Fractal and multifractal analysis: A review

Medical Image Analysis

Volumn 13, Issue 4, 2009, Pages 634-649

  Lopes, R ^a,b   Betrouni, N ^a  

^a LILLE UNIVERSITY HOSPITAL   (France)

^b ECOLE CENTRALE DE LILLE   (France)

Author keywords

Characterization; Fractal analysis; Fractal dimension; Multifractal analysis; Multifractal spectrum; Texture

Indexed keywords

FRACTAL ANALYSIS; FRACTAL FEATURE; MULTI FRACTALS; MULTIFRACTAL ANALYSIS; MULTIFRACTAL SPECTRUM; TEXTURE ANALYSIS;

PARTIAL DISCHARGES; PATTERN RECOGNITION; SIGNAL ANALYSIS; TEXTURES;

FRACTAL DIMENSION;

ALGORITHM; ANALYTIC METHOD; ANISOTROPY; ARTICLE; BONE RADIOGRAPHY; BRAIN RADIOGRAPHY; BREAST CALCIFICATION; COMPUTER ASSISTED TOMOGRAPHY; CONTRAST ENHANCEMENT; ECHOGRAPHY; ELECTROCARDIOGRAPHY; ELECTROENCEPHALOGRAPHY; FRACTAL ANALYSIS; GEOMETRY; HUMAN; IMAGE ANALYSIS; IMAGE DISPLAY; IMAGE PROCESSING; IMAGE QUALITY; IMAGING SYSTEM; MAMMOGRAPHY; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; MULTIFRACTAL ANALYSIS; NUCLEAR MAGNETIC RESONANCE IMAGING; OSTEOPOROSIS; POSITRON EMISSION TOMOGRAPHY; PRIORITY JOURNAL; SINGLE PHOTON EMISSION COMPUTER TOMOGRAPHY; STATISTICAL ANALYSIS; STATISTICAL PARAMETERS; STATISTICAL SIGNIFICANCE; TRABECULAR BONE; X RAY;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; FRACTALS; IMAGE ENHANCEMENT; IMAGE INTERPRETATION, COMPUTER-ASSISTED; IMAGING, THREE-DIMENSIONAL; PATTERN RECOGNITION, AUTOMATED; REPRODUCIBILITY OF RESULTS; SENSITIVITY AND SPECIFICITY;

EID: 70149091472     PISSN: 13618415     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.media.2009.05.003     Document Type: Article

References (206)

1. Abadi M., and Grandchamp E. Texture features and segmentation based on multifractal approach. The 11th Iberoamerican Congress on Pattern Recognition 4225 (2006) 297-305
2. Anderson J., Karagounis L., Stein K., Moreno F., Ledingham R., and Hallstrom A. Predictive value for future arrhythmic events of fractal dimension, a measure of time clustering of ventricular premature complexes, after myocardial infarction. Journal American College of Cardiology 30 1 (1997) 226-232
3. Appleby S. Multifractal characterization of the distribution pattern of the human population. Geographical Analysis 28 2 (1996) 147-160
4. Arneodo A., Bacry E., and Muzy J. Oscillating singularities in locally self-similar functions. Physical review Letters 74 24 (1995) 4823-4826
5. Arneodo A., Decoster N., Kestener P., and Roux S. A wavelet-based method for multifractal image analysis : from theoretical concepts to experimental applications. In: Advances in Imaging and Electron Physics, P.W. Hawkes, Academic Press (Eds), Vol. 126 (2003) 1-98
6. Arneodo A., Bacry E., Jaffard S., and Muzzy J. Singularity spectrum of multifractal functions involving oscillating singularities. Journal of Fourier Analysis and Applications 4 2 (1998) 159-174
7. Asvestas P., Matsopoloulos G., and Nikita K. A power differentiation method of fractal dimension estimation for 2-D signals. Journal of Visual Communication and Image Representation 9 4 (1998) 392-400
8. Bachelier L. Théorie de la spéculation. Annales scientifiques de l'École Normale Supérieure 17 3 (1900) 21-86
9. Badii R., and Politi A. Hausdorff dimension and uniformity factor of strange attractors. Physical Review Letter 52 19 (1984) 1661-1664
10. Badii R., and Politi A. Statistical description of chaotic attractors: the dimension function. Journal of Statistical Physics 40 5-6 (1985) 725-750
11. Bassingthwaighte J., Van Beek J., and King R. Fractal branchings: The basis of myocardial flow heterogeneities?. Annals of the New York Academy of Science 591 1 (1990) 392-401
12. Benhamou C., Lespesailles E., Jacquet G., Harba R., Jennane R., Loussot T., Tourliere D., and Ohley W. Fractal organization of trabecular bone images on calcaneus radiographs. Journal of Bone and Mineral Research 9 12 (1994) 1909-1918
13. Bhatti A., Nahavandi S., and Frayman Y. 3D depth estimation for visual inspection using wavelet transform modulus maxima. Computers and Electrical Engineering 33 1 (2007) 48-57
14. Bisoi A., and Mishra J. On calculation of fractal dimension of images. Pattern Recognition Letters 22 6-7 (2001) 631-637
15. Biswas M., Ghose T., Guha S., and Biswas P. Fractal dimension estimation for texture images: A parallel approach. Pattern Recognition Letters 19 3-4 (1998) 309-313
16. Blanton R., Levitt J., Thompson P., Narr K., Capetillo-Cunliffe L., Nobel A., and Singerman J. Mapping cortical asymmetry and complexity patterns in normal children. Psychiatry Research 107 1 (2001) 29-43
17. Block A., Bloh W., and von Schellnhuber H. Efficient boxcounting determination of generalized fractal dimensions. Physical Review A 42 4 (1990) 1869-1874
18. Brodu N. Multifractal Feature Vectors for Brain-Computer Interfaces. Proceedings of the IEEE World Congress on Computational Intelligence 2883 (2005) 2890
19. Broniatowski M., and Mignot P. A self-adaptive technique for the estimation of the multifractal spectrum. Statistics and Probability Letters 54 2 (2001) 125-135
20. Bullmore E., Brammer M., Harvey I., Persaud R., Murray R., and Ron M. Fractal analysis of the boundary between white matter and cerebral cortex in magnetic resonance images: A controlled study of schizophrenic and manic-depressive patients. Psychological Medicine 24 3 (1994) 771-781
21. Byng J., Boyd N., Fishell E., Jong R., and Yaffe M. Automated analysis of mammographic densities. Physics in Medicine and Biology 41 5 (1996) 909-923
22. Caldwell C., Stapleton S., Holdsworth D., Jong R., Weiser W., Cooke G., and Yaffe M. Characterization of mammographic parenchyma pattern by fractal dimension. Physics in Medicine and Biology 35 2 (1990) 235-247
23. Caldwell C., Moran E., and Bogoch E. Fractal dimension as a measure of altered trabecular bone in experimental inflammatory arthritis. Journal of Bone and Mineral Research 13 6 (1998) 978-985
24. Caligiuri P., Giger M., and Favus M. Multifractal radiographic analysis of osteoporosis. Medical Physics 21 4 (1994) 503-508
25. Caswell Schuckers S. Approximate entropy as a measure of morphologic variability for ventricular tachycardia and fibrillation. Computers in Cardiology 1998 13-16 (1998) 265-268
26. Chappard D., Chennebault A., Moreau M., Legrand E., Audran M., and Basle M.F. Texture analysis of X-ray radiographs is a more reliable descriptor of bone loss than mineral content in a rat model of localized disuse induced by the clostridium botulinum toxin. Bone 28 1 (2001) 72-79
27. Chaudhuri B., and Sarkar N. Texture Segmentation Using Fractal Dimension. IEEE Transactions on Pattern Analysis and Machine Intelligence 17 1 (1995) 72-77
28. Chen S., Keller J., and Crownover R. On the calculation of fractal features from images. IEEE Transactions on Pattern Analysis and Machine Intelligence 15 10 (1993) 1087-1090
29. Chhabra A., and Jensen R. Direct determination of the f(alpha) singularity spectrum. Physical Review Letter 62 12 (1989) 1327-1330
30. Chung H. Fractal Analysis of Nuclear medicine Images Again: Validity and Interpretation of Results from New Analysis Methods. The Journal of Nuclear Medicine 44 2 (2003) 316-317
31. Chung H., Wu W., and Chung H. A reinvestigation of the extended counting method for fractal analysis of the pial vasculature. Journal of Cerebral Blood Flow & Metabolism 22 3 (2002) 361-363
32. Clarke K. Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method. Computers & Geosciences 12 5 (1986) 713-722
33. Cook M., Free S., Manford M., Fish D., Shorvon S., and Stevens J. Fractal description of cerebral cortical patterns in frontal lobe epilepsy. European Neurology 35 6 (1995) 327-335
34. Cooper C. The epidemiology of fragility fractures: is there a role for bone quality?. Calcified Tissue International 53 Suppl. 1 (1993) 23-26
35. Cross S., Howat A., Stephenson T., Cotton D., and Underwood J. Fractal geometric analysis of material from molar and non-molar pregnancies. Journal of Pathology 173 2 (1994) 115-118
36. Cross S., Start R., Silcocks P., Bull A., Cotton D., and Underwood J. Quantitation of the renal arterial tree by fractal analysis. Journal of Pathology 170 4 (1993) 479-484
37. Cuevas E. F(alpha) multifractal spectrum at strong and weak disorder. Physical Review B 68 (2003) 24206-24212
38. De Bartolo S., Gaudio R., and Gabriele S. Multifractal analysis of river networks: sandbox approach. Water Resources Research 40 2 (2004) W02201
39. De Jong S.M., and Burrough P.A. A fractal approach to the classification of Mediterranean vegetation types in remotely sensed images. Photogrammetric Engineering & Remote Sensing 61 8 (1995) 1041-1053
40. Dempster D., Ferguson P., Mellish R., Cochran G., Xie F., Fey C., Horbert W., Parisien M., and Lindsay R. Relationships between bone structure in the iliac crest and bone structure and strength in the lumpar spine. Osteoporosis International 3 (1993) 90-96
41. Dougherty G. A comparison of the texture of computed tomography and projection radiography images of vertebral trabecular bone using fractal signature and lacunarity. Medical Engineering and Physics 23 9 (2001) 313-321
42. Dougherty G., and Henebry G. Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images. Medical Engineering and Physics 23 6 (2001) 369-380
43. Dubuisson M., and Dubes R. Efficacy of fractal features in segmenting images of natural textures. Pattern Recognition Letters 15 4 (1994) 419-431
44. Duncan J., and Ayache N. Medical image analysis: progress over two decades and the challenges ahead. IEEE Transactions on Pattern Analysis and Machine Intelligence 22 1 (2000) 85-105
45. Enescu B., Ito K., and Struzik Z. Wavelet-based multiscale resolution analysis of real and simulated time-series of earthquakes. Geophysical Journal International 164 1 (2006) 63-74
46. Espinal F., and Chandran R. Wavelet-based fractal signature for texture classification. Proc. SPIE of Wavelet Applications V 3391 (1998) 602-611
47. Esteller, R., Echauz, J., Tcheng, T., 2004. Comparison of line length feature before and after brain electrical stimulation in epileptic patients. Proceedings of the 26th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 7, 4710-4713.
48. Ezekiel S. Medical Image Segmentation using multifractal analysis. Proceedings of the Applied Informatics 378 (2003) 220-224
49. Feeny B. Fast multifractal analysis by recursive box covering. International Journal of Bifurcation and Chaos 10 9 (2000) 2277-2287
50. Free S., Sisodiya S., Cook M., Fish D., and Shorvon S. Three-dimensional fractal analysis of the white matter surface from magnetic resonance images of the human brain. Cerebral Cortex 6 6 (1996) 830-836
51. Frish, U., 1995. Turbulence: the Legacy of A.N Kolmogorov. Cambridge University Press (Eds), Cambridge (UK), 296 pp.
52. Glenny R., and Robertson H. Fractal modeling of pulmonary blood flow heterogeneity. Journal of Applied Physiology 70 3 (1991) 1024-1030
53. Goodchild M. Fractals and the accuracy of geographical measures. Mathematical Geology 12 2 (1980) 85-98
54. Gorski A.Z., and Skrzat J. Error estimation of the fractal dimension measurements of cranial sutures. Journal of Anatomy 208 3 (2006) 353-359
55. Grassberger P., Badii R., and Politi A. Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors. Journal of Statistical Physics 51 1-2 (1988) 135-178
56. Grassberger P., and Procaccia I. Characterization of strange attractors. Physical Review Letter 50 5 (1983) 346-349
57. Guillemet, H. Détection et caractérisation des foyers de micro calcifications en mammographie numérique. 1996. Ecole Nationale Supérieure des Télécommunications, Paris, France, 171 pp.
58. Guo Q., Shao J., Guo F., and Ruiz V. Classification of mammographic masses using geometric symmetry and fractal analysis. International Journal of Computer Assisted Radiology and Surgery 2 Suppl. 1 (2007) 336-338
59. Halsey T., Jensen M., Kadanoff L., Procaccia I., and Shraiman B.I. Fractal measures and their singularities: the characterization of strange sets. Physical Review A 33 2 (1986) 1141-1151
60. Haralick R., Shanmugan K., and Dinstein I. Textural features for image classification. IEEE Transactions on Systems, Man and Cybernetics 3 (1973) 610-621
61. Hou X., Gilmore R., Mindlin G., and Solari H. An efficient algorithm for fast O(N*ln(N)) box-counting. Physics Letters A 151 1-2 (1990) 43-46
62. Hsu W.Y., Lin C.C., Ju M.S., and Sun Y.N. Wavelet-based fractal features with active segment selection: application to single-trial EEG data. Journal of Neuroscience Methods 163 (2007) 145-160
63. Huang, Y., Yu, S., 2007. Recognition of micro-calcifications in digital mammograms based on Markov random fiels and deterministic fractal modeling. Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 3922-3925.
64. Ida T., and Sambonsugi Y. Image segmentation and contour detection using fractal coding. IEEE transactions on circuits and systems for video technology 8 8 (1998) 968-975
65. Iftekharuddin, K., Jia, W., Marsh, R., 2000. A fractal analysis approach to identification of tumor in brain MR images. Proceedings of the 22nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society 4, 3064-3066.
66. Iftekharuddin K., Jia W., and Marsh R. A fractal analysis of tumor in brain MR images. Machine Vision and Applications 13 5 (2003) 352-362
67. Imre A.R., and Bogaert J. The fractal dimension as a measure of the quality of habitats. Acta Biotheoretica 52 1 (2004) 41-56
68. Jaffard, S., Abry, P., Lashermes, B., 2005. Wavelet leaders in multifractal analysis. In: Wavelet analysis and Applications, T. Qian, M.I. Vai, X. Yuesheng, Birkhäuser Verlag (Eds), 219-264.
69. Jaffard S. Multifractal formalism for functions. SIAM Journal of Mathematical Analysis 28 (1997) 944-970
70. Jain R., Munn L., and Fukumura D. Dissecting tumour pathopsysiology using intravital microscopy. Nature Reviews Cancer 2 4 (2002) 266-276
71. Jennane R., Harba R., Lemineur G., Bretteil S., Estrade A., and Benhamou J. Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection. Medical Image Analysis 11 1 (2006) 91-98
72. Jin X., Ong S., Jayasooriah,. A practical method for estimating fractal dimension. Pattern Recognition Letters 16 5 (1995) 457-464
73. John, S., Bradford, A., 1993. The electroencephalogram: Its pattern and origins, MIT Press (Eds), London.
74. Jun, Y., Yoon, Y., Yoon, H., 1994. ECG data compression using fractal interpolation. Proceedings of the 16th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 1 (3-6), 161-162.
75. Kalmanti E., and Maris T.G. Fractal dimension as an index of brain cortical changes throughout life. In Vivo 21 4 (2007) 641-646
76. Kaplan L. Extended fractal analysis for texture classification and segmentation. IEEE Transactions on Image Processing 8 11 (1999) 1572-1585
77. Kedzia A., Rybaczuk M., and Andrzejak R. Fractal dimensions of human brain cortex vessels during the fetal period. Medical Science Monitor 8 3 (2002) 46-51
78. Keller J., Crownover R., and Chen S. Texture description and segmentation through fractal geometry. Computer Vision Graphics and Image Processing 45 2 (1989) 150-160
79. Kenkel N., and Walker D. Fractals in the biological sciences. Coenoses 1 (1996) 77-100
80. Keough K., Hyam P., Pink D., and Quinn B. Cell surfaces and fractal dimensions. Journal of Microscopy 163 1 (1991) 95-99
81. Kestener P., and Arneodo A. A three-dimensional wavelet based multifractal method : about the need of revisiting the multifractal description of turbulence dissipation data. Physical Review Letter 91 19 (2003) 194501.1-194501.4
82. Kestener P., Lina J., Saint-Jean P., and Arneodo A. Wavelet-based multifractal formalism to assist in diagnosis in digitized mammograms. Image Analysis and Stereology 20 3 (2004) 169-175
83. Khalil A., Joncas G., Nekka F., Kestener P., and Arneodo A. Morphological analysis of H I features. II. Wavelet-based multifractal formalism. Astrophysical Journal 165 2 (2006) 512-550
84. Khosrovi P., Kahn A., Genant H., and Majumdar S. Characterization of trabecular bone structure from radiographs using fractal analysis. Journal of Bone and Mineral Research 9 (1994) 156-157
85. Kikuchi A., Unno N., Horikoshi T., Shimizu T., Kozuma S., and Taketani Y. Changes in fractal features of fetal heart rate during pregnancy. Early Human Development 81 (2005) 655-661
86. Klapetek P., and Ohlidal I. Applications of the wavelet transform in AFM data analysis. Acta Physica Slovaca 55 3 (2005) 295-303
87. Klinkenberg B., and Goodchild M. The fractal properties of topography: a comparison of methods. Earth Surface Processes and Landforms 17 (1992) 217-234
88. Kolibal J., and Monde J. Fractal image error analysis. Computers & Geosciences 4 8 (1998) 785-795
89. Kolmogorov A. The local structure of turbulence in incompressible viscous fluid for very large reynolds number. Comptes Rendus de l'Académie des sciences 30 (1941) 9-13
90. Krenz G., Linehan J., and Dawson C. A fractal continuum model of the pulmonary arterial tree. Journal of Applied Physiology 72 6 (1992) 2225-2237
91. Kulish V., Sourin A., and Sourina O. Human electroencephalograms seen as fractal time series: Mathematical analysis and visualization. Computers in Biology and Medicine 36 (2006) 291-302
92. Lam, N., De Cola, L., 2002. Fractals in Geography. The Blackburn Press (Eds), Caldwell, NJ, 308 pp.
93. Lam N., Qiu H., and Quattochi D. An evaluation of fractal surface measurement methods using ICAMS (Image Characterization and Modeling System). ACSM/ASPRS Annual Convention 5 (1997) 377-386
94. Lam N., Qiu H., Quattochi D., and Emerson C. An evaluation of fractal methods for characterizing image complexity. Cartography and Geographic Information Science 29 1 (2002) 25-35
95. Landini G., and Rippin J.W. Quantification of nuclear pleomorphism using an asymptotic fractal model. Analytical & Quantitative Cytology & Histology 18 2 (1996) 167-176
96. Lanyi, M., 1986. Diagnosis and differential diagnosis of breast calcifications. Springer-Verlag (Eds), Berlin, Heidelberg, New York, 252 pp.
97. Lashermes B., Jaffard S., and Abry P. Wavelet leader based multifractal analysis. Acoustics, Speech, and Signal Processing 4 18 (2005) 161-164
98. Lee W., Chen Y., Chen Y., and Hsieh K. Unsupervised segmentation of ultrasonic liver images by multiresolution fractal feature vector. Information Sciences 175 3 (2005) 177-199
99. Lespessailles E., Jullien A., Eynard E., Harba R., Jacquet G., Ildefonse J., Ohley W., and Benhamou C.L. Biomechanical properties of human os calcanei: relationships with bone density and fractal evaluation of bone microarchitecture. Journal of Biomechanics 31 9 (1998) 817-824
100. Lévy-Véhel J. Fractal approches in signal processing. Fractals 3 (1995) 755-775
101. Lévy-Véhel, J., Mignot, P., Berroir, J., 1992. Texture and Multifractals: New tools for image analysis. Technical report of INRIA-Rocquencourt, vol. 1706.
102. Lévy-Véhel, J., Vojak, R., 1995. Multifractal analysis of Choquet capacities: Preliminary Results. Technical report of INRIA-Rocquencourt, vol. 2576.
103. Li, H., Liu, K., Lo, S., 1996. Enhancement of micro-calcifications on mammograms using a fractal modelling approach. Proceedings of the 18th Annual International Conference of the IEEE Engineering in Medecine and Biology Society 3, 1111-1112.
104. Li H., Giger M.L., Olopade O.I., and Lan L. Fractal analysis of mammographic parenchymal patterns in breast cancer risk assessment. Academic Radiology 14 5 (2007) 513-521
105. Li X., Polygiannakis J., Kapiris P., Peratzakis A., Eftaxias K., and Yao X. Fractal spectral analysis of pre-epileptic seizures in terms of criticality. Journal of Neural Engineering 2 2 (2005) 11-16
106. Link T., Majumdar S., Konermann W., Meier N., Lin J., Newitt D., Ouyang X., Peters P., and Genant H. Texture analysis of direct magnification radiographs of vertebral specimens: correlation with bone mineral density and biomechanical properties. Academic Radiology 4 (1997) 167-176
107. Liu J., Zhang L., and Yue G. Fractal dimension in human cerebellum measured by magnetic resonance imaging. Biophysical Journal 85 6 (2003) 4041-4046
108. Lopes R., Betrouni N., Szurhaj W., and Steinling M. 3D Fractal analysis for epilepsy characterization on SPECT images. European Journal of Nuclear Medicine 34 2 (2007) 394
109. Lopes R., Dubois P., Dewalle A., Maouche S., and Betrouni N. 3D multifractal analysis of cerebral tomoscintigraphy images. International journal of Computer Assisted Radiology and Surgery 2 Supp1 (2007) 17-18
110. Lopes R., Dubois P., Makni N., Szurhaj W., Maouche S., and Betrouni N. Classification of brain SPECT imaging using 3D local multifractal spectrum for epilepsy detection. International Journal of Computer Assisted Radiology and Surgery 3 3-4 (2008) 341-346
111. Maeda, J., Novianto, S., Miyashita, A., Saga, S., Suzuki, Y., 1998. Fuzzy region-growing segmentation of natural images using local fractal dimension. Proceedings of the 14th International Conference on Pattern Recognition 2, 991.
112. Majumdar S., Lin J., Link T., Millard J., Augat P., Ouyang X., Newitt D., Gould R., Kothari M., and Genant H. Fractal analysis of radiographs: Assessment of trabecular bone structure and prediction of elastic modulus and strength. Medical Physics 26 7 (1999) 1330-1340
113. Mandelbrot B. The variation of certain speculative prices. J. of Business 36 (1963) 394-419
114. Mandelbrot, B., 1975. Les objets fractals : forme, hasard et dimension. Flammarion (Eds), Paris, 208 pp.
115. Mandelbrot, B., 1977. Fractals: Form. Chance and Dimension, Freeman (Eds), 365 pp.
116. Mandelbrot, B., 1983. The Fractal Geometry of Nature. Freeman (Eds), 468 pp.
117. Mandelbrot B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156 (1967) 636-638
118. Mandelbrot B. Self-affine fractals and the fractal dimension. Physica Scripta 32 (1985) 257-260
119. Mansury Y., and Deisboeck T. Simulating "structure-function" patterns of malignant brain tumors. Physica A 331 1-2 (2004) 219-232
120. Martinez P., Schertzer D., and Pham K. Texture modelisation by multifractal processes for SAR image segmentation. Radar 97 4 (1997) 135-139
121. Mavroforakis M., Georgiou H., Dimitropoulos N., Cavouras D., and Theodoridis S. Mammographic masses characterization based on localized texture and dataset fractal analysis using linear, neural and support vector machine classifiers. Artificial Intelligence In Medicine 37 (2006) 145-162
122. Messent E., Ward R., Tonkin C., and Buckland-Wright C. Tibial cancellous bone changes in patients with knee osteoarthritis. A short-term longitudinal study using fractal signature analysis. OsteoArthritis and Cartilage 13 (2005) 463-470
123. Meyer, Y., 1998. Wavelets, vibrations and scaling. CRM Monograph Ser. 9, American Mathematical Society, Providence, RI.
124. Molteno T. Fast O(N) box-counting algorithm for estimating dimensions. Physical Review E 48 5 (1993) 3263-3266
125. Munoz, D.A., Angulo, B.F., Del Rio Correa, J.L., 2005. Changes in multifractality with aging and heart failure in heartbeat interval time series. Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 7, 6981-6984.
126. Murali, S., 2005. Fractal and FFT power spectra analysis of human electroencephalograms (brain waves) for six different odors, M.Sc. dissertation, Nanyang Technological University, Singapore.
127. Nagao M., and Murase K. Measurement of heterogeneous distribution on Technegas SPECT images by three-dimensional fractal analysis. Annals of Nuclear Medicine 16 6 (2002) 369-376
128. Nguyen, T., Rangayyan, R., 2005. Shape Analysis of Breast Masses in Mammograms via the Fractal Dimension. Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medecine and Biology Society 3, 3210-3213.
129. Normant F., and Tricot C. Methods for evaluating the fractal dimension of curves using convex hulls. Physical Review A 43 12 (1991) 6518-6525
130. Novianto S., Suzuki Y., and Maeda J. Near optimum estimation of local fractal dimension for image segmentation. Pattern Recognition Letters 24 1-3 (2003) 365-374
131. Oczeretko E., Juczewska M., and Kasacka I. Fractal geometric analysis of lung cancer angiogenic patterns. Folia Histochemica et Cytobiologica 39 Suppl. 2 (2001) 75-76
132. Oiwa N., and Glazier J. Self-Similar Mitochondrial DNA. Cell. Biochem. Biophys. 41 1 (2004) 41-62
133. Osborne A.R., and Provenzale A. Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35 (1989) 357-381
134. Ott S. Should women get screening bone mass measurements?. Annals of Internal Medicine 104 (1986) 874-876
135. Ouyang X., Majumdar S., Link T., Lu Y., Augat P., Lin J., Newitt D., and Genant H. Morphometric texture analysis of spinal trabecular bone structure assessed using orthogonal radiographic projections. Medical Physics 25 10 (1998) 2037-2045
136. Owis M., Abou-Zied A., Youssef A., and Kadah Y. Study of features based on nonlinear dynamical modeling in ECG arrhythmia detection and classification. IEEE Transactions on Biomedical Engineering 49 7 (2002) 733-736
137. Papaloucas C., Ward R., Tonkin C., and Buckland-Wright C. Cancellous bone changes in hip osteoarthritis: a short-term longitudinal study using fractal signature analysis. OsteoArthritis and Cartilage 13 11 (2005) 998-1003
138. Parkinson I., and Fazzalari N. Cancellous bone structure analysis using image analysis. Australasian Physical and Engineering Sciences in Medicine 470 (1994) 64-70
139. Peleg S., Naor J., Hartley R., and Avnir D. Multiple resolution texture analysis and classification. IEEE Transactions on Pattern Analysis and Machine Intelligence 6 6 (1984) 661-674
140. Penn A., Bolinger L., Schnall M., and Loew M. Discrimination of MR images of breast masses with fractal-interpolation function models. Academic Radiology 6 3 (1999) 156-163
141. Pentland A. Fractal-based description of natural scenes. IEEE Transactions on Pattern Analysis and Machine Intelligence 6 6 (1984) 661-674
142. Pereda E., Gamundi A., Rial R., and Gonzalez J. Non-linear behaviour of human EEG: fractal exponent versus correlation dimension in awake and sleep stages. Neuroscience Letters 250 2 (1998) 91-94
143. Pereira, D., Zambrano, C., Martin-Landrove, M., 2000. Evaluation of malignancy in tumors of the central nervous system using fractal dimension. Proceedings of the 22nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society 3, 1775-1778.
144. Perrier E., Tarquis A., and Dathe A. A program for fractal and multifractal analysis of 2D binary images : Computer algorithms versus mathematical theory. Geoderma 134 3-4 (2006) 284-294
145. Pham D., Xu C., and Prince J. Current methods in medical image segmentation. Annual Review of Biomedical Engineering 2 (1999) 315-337
146. Phothisonothai M., and Nakagawa M. Fractal-based EEG data analysis of body parts movement imagery tasks. Journal of Physiological Sciences 57 4 (2007) 217-226
147. Pickover C., and Khorasani A. Fractal characterization of speech waveforms graphs. Computer Graphics 1 1 (1986) 51-61
148. Popivanov D., Jivkova S., Stomonyakov V., and Nicolova G. Effect of independant component analysis on multifractality of EEG during visual-motor task. Signal Processing 85 11 (2005) 2112-2123
149. Pratt W., Faugeras O., and Gagalowicz A. Visual discrimination of stochastic texture fields. IEEE Transactions on Systems. Man and Cybernetics 8 11 (1978) 796-804
150. Prigarin S., Hahn K., and Winkler G. Comparative analysis of two numerical methods to measure Hausdorff dimension of the fractional Brownian motion. Numerical Analysis and Applications 1 2 (2008) 163-178
151. Pruess S. Some remarks on the numerical estimation of fractal dimension. In: Fractals in the Earth Sciences, C.C. Barton and P.R. La Pointe, Plenum Press (Eds), New-York, Chap. 3 (2007) 65-75
152. Qiu H., Lam N., Quattochi D., and Gamon J. Fractal characterization of hyperspectral imagery. Photogrammetric Engineering and Remote Sensing 65 1 (1999) 63-71
153. Rangayyan R., and Nguyen T. Pattern classification of breast masses via fractal analysis of their contours. Proceedings of Computer Assisted radiology and Surgery (CARS) 1281 (2005) 1041-1046
154. Renyi A. On a new axiomatic theory of probability. Acta Mathematica Hungarica 6 3-4 (1955) 285-335
155. Russel D., Hanson J., and Ott E. Dimension of strange attractors. Physical Review Letters 45 14 (1980) 1175-1178
156. Ruttiman U., Webber R., and Hazelrig J. Fractal dimension from radiographs of peridental alveolar bone. Oral Pathology 74 1 (1992) 98-110
157. Rybaczuk M., Kedzia A., and Blaszcyk E. Fractal description of cerebellum surface during fetal period. Folia Morphologica 55 4 (1996) 434-436
158. Sandau K., and Kurz H. Measuring fractal dimension and complexity-an alternative approach with an application. Journal of Microscopy 186 2 (1997) 164-176
159. Sanderson B., and Goulding A. The fractal dimension of relative Lagrangian motion. Tellus 42A (1990) 550-556
160. Sato K., Sugawara K., Narita Y., and Namura I. Consideration of the method of image diagnosis with respect to frontal lobe antrophy. Proceedings of IEEE Transactions on Nuclear Science 43 (1996) 3230-3239
161. Shan Z., Liu J., Glass J., Gajjar A., Li C., and Reddick W. Quantitative morphological evaluation of white matter in survivors of chilhood medulloblastoma. Magnetic Resonance Imaging 24 8 (2006) 1015-1022
162. Shelberg M., Lam N., and Moellering H. Measuring the fractal dimension of surfaces. Proceedings of the Sixth International Symposium on Computer-Assisted Cartography Auto-Carto 6 (1983) 319-328
163. Shimizu Y., Barth M., Windischberger C., Moser E., and Thurner S. Wavelet-based multifractal analysis of fMRI time series. NeuroImage 22 3 (2004) 1195-1202
164. Sisodiya S., and Free S. Disproportion of cerebral surface areas and volumes in cerebral dysgenesis. MRI-based evidence for connectional abnormalities. Brain 120 2 (1997) 271-281
165. Small M., Yu D., Harrison R., Grubb N., and Fox K. Uncovering non-linear structure in human ECG recordings. Chaos, Solitons & Fractals 13 (2002) 1755-1762
166. Smith T., and Behar T. Comparative fractal analysis of cultured glia derived from optic nerve and brain demonstrate different rates of morphological differentiation. Brain Research 634 2 (1994) 181-190
167. Smith T., Brauer K., and Reichenbach A. Quantitative phylogenetic constancy of cerebellar Purkinje cell morphological complexity. The Journal of Comparative Neurology 331 3 (1993) 402-406
168. Smrcka P., Bittner R., Vysoky P., and Hana K. Fractal and multifractal properties of heartbeat interval series in extremal states of the human organism. Measurement Science Review 3 (2003) 13-15
169. Soille P., and Rivest J. On the validity of fractal dimension measurements in image analysis. Journal of Visual Communication and Image Representation 7 3 (1996) 217-229
170. Soltys Z., Ziaja M., Pawlinski R., Setkowicz Z., and Janeczko K. Morphology of reactive microglia in the injured cerebral cortex. Fractal analysis and complementary quantitative methods. Journal of Neuroscience Research 63 1 (2001) 90-97
171. Spasic S., Kalauzi A., Grbic G., Martac L., and Culic M. Fractal analysis of rat brain activity after injury. Medical and Biological Engineering and Computing 43 3 (2005) 345-348
172. Stojic T., Reljin I., and Reljin B. Adaptation of multifractal analysis to segmentation of microcalcifications in digital mammograms. Physica A 367 (2006) 494-508
173. Stosic T., and Stosic B.D. Multifractal analysis of human retinal vessels. IEEE Transactions on Medical Imaging 25 8 (2006) 1101-1107
174. Sun W. Three new implementations of the triangular prism method for computing the fractal dimension of remote sensing images. Photogrammetric Engineering and Remote Sensing 72 4 (2006) 373-382
175. Suri, J., Setarehdan, S., Sing, S., 2002. Advanced algorithmic approaches to medical image segmentation, Springer (Eds), New-York, 668 pp.
176. Tabar L., Duffy S., and Burhenne L. New Swedish breast cancer detection results for women aged 40-49. Cancer 72 suppl. 4 (1993) 1437-1448
177. Takahashi T., Murata T., Narita K., Hamada T., Kosaka H., Omori M., Takahashi K., Kimura H., Yoshida H., and Wada Y. Multifractal analysis of deep white matter microstructural changes on MRI in relation to early-stage atherosclerosis. NeuroImage 32 3 (2006) 1158-1166
178. Taleb-Ahmed A., Dubois P., and Duquenoy E. Analysis methods of CT-scan images for the characterization of the bone texture: First results. Pattern Recognition Letters 24 12 (2003) 1971-1982
179. Tanki N., Murase K., and Nagao M. A new parameter enhancing breast cancer detection in computer-aided diagnosis of X-ray mammograms. Igaku Butsuri 26 4 (2006) 207-215
180. Taylor P., Hajnal S., Dilhuydy M., and Barreau B. Measuring image texture to separate "difficult" from "easy" mammograms. British Journal of Radiology 67 (1994) 456-463
181. Tél T., Fulop A., and Vicsek T. Determination of fractal dimensions for geometrical multifractals. Physica A 159 (1989) 155-166
182. Thakor N., Zhu Y., and Pan K. Ventricular tachycardia and fibrillation detection by a sequential hypothesis testing algorithm. IEEE Transactions on Biomedical Engineering 37 9 (1990) 837-843
183. Thompson P., Schwartz C., Lin R., Khan A., and Toga A. Three-dimensional statistical analysis of sulcal variability in the human brain. Journal of Neuroscience 16 13 (1996) 4261-4274
184. Touchette H., and Beck C. Nonconcave entropies in multifractals and the thermodynamic formalism. Journal of Statistical Physics 125 2 (2005) 455-471
185. Tourassi G. Journey toward computer-aided diagnosis: Role of image texture analysis. Radiology 213 2 (1999) 317-320
186. Uitewaal P., Lips P., and Netelenbos J. An analysis of bone structure in patients with hip fracture. Journal of Bone and Mineral Research 3 2 (1987) 63-67
187. Veldkamp W., Karssemeijer N., Otten J., and Hendriks J. Automated classification of clustered microcalcifications into malignant and benign types. Medical Physics 27 11 (2000) 2600-2608
188. Veneziano D., Moglen G., and Bras R. Multifractal analysis: pitfalls of standard procedures and alternatives. Physical Review E 52 2 (1995) 1387-1398
189. Vicsek T. Mass multifractals. Physica A 168 (1990) 490-497
190. Wang G., Huang H., Xie H., Wang Z., and Hu X. Multifractal analysis of ventricular fibrillation and ventricular tachycardia. Medical Engineering & Physics 29 3 (2007) 375-379
191. Wang G., Ren X., Li L., and Wang Z. Multifractal analysis of surface EMG signals for assessing muscle fatigue during static contractions. Journal of Zhejiang University - Science A 8 (2006) 910-915
192. Wang J., Ning X., Ma Q., Bian C., Xu Y., and Chen Y. Multiscale multifractality analysis of a 12-lead electrocardiogram. Physical Review E 71 6 (2005) 062902.1-062902.4
193. Wax A., Yang C., Muller M., Nines R., Boone C., Steele V., Stoner G., Dasari R., and Feld M. In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry. Cancer Research 63 (2003) 3556-3559
194. Wilkie J.R., Giger M.L., Chinander M.R., Vokes T.J., Li H., Dixon L., and Jaros V. Comparison of radiographic texture analysis from computed radiography and bone densitometry systems. Medical Physics 31 4 (2004) 882-891
195. Woyshville M., and Calabrese J. Quantification of occipital EEG changes in Alzheimer's disease utilizing a new metric; the fractal dimension. Biological Psychology 35 6 (1994) 291-302
196. Xia Y., Feng D., and Zhao R. Morphology-based multifractal estimation for texture segmentation. IEEE Transactions on Image Processing 15 3 (2006) 614-624
197. Yasar F., and Akgunlu F. The differences in panoramic mandibular indices and fractal dimension between patients with and without spinal osteoporosis. Dentomaxillofacial Radiology 35 1 (2006) 1-9
198. Yi W.J., Heo M.S., Lee S.S., Choi S.C., Huh K.H., and Lee S.P. Direct measurement of trabecular bone anisotropy using directional fractal dimension and principal axes of inertia. Oral surgery, oral medicine, oral pathology, oral radiology, and endodontics 104 1 (2007) 110-116
199. Yoshikawa T., Murase K., Oku N., Kitagawa K., Imaizumi M., Takasawa M., Nishikawa T., Matsumoto M., Hatazawa J., and Hori M. Statistical Image Analysis of Cerebral Blood Flow in Vascular Dementia with Small-Vessel Disease. Journal of Nuclear medicine 44 4 (2003) 505-511
200. Yu Z.G., Anh V., and Lau K.S. Measure representation and multifractal analysis of complete genomes. Phys. Rev. E Stat. Nonlin. Soft Matter Physics 64 (2001) 031903
201. Yum M.K., and Kim J.S. Increased intermittency and decreased nonstationarity of heart rates during the daytime in patients with neurocardiogenic syncope. Journal of Cardiovascular Electrophysiology 13 8 (2002) 788-793
202. Zhang H., Zhu Y.S., and Xu Y. Complexity information based analysis of pathological ECG rythm for ventricular tachycardia and ventricula fibrillation. International Journal of Bifurcation and Chaos 12 (2002) 2293-2303
203. Zhang L., Dean D., Liu J., Sahgal V., Wang X., and Yue G. Quantifying degeneration of white matter in normal aging using fractal dimension. Neurobiology of Aging 28 10 (2006) 1543-1555
204. Zhang X.S., Zhu Y.S., Thakor N.V., and Wang Z.Z. Detecting ventricular tachycardia and fibrillation by complexity measure. IEEE Transactions on Biomedical Engineering 46 5 (1999) 548-555
205. Zhuang X., and Meng Q. Local fuzzy fractal dimension and its application in medical image processing. Artificial Intelligence in Medicine 32 1 (2004) 29-36
206. Zook J., and Iftekharuddin K. Statistical analysis of fractal-based brain tumor detection algorithms. Magnetic Resonance Imaging 23 5 (2005) 671-678