From the Slit-Island Method to the Ising model: Analysis of irregular grayscale objects

Przemysław Mazurek ; Dorota Oszutowska-Mazurek

International Journal of Applied Mathematics and Computer Science, Tome 24 (2014), p. 49-63 / Harvested from The Polish Digital Mathematics Library

Résumé

The Slit Island Method (SIM) is a technique for the estimation of the fractal dimension of an object by determining the area-perimeter relations for successive slits. The SIM could be applied for image analysis of irregular grayscale objects and their classification using the fractal dimension. It is known that this technique is not functional in some cases. It is emphasized in this paper that for specific objects a negative or an infinite fractal dimension could be obtained. The transformation of the input image data from unipolar to bipolar gives a possibility of reformulated image analysis using the Ising model context. The polynomial approximation of the obtained area-perimeter curve allows object classification. The proposed technique is applied to the images of cervical cell nuclei (Papanicolaou smears) for the preclassification of the correct and atypical cells.

Détail

Publié le : 2014-01-01

Zbl 1292.68149

EUDML-ID : urn:eudml:doc:271912

Bibliographie

[000] Adam, R., Silva, R., Pereira, F., Leite, N., Lorand-Metze, I. and Metze, K. (2006). The fractal dimension of nuclear chromatin as a prognostic factor in acute precursor B lymphoblastic leukemia, Cellular Oncology 28(1-2): 55-59.

[001] Atkinson, P. (2002). Spatial statistics, in A. Stein, F. Van Der Meer and B. Gorte (Eds.), Spatial Statistics for Remote Sensing, Kluwer Academic Publishers, Boston, MA, pp. 57-82.

[002] Barnsley, M., Devaney, R., Mandelbrot, B., Peitgen, H.-O., Saupe, S. and Voss, R. (1988). The Science of Fractal Images, Springer-Verlag, Heidelberg. | Zbl 0683.58003

[003] Bedin, V., Adam, R., de Sa, B., Landman, G. and Metze, K. (2010). Fractal dimension of chromatin is an independent prognostic factor for survival in melanoma, BMC Cancer 10(260), 6 pages.

[004] Binney, J., Dowrick, N., Fisher, A. and Newman, M. (1992). The Theory of Critical Phenomena. An Introduction to the Renormalization Group, Clarendon Press, Oxford. | Zbl 0771.00009

[005] Chosia, M. and Domagała, W. (2010). CCervical Cytodiagnosis, Pro Pharmacia Futura, Warsaw, (in Polish).

[006] Cibas, E. and Ducatman, B. (2009). Cytology. Diagnostic Principles and Clinical Correlates, Saunders Elsevier, Philadelphia, PA.

[007] Clarke, K. (1986). Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method, Computer and Geoscience 12(5): 713-722.

[008] Costa, L. and Cesar Jr., R. (2009). Shape Classification and Analysis. Theory and Practice, CRC Press, Boca Raton, FL.

[009] Dey, P. and Banik, T. (2012). Fractal dimension of chromatin texture of squamous intraepithelial lesions of cervix, Diagnostic Cytopathology 40(2): 152-154.

[010] Engler, O. and Randle, V. (2010). Introduction to Texture Analysis: Macrotexture, Microtexture, and Orientation Mapping, CRC Press, Amsterdam.

[011] Ferro, D., Falconi, M., Adam, R., Ortega, M., Lima, C., de Souza, C., Lorand-Metze, I. and Metze, K. (2011). Fractal characteristics of May-Grünwald-Giemsa stained chromatin are independent prognostic factors for survival in multiple myeloma, PLoS ONE 6(6): 1-8.

[012] Filipczuk, P., Wojtak, W. and Obuchowicz, A. (2012). Automatic nuclei detection on cytological images using the firefly optimization algorithm, in E. Piętka and J. Kawa (Eds.), Information Technologies in Biomedicine, Lecture Notes in Computer Science, Vol. 7339, Springer, Heidelberg, pp. 85-92.

[013] Geman, S. and McClure, D. (1987). Statistical methods for tomographic image reconstruction, Bulletin of the International Statistical Institute LII(4): 5-21.

[014] Glauber, R. (1963). Time-dependent statistics of the Ising model, Journal of Mathematical Physics A 4(2): 1299-1303. | Zbl 0145.24003

[015] Goodchild, M. (1980). Fractals and the accuracy of geographical measures, Mathematical Geology 12(2): 85-98.

[016] Goutzanis, L., Papadogeorgakis, N., Pavlopoulos, P., Katti, K., Petsinis, V., Plochoras, I., Pantelidaki, C., Kavantzas, N., Patsouris, E. and Alexandridis, C. (2008). Nuclear fractal dimension as a prognostic factor in oral squamous cell carcinoma, Oral Oncology 44(4): 345-353.

[017] Harte, D. (2001). Multifractals. Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL. | Zbl 1016.62111

[018] Hoda, R. and Hoda, S. (2007). Fundamentals of Pap Test Cytology, Humana Press, Totowa.

[019] Hrebień, M., Korbicz, J. and Obuchowicz, A. (2007). Hough transform, (1+1) search strategy and watershed algorithm in segmentation of cytological images, in M. Kurzyński, E. Puchała, M. Woźniak and A. Żołnierek (Eds.), Computer Recognition Systems, Advances in Soft Computing, Vol. 45, Springer, Berlin, pp. 550-557.

[020] Hrebień, M., Steć, P., Nieczkowski, T. and Obuchowicz, A. (2008). Segmentation of breast cancer fine needle biopsy cytological images, International Journal of Applied Mathematics and Computer Science 18(2): 159-170, DOI: 10.2478/v10006-008-0015-x. | Zbl 1228.92045

[021] Huang, Z., Tian, J. and Wang, Z. (1990). A study of the slit island analysis as a method for measuring fractal dimension of fractured surface, Scripta Metall Mater 24(6): 967-972.

[022] Jeleń, L., Fevens, T. and Krzyżak, A. (2008). Classification of breast cancer malignancy using cytological images of fine needle aspiration biopsies, International Journal of Applied Mathematics and Computer Science 18(1): 75-83, DOI: 10.2478/v10006-008-0007-x.

[023] Kaye, B. (1994). A Random Walk through Fractal Dimensions, VCH, Weinham/New York, NY. | Zbl 0805.58004

[024] Kowal, M., Filipczuk, P., Marciniak, A. and Obuchowicz, A. (2013). Swarm optimization and multi-level thresholding of cytological images for breast cancer diagnosis, in R. Burduk, K. Jackowski, M. Kurzyński, M. Woźniak and A. Żołnierek (Eds.), CORES 2013, Advances in Intelligent Systems and Computing, Vol. 226, Springer-Verlag, Heidelberg, pp. 611-620. P. Mazurek and D. Oszutowska-Mazurek

[025] Kuehnel, W. (2003). Color Atlas of Cytology, Histology, and Microscopic Anatomy, Thieme, New York, NY.

[026] Loncaric, A. (1998). A survey of shape analysis techniques, Pattern Recognition 31(8): 983-1001.

[027] Lu, C. (1995). On the validity of the slit islands analysis in the measure of fractal dimension of fracture surfaces, International Journal of Fracture 69(4): 77-80.

[028] Mandelbrot, B. (1983). The Fractal Geometry of the Nature, W.H. Freeman and Company, New York, NY. | Zbl 0925.28001

[029] Mandelbrot, B., Passoja, D. and Paullay, A. (1984). Fractal character of fracture surfaces of metals, Nature 308: 721-722.

[030] McKenna, S. (1994). Automated Analysis of Papanicolaou Smears, Ph.D. thesis, University of Dundee, Dundee.

[031] Metze, K. (2010). Fractal dimension of chromatin and cancer prognosis, Epigenomics 2(5): 601-604.

[032] Metze, K. (2013). Fractal dimension of chromatin: Potential molecular diagnostic applications for cancer prognosis, Expert Review of Molecular Diagnostics 13(7): 719-735.

[033] Mingqiang, Y., Kidiyo, K. and Ronsin, J. (2008). A survey of shape feature extraction techniques, in P.-Y. Yin (Ed.), Pattern Recognition Techniques, Technology and Applications, InTech, Rijeka, pp. 43-90.

[034] Nielsen, B., Albregtsen, F. and Danielsen, H. (2005). Fractal analysis of monolayer cell nuclei from two different prognostic classes of early ovarian cancer, in G.A. Losa, D. Merlini, T.F. Nonnenmacher and E.R. Weibel (Eds.), Fractals in Biology and Medicine, Vol. 4, Birkhäuser, Boston, MA, pp. 175-186.

[035] Obuchowicz, A., Hrebień, M., Nieczkowski, T. and Marciniak, A. (2008). Computational intelligence techniques in image segmentation for cytopathology, in T.G. Smoliński, M.G. Milanova and A.-E. Hassanien (Eds.), Computational Intelligence in Biomedicine and Bioinformatics, Studies in Computational Intelligence, Vol. 151, Springer, Berlin, pp. 169-199.

[036] Oszutowska, D. and Purczyński, J. (2012). Estimation of the fractal dimension using tiled triangular prism method for biological non-rectangular objects, Electrical Review R.88 (10b): 261-263.

[037] Oszutowska-Mazurek, D. (2013). Parameter Estimation of Microscopic Objects Using Algorithms of Digital Image Processing for the Purpose of Cytomorphometry, Ph.D. thesis, West-Pomeranian University of Technology, Szczecin, (in Polish).

[038] Oszutowska-Mazurek, D. and Mazurek, P. (2012). Analysis of influence of cell nuclei segmentation in Papanicolaou smears on fractal dimension measurements, Measurement Automation and Monitoring 58(6): 498-501, (in Polish).

[039] Oszutowska-Mazurek, D., Mazurek, P., Sycz, K. and Wójciuk, G.-W. (2012). Estimation of fractal dimension according to optical density of cell nuclei in Papanicolaou smears, in E. Piętka and J. Kawa (Eds.), Information Technologies in Biomedicine, Lecture Notes in Computer Science, Vol. 7339, Springer, Heidelberg, pp. 456-463.

[040] Oszutowska-Mazurek, D., Mazurek, P., Sycz, K. and Wójciuk, G.-W. (2013). Variogram based estimator of fractal dimension for the analysis of cell nuclei from the Papanicolaou smears, in R.S. Choraś (Ed.), Image Processing and Communications Challenges 4, Advances in Intelligent Systems and Computing, Springer-Verlag, Heidelberg, pp. 47-54.

[041] Parker, J. (1997). Algorithms for Image Processing and Computer Vision, Wiley, Indianapolis, IN.

[042] Peitgen, H., Jürgens, H. and Saupe, D. (1991). Fractals for the Classrooms, Vol. 1, Springer-Verlag, Heidelberg. | Zbl 0842.58057

[043] Peitgen, H., Jürgens, H. and Saupe, D. (1992). Fractals for the Classrooms, Vol. 2, Springer-Verlag, Heidelberg. | Zbl 0746.58005

[044] Sawaya, G. and Sox, H. (2007). Trials that matter: Liquid-based cervical cytology: Disadvantages seem to outweigh advantages, Annals Internal Medicine 147(9): 668-669.

[045] Sedivy, R., Windischberger, C., Svozil, K., Moser, E. and Breitenecker, G. (1999). Fractal analysis: An objective method for identifying atypical nuclei in dysplastic lesions of the cervix uteri, Gynecologic Oncology 75: 78-83.

[046] Seuront, L. (2010). Fractals and Multifractals in Ecology and Aquatic Science, CRC Press, Boca Raton, FL. | Zbl 1309.92009

[047] Skomski, R. (2008). Simple Models of Magnetism, Oxford University Press, Oxford.

[048] Smereka, M. and Dulęba, I. (2008). Circular object detection using a modified Hough transform, International Journal of Applied Mathematics and Computer Science 18(1): 85-91, DOI: 10.2478/v10006-008-0008-9.

[049] Śmietański, J., Tadeusiewicz, R. and Łuczyńska, E. (2010). Texture analysis in perfusion images of prostate cancer-A case study, International Journal of Applied Mathematics and Computer Science 20(1): 149-156, DOI: 10.2478/v10006-010-0011-9.

[050] Solomon, D. and Nayar, R. (2004). The Bethesda System for Reporting Cervical Cytology, Springer, New York, NY.

[051] Steven, I. (1993). Linear Richardson plots from non-fractal data sets, Dutch Mathematical Geology 25(6): 737-751.

[052] Styer, D. (2007). Statistical Mechanics, Oberlin College, Oberlin.

[053] Sun, W. (2006). Three new implementations of the triangular prism method for computing the fractal dimension of remote sensing images, Photogrammetric Engineering & Remote Sensing 72(4): 372-382.

[054] Sykes, P., Harker, D., Miller, A., Whitehead, M., Neal, H., Wells, J. and Peddied, D. (2008). A randomised comparison of SurePath liquid-based cytology and conventional smear cytology in a colposcopy clinic setting, General Gynaecology 115(11): 1375-1381.

[055] Voss, R. (1988). Fractals in nature: From characterization to simulation, in H.-O. Peitgen and D. Saupe (Eds.), The Science of Fractal Images, Springer-Verlag, Heidelberg, pp. 21-70.

[056] Walker, R. (1997). Adaptive Multi-Scale Texture Analysis with Applications to Automated Cytology, Ph.D. thesis, University of Queensland, Brisbane.

[057] Wen, R. and Sinding-Larsen, R. (1997). Uncertainty in fractal dimension estimated from power spectra and variogram, Mathematical Geology 29(6): 727-753.

[058] Zhou, G. and Lam, N.-N. (2005). A comparison of fractal dimension estimators based on multiple surface generation algorithms, Computers & Geosciences 31(10): 1260-1269.

[059] Zieliński, K. and Strzelecki, M. (2002). Computer Biomedical Image Analysis. Introduction to Morphometry and Quantitative Pathology, PWN, Warsaw, (in Polish).