{x}_{11}& {x}_{12}& {x}_{13}\\ {x}_{21}& {x}_{22}& {x}_{23}\\ .& .& .\end{array}& \begin{array}{ccc}..& .& {x}_{1N}\\ ..& .& {x}_{2N}\\ ..& .& .\end{array}\\ \begin{array}{ccc}.& .& .\\ .& .& .\\ {x}_{M1}& {x}_{M2}& {x}_{M3}\end{array}& \begin{array}{ccc}..& .& .\\ ..& .& .\\ ..& .& {x}_{MN}\end{array}\end{array}\right]$$

- (1)
- Compute average superpixel.$${S}_{a}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{S}_{i}$$
_{i}is the ith superpixel and S_{a}average superpixel. - (2)
- Determine the covariance of superpixels (C
_{s})$${C}_{s}=\frac{1}{N+1}\left(Y-{Y}_{a}{}^{t}\right){\left(Y-{Y}_{a}{}^{t}\right)}^{T}$$_{a}^{t}is the mean of transpose of Y. - (3)
- Calculate the eigensuperpixels (eigenvectors) and eigenvalues of the covariance matrix$${C}_{s}=P\Sigma {P}^{T}$$
_{1}, λ_{2}, …, λ_{N}, where, λ_{1}≥ λ_{2}≥ λ_{3}…≥ λ_{N}.The magnitude of eigenvalue shows the variance of the data in the direction of its corresponding eigensuperpixel. For N superpixels in Equation (3) above, total variance of intensities of the M- dimensional superpixels can be computed in terms of eigenvalues from Equation (7).$$V={\displaystyle \sum}_{k=1}^{N}{\mathsf{\lambda}}_{k}$$ - (4)
- Project the superpixels onto eigensuperpixels that contain most of variance of the data. In Equation (6), the number of principal components is same as the number of superpixels. As stated in [17], the eigenvectors or principal components that contain at least 95% of the variance of superpixels can represent the whole image with confidence and this is computed as shown in Equation (8). It reduces the dimensional space, as most of the information is contained in the first two or three largest eigenvalues.$$\sum}_{k=1}^{K}{\mathsf{\lambda}}_{k}\le 0.95$$$${Y}_{proj}={P}_{K}^{T}Y$$
_{k}is eigenvectors matrix that contains at least 95% of the variation in the image and P_{proj}is projection of superpixel matrix to P_{k}. - (5)
- Calculate the distance of each superpixel to average superpixel. Computing distance should consider the distribution of superpixels in the principal component coordinate system [12]. To incorporate this concept, we computed the distance along the principal components. Mathematically, this will be computing L
_{1}norm distance in the principal components coordinate system as shown in Equation (10) below.$$D\left({S}_{i}\right)=\left|\right|S{\prime}_{i}|{|}_{1}$$_{i}is coordinate of S_{i}relative to S_{a}in the principal component coordinate system, and D is L_{1}norm distance.

#### 2.3. Tumor Detection and Contouring

_{i}is the set of points that belong to cluster i, µ

_{i}is center of ith cluster, X is distance vector extracted above and D is square of the Euclidean distance.

## 3. Result and Discussion

_{1}norm distance of superpixels along principal components coordinate system from average superpixel. It has superpixel index as horizontal axis and distance as a vertical axis. For the input image in Figure 2a is 233 pixels by 328 pixels, the distance is 692 dimension vector. If distance was computed from all pixels to average pixels intensity the result will be a 76424-dimensional vector, which needs larger memory and high computation time. As depicted in Figure 5, most of the superpixels are within L

_{1}distance of 500, while 4 superpixels have a distance greater than 1500.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Implementation overview. PET: Positron emission tomography; PCA: principal component analysis.

**Figure 3.**Scatter plot of projection of superpixels of the enhanced image onto the principal components classification is small.

**Figure 5.**Superpixel k-means clustering and Heat map plot. (

**a**) Superpixel k-means clustering; (

**b**) heat map plot.

**Table 1.**Size of images, superpixels, distance vector after Principal Component Analysis (PCA), and execution time.

Size | No. Superpixels | Distance Vector Dimension | Execution Time (s) | |
---|---|---|---|---|

Image 1 | 233 × 328 | 692 | 692 | 2.2 |

Image 2 | 233 × 328 | 500 | 500 | 2.4 |

Image 3 | 681 × 572 | 660 | 660 | 2.55 |

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**MDPI and ACS Style**

Hagos, Y.B.; Minh, V.H.; Khawaldeh, S.; Pervaiz, U.; Aleef, T.A.
Fast PET Scan Tumor Segmentation Using Superpixels, Principal Component Analysis and K-Means Clustering. *Methods Protoc.* **2018**, *1*, 7.
https://doi.org/10.3390/mps1010007

**AMA Style**

Hagos YB, Minh VH, Khawaldeh S, Pervaiz U, Aleef TA.
Fast PET Scan Tumor Segmentation Using Superpixels, Principal Component Analysis and K-Means Clustering. *Methods and Protocols*. 2018; 1(1):7.
https://doi.org/10.3390/mps1010007

**Chicago/Turabian Style**

Hagos, Yeman Brhane, Vu Hoang Minh, Saed Khawaldeh, Usama Pervaiz, and Tajwar Abrar Aleef.
2018. "Fast PET Scan Tumor Segmentation Using Superpixels, Principal Component Analysis and K-Means Clustering" *Methods and Protocols* 1, no. 1: 7.
https://doi.org/10.3390/mps1010007