Local fractal and multifractal features for volumic texture characterization
Research highlights
► Algorithms for estimating fractal and multifractal features in 3D medical images. ► Optimal parameters tuning using synthetic data. ► Definition of a general classification framework using these features. ► Validation on clinical data in 2 different applications.
Introduction
The diagnostic interpretation of medical images is a multi-step task where the aim is the detection of potential abnormalities. This goal is accurately achieved when the clinician integrates two processes: the first one is the image perception to recognize unique image patterns and the second one is the identification of the relationship between perceived patterns and possible diagnoses. The success of these two steps heavily relies on the clinician skill. Texture features are a rich source of visual information and are key components in image analysis and understanding. It was shown [1] that there is a link between texture perception and texture classification.
In light of this, many studies were conducted to develop algorithms that can quantify the textural properties of an image. Moreover, the exciting evolution of both texture analysis algorithms and computer technology revived researchers' interest in applications for medical imaging in recent years [2]. During the past decade, results from numerous published articles have shown the ability of texture analysis algorithms to extract diagnostically meaningful information from medical images.
The applicability and the relevance of fractal geometry in medical image analysis are justified by the fact that, as they are imaged with a finite resolution, biological objects exhibit hard properties of self-similarity. Indeed, the images are not only spectrally and spatially complex, but they often exhibit some similarities at different spatial scales. This assertion induces that spatially complex patterns could be described by simple texture features. In fact, the problem of features definition for texture analysis in image understanding and pattern recognition was an increasing research domain for many years [3], [4]. Precisely, fractal geometry provides a means of describing and characterizing the complexity of the images or more precisely of their texture components using two features: fractal dimension (FD) and multifractal spectrum or the Hölder coefficient.
For small regions characterization, the use of fractal features is faced with two limitations. First, FD gives overall information on the texture and is unable to distinguish between key textures, such as edges and corners, contrary to multifractal features. In a recent study, Iftekharuddin et al. [5] exploited the fractal models in analyzing brain tumor in MR images. The authors reported that the dynamics of tumor growth followed a fractal process. But, they only used two features, the FD computed on 8×8 sub-images and the pixel gray level. Even if they distinguished tumor regions from non-tumor regions, they had no satisfying specificity in tumor classification.
Moreover, the features are restricted to 2D cases, while the 3D information is very important in medical imaging referring to volumic objects. Chen et al. [6] studied breast lesions on MR images by 3D gray-level co-occurrence matrix. They showed that the classification performance of volumetric texture features was significantly better than those based on 2D analysis.
The purpose of this paper is to investigate the combined use of 3D local fractal and multifractal features to aid diagnosis in medical images. We used three methods to estimate the FD and two methods to estimate the Hölder coefficient for each voxel. As usual, several parameters must be defined: we used synthetic models to optimize these parameters. Then, 3D texture features based on fractal geometry and those currently used were jointly applied to two supervised classifiers: AdaBoost and support vector machines (SVM).
The organization of this paper is as follows. Section 2 proposes a 3D extension of fractal and multifractal methods, and the algorithm of the supervised classification. Section 3 conducts some experiments for investigating fractal geometry based features. Finally, Section 4 draws the discussion of the study.
Section snippets
3D fractal dimension computing
One difficulty in the use of fractal analysis is in the choice of the computation algorithm. Many methods were proposed to compute the FD, each one having its own theoretic basis. Although the applied algorithms are different, they all follow the same basis summarized by the 3 steps [7]:
- –
Measuring the quantities of the object using different step sizes.
- –
Plotting log(measured quantities) versus log(step sizes) and fit a least-squares regression line through the data points.
- –
Estimating FD as the
Algorithms optimization and parameters tuning
The FD and MS computations methods have many freedom degrees. Optimization of these parameters is very important in the estimation accuracy. In order to optimize these methods, we used synthetic models generated by the fractional Brownian motion (fBm) model. The theoretical value of the FD of each nD fBm was known theoretically and was equal to n+1−H.
A 2D 256×256 model and a 3D 128×128×128 were generated. They were composed of 2 fBm functions (H=0.3 and H=0.6), i.e. two textures (Fig. 1a). The
Discussion
The efficiency of fractal geometry analysis when applied on 1D or 2D signals was already demonstrated in many studies where correlations were established between the fractal features and the studied phenomena: EEG/ECG signals [32], brain [33], bone [34], mammography [35] and molecular images [36].
The use of fractal geometry features involves the estimation of the fractal dimension and the multifractal spectrum. Although fractal dimension brings only overall information about the homogeneity of
Conclusion
The purpose of this research was to show the feasibility of the 3D fractal geometry based classification method. The obtained results suggest that a carefully optimized input set of fractal and multifractal features might provide even better results than classical texture features.
Acknowledgment
The authors thank Professor Marc Steinling from the Nuclear Medicine Department and Doctor William Szurhaj from the Neurophysiology Department of the University Hospital of Lille for their interest in this work and for their clinical diagnosis.
Renaud Lopes (born in 1983) obtained a master degree in automation from the Université des Sciences et Technologies de Calais in 2006. He is PhD student in automation from the Université des Sciences et Technologies de Lille since 2006.
His research interests are fractal geometry, texture analysis and classification. He is a member of the Inserm U703 team (Interventional Therapies Guided by Image and Simulation).
References (41)
- et al.
‘Fractal-based brain tumour detection in multimodal MRI’
Applied Mathematics and Computation
(2009) - et al.
Fractal and multifractal analysis: a review
Medical Image Analysis
(2009) - et al.
Statistical analysis of fractal-based brain tumor detetction algorithms
Magnetic Resonance Imaging
(2005) - et al.
Near optimum estimation of local fractal dimension for image segmentation
Pattern Recognition Letters
(2003) - et al.
Predictive value for future arrhythmic events of fractal dimension, a measure of time clustering of ventricular premature complexes after myocardial infraction
Journal of the American College of Cardiology
(1997) - et al.
Endogenous multifractal brain dynamics are modulated by age, cholinergic blockade and cognitive performance
Journal of Neuroscience Methods
(2008) - et al.
Design and implementation of an estimator of fractal dimension using fuzzy techniques
Pattern Recognition
(2001) - et al.
Mammographic masses characterization based on localized texture and dataset fractal analysis using linear, neural and support vector machine classifiers
Artificial Intelleligence in Medicine
(2006) Journey toward computer-aided diagnosis: role of image texture analysis
Radiology
(1999)- et al.
Texture Analysis for Magnetic Resonance Imaging
(2006)
Textural features for image classification
IEEE Transactions on Systems, Man and Cybernetics
Visual discrimination of stochastic texture fields,
IEEE Transactions on Systems, Man and Cybernetics
Volumetric texture analysis of breast lesions on contrast-enhanced magnetic resonance images
Magnetic Resonance in Medicine
The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers
Wavelet analysis and synthesis of fractional Brownian motion
IEEE Transactions on Information Theory
Über die begriffe länge, öberflache und volumen
Jahb. Dtsch. Math.
Multiple resolution texture analysis and classification
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fractional Brownian motion, fractional noises and applications
S.I.A.M. Review
Cited by (56)
Gradient-assisted deep model for brain tumor segmentation by multi-modality MRI volumes
2023, Biomedical Signal Processing and ControlAn approach to determine the features of dental X-ray images based on the fractal dimension
2021, Procedia Computer ScienceA fractal based approach to evaluate the progression of esophageal squamous cell dysplasia
2019, Biomedical Signal Processing and ControlFeatures based on the percolation theory for quantification of non-Hodgkin lymphomas
2017, Computers in Biology and MedicineCitation Excerpt :Moreover, these methods have provided interesting classification rates with an expressive number of features. Although there is not a single and universal texture descriptor capable of performing the quantification of any kind of image, some researches have indicated that fractal techniques can provide better results for histological image quantification when compared to the previously mentioned [19–22], mainly due to the presence of stochastic properties and self-similarities. Amongst the main fractal features present in the literature, fractal dimension (FD), lacunarity (LAC) and percolation-based features stand out.
Characterization of synthetic porous media images by using fractal and multifractal analysis
2023, GEM - International Journal on GeomathematicsA MULTIFRACTAL APPROACH FOR TEXTURE CLASSIFICATION APPLIED TO THE KIDNEY ULTRASOUND IMAGES
2023, Journal of Theoretical and Applied Information Technology
Renaud Lopes (born in 1983) obtained a master degree in automation from the Université des Sciences et Technologies de Calais in 2006. He is PhD student in automation from the Université des Sciences et Technologies de Lille since 2006.
His research interests are fractal geometry, texture analysis and classification. He is a member of the Inserm U703 team (Interventional Therapies Guided by Image and Simulation).
Patrick Dubois (born in 1948) Lecturer. He received an Engineer Degree in 1972 and PhD in 1974. Its Biomedical Engineering and Medical research topics are medical imaging processing, interventional imaging and image-guided conformal radiotherapy.
Imen Bhouri (Lecturer). She is member of the Science Faculty of Monastir, Tunisia. Her research interest is centred on the fractal geometry.
Mohamed Hédi Bedoui (Professor). He is a member of the Medicine Faculty of Monastir, Tunisia. His research interest is in biophysics.
Salah Maouche (born in 1952) received the B.E. degree in engineering from Ecole Nationale Polytechnique of Algiers, Algeria, in 1978. He received the Ph.D. and D.Sc degrees in automatic control from University of Lille, France, in 1982 and 1995, respectively. He is currently professor in Automatic Control at the Université des Sciences et Technologies de Lille and he is responsible of the Decision Engineering team of the automatic control laboratory LAGIS at the university of Lille. His research interests are in optimization and decision making, which attempts to apply to transportation and conformal radiotherapy problems. He is headmaster of the UFR IEEA, research and training unit of computing, Electronics, Electrotechnics and Automatics, at the university of Lille.
Nacim Betrouni (born in 1976) received a computer engineer degree from the university of Tizi Ouzou, Algeria in 1999. He obtained a master degree in electrical engineering and automation from Université des Sciences et Technologies de Lille, France in 2001 and a Ph.D. degree in 2004. He is a research scientist in the French Institute of Health and Medical Research (INSERM U703). His research interests include medical images processing, segmentation, registration, data fusion and image-guided therapy.