1. Propriedades da Entropia Paulo Sérgio Rodrigues PEL205

2. Entropia Não-Extensiva Constantino Tsallis

3. Entropia Não-Extensiva

4. Additive property of Shannon Entropy Tsallis Entropy formula Pseudo-Additive property of Tsallis Entropy

5. Background and Foreground distribution Background and Foreground Tsallis Entropy

6. Pseudo-Additivity for Background and Foreground distribution Here, topt is ideal partition (that maximizes) the pseudo additivity of Tsallis Entropy

7. A new partition of Background and Foreground for new application of Tsallis entropy

8. Respectivelly news Tsallis entropy for the new background and foregrounds

9. General Equation of Pseudo-additivity for one recurssion

10. Here, topt is ideal partition (that maximizes) the pseudo additivity of Tsallis Entropy for the new partition

11. Ultrasound original Benign Tumor Left Column: 1 recurssion; Right column: 3 recurssions row 1: q = 0.00001; row 2: q = 1.0 (Shannon) ; row 3: q = 4 Visual Segmentation Results

12. Left Column: 1 recurssion; Right column: 3 recurssions row 1: q = 0.00001; row 2: q = 1.0 (Shannon) ; row 3: q = 4 Ultrasound original Malignant Tumor Visual Segmentation Results

13. Left upper: NESRA with 16 clusters (3 recurssions); right upper: fuzzy c-means with 16 clusters Left bellow: k-means with 8 clusters; right bellow: SOM with 16 neurons Visual Segmentation Results Benign Tumor

14. Left upper: NESRA with 16 clusters (3 recurssions); right upper: fuzzy c-means with 16 clusters Left bellow: k-means with 8 clusters; right bellow: SOM with 16 neurons Visual Segmentation Results Malignant Tumor

15. Results of application of three approaches for image segmentation: column 1: proposed (NESRA) method; column 2: bootstrap; column 3: fuzzy c-means Some Natural Image Results NESRABootstrapFuzzy C-means

16. Results of application of three approaches for image segmentation: column 1: proposed (NESRA) method; column 2: bootstrap; column 3: fuzzy c-means Some Natural Image Results NESRABootstrapFuzzy C-means

17. Results of application of three approaches for image segmentation: column 1: k-means; column 2: SOM; column 3: watershed Some Natural Image Results K-meansSOMWatershed

18. Results of application of three approaches for image segmentation: column 1: k-means; column 2: SOM; column 3: watershed Some Natural Image Results K-meansSOMWatershed

19. The synthetic image used to compare the robustness of the methods and increasing application of gaussian noise. The two concentric circles have radius 100 and 50, and the intensities for the background, outer and inner circles are 150, 100 and 50 respectively. The letfmost image is the original image; the three others, from left to right, have μ =0 and σ 2 = 0.01, 0.05 and 0.1 gaussian noise respectively. Synthetic Image Results

20. The result segmentation of the six considered algorithms in this paper. In this illustration, for all the original image we have applied a gaussian noise with zero μ and σ 2 = 0.1 which is the highest noise used, and after, a 9 x 9 2D adaptive filter was used for smoothing the noise. In the specific case of NESRA algorithm we use the parameter q = 0.001 since it generates the best visual result with more homogeneous and noiseless regions. Synthetic Image Results NESRA Bootstrap Fuzzy C-meansK-means SOMWatershed

21. The estimated (black ones) and original (white ones) curves superimposed over the original image corresponding to the segmentations of synthetic image. Only the watershed was traced manually since we do not have good precision of the boundary in this case. NESRA Bootstrap Fuzzy C-meansK-means SOMWatershed

22. Comparative performance of the five used methods as a function of increasing gaussian noise. The x-line is the σ 2 and y-line is Robustness Outer Circle

23. Comparative performance of the five used methods as a function of increasing gaussian noise. The x-line is the σ 2 and y-line is Robustness Inner Circle

24. Comparative performance for the five used methods according to the estimated area inside inner, outer and background regions. The performance percentage is an average of the estimated area of the three regions. The x-line is the σ 2 and y-line is the average of estimated area (for the three regions) divided by real area. Performance in Achieving Homogeneous Regions