# Pathological Brain Detection by a Novel Image Feature—Fractional Fourier Entropy

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## Abstract

**:**

## 1. Background

## 2. State-of-the-Art

## 3. Materials

**Figure 1.**Samples of pathological brain images: (

**a**) healthy brain; (

**b**) AD with visual agnosia; (

**c**) Meningioma; (

**d**) AD; (

**e**) Glioma; (

**f**) Huntington’s disease; (

**g**) Herpes encephalitis; (

**h**) Pick’s disease; (

**i**) Multiple sclerosis; (

**j**) Cerebral toxoplasmosis; (

**k**) Sarcoma; (

**l**) Subdural hematoma.

## 4. Feature Extraction and Selection

#### 4.1. Basic Concept

_{α}:

#### 4.2. Fractional Fourier Domain

^{2}(u). We can observe that the FRFT output is in an intermediate domain between time and frequency, viz., a unified time-frequency domain.

**Figure 2.**Illustration of how FRFT changes with α, whose value varies from zero to one (the real and imaginary parts are shown in black and blue lines, respectively).

#### 4.3. Weighted-Type FRFT

#### 4.4. Shannon Entropy

_{1}, x

_{2}, …, x

_{n}) with a probability mass function P(X), we have entropy H defined in the form of:

_{i}) is equal to zero, then, we force the 0log

_{a}0=0.

#### 4.5. Fractional Fourier Entropy

_{x}and α

_{y}. The entropy over all spectra of 25 domains are extracted and aligned into a set of vectors:

_{(α, β)}represents a FRFT performed with α-angle along the x-axis and β-angle along the y-axis. I is the brain image. In this study, both angles were set to change within the range from 0.6 to 1 with equal increases of 0.1. The range of [0, 0.5] was not considered since FRFT with angles near 0 will yield an identity operation.

#### 4.6. Feature Selection

^{2}denotes the variance of a particular feature, n the sample size, w the WTT score. The null hypothesis in this work is that the FRFE values of both pathological and healthy brains have the same means (equal variances are not of concern). The alternative hypothesis is that they have unequal means. WTT was performed at the confidence interval of 95%. Then, the selected FRFEs are used as input features for following classification.

_{p}and C

_{q}represents the covariance matrixes of the characteristic vectors in class p and class q, respectively.

## 5. Classification

#### 5.1. Support Vector Machine (SVM)

_{n}denotes a p-dimensional data point, and y

_{i}denotes the corresponding class, with a value of either −1 or +1, denoting the sample target is either class 1 or class 2. Our aim is to build a hyperplane, which separates the first class from the second class is the desired SVM. Usually the hyperplane is (p−1)-dimensional.

**w**and

**b**denotes the weights and biases. Hence, SVM can be written as:

**ξ**= (ξ

_{1}, …, ξ

_{i}, …, ξ

_{N}) are introduced to measure the misclassification degree of sample x

_{i}. Then, the optimal hyperplane corresponding to the SVM is yielded by solving:

#### 5.2. NPSVM I—Generalized Eigenvalue Proximal SVM

_{1}and samples from class 2 are denoted as X

_{2}, GEPSVM builds the two nonparallel planes by:

#### 5.3. NPSVM II—Twin Support Vector Machine

_{i}(i = 1,2) are the same as in Equation (19), and c

_{i}(i = 1,2) are positive parameters. The constraint requires the hyperplane to be at a distance of more than one from points of the other class. The first and second terms in the equations above represent the sum of squared distances from the hyperplane to one class, and the sum of error variables, respectively.

## 6. Experimental Design

#### 6.1. K-Fold Stratified Cross Validation

Data | Training | Validation | Total | Fold No. | |||
---|---|---|---|---|---|---|---|

P | H | P | H | P | H | ||

D66 | 40 | 15 | 8 | 3 | 48 | 18 | 6 |

D160 | 112 | 16 | 28 | 4 | 140 | 20 | 5 |

D255 | 176 | 28 | 44 | 7 | 220 | 35 | 5 |

#### 6.2. Implementation

Offline learning | |

Step I | Feature Extraction: Fractional Fourier Entropy (FRFE) were performed on all ground-truth images: Twenty-five different WFRFT were carried out with α and β from the set of [0.6, 0.7, 0.8, 0.9, 1.0], respectively. Entropy was extracted based on the 25 fractional Fourier spectrums. |

Step II | Feature Selection: Welch’s t-test (WTT) was employed to select the most important FRFEs among the 25 ones the 95% confidence interval. |

Step III | Classifier Training: Those chosen FRFEs with their class labels, were fed into train SVM and two NPSVMs. |

Step IV | Classifier Evaluation: Evaluate the classification performance based on a 10 times K-fold SCV, and report which classifiers performs best. |

Online prediction | |

Step I | Feature Extraction: A new query image is decomposed with 25 FRFE results extracted |

Step II | Feature Selection: Select the most important FRFEs from the 25 ones. |

Step III | Query Image Prediction: Input the selected FRFEs of the query image to the reported best classifier, so as to obtain whether the query brain is pathological or healthy. |

## 7. Results and Discussion

#### 7.1. WFRFT Result

#### 7.2. FRFE Results

β = 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |

α =0.6 | 6.14 ± 0.15 | 5.96 ± 0.14 | 5.79 ±0.15 | 5.69 ± 0.16 | 5.65 ± 0.18 |

6.07 ± 0.43 | 5.92 ± 0.41 | 5.74 ± 0.36 | 5.57 ± 0.28 | 5.50 ± 0.28 | |

0.7 | 5.99 ± 0.14 | 5.85 ± 0.14 | 5.73 ± 0.15 | 5.67 ± 0.16 | 5.64 ± 0.18 |

5.94 ± 0.41 | 5.81 ± 0.38 | 5.66 ± 0.31 | 5.53 ± 0.27 | 5.48 ± 0.26 | |

0.8 | 5.88 ± 0.14 | 5.78 ± 0.15 | 5.70 ± 0.16 | 5.66 ± 0.17 | 5.63 ± 0.18 |

5.79 ± 0.36 | 5.68 ± 0.31 | 5.58 ± 0.27 | 5.50 ± 0.24 | 5.46 ± 0.25 | |

0.9 | 5.82 ± 0.15 | 5.76 ± 0.16 | 5.70 ± 0.16 | 5.66 ± 0.17 | 5.63 ± 0.18 |

5.68 ± 0.32 | 5.61 ± 0.30 | 5.54 ± 0.26 | 5.48 ± 0.25 | 5.45 ± 0.24 | |

1.0 | 5.75 ± 0.16 | 5.71 ± 0.16 | 5.68 ± 0.17 | 5.65 ± 0.17 | 5.59 ± 0.18 |

5.58 ± 0.30 | 5.53 ± 0.27 | 5.49 ± 0.25 | 5.45 ± 0.24 | 5.41 ± 0.24 |

#### 7.3. Feature Selection

β | ||||||

0.6 | 0.7 | 0.8 | 0.9 | 1.0 | ||

α | 0.6 | X | X | X | X | S |

0.7 | X | X | X | X | S | |

0.8 | X | X | X | S | S | |

0.9 | X | X | S | S | S | |

1.0 | S | S | S | S | S |

#### 7.4. Feature Comparison

Method | Feature # | Accuracy | ||
---|---|---|---|---|

D66 | D160 | D255 | ||

“Wavelet Entropy + NBC” [22] | 7 | 92.58 | 91.87 | 90.51 |

FRFE + WTT + NBC (Proposed) | 12 | 97.12 | 95.94 | 95.69 |

Method | Feature # | Accuracy | ||
---|---|---|---|---|

D66 | D160 | D255 | ||

“Wavelet Energy + SVM” [26] | 7 | 82.58 | 80.13 | 77.76 |

FRFE + WTT + SVM (Proposed) | 12 | 100 | 99.69 | 98.98 |

#### 7.5. SVM versus Non-Parallel SVMs

Method | Classifier | D66 | D160 | D255 |
---|---|---|---|---|

FRFE + WTT + SVM | SVM | 100 | 99.69 | 98.98 |

FRFE + WTT + GEPSVM | GEPSVM | 100 | 100 | 99.18 |

FRFE + WTT + TSVM | TSVM | 100 | 100 | 99.57 |

#### 7.6. Best Proposed Approach

CM on Validation Set | Correct Cases | Accuracy | |
---|---|---|---|

Fold 1 | $\left[\begin{array}{cc}44& 0\\ 0& 7\end{array}\right]$ | 51 | 100.00 |

Fold 2 | $\left[\begin{array}{cc}44& 0\\ 0& 7\end{array}\right]$ | 51 | 100.00 |

Fold 3 | $\left[\begin{array}{cc}43& 1\\ 0& 7\end{array}\right]$ | 50 | 98.04 |

Fold 4 | $\left[\begin{array}{cc}44& 0\\ 0& 7\end{array}\right]$ | 51 | 100.00 |

Fold 5 | $\left[\begin{array}{cc}44& 0\\ 0& 7\end{array}\right]$ | 51 | 100.00 |

Sum | $\left[\begin{array}{cc}219& 1\\ 0& 35\end{array}\right]$ | 254 | 99.61% |

Run | Fold 1 | Fold 2 | Fold 3 | Fold 4 | Fold 5 | Sum |
---|---|---|---|---|---|---|

Run 1 | 51 (100.00%) | 51 (100.00%) | 50 (98.04%) | 51 (100.00%) | 51 (100.00%) | 254 (99.61%) |

Run 2 | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 50(98.04%) | 51 (100.00%) | 254 (99.61%) |

Run 3 | 50 (98.04%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 254 (99.61%) |

Run 4 | 50 (98.04%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 254 (99.61%) |

Run 5 | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 50 (98.04%) | 254 (99.61%) |

Run 6 | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 50 (98.04%) | 254 (99.61%) |

Run 7 | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 50 (98.04%) | 51 (100.00%) | 254 (99.61%) |

Run 8 | 51 (100.00%) | 51 (100.00%) | 50 (98.04%) | 50 (98.04%) | 51 (100.00%) | 253 (99.22%) |

Run 9 | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 50 (98.04%) | 51 (100.00%) | 254 (99.61%) |

Run 10 | 50 (98.04%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 51 (100.00%) | 254 (99.61%) |

Sum | 2539 (99.57%) |

Method | Accuracy | Sensitivity | Specificity | Precision |
---|---|---|---|---|

FRFE + WTT + TSVM | 99.57% | 99.59% | 99.43% | 99.91% |

#### 7.7. Comparison with State-of-the-Art

## 8. Discussion

Existing Algorithms | Feature # | Run # | Accuracy | ||
---|---|---|---|---|---|

D66 | D160 | D255 | |||

DWT+SOM [12] | 4761 | 5 | 94.00 | 93.17 | 91.65 |

DWT+SVM [12] | 4761 | 5 | 96.15 | 95.38 | 94.05 |

DWT + SVM + RBF [12] | 4761 | 5 | 98.00 | 97.33 | 96.18 |

DWT + PCA + ANN [13] | 7 | 5 | 97.00 | 96.98 | 95.29 |

DWT + PCA + KNN [13] | 7 | 5 | 98.00 | 97.54 | 96.79 |

DWT + PCA + SCG-FNN [14] | 19 | 5 | 100.00 | 99.27 | 98.82 |

DWT + PCA + SVM [15] | 19 | 5 | 96.01 | 95.00 | 94.29 |

DWT + PCA + SVM + HPOL [15] | 19 | 5 | 98.34 | 96.88 | 95.61 |

DWT + PCA + SVM + IPOL [15] | 19 | 5 | 100.00 | 98.12 | 97.73 |

DWT + PCA + SVM + RBF [15] | 19 | 5 | 100.00 | 99.38 | 98.82 |

WE + SWP + PNN [16] | 3 | 10 | 100.00 | 99.94 | 98.86 |

RT + PCA + LS-SVM [19] | 9 | 5 | 100.00 | 100.00 | 99.39 |

PCNN + DWT + PCA + BPNN [21] | 7 | 10 | 100.00 | 98.88 | 98.24 |

DWPT + SE + GEPSVM [23] | 16 | 10 | 99.85 | 99.62 | 98.78 |

DWPT + TE + GEPSVM [23] | 16 | 10 | 100.00 | 100.00 | 99.33 |

SWT + PCA + IABAP-FNN [30] | 7 | 10 | 100.00 | 99.44 | 99.18 |

SWT + PCA + ABC-SPSO-FNN [30] | 7 | 10 | 100.00 | 99.75 | 99.02 |

SWT + PCA + HPA-FNN [30] | 7 | 10 | 100.00 | 100.00 | 99.45 |

WE + HMI + GEPSVM [33] | 14 | 10 | 100.00 | 99.56 | 98.63 |

WE + HMI + GEPSVM + RBF [33] | 14 | 10 | 100.00 | 100.00 | 99.45 |

FRFE + WTT + TSVM (Proposed) | 12 | 10 | 100 | 100.00 | 99.57 |

## 9. Conclusions and Future Research

## Acknowledgments

## Author Contributions

## Conflict of interest

## Nomenclature

Abbreviation | Definition |

(A)(BP)(F)(PC)NN | (Artificial) (Back-propagation) (Feed-forward) (Pulse-coupled) neural network |

(D)(S)W(P)T | (Discrete) (Stationary) wavelet (packet) transform |

(H)(I)POL | (Homogeneous) (Inhomogeneous) Polynomial |

(I)(D)(S)FT | (Inverse) (Discrete) (Standard) Fourier Transform |

(k)(F)(LS)(GEP)(NP)SVM | (kernel) (Fuzzy) (Lease-Square) (Generalized eigenvalue proximal) (Non-Parallel) Support vector machine |

(S)(C)ABC(-SPSO) | (Scaled) (Chaotic) Artificial bee colony (-Standard PSO) |

(W)(P)(S)(T)E | (Wavelet) (Packet) (Shannon) (Tsallis) entropy |

CAD | Computer-aided diagnosis |

CM | Confusion matrix |

FRF(T)(E)(D) | Fractional Fourier (Transform) (Entropy) (Domain) |

HPA | Hybridization of PSO and ABC |

IABAP | Integrated algorithm based on ABC and PSO |

IC | Information Content |

KNN | K-nearest neighbors |

MD | Mahalanobis distance |

MR(I) | Magnetic resonance (imaging) |

NBC | Naive Bayesian Classifier |

PCA | Principal Component Analysis |

PNN | Probabilistic neural network |

PSO | Particle Swarm Optimization |

QP | Quadratic programming |

RBF | Radial Basis Function |

RQ | Rayleigh Quotient |

RT | Ripplet transform |

SCV | Stratified cross validation |

SD | Standard deviation |

SOM | Self-organizing map |

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## Share and Cite

**MDPI and ACS Style**

Wang, S.; Zhang, Y.; Yang, X.; Sun, P.; Dong, Z.; Liu, A.; Yuan, T.-F.
Pathological Brain Detection by a Novel Image Feature—Fractional Fourier Entropy. *Entropy* **2015**, *17*, 8278-8296.
https://doi.org/10.3390/e17127877

**AMA Style**

Wang S, Zhang Y, Yang X, Sun P, Dong Z, Liu A, Yuan T-F.
Pathological Brain Detection by a Novel Image Feature—Fractional Fourier Entropy. *Entropy*. 2015; 17(12):8278-8296.
https://doi.org/10.3390/e17127877

**Chicago/Turabian Style**

Wang, Shuihua, Yudong Zhang, Xiaojun Yang, Ping Sun, Zhengchao Dong, Aijun Liu, and Ti-Fei Yuan.
2015. "Pathological Brain Detection by a Novel Image Feature—Fractional Fourier Entropy" *Entropy* 17, no. 12: 8278-8296.
https://doi.org/10.3390/e17127877