# Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Datasets

#### 2.1.1. Extended MNIST (EMNSIT) Dataset

#### 2.1.2. Cats vs. Dogs (CVD) Dataset

#### 2.1.3. Malaria Cell (MC) Dataset

#### 2.2. Methodology

#### 2.2.1. Preprocessing

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_{1}) for i

^{th}node inside image.

_{2}, y

_{2}) for j

^{th}node inside image.

#### 2.2.2. Modified Weighted Laplacian (MWL) Matrix

#### 2.2.3. Gershgorin Circle Feature Extraction

#### 2.2.4. Classification

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An overview of the GCFE methodology from image preprocessing to classification using the modified weighted Laplacian approach.

**Figure 2.**Flowchart of the proposed matrix transformation (pairwise graph) and GCFE for sample image size.

**Figure 3.**Representation of deep-learning architectures utilized in this study for feature classification. (

**a**) 2D-CNN model. (

**b**) 1D-CNN model.

**Figure 4.**GCFE performance metric for all datasets with different image patch sizes and 2D-grid graph classified using 2D CNN.

**Figure 5.**Comparison of mean ACC performance across feature extraction methods along with average Z-score for two graph types on E_Balanced dataset. (

**a**) with 2D-grid graph. (

**b**) with pairwise graph.

**Figure 6.**Accuracy vs. total computational time (generating graph to feature reduction) in log scale between various feature reduction methods on E_MNIST dataset with different image patch sizes.

**Figure 7.**Number of training parameters (scaled by a factor of ${10}^{-6}$) of 2D CNN model for GCFE and standard Laplacian (SLap) features.

Datasets | Instances | Classes | Type | Distribution of Classes |
---|---|---|---|---|

EMNIST | ||||

E_Balanced | 131,600 | 47 | 1D | Balanced |

E_ByClass | 814,255 | 62 | 1D | Imbalanced |

E_ByMerge | 814,255 | 47 | 1D | Imbalanced |

E_Digits | 280,000 | 10 | 1D | Imbalanced |

E_Letter | 145,600 | 26 | 1D | Imbalanced |

E_MNIST | 70,000 | 10 | 1D | Imbalanced |

CVD | 25,000 | 2 | 3D | Balanced |

MC | 27,558 | 2 | 3D | Balanced |

**Table 2.**Overview of GCFE comparison approaches across diverse datasets and graph structures using different classification architectures and performance metrics.

Approach No. | Dataset | Type of Comparison | Type of Graph | Classification Architecture | Performance Metric |
---|---|---|---|---|---|

1 | E_Balanced, E_ByClass, E_ByMerge, E_Digits, E_Letter, E_MNIST, CVD, MC | Comparison between GCFE of all datasets with different image patch sizes. | 2D-Grid | 2D-CNN | Accuracy |

2 | E_Balanced, CVD, MC | Comparison between GCFE, Laplacian, I-PCA, Kernel-PCA, and spectral embedding with different image patch sizes. | 2D-Grid, pairwise | 2D-CNN, 1D-CNN | Accuracy, Z-Score |

3 | E_MNIST | Comparison between GCFE, Isomap, LLE, MLLE, and Hessian Eigenmap with different image patch sizes. | K-NN | 1D-CNN | Accuracy |

**Table 3.**Comparison of proposed GCFE with other methods by measuring accuracy performance and computational time.

Datasets | GCFE (2D CNN) | Laplacian (2D CNN) | GCFE (1D CNN) | I-PCA | Kernel-PCA (RBF) | Spectral Embed. | Raw Image |
---|---|---|---|---|---|---|---|

Graph type—2D-Grid | |||||||

E_Balanced_P2 | 84.329 t = ≈6 | 84.553 | 83.797 t = ≈6 | 76.468 t = ≈225 | 78.138 t* = ≈12 | 75.787 t* = ≈3 | 86.617 |

E_Balanced_P4 | 83.978 t = ≈6 | 85.010 | 84.691 t = ≈6 | 76.499 t = ≈926 | 79.776 t* = ≈107 | 76.329 t* = ≈54 | 86.117 |

E_Balanced_P7 | 84.117 t = ≈16 | 84.595 | 84.329 t = ≈16 | 78.223 t = ≈2010 | 80.585 t* = ≈1163 | 77.755 t* = ≈1136 | 85.659 |

CVD_P2 | 68.681 t = ≈32 | 71.518 | 66.045 t = ≈32 | 62.722 t = ≈10,475 | ~ | 61.318 t* = ≈1182 | 70.773 |

MC_P2 | 94.581 t = ≈38 | 93.783 | 92.646 t = ≈38 | 63.691 t = ≈6944 | ~ | 62.796 t* = ≈1256 | 92.186 |

Graph type—Pairwise | |||||||

E_Balanced_P2 | 84.744 t = ≈5 | 85.372 | 84.989 t = ≈5 | 78.595 t = ≈194 | 79.287 t* = ≈11 | 77.638 t* = ≈1 | 86.617 |

E_Balanced_P4 | 84.276 t = ≈5 | 85.255 | 84.148 t = ≈5 | 77.297 t = ≈917 | 80.553 t* = ≈111 | 95.86 t* = ≈55 | 86.117 |

E_Balanced_P7 | 83.237 t = ≈8 | 84.074 | 83.808 t = ≈8 | 78.755 t = ≈1912 | 79.351 t* = ≈1122 | 79.595 t* = ≈1182 | 85.659 |

CVD_P2 | 69.684 t = ≈27 | 70.773 | 69.025 t = ≈27 | ~ | ~ | 63.954 t* = ≈1337 | 70.773 |

MC_P2 | 92.597 t = ≈33 | 92.938 | 92.670 t = ≈33 | ~ | ~ | 63.570 t* = ≈1407 | 92.186 |

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**MDPI and ACS Style**

Patel, S.A.; Yildirim, A.
Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications. *J. Imaging* **2024**, *10*, 121.
https://doi.org/10.3390/jimaging10050121

**AMA Style**

Patel SA, Yildirim A.
Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications. *Journal of Imaging*. 2024; 10(5):121.
https://doi.org/10.3390/jimaging10050121

**Chicago/Turabian Style**

Patel, Sahaj Anilbhai, and Abidin Yildirim.
2024. "Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications" *Journal of Imaging* 10, no. 5: 121.
https://doi.org/10.3390/jimaging10050121