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A FRACTAL ANALYSIS APPROACH TO IDENTIFICATION OF TUMOR IN BRAIN MR IMAGES Khan M. Iftekharuddin, Wei Jia*, and Ronald Marsh* ISIP lab, ECE Department, The University of Memphis, Memphis, TN 38152. *Dept. of Computer Science, North Dakota State University, Fargo, ND 58105. This research is partly supported by ND EPSCoR through a biomedical SEED grant. Introduction • The purpose of this study is to apply fractal analysis to identify tumor in brain MR images. • Three models are developed to detect tumor in MR brain images using fractal dimension analysis. • A multimedia web-based application is developed for tumor detection application Introduction • Three models are: – Piecewise-threshold-box-counting (PTBC), – Piecewise modified box-counting (PMBC), – Piecewise triangular prism surface area (PTPSA). Fractal • Seminal works from Hilbert, Minkowski, Cantor, Mandelbrot, (Hausdorff, Lyapunov, Ken Wilson, …) • VP A. Gore is “fascinated by fractals” – Time Mag., 8/21/00, p. 41 What is Fractal Geometry? • Fractal is an irregular geometric object with an infinite nesting of structure at all scales. • Fractal objects can be found everywhere in the nature, such as trees, ferns, clouds, snow flakes, mountains, bacteria, and coastlines. Application of Fractal Analysis • Identification of corn roots stressed by nitrogen fertilizer, • Determination of steers body temperature fluctuations in hot and cool chambers, • Estimation of surface roughness of textural images. Application of Fractal Analysis • Medical images: – Detection of micro-calcifications in mammograms, – Prediction of osseous changes in ankle fractures, – Diagnosis of small peripheral lung tumors, – Identification of breast tumors in digitized mammograms. Properties of Fractal Object • Three most important properties of fractals are: – self-similarity, – chaos, – non-integer fractal dimension (FD). Example Fractal Geometry – Self Similarity Example Fractal Geometry – Chaos Fractal Dimension • The equation for fractal dimension (FD) is as follows: ln (number of self-similar pieces, N) FD = lim r->00+ ln (magnification factor, 1/r) Non-integer Fractal Dimension • The Fractal Dimension for Koch Curve is: For N = 4, the magnification (height/width ) is reduces by 1/3 ( = r) • FD = ln 4 / ln 3 = 1.2618… Non-integer Fractal Dimension • The Fractal Dimension for Sierpinski triangle is: • Each triangle is divided into 3 (= N) equal triangles for each iteration and the height/width are reduced by ½ ( = r). • FD = ln 3 / ln 2 = 1.5849… Methods to Estimate Fractal Dimension • There are a wide variety of computer algorithms for estimating the fractal dimension of a structure such as, – Box-counting algorithm (BC). – Modified Box-counting algorithm (MBC). – Triangular Prism Surface Area (TPSA). BC Method to Estimate Fractal Dimension – Box-counting algorithm. 1. Box size r = 3, 5, 7, 9, 11, 13 pixels. 2. Number of boxes occupied (N). 3. A linear regression of the ln N versus ln 1/r to find the slope (FD): BC Method to Estimate Fractal Dimension Box size(pixels) No. of Occupied boxes r = 40, N = 16 r = 30, N = 24 …. …. r = 20, N = 31 BC Method to Estimate Fractal Dimension BC Method to Estimate Fractal Dimension Test Clouds images Box-Counting Algorithm Analyzing Clouds Images The estimation the fractal dimensions of clouds of 2.3, 2.5, and 2.8 using box-counting algorithm for whole image. BC D = 2.3 D = 2.5 D = 2.8 Whole 2.034 2.034 2.034 MBC Method to Estimate Fractal Dimension – Modified box-counting (MBC) method for measurement surface fractal dimension. Image Intensity max ( ri) - min (ri) N = floor { }+1 r MBC Algorithm Analyzing Clouds Images The estimation the fractal dimensions of clouds of 2.3, 2.5, and 2.8 using modified box-counting algorithm for whole image. M BC D = 2.3 D = 2.5 D = 2.8 W hole 2.17 2.27 2.40 TPSA Method to Estimate Fractal Dimension – Triangular Prism Surface Area (TPSA). • The connections of the pixels grayscale values p1, p2, p3, p4 and pc produces four triangles. • N = Sum of the top areas. TPSA Algorithm Analyzing Clouds Images The estimation the fractal dimensions of clouds of 2.3, 2.5, and 2.8 using Triangular Prism Surface Area Procedure algorithm for whole image. TPSA D = 2.3 D = 2.5 D = 2.8 Whole 2.44 2.60 2.81 Developed Algorithms • Piecewise-threshold-box-counting (PTBC), • Piecewise modified box-counting (PMBC), • Piecewise triangular prism surface area (PTPSA). PTBC Algorithm for Brain MRI Detection Load .pgm MR image Divide the image into sub-images Histogram the sub- Divide the sub-images into images intensity different Intensity period Count the occupied box number (N) of box size (r) Cumulative histogram Calculate FD using ln(N)/ln(1/r) No No Is it the last Yes Last threshold sub-images Yes Plot sub-image’s FD versus cumulative histogram Brain MR Images (source: Harvard med school web) m35 m40 m45 m35b m40b m45b m35w m40w m45w PTBC Algorithm Analyzing MR Images PMBC and PTPSA Algorithms for Brain MRI Detection Load .pgm Load .pgm normal MRI test MRI Divide the image into Divide the image into sub-images sub-images PMBC or PTPSA PMBC or PTPSA PMBC PTPSA PMBC PTPSA Box Size Box Size Box Size Box Size r = 3, 5,.. 13 r = 3, 5,.. 13 r = 3, 5,.. 13 r = 3, 5,.. 13 Count sub-image Count sub-image N = floor { (max – min)/r } +1 N = floor { (max – min)/r } +1 Count sub-image Count sub-image N = sum of top area N = sum of top area Calculate sub-image Calculate sub-image FD = ln N versus ln 1/r FD = ln N versus ln 1/r Same No Last sub-image Yes Yes Last sub-image No Compare FD? Not same Record FD and position Plot tumor position Illustration of the tumor positions and differences in FD between the normal and tumor MR images using PMBC algorithm (8 x 8) m35b m35 and m35b m35 and m35bw m35w m40b m40 and m40b m40 and m40bw m40w m45w m45b m45 and m45b m45 and m45bw Illustration of the tumor positions and differences in FD between the normal and tumor MR images using PTPSA algorithm m35b m35 and m35b m35 and m35bw m35w m40b m40 and m40b M40 and m40bw m40w m45w m45b m45 and m45b m45 and m45bw Real Brain MR Images (source: ACR CD) 306-a 306-b 401-a 401-b 503-a 503-b The results of PTBC algorithm test on real MR image 306-a and 306-b, 401-a and 401-b, 503-a and 503-b divided into 2 x 2 pieces Illustration of the tumor positions and differences in FD between MR images using PMBC and PTPSA algorithms 306-a 306-b PMBC_306 PTPSA_306 401-a 401-b PMBC_401 PTPSA_401 503-a 503-b PMBC_503 PTPSA_503 Tumor Detection – Multimedia Interface http://engronline.ee.memphis.edu/iftekhar/ISIP_det.htm Tumor Detection – Database Support Conclusion • The original box-counting (BC) method offers correct results for 1-D signal only. • However, BC fails to yield correct result for 2-D image. • The PTBC method can detect the tumor in MR images, though it is hard to locate the exact position of tumor. • The PMBC and PTPSA methods can detect and locate the tumor correctly when applied to the brain tumor MR images. • The PMBC algorithm is more sensitive and offers better result to detect and locate the tumor. Conclusion • This program is first developed in C on Unix OS. We then develop an easy and friendly user interface in java. • We automate the tumor identification process by building a reference FD database to compare with test brain images. • We improve the algorithms such that we divide each MR image into two halves – use one half as reference for the other. • The research is presented at the World Congress on Medical Physics and Biomedical Engineering, Chicago, June, 2000. Future Work • Develop a more robust algorithm for tumor detection to detect hard-to-recognize feature. • Combination of multiresolution-based wavelet to fractional Brownian motion analysis? • Extend to volume rendering/registration. • Expand web-based approach to support remote learning, surgery and research. Reference • 1. B. B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco, (1983). • 2. D. Comis, “Fractals--A bridge to the future for soil science,” Agricultural Research Magazine. 46(4), pp. 10-13, (1998). • 3. N. Sarkar and B. B. Chaudhuri, “An efficient approach to estimate fractal dimension of textural images,” Pattern Recognition. 23(9), pp. 1035-1041, (1992). • 4. S. Davies and P. Hall, “Fractal analysis of surface roughness by using spatial data,” Journal of The Royal Statistical Society Series, B Statistical Methodology. 61(1), pp. 3-29, (1999). • 5. D. Osman, D. Newitt, A. Gies, T. Budinger, V. Truong, and S. Majumdar, “Fractal based image analysis of human trabecular bone using the box counting algorithm: impact of resolution and relationship to standard measures of trabecular bone structure,” Fractals. 6(3), pp. 275-283, (1998). • 6. C. B. Caldwell, S. J. Stapleton, D. W. Hodsworth, R. A. Jong, W. J. Weiser, G. Cooke, and M. J. Yaffe, “Characterisation of mammographic parenchymal pattern by fractal dimension,” Physics in Medicine & Biology. 35(2), pp. 235-247, (1990). • 7. R. L. Webber, T. E. Underhill, R. A. Horton, R. L. Dixon, T. L. Pope. Jr, “Predicting osseous changes in ankle fractures,” IEEE Engineering In Medicine And Biology Magazine. 12(1), pp. 103-110, (1993). • 8. N. Mihara, K. Kuriyama, S. Kido, C. Kuroda, T. Johkoh, H. Naito, and H. Nakamura, “The usefulness of fractal geometry for the diagnosis of small peripheral lung tumors,” Nippon Igaku Hoshasen Gakkai Zasshi. 58(4), pp. 148-151, (1998). • 9. S. Pohlman, K. A. Powell, N. A. Obuchowski, W. A. Chilcote, and S. Grundfest-Broniatowski, “Quantitative classification of breast tumor in digitized mammograms,” Medical Physics. 23(8), pp. 1337-1345, (1996). • 10. V. Swarnakar, R. S. Acharya, C. Sibata and K. Shin, “Fractal based characterization of structural changes in biomedical images.” SPIE. 2709, pp. 444-455, (1996). • 11. A. Bru, J. M. Pastor, I. Fernaud, I. Bru, S. Melle, C. Berenguer, “Super-rough dynamics on tumor growth,” Physical Review Letters. 81(18), pp. 4008-4011, (1998). • 12. K. J. Falconer, Chapter 2: Hausdorff measure and dimension, In “Fractal Geometry Mathematical Foundations And Applications.” Thomson Press Ltd, (1990). • 13. K. C. Clarke, “Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method.” Computers and Geosciences 12( 5), pp. 713-722, (1986). • 14. B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications.” SIAM Review. 10(40), pp. 422-437, (1968).

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