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Validation of the 2-D Wavelet Packet Analysis method to compute the Hurst Exponent and in turn compute the global Fractal Dimension of the object.

The Fractal Dimension is based upon the Coefficient Index Position number (which maps out Energy) which returned the highest R squared (r^2) result. This is highlighted in BOLD in the table.

2-D Structure s8 wavelet

Energy, slope, r^2

s10 wavelet

Energy, slope, r^2

s12 wavelet

Energy, slope, r^2

Fungus DTheor=1.600

Ritz et al.

20. 5.576e+003 -1.3462 0.9190

21. 3.382e+003 -1.3542 0.9249

22. 1.730e+003 -1.3674 0.9300

23. 7.465e+002 -1.3874 0.9336

24. 2.444e+002 -1.4173 0.9343

25. 5.306e+001 -1.4612 0.9296

26. 3.965e+000 -1.5318 0.9107

Mean: D=1.583

20. 5.538e+003 -1.3480 0.9180

21. 3.447e+003 -1.3554 0.9240

22. 1.849e+003 -1.3672 0.9292

23. 8.353e+002 -1.3853 0.9331

24. 2.963e+002 -1.4123 0.9347

25. 7.120e+001 -1.4522 0.9316

26. 1.029e+000 -1.5099 0.9203

Mean: D=1.588

20. 5.631e+003 -1.3482 0.9186

21. 3.411e+003 -1.3557 0.9246

22. 1.883e+003 -1.3671 0.9298

23. 9.168e+002 -1.3837 0.9339

24. 3.171e+002 -1.4097 0.9356

25. 8.053e+001 -1.4480 0.9331

26. 1.451e+001 -1.5011 0.9239

Mean: D=1.590

Neurite DTheor=1.210 15. 2.880e+003 -1.6183 0.9102

16. 1.480e+003 -1.6460 0.9198

17. 7.466e+002 -1.6750 0.9271

18. 3.103e+002 -1.7093 0.9322

19. 1.194e+002 -1.7489 0.9352

20. 3.236e+001 -1.7993 0.9349

21. 3.708e+000 -1.8739 0.9263

Mean: D=1.806

15. 2.954e+003 -1.6330 0.9027

16. 1.676e+003 -1.6572 0.9134

17. 7.382e+002 -1.6870 0.9214

18. 3.310e+002 -1.7200 0.9273

19. 1.165e+002 -1.7605 0.9307

20. 3.444e+001 -1.8099 0.9313

21. 5.849e+001 -1.8762 0.9265

Mean: D=1.190

16. 1.692e+003 -1.6443 0.9188

17. 9.587e+002 -1.6676 0.9266

18. 4.081e+002 -1.6970 0.9323

19. 1.464e+002 -1.7344 0.9356

20. 4.700e+001 -1.7796 0.9364

21. 1.280e+001 -1.8337 0.9343

22. 1.675e+000 -1.9075 0.9256

Mean: D=1.220

Vicsek Snowflake DTheor=1.465 19. 2.364e+003 -1.4915 0.9329

20. 1.256e+003 -1.5079 0.9379

21. 6.341e+002 -1.5278 0.9417

22. 2.863e+002 -1.5525 0.9442

23. 1.124e+002 -1.5831 0.9449

24. 2.865e+001 -1.6252 0.9420

25. 2.995e+000 -1.6899 0.9297

Mean: D=1.417

20. 1.413e+002 -1.5033 0.9377

21. 7.324e+002 -1.5212 0.9417

22. 3.150e+002 -1.5450 0.9443

23. 1.118e+002 -1.5766 0.9448

24. 3.386e+001 -1.6168 0.9425

25. 5.414e+001 -1.6736 0.9339

Mean: D=1.423

19. 2.347e+003 -1.4944 0.9328

20. 1.354e+003 -1.5089 0.9379

21. 7.384e+002 -1.5259 0.9420

22. 3.417e+002 -1.5476 0.9449

23. 1.399e+002 -1.5751 0.9463

24. 4.242e+001 -1.6118 0.9449

25. 9.003e+000 -1.6612 0.9391

Mean: D=1.425

2-D Structure s8 wavelet

Energy, slope, r^2

s10 wavelet

Energy, slope, r^2

s12 wavelet

Energy, slope, r^2

Snowflake DTheor=1.850 25. 5.730e+003 -1.1786 0.9521

26. 3.766e+003 -1.1820 0.9550

27. 2.161e+003 -1.1889 0.9575

28. 1.033e+003 -1.2013 0.9590

29. 3.163e+002 -1.2239 0.9572

30. 3.831e+001 -1.2663 0.9443

Mean: D=1.799

25. 5.994e+003 -1.1814 0.9570

26. 3.913e+003 -1.1839 0.9596

27. 2.237e+003 -1.1902 0.9619

28. 1.053e+003 -1.2021 0.9631

29. 3.472e+002 -1.2234 0.9615

30. 4.795e+001 -1.2631 0.9500

Mean: D=1.798

25. 6.063e+003 -1.1818 0.9532

26. 3.970e+003 -1.1842 0.9560

27. 2.284e+003 -1.1903 0.9585

28. 1.102e+003 -1.2017 0.9601

29. 3.799e+002 -1.2220 0.9591

30. 5.119e+001 -1.2611 0.9483

Mean: D=1.798

Van Damm Plaster dissolution DTheor=1.600 22. 2.800e+003 -1.3237 0.9604

23. 1.574e+003 -1.3384 0.9622

24. 7.880e+002 -1.3572 0.9622

25. 3.825e+002 -1.3796 0.9624

26. 1.563e+002 -1.4074 0.9603

27. 4.172e+001 -1.4454 0.9541

28. 5.423e+000 -1.5014 0.9389

Mean: D=1.620

22. 2.622e+003 -1.3280 0.9623

23. 1.666e+003 -1.3407 0.9642

24. 9.073e+002 -1.3564 0.9653

25. 4.225e+002 -1.3768 0.9652

26. 1.571e+002 -1.4043 0.9629

27. 4.703e+001 -1.4404 0.9573

28. 8.546e+000 -1.4903 0.9453

Mean: D=1.623

21. 4.273e+003 -1.3151 0.9579

22. 2.613e+003 -1.3277 0.9603

23. 1.505e+003 -1.3427 0.9620

24. 8.352e+002 -1.3599 0.9631

25. 4.021e+002 -1.3812 0.9630

26. 1.701e+002 -1.4075 0.9613

27. 5.041e+001 -1.4427 0.9563

Mean: D=1.619

Kock curve DTheor=1.500 21. 3.324e+003 -1.3118 0.9601

22. 1.632e+003 -1.3312 0.9612

23. 7.722e+002 -1.3550 0.9609

24. 3.186e+002 -1.3845 0.9587

25. 1.030e+002 -1.4222 0.9534

26. 2.369e+001 -1.4713 0.9431

27. 1.860e+000 -1.5446 0.9192

Mean: D=1.455

22. 2.424e+003 -1.3370 0.9653

23. 1.130e+003 -1.3540 0.9663

24. 4.386e+002 -1.3788 0.9651

25. 1.675e+002 -1.4099 0.9617

26. 4.540e+001 -1.4510 0.9543

27. 6.190e+000 -1.5090 0.9380

Mean: D=1.491

22. 3.481e+003 -1.3199 0.9634

23. 1.780e+003 -1.3371 0.9647

24. 9.996e+002 -1.3562 0.9653

25. 1.304e+002 -1.4161 0.9594

26. 3.410e+001 -1.4607 0.9505

27. 5.772e+000 -1.5195 0.9351

28. 2.342e-001 -1.6088 0.9014

Mean: D=1.520

Real Fractal aggregate: Pycnoporus cinnabarinus DTheor=1.691 22. 4.526e+003 -1.2808 0.9427

23. 2.624e+003 -1.2905 0.9464

24. 1.270e+003 -1.3057 0.949

25. 2.022e+002 -1.3283 0.9496

26. 1.366e+002 -1.3624 0.9459

27. 1.855e+001 -1.4153 0.9324

Mean: D=1.672

22. 4.520e+003 -1.2931 0.9463

23. 2.730e+003 -1.2916 0.9498

24. 1.395e+003 -1.3049 0.9524

25. 6.033e+002 -1.3246 0.9535

26. 1.770e+002 -1.3551 0.9508

27. 3.103e+001 -1.4014 0.9406

28. 1.710e+000 -1.4761 0.9123

Mean: D=1.675

22. 4.514e+003 -1.2829 0.9427

23. 2.646e+003 -1.2921 0.9464

24. 1.434e+003 -1.3051 0.9493

25. 6.085e+002 -1.3249 0.9506

26. 1.750e+002 -1.3556 0.9481

27. 3.261e+001 -1.4014 0.9386

28. 2.260e+000 -1.4726 0.9136

Mean: D=1.675