# A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images

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## Abstract

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## 1. Introduction

- We propose a histogram-based technique to automatically detect the COVID-19 disease from CXRs and CT images. The histogram evaluation is a simple and very fast operation. In addition, in order to construct a target histogram, very few images are needed, in contrast to DNNs that have to use a huge number of scans to be trained.
- We investigate different inter-histogram distances to evaluate how an unknown scan is far from the target one. These distances are used to label the test images as normal (reference class) or anomaly (NCP), depending on whether they are less or greater than suitably set thresholds. Although all the proposed distances can be computed with the same computational cost, we expect that some of them work better in practice.
- We evaluate numerical results on benchmark datasets available in the open literature on CXRs and CT scans and compare the proposed approach to other state-of-the-art DNN-based architectures. We observed that the proposed approach, although very simple to implement, is able to obtain excellent results.

## 2. Related Work

## 3. Materials and Methods

#### 3.1. Proposed Approach

#### 3.1.1. The Evaluation of the Target Histogram

#### 3.1.2. The Considered Inter-Histogram Distances

- Cosine distance: It is formally defined as:$${d}_{c}=1-\frac{{\sum}_{i\in \mathcal{I}}{p}_{i}{q}_{i}}{\sqrt{{\sum}_{i\in \mathcal{I}}{p}_{i}^{2}}\sqrt{{\sum}_{i\in \mathcal{I}}{q}_{i}^{2}}},$$
- Kullback–Leibler (KL) divergence [106]: It is defined as:$${d}_{KL}=\sum _{i\in \mathcal{I}}{p}_{i}\mathrm{log}\phantom{\rule{-0.166667em}{0ex}}\left(\frac{{p}_{i}}{{q}_{i}}\right),$$
- Bhattacharyya distance [107]: It is defined as:$${d}_{B}=-\mathrm{log}\phantom{\rule{-0.166667em}{0ex}}\left(\sum _{i\in \mathcal{I}}\sqrt{{p}_{i}{q}_{i}}\right).$$The Bhattacharyya distance, such as the KL divergence, is always non-negative, while it is vanishing when the two distributions are equal.
- ${\chi}^{2}$ distance: It is defined as:$${d}_{{\chi}^{2}}=\sum _{i\in \mathcal{I}}\frac{{\left({p}_{i}-{q}_{i}\right)}^{2}}{{p}_{i}+{q}_{i}}.$$In addition, the ${\chi}^{2}$ distance is a non-negative measure.

#### 3.2. The Considered Datasets

#### 3.3. Built-Up Simulation Environment

#### 3.4. The Considered Performance Metrics

- True Positive (TP): correct positive assignments;
- True Negative (TN): correct negative assignments;
- False Positive (FP): incorrect positive assignments; and
- False Negative (FN): incorrect negative assignments.

## 4. Results and Discussion

#### 4.1. Evaluation of the Proposed Approach

- although all the considered distances provide good results for the CT datasets, the cosine distance is able to reach 100%;
- in case of CXR dataset, all the considered distances obtain the accuracy of 100%; and
- for both the datasets, the cosine distance provides the lower values of the standard deviation ${\sigma}_{d}$.

#### 4.2. Sensitivity of the Proposed Approach to the Parameter Settings

#### 4.3. Performance Comparison with an Alternative Histogram-Based Benchmark Approach

#### 4.4. Performance Comparisons with the State-of-the-Art DNN-Based Approaches

#### 4.5. Performance Robustness of the Considered Approaches

#### 4.6. Limitations of the Proposed Approach

## 5. Conclusions and Hints for Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AI | Artificial Intelligence |

AUC | Area Under the Curve |

CAD | Computer-Aided Diagnosis |

CNN | Convolutional Neural Network |

CM | Confusion Matrix |

CP | Common Pneumonia |

CT | Computed Tomography |

CXR | Chest X-ray |

DL | Deep Learning |

DNN | Deep Neural Network |

FN | False Negative |

FP | False Positive |

HOG | Histogram Orientation Gradients |

KL | Kullback–Leibler |

ML | Machine Learning |

MRI | Magnetic Resonance Imaging |

NCP | Novel COVID-19 Pneumonia |

ROC | Receiver Operating Characteristic |

TN | True Negative |

TP | True Positive |

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**Figure 1.**Proposed approach: (

**a**) evaluation of the target histogram, and (

**b**) inference phase for an unknown image.

**Figure 2.**Examples of some chest CT and CXR images: (

**a**) CT normal, (

**b**) CT common pneumonia (CP), (

**c**) CT novel coronavirus pneumonia (NCP), (

**d**) RX normal, (

**e**) RX CP, and (

**f**) RX NCP.

**Figure 3.**The obtained distances on the considered CT and CXRs datasets, and related threshold $TH$ (the red lines) set using (2). The distances are: (

**a**) Cosine for CT, (

**b**) Cosine for CXRs, (

**c**) Kullback–Leibler divergence for CT, (

**d**) Kullback–Leibler divergence for CXRs, (

**e**) Bhattacharyya distance for CT, and (

**f**) Bhattacharyya distance for CXRs. The ${\chi}^{2}$ distance behaves as in the KL and Bhattacharyya distances; hence, it is not explicitly shown. The results have been obtained by selecting ${N}_{T}=500$ reference images and a number ${N}_{bin}=50$ of histogram bins. The threshold in (2) has been set to $\eta =2$ for the CT dataset and to $\eta =1.4$ for CXRs.

**Figure 4.**Accuracy (in %) as a function of the $\eta $ parameter: (

**a**) CT dataset and (

**b**) CXR dataset. The best selected values ($\eta =2$ for CT dataset and $\eta =1.4$ for CXR dataset) are shown with a red cross. The results have been evaluated by using the cosine distance, a number ${N}_{T}=500$ of target images, and a number ${N}_{bin}=50$ of histogram bins.

**Figure 6.**The confusion matrices obtained by the considered state-of-the-art supervised approaches: (

**a**) AlexNet for CT, (

**b**) AlexNet for CXRs, (

**c**) GoogLeNet for CT, (

**d**) GoogLeNet for CXRs, (

**e**) ResNet18 for CT, and (

**f**) ResNet18 for CXRs.

Family | Approach | Work | Image Type |
---|---|---|---|

Manual screening | CAD | [57] | CT |

[58] | CT | ||

[59] | CT | ||

[60] | CT | ||

[61] | CXR | ||

Hand-crafted features | Texture-based features | [33] | Cells |

Edge-based features | [34] | Histopathological | |

[35] | Cytological | ||

Graph mining | [36] | Histopathological | |

Color-based features | [37] | Cells | |

[38] | Histopathological | ||

[39] | Histopathological | ||

Deep learning | Review | [72] | CXR |

[73] | CT | ||

[74] | CT | ||

[75] | CXR + CT | ||

[65] | CXR + CT | ||

[76] | CXR + CT | ||

[77] | CXR + CT | ||

[78] | CXR + CT | ||

Segmentation | [79] | CXR | |

[80] | CT | ||

[81] | CT | ||

[82] | CT | ||

[56] | CT | ||

[83] | CXR | ||

[84] | CXR | ||

Classification | [70] | CT | |

[69] | CT | ||

[85] | CT | ||

[86] | CT | ||

[87] | CXR | ||

[20] | CXR | ||

[88] | CXR | ||

Unsupervised | [89] | Histopathological | |

[90] | CT | ||

[91] | CT | ||

[92] | CXR | ||

Histogram | Histogram + DL | [93] | CXR |

[94] | CXR | ||

[95] | Skin | ||

[96] | CT | ||

[97] | MRI | ||

[98] | Skin | ||

[99] | Skin | ||

[101] | MRI | ||

[102] | MRI | ||

[103] | CT |

Computed Tomography | X-rays | ||||
---|---|---|---|---|---|

Type | Target | Test | Target | Test | |

COVID | 3500 | 500 | 84 | 100 | |

Non-COVID | 3500 | 500 | 580 | 97 |

Performance Metrics | Formula |
---|---|

Precision | $TP/(TP+FP)$ |

Recall | $TP/(TP+FN)$ |

F-measure | $2TP/(2TP+FP+FN)$ |

Accuracy | $(TP+TN)/(TP+FN+FP+TN)$ |

TP rate | $TP/(TP+FN)$ |

FP rate | $FP/(FP+TN)$ |

**Table 4.**Results obtained by the proposed approach under the two considered datasets. The results have been obtained by using ${N}_{T}=500$ reference images from the CP class and a number ${N}_{bin}=50$ of histogram bins.

Model | ${\mathit{d}}_{\mathit{m}}$ | ${\mathit{\sigma}}_{\mathit{d}}$ | $\mathit{TH}$ | Accuracy | Precision | Recall | F-Measure |
---|---|---|---|---|---|---|---|

Computed Tomography
($\mathbf{\eta}=\mathbf{2}$) | |||||||

Proposed (Cosine) | 0.0544 | 0.0664 | 0.1872 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Proposed (KL) | 0.1666 | 0.1218 | 0.4102 | 0.9870 | 0.9870 | 0.9870 | 0.9870 |

Proposed (Bhattacharyya) | 0.0027 | 0.1037 | 0.2101 | 0.9870 | 0.9870 | 0.9870 | 0.9870 |

Proposed
(${\mathbf{\chi}}^{\mathbf{2}}$) | 0.2272 | 0.1119 | 0.4652 | 0.9860 | 0.9860 | 0.9860 | 0.9860 |

X-Rays
($\mathbf{\eta}=\mathbf{1.4}$) | |||||||

Proposed (Cosine) | 0.0339 | 0.0325 | 0.0794 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Proposed (KL) | 0.0011 | 0.0564 | 0.0801 | 0.9949 | 0.9950 | 0.9949 | 0.9949 |

Proposed (Bhattacharyya) | 0.0027 | 0.0447 | 0.0653 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Proposed
(${\mathbf{\chi}}^{\mathbf{2}}$) | 0.0016 | 0.0450 | 0.0645 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

**Table 5.**Results obtained by the proposed approach under the two considered datasets. The results have been obtained by using ${N}_{T}=500$ reference images from the N class and a number ${N}_{bin}=50$ of histogram bins.

Model | ${\mathit{d}}_{\mathit{m}}$ | ${\mathit{\sigma}}_{\mathit{d}}$ | $\mathit{TH}$ | Accuracy | Precision | Recall | F-Measure |
---|---|---|---|---|---|---|---|

Computed Tomography
($\mathbf{\eta}=\mathbf{2}$) | |||||||

Proposed (Cosine) | 0.0458 | 0.0436 | 0.1330 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Proposed (KL) | 0.1231 | 0.1044 | 0.3319 | 0.9949 | 0.9950 | 0.9949 | 0.9949 |

Proposed (Bhattacharyya) | 0.0048 | 0.0983 | 0.2014 | 0.9870 | 0.9870 | 0.9870 | 0.9870 |

Proposed
(${\mathbf{\chi}}^{\mathbf{2}}$) | 0.1783 | 0.0970 | 0.3723 | 0.9747 | 0.9748 | 0.9747 | 0.9747 |

X-Rays
($\mathbf{\eta}=\mathbf{1.4}$) | |||||||

Proposed (Cosine) | 0.0283 | 0.0292 | 0.0692 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Proposed (KL) | 0.0018 | 0.0509 | 0.0731 | 0.9898 | 0.9898 | 0.9898 | 0.9898 |

Proposed (Bhattacharyya) | 0.0023 | 0.0415 | 0.0604 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Proposed
(${\mathbf{\chi}}^{\mathbf{2}}$) | 0.0015 | 0.0398 | 0.0572 | 0.9870 | 0.9870 | 0.9870 | 0.9870 |

**Table 6.**Results obtained by the proposed approach for different number ${N}_{bin}$ of histogram bins. The reported performance metrics refer to the cosine distance at ${N}_{T}=500$ reference images.

${\mathit{N}}_{\mathit{bins}}$ | ${\mathit{d}}_{\mathit{m}}$ | ${\mathit{\sigma}}_{\mathit{d}}$ | $\mathit{TH}$ | Accuracy | Precision | Recall | F-Measure |
---|---|---|---|---|---|---|---|

Computed Tomography
($\mathbf{\eta}=\mathbf{2}$) | |||||||

5 | 0.0222 | 0.0169 | 0.0560 | 0.7840 | 0.7927 | 0.7840 | 0.7824 |

10 | 0.0331 | 0.0258 | 0.0847 | 0.9570 | 0.9604 | 0.9570 | 0.9569 |

25 | 0.0453 | 0.0441 | 0.1335 | 0.9880 | 0.9883 | 0.9880 | 0.9880 |

50 | 0.0544 | 0.0664 | 0.1872 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

100 | 0.0558 | 0.0832 | 0.2222 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

250 | 0.0646 | 0.1225 | 0.3096 | 0.9960 | 0.9960 | 0.9960 | 0.9960 |

500 | 0.0764 | 0.1454 | 0.3672 | 0.9960 | 0.9960 | 0.9960 | 0.9960 |

X-Rays
($\mathbf{\eta}=\mathbf{1.4}$) | |||||||

5 | 0.0005 | 0.0045 | 0.0068 | 0.9260 | 0.9261 | 0.9260 | 0.9260 |

10 | 0.0133 | 0.0143 | 0.0333 | 0.9880 | 0.9883 | 0.9880 | 0.9880 |

25 | 0.0236 | 0.0259 | 0.0598 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

50 | 0.0339 | 0.0325 | 0.0794 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

100 | 0.0076 | 0.0380 | 0.0608 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

250 | 0.0122 | 0.0430 | 0.0724 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

500 | 0.0135 | 0.0568 | 0.0931 | 0.9898 | 0.9898 | 0.9898 | 0.9898 |

**Table 7.**Results obtained by the proposed approach at different number ${N}_{T}$ of target images. The metrics have been evaluated by using the cosine distance and ${N}_{bin}=50$ histogram bins.

${\mathit{N}}_{\mathit{T}}$ | ${\mathit{d}}_{\mathit{m}}$ | ${\mathit{\sigma}}_{\mathit{d}}$ | $\mathit{TH}$ | Accuracy | Precision | Recall | F-Measure |
---|---|---|---|---|---|---|---|

Computed Tomography
($\mathbf{\eta}=\mathbf{2}$) | |||||||

5 | 0.0351 | 0.0911 | 0.2173 | 0.8968 | 0.8968 | 0.8968 | 0.8968 |

10 | 0.0312 | 0.0911 | 0.2134 | 0.9880 | 0.9883 | 0.9880 | 0.9880 |

25 | 0.0383 | 0.0852 | 0.2087 | 0.9960 | 0.9960 | 0.9960 | 0.9960 |

50 | 0.0414 | 0.0787 | 0.1988 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

100 | 0.0460 | 0.0728 | 0.1916 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

250 | 0.0501 | 0.0704 | 0.1909 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

500 | 0.0544 | 0.0664 | 0.1872 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

3500 | 0.0732 | 0.0566 | 0.1864 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

X-Rays
($\mathbf{\eta}=\mathbf{1.4}$) | |||||||

5 | 0.0022 | 0.0807 | 0.1152 | 0.7817 | 0.8474 | 0.7817 | 0.7700 |

10 | 0.0031 | 0.0724 | 0.1045 | 0.9188 | 0.9300 | 0.9188 | 0.9188 |

25 | 0.0076 | 0.0722 | 0.1087 | 0.9137 | 0.9262 | 0.9137 | 0.9129 |

50 | 0.0125 | 0.0669 | 0.1062 | 0.9645 | 0.9669 | 0.9645 | 0.9644 |

100 | 0.0179 | 0.0492 | 0.0868 | 0.9898 | 0.9901 | 0.9898 | 0.9898 |

250 | 0.0257 | 0.0329 | 0.0718 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

500 | 0.0339 | 0.0325 | 0.0794 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

**Table 8.**Numerical results obtained by using the target histogram evaluated as in Figure 5. The results have been obtained by using the cosine distance, ${N}_{T}=500$ reference images, and a number ${N}_{bin}=50$ of histogram bins.

Architecture | Accuracy | Precision | Recall | F-Measure |
---|---|---|---|---|

CT | 0.5460 | 0.5902 | 0.5460 | 0.4826 |

CXR | 0.9137 | 0.9266 | 0.9137 | 0.9137 |

**Table 9.**Performance of the tested benchmark DNNs. These architectures have been trained and tested on the whole datasets in Table 2.

Architecture | Accuracy | Precision | Recall | F-Measure | AUC |
---|---|---|---|---|---|

Computed Tomography | |||||

AlexNet | 0.7110 | 0.8601 | 0.7110 | 0.7343 | 0.9460 |

GoogLeNet | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

ResNet18 | 0.9130 | 0.9281 | 0.9130 | 0.9137 | 0.9130 |

X-Rays | |||||

AlexNet | 0.9340 | 0.9341 | 0.9340 | 0.9340 | 0.9340 |

GoogLeNet | 0.9746 | 0.9794 | 0.9694 | 0.9744 | 0.9750 |

ResNet18 | 0.9695 | 0.9697 | 0.9695 | 0.9695 | 0.9700 |

**Table 10.**Computational complexity of the tested models (“M” stands for millions of parameters). The training time, in minutes, refers to data sets composed of images of size $300\times 200$ pixels for the CT dataset and $320\times 390$ pixels for the CXR dataset. The number of images is presented in Table 2.

Model | # param. | Training Time CT | Training Time CXRs |
---|---|---|---|

Proposed | 1 | 0.31 | 0.087 |

AlexNet | 58 M | 223 | 62 |

GoogLeNet | 6 M | 258 | 73 |

ResNet18 | 11 M | 290 | 84 |

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Scarpiniti, M.; Sarv Ahrabi, S.; Baccarelli, E.; Piazzo, L.; Momenzadeh, A.
A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images. *Appl. Sci.* **2021**, *11*, 8867.
https://doi.org/10.3390/app11198867

**AMA Style**

Scarpiniti M, Sarv Ahrabi S, Baccarelli E, Piazzo L, Momenzadeh A.
A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images. *Applied Sciences*. 2021; 11(19):8867.
https://doi.org/10.3390/app11198867

**Chicago/Turabian Style**

Scarpiniti, Michele, Sima Sarv Ahrabi, Enzo Baccarelli, Lorenzo Piazzo, and Alireza Momenzadeh.
2021. "A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images" *Applied Sciences* 11, no. 19: 8867.
https://doi.org/10.3390/app11198867