# Preclinical Diagnosis of Magnetic Resonance (MR) Brain Images via Discrete Wavelet Packet Transform with Tsallis Entropy and Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM)

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## Background

## Methods

## Results

## Conclusions

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Benchmark Dataset

- Changing the class distribution: resampling, instance reweighting, metacost;
- Boost methods: AdaBoost/Adacost, cost boosting, asymmetric boosting;
- Modifying the learning algorithms: modifying the decision tree, modifying neural networks, modifying SVMs, modifying naive Bayes classifier;
- Direct cost-sensitive learning: Laplace correction, smoothing, curtailment, Platt calibration, and Isotonic regression;
- Other methods: Cost-sensitive CBR, Cost-sensitive specification, Cost-sensitive genetic programming.

#### 2.2. CV Setting

#### 2.3. Discrete Wavelet Transform

_{s}, f

_{t}) is calculated from the mother wavelet ψ(t) by translation and dilation: f

_{s}is the scale factor, f

_{t}the translation factor (both real positive numbers), and C the coefficients of WT. There are several different kinds of wavelets which have gained popularity throughout the development of wavelet analysis.

_{s}and f

_{t}to a discrete lattice (f

_{s}= 2^f

_{t}& f

_{s}> 0) to give the DWT, which can be expressed as follows:

#### 2.4. Discrete Wavelet Packet Transform

^{S}sequences can be produced in the S-th level. The fast decomposition equation to next level is:

^{j}different sets of coefficients as opposed to (3j + 1) sets for the DWT. However; due to the downsampling process the overall number of coefficients of DWPT is still the same of those of DWT; so there is no redundancy.

#### 2.5. Shannon and Tsallis Entropy

_{q}(A + B) < S

_{q}(A) + S

_{q}(B); for q = 1, the TE reduces to an standard extensive entropy where S

_{q}(A + B) = S

_{q}(A) + S

_{q}(B); for q > 1, the TE becomes a super-extensive entropy where S

_{q}(A + B) > S

_{q}(A) + S

_{q}(B).

#### 2.6. Feature Extraction

Step 1 Import MR image. |

Step 2 Carry out 2-level DWPT decomposition. |

Step 3 Calculate the entropy of each subband. |

Step 4 Output 16-element entropy vector. |

#### 2.7. Generalized Eigenvalue Proximal SVM

_{1}and X

_{2}, respectively. GEPSVM aims to determine two nonparallel planes:

**w**, b) that determine the first plane in Equation (11):

_{1}corresponding to the smallest eigenvalue λ

_{min}of Equation (19). Hence,

**w**

_{1}and b

_{1}can be obtained through Equation (13), and used to determine the plane in Equation (11). Next, we create another optimization problem analogous to Equation (14) by interchanging the roles of X

_{1}and X

_{2}. The eigenvector z

_{2}* corresponding to the smallest eigenvalue of the second generalized eigenvalue problem will yield the second plane which is close to points of class 2.

#### 2.8. Kernel Technique

#### 2.9. Implementation of the Proposed Method

Phase I: Offline learning (users are scientists) | |
---|---|

Step 1. | Feature Extraction: Users decompose images by DWPT, and extract Tsallis entropies from all subbands |

Step 2. | Classifier Training: The set of features along with the corresponding labels, were used to train the classifier. 10 repetition of K-fold stratified CV was employed for get the out-of-sample evaluation |

Step 3. | Evaluation: Report the classification performance. |

Step 3. | Parameter Selection: Above three steps iterated with q varies from 0.1 to 1 with increment of 0.1. Select the optimal q that corresponds to the highest classification accuracy. |

Phase II: Online prediction (users are doctors and radiologists) | |

Step 1. | Feature Extraction: Users presented to the system the query image to be classified. Its feature was obtained as in the offline learning phase. |

Step 2. | Predict: Input the features of the query image to previously trained classifier. The classifier labeled the input query image as normal or abnormal. |

#### 2.10. Evaluation

- (i) DWPT + SE + GEPSVM;
- (ii) DWPT + TE + GEPSVM;
- (iii) DWPT + SE + GEPSVM + RBF;
- (iv) DWPT + TE + GEPSVM + RBF;

## 3. Experimental Results

#### 3.1. DWPT Result

#### 3.2. Classification Comparison

#### 3.3. Setting of Parameter q

#### 3.4. Computational Burden Analysis

#### 3.5. Discussion

#### 3.5.1. Discussion of Results

#### 3.5.2. Discussion on the Proposed Method

## 4. Conclusions and Future Research

## Nomenclature

t | Time |

x | 1D signal |

I | 2D image |

ψ | Wavelet function |

f_{s} | scale factor |

f_{t} | translation factor |

C | Coefficients of wavelet decomposition |

w | weight |

b | bias |

z | Combination of weight and bias |

N | Sample number |

n | Index of sample |

p | Attribute number |

X | Sample matrix |

X_{1} | Sample matrix belonging to class 1 |

X_{2} | Sample matrix belonging to class 2 |

y | Class label (y = 1 for class 1 |

y | = −1 for class 2) |

e | Vector of ones (dimension varies) |

δ | Tikhonov factor |

λ | eigenvalue |

K | Folds of CV |

q | Entropic parameter of TE |

## Acknowledgments

## Author Contributions

## Conflict of Interest

## References

- Goh, S.; Dong, Z.; Zhang, Y.; DiMauro, S.; Peterson, B.S. Mitochondrial dysfunction as a neurobiological subtype of autism spectrum disorder: Evidence from brain imaging. JAMA Psychiatry
**2014**, 71, 665–671. [Google Scholar] - Zhang, Y.; Wang, S.; Ji, G.; Dong, Z. Exponential wavelet iterative shrinkage thresholding algorithm with random shift for compressed sensing magnetic resonance imaging. IEEJ Trans. Electr. Electron. Eng.
**2015**, 10, 116–117. [Google Scholar] - Zhang, Y.D.; Dong, Z.C.; Ji, G.L.; Wang, S.H. An improved reconstruction method for CS-MRI based on exponential wavelet transform and iterative shrinkage/thresholding algorithm. J. Electromagn. Waves Appl
**2014**, 28, 2327–2338. [Google Scholar] - Levman, J.E.D.; Warner, E.; Causer, P.; Martel, A.L. A Vector machine formulation with application to the computer-aided diagnosis of breast cancer from DCE-MRI screening examinations. J. Digit. Imaging
**2014**, 27, 145–151. [Google Scholar] - Chaplot, S.; Patnaik, L.M.; Jagannathan, N.R. Classification of magnetic resonance brain images using wavelets as input to support vector machine and neural network. Biomed. Signal Process. Control
**2006**, 1, 86–92. [Google Scholar] - Maitra, M.; Chatterjee, A. A Slantlet transform based intelligent system for magnetic resonance brain image classification. Biomed. Signal Process. Control
**2006**, 1, 299–306. [Google Scholar] - El-Dahshan, E.S.A.; Hosny, T.; Salem, A.B.M. Hybrid intelligent techniques for MRI brain images classification. Digit. Signal Process
**2010**, 20, 433–441. [Google Scholar] - Zhang, Y.; Wu, L.; Wang, S. Magnetic resonance brain image classification by an improved artificial bee colony algorithm. Progress Electromagn. Res.
**2011**, 116, 65–79. [Google Scholar] - Zhang, Y.; Dong, Z.; Wu, L.; Wang, S. A hybrid method for MRI brain image classification. Expert Syst. Appl
**2011**, 38, 10049–10053. [Google Scholar] - Ramasamy, R.; Anandhakumar, P. Brain tissue classification of MR images using fast Fourier transform based expectation-maximization Gaussian mixture model. In Advances in Computing and Information Technology; Springer: Berlin/Heidelberg, Germany, 2011; pp. 387–398. [Google Scholar]
- Zhang, Y.; Wu, L. An MR brain images classifier via principal component analysis and Kernel support vector machine. Progress Electromagn. Res.
**2012**, 130, 369–388. [Google Scholar] - Saritha, M.; Paul Joseph, K.; Mathew, A.T. Classification of MRI brain images using combined wavelet entropy based spider web plots and probabilistic neural network. Pattern Recognit. Lett.
**2013**, 34, 2151–2156. [Google Scholar] - Das, S.; Chowdhury, M.; Kundu, M.K. Brain MR image classification using multiscale geometric analysis of ripplet. Progress Electromagn. Res.
**2013**, 137, 1–17. [Google Scholar] - Kalbkhani, H.; Shayesteh, M.G.; Zali-Vargahan, B. Robust algorithm for brain magnetic resonance image (MRI) classification based on GARCH variances series. Biomed. Signal Process. Control
**2013**, 8, 909–919. [Google Scholar] - Zhang, Y.; Wang, S.; Dong, Z. Classification of Alzheimer disease based on structural Magnetic resonance imaging by Kernel support vector machine decision tree. Progress Electromagn. Res.
**2014**, 144, 171–184. [Google Scholar] - Qin, Z.X.; Zhang, C.Q.; Wang, T.; Zhang, S.C. Cost sensitive classification in data mining. Adv. Data Min. Appl 2010, Pt I
**2010**, 6440, 1–11. [Google Scholar] - Zhang, Y.; Wang, S.; Phillips, P.; Ji, G. Binary PSO with mutation operator for feature selection using decision tree applied to spam detection. Knowl.-Based Syst.
**2014**, 64, 22–31. [Google Scholar] - Kong, Y.H.; Zhang, S.M.; Cheng, P.Y. Super-resolution reconstruction face recognition based on multi-level FFD registration. Optik
**2013**, 124, 6926–6931. [Google Scholar] - Liao, S.; Shen, D.G.; Chung, A.C.S. A Markov random field groupwise registration framework for face recognition. IEEE Trans. Pattern Anal. Mach. Intell.
**2014**, 36, 657–669. [Google Scholar] - Shi, J.G.; Qi, C. From local geometry to global structure: Learning latent subspace for low-resolution face image recognition. IEEE Signal Process. Lett.
**2015**, 22, 554–558. [Google Scholar] - Fan, Z.Z.; Ni, M.; Zhu, Q.; Liu, E. Weighted sparse representation for face recognition. Neurocomputing
**2015**, 151, 304–309. [Google Scholar] - Ribbens, A.; Hermans, J.; Maes, F.; Vandermeulen, D.; Suetens, P.; Alzheimers Dis, N. Unsupervised segmentation, clustering, and groupwise registration of heterogeneous populations of brain MR images. IEEE Trans. Med. Imaging
**2014**, 33, 201–224. [Google Scholar] - Schwarz, D.; Kasparek, T. Brain morphometry of MR images for automated classification of first-episode schizophrenia. Inf. Fusion
**2014**, 19, 97–102. [Google Scholar] - Fang, L.; Wu, L.; Zhang, Y. A novel demodulation system based on continuous wavelet transform. Math. Probl. Eng.
**2015**, 2015. [Google Scholar] [CrossRef] - Zhou, R.; Bao, W.; Li, N.; Huang, X.; Yu, D. Mechanical equipment fault diagnosis based on redundant second generation wavelet packet transform. Digit. Signal Process
**2010**, 20, 276–288. [Google Scholar] - Campos, D. Real and spurious contributions for the Shannon, Rényi and Tsallis entropies. Physica A
**2010**, 389, 3761–3768. [Google Scholar] - Tsallis, C. Nonadditive entropy: The concept and its use. Eur. Phys. J. A.
**2009**, 40, 257–266. [Google Scholar] - Zhang, Y.; Wu, L. Optimal multi-level thresholding based on maximum Tsallis entropy via an artificial bee colony approach. Entropy
**2011**, 13, 841–859. [Google Scholar] - Tsallis, C. The nonadditive entropy S-q and its applications in physics and elsewhere: Some remarks. Entropy
**2011**, 13, 1765–1804. [Google Scholar] - Amaral-Silva, H.; Wichert-Ana, L.; Murta, L.O.; Romualdo-Suzuki, L.; Itikawa, E.; Bussato, G.F.; Azevedo-Marques, P. The superiority of Tsallis entropy over traditional cost functions for brain MRI and SPECT registration. Entropy
**2014**, 16, 1632–1651. [Google Scholar] - Venkatesan, A.S.; Parthiban, L. A Novel nature inspired fuzzy Tsallis entropy segmentation of magnetic resonance images. Neuroquantology
**2014**, 12, 221–229. [Google Scholar] - Khader, M.; Ben Hamza, A. Nonrigid image registration using an entropic similarity. IEEE Trans. Inf. Technol. Biomed.
**2011**, 15, 681–690. [Google Scholar] - Hussain, M. Mammogram enhancement using lifting dyadic wavelet transform and normalized Tsallis entropy. J. Comput. Sci. Technol
**2014**, 29, 1048–1057. [Google Scholar] - Liu, Z.G.; Hu, Q.L.; Cui, Y.; Zhang, Q.G. A new detection approach of transient disturbances combining wavelet packet and Tsallis entropy. Neurocomputing
**2014**, 142, 393–407. [Google Scholar] - Chen, J.K.; Li, G.Q. Tsallis wavelet entropy and its application in power signal analysis. Entropy
**2014**, 16, 3009–3025. [Google Scholar] - Mangasarian, O.L.; Wild, E.W. Multisurface proximal support vector machine classification via generalized eigenvalues. IEEE Trans. Pattern Anal. Mach. Intell
**2006**, 28, 69–74. [Google Scholar] - Tsallis, C. An introduction to nonadditive entropies and a thermostatistical approach to inanimate and living matter. Contemp. Phys.
**2014**, 55, 179–197. [Google Scholar] - Sturzbecher, M.J.; Tedeschi, W.; Cabella, B.C.T.; Baffa, O.; Neves, U.P.C.; de Araujo, D.B. Non-extensive entropy and the extraction of BOLD spatial information in event-related functional MRI. Phys. Med. Biol
**2009**, 54, 161–174. [Google Scholar] - Cabella, B.C.T.; Sturzbecher, M.J.; de Araujo, D.B.; Neves, U.P.C. Generalized relative entropy in functional magnetic resonance imaging. Physica A
**2009**, 388, 41–50. [Google Scholar] - Diniz, P.R.B.; Murta, L.O.; Brum, D.G.; de Araujo, D.B.; Santos, A.C. Brain tissue segmentation using q-entropy in multiple sclerosis magnetic resonance images. Braz. J. Med. Biol. Res.
**2010**, 43, 77–84. [Google Scholar][Green Version] - Dong, Z.; Zhang, Y.; Liu, F.; Duan, Y.; Kangarlu, A.; Peterson, B.S. Improving the spectral resolution and spectral fitting of 1H MRSI data from human calf muscle by the SPREAD technique. NMR Biomed.
**2014**, 27, 1325–1332. [Google Scholar]

**Figure 1.**Samples of brain MR images. (

**a**) Normal brain; (

**b**) Glioma; (

**c**) Meningioma; (

**d**) AD; (

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

**f**) Pick’s disease; (

**g**) Sarcoma; (

**h**) Huntington’s disease; (

**i**) Chronic subdural hematoma; (

**j**) Cerebral toxoplasmosis; (

**k**) Herpes encephalitis; (

**l**) Multiple sclerosis.

**Figure 2.**Schematic Diagram of DWT. The downward arrow denotes DS operation. (

**a**) 2-level 1D-DWT; (

**b**) 2-level 2D-DWT.

**Figure 5.**Decomposition comparison between DWT and DWPT. (

**a**) normal brain; (

**b**) 2-level DWT of normal brain; (

**c**) 2-level DWPT of normal brain; (

**d**) AD brain; (

**e**) 2-level DWT of AD brain; (

**f**) 2-level DWPT of AD brain.

Dataset | Total
| Training
| Validation
| K-Fold
| |||
---|---|---|---|---|---|---|---|

Normal | Abnormal | Normal | Abnormal | Normal | Abnormal | ||

Dataset-66 | 18 | 48 | 15 | 40 | 3 | 8 | 6- |

Dataset-160 | 20 | 140 | 16 | 112 | 4 | 28 | 5- |

Dataset-255 | 35 | 220 | 28 | 177 | 7 | 43 | 5- |

Dataset-66 | Dataset-160 | Dataset-255 | ||
---|---|---|---|---|

Existing Approaches [13] (5 Repetitions) | DWT+SOM [5] | 94.00 | 93.17 | 91.65 |

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

DWT + SVM + POLY [5] | 98.00 | 97.15 | 96.37 | |

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

DWT + PCA + FP-ANN [7] | 97.00 | 96.98 | 95.29 | |

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

DWT + PCA + SVM [11] | 96.01 | 95.00 | 94.29 | |

DWT + PCA + SVM + HPOL [11] | 98.34 | 96.88 | 95.61 | |

DWT + PCA + SVM + IPOL [11] | 100.00 | 98.12 | 97.73 | |

DWT + PCA + SVM + GRB [11] | 100.00 | 99.38 | 98.82 | |

DWT + SE + SWP + PNN [12] | 100.00 | 99.88 | 98.90 | |

RT + PCA + LS-SVM [13] | 100.00 | 100.00 | 99.39 | |

Proposed approaches (10 repetitions) | DWPT + SE + GEPSVM | 99.85 | 99.62 | 98.78 |

DWPT + TE + GEPSVM | 100.00 | 100.00 | 99.33 | |

DWPT + SE + GEPSVM + RBF | 100.00 | 99.88 | 99.33 | |

DWPT + TE + GEPSVM + RBF | 100.00 | 100.00 | 99.53 |

Dataset | Sensitivity | Specificity | Precision | Accuracy |
---|---|---|---|---|

Dataset-66 | 100.00 | 100.00 | 100.00 | 100.00 |

Dataset-160 | 100.00 | 100.00 | 100.00 | 100.00 |

Dataset-255 | 100.00 | 97.14 | 99.55 | 99.61 |

**Table 4.**Detailed Data of Figure 6.

Method | q = 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
---|---|---|---|---|---|---|---|---|---|---|

DWPT + TE + GEPSVM | 99.02 | 99.02 | 99.06 | 99.06 | 99.18 | 99.11 | 99.29 | 99.33 | 98.94 | 98.82 |

DWPT + TE + GEPSVM + RBF | 99.29 | 99.33 | 99.33 | 99.41 | 99.37 | 99.37 | 99.41 | 99.53 | 99.49 | 99.33 |

Step | Time (s) | |
---|---|---|

Offline Learning | DWPT decomposition | 4.0565 |

Entropy calculation | 3.4961 | |

Classifier training | 0.8904 | |

Online Prediction | DWPT decomposition | 0.0817 |

Entropy calculation | 0.0213 | |

Brain classification | 0.0029 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Dong, Z.; Wang, S.; Ji, G.; Yang, J. Preclinical Diagnosis of Magnetic Resonance (MR) Brain Images via Discrete Wavelet Packet Transform with Tsallis Entropy and Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM). *Entropy* **2015**, *17*, 1795-1813.
https://doi.org/10.3390/e17041795

**AMA Style**

Zhang Y, Dong Z, Wang S, Ji G, Yang J. Preclinical Diagnosis of Magnetic Resonance (MR) Brain Images via Discrete Wavelet Packet Transform with Tsallis Entropy and Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM). *Entropy*. 2015; 17(4):1795-1813.
https://doi.org/10.3390/e17041795

**Chicago/Turabian Style**

Zhang, Yudong, Zhengchao Dong, Shuihua Wang, Genlin Ji, and Jiquan Yang. 2015. "Preclinical Diagnosis of Magnetic Resonance (MR) Brain Images via Discrete Wavelet Packet Transform with Tsallis Entropy and Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM)" *Entropy* 17, no. 4: 1795-1813.
https://doi.org/10.3390/e17041795