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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 d4 wavelet

Energy, slope, r^2

d6 wavelet

Energy, slope, r^2

d8 wavelet

Energy, slope, r^2

Fungus DTheor=1.600

Ritz et al.

20. 6.458e+003 -1.3355 0.9164

21. 3.538e+003 -1.3437 0.9226

22. 1.896e+003 -1.3562 0.9278

23. 5.445e+002 -1.3826 0.9303

24. 8.643e+001 -1.4293 0.9249

Mean: D=1.617

20. 5.845e+003 -1.3408 0.9168

21. 3.578e+003 -1.3482 0.9229

22. 1.952e+003 -1.3597 0.9282

23. 8.703e+002 -1.3779 0.9322

24. 2.740e+002 -1.4069 0.9333

25. 4.828e+001 -1.4532 0.9275

26. 1.481e+000 -1.5389 0.8976

Mean: D=1.593

20. 6.040e+003 -1.3453 0.9193

21. 3.862e+003 -1.3509 0.9252

22. 2.046e+003 -1.3615 0.9304

23. 9.547e+002 -1.3782 0.9344

24. 3.356e+002 -1.4040 0.9361

25. 8.061e+001 -1.4430 0.9331

26. 6.483e+000 -1.5085 0.9165

Mean: D=1.596

Neurite DTheor=1.210 14. 5.062e+003 -1.5909 0.8880

15. 2.595e+003 -1.6228 0.9020

16. 1.174e+003 -1.6582 0.9122

17. 4.084e+002 -1.7025 0.9187

18. 1.032e+002 -1.7601 0.9207

19. 1.201e+000 -1.8938 0.8904

Mean: D=1.240

14. 4.828e+003 -1.5883 0.8938

15. 2.971e+003 -1.6159 0.9071

16. 1.618e+003 -1.6429 0.9172

17. 7.964e+002 -1.6720 0.9248

18. 2.666e+002 -1.7115 0.9294

19. 7.432e+001 -1.7621 0.9304

20. 8.931e+000 -1.8379 0.9229

Mean: D=1.238

15. 3.517e+003 -1.6083 0.9094

16. 1.941e+003 -1.6310 0.9193

17. 9.046e+002 -1.6584 0.9268

18. 4.142e+002 -1.6893 0.9323

19. 1.393e+002 -1.7292 0.9351

20. 2.816e+001 -1.7858 0.9329

21. 2.703e+000 -1.8690 0.9210

Mean: D=1.271

Vicsek Snowflake DTheor=1.465 18. 4.654e+003 -1.4496 0.9248

19. 2.739e+003 -1.4635 0.9310

20. 1.435e+003 -1.4809 0.9360

21. 6.054e+002 -1.5054 0.9393

22. 1.915e+002 -1.5402 0.9396

23. 3.697e+001 -1.5912 0.9341

24. 1.673e+000 -1.6782 0.9103

Mean: D=1.460

17. 5.720e+003 -1.4639 0.9196

18. 3.600e+003 -1.4783 0.9268

19. 2.200e+003 -1.4933 0.9328

20. 1.227e+002 -1.5104 0.9377

21. 6.330e+002 -1.5306 0.9415

22. 2.713e+002 -1.5563 0.9438

23. 7.928e+001 -1.5927 0.9430

24. 1.303e+001 -1.6467 0.9358

Mean: D=1.444

18. 3.923e+003 -1.4669 0.9201

19. 2.567e+003 -1.4794 0.9323

20. 1.497e+003 -1.4937 0.9374

21. 7.840e+002 -1.5113 0.9415

22. 3.502e+002 -1.5343 0.9442

23. 1.294e+002 -1.5646 0.9449

24. 3.573e+001 -1.6052 0.9424

25. 4.320e+000 -1.6665 0.9314

Mean: D=1.435

2-D Structure d4 wavelet

Energy, slope, r^2

d6 wavelet

Energy, slope, r^2

d8 wavelet

Energy, slope, r^2

Snowflake DTheor=1.850 24. 9.144e+003 -1.1689 0.9453

25. 6.344e+003 -1.1695 0.9485

26. 4.072e+003 -1.1726 0.9516

27. 2.435e+003 -1.1789 0.9544

28. 1.124e+003 -1.1911 0.9561

29. 3.448e+002 -1.2135 0.9546

30. 2.668e+001 -1.2614 0.9375

Mean: D=1.809

24. 8.627e+003 -1.1746 0.9486

25. 5.979e+003 -1.1756 0.9517

26. 3.911e+003 -1.1788 0.9546

27. 2.286e+003 -1.1854 0.9571

28. 1.099e+003 -1.1974 0.9587

29. 3.378e+002 -1.2195 0.9571

30. 3.484e+001 -1.2636 0.9429

Mean: D=1.803

24. 8.568e+003 -1.1817 0.9503

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 21. 5.535e+003 -1.2886 0.9580

22. 3.443e+003 -1.2989 0.9606

23. 1.954e+003 -1.3122 0.9626

24. 9.610e+002 -1.3301 0.9633

25. 3.695e+002 -1.3554 0.9619

26. 9.210e+001 -1.3929 0.9554

27. 1.067e+001 -1.4510 0.9378

Mean: D=1.670

22. 2.951e+003 -1.3619 0.9600

23. 1.713e+003 -1.3309 0.9619

24. 9.520e+002 -1.3474 0.9631

25. 4.576e+002 -1.3631 0.9631

26. 1.679e+002 -1.3959 0.9608

27. 3.927e+001 -1.4359 0.9534

28. 3.479e+000 -1.4987 0.9334

Mean: D=1.637

22. 2.840e+003 -1.3227 0.9630

23. 1.816e+003 -1.3344 0.9650

24. 9.965e+002 -1.3491 0.9662

25. 4.665e+002 -1.3687 0.9662

26. 1.732e+002 -1.3954 0.9640

27. 5.317e+001 -1.4307 0.9586

28. 7.771e+000 -1.4827 0.9449

Mean: D=1.631

Kock curve DTheor=1.500

21. 3.510e+003 -1.2989 0.9634

22. 1.881e+003 -1.3168 0.9645

23. 5.945e+002 -1.3462 0.9620

24. 1.351e+002 -1.3900 0.9531

25. 7.068e-002 -1.5374 0.8497

Mean: D=1.463

22. 1.836e+003 -1.3327 0.9639

23. 8.978e+002 -1.3544 0.9639

24. 2.513e+002 -1.3884 0.9596

25. 5.705e+001 -1.4352 0.9496

26. 7.369e+000 -1.5005 0.9300

27. 2.953e-003 -1.6610 0.8276

Mean: D=1.500

22. 2.758e+003 -1.3238 0.9647

23. 1.308e+003 -1.3394 0.9660

24. 4.810e+002 -1.3638 0.9648

25. 1.598e+002 -1.3968 0.9606

26. 3.783e+001 -1.4415 0.9512

27. 2.615e+000 -1.5129 0.9262

28. 1.426e-003 -1.6707 0.8270

Mean: D=1.487

Real Fractal aggregate: Pycnoporus cinnabarinus DTheor=1.691 22. 5.688e+003 -1.2685 0.9432

23. 3.233e+003 -1.2757 0.9470

24. 1.321e+003 -1.2916 0.9491

25. 3.590e-002 -1.3206 0.9478

26. 2.075e+001 -1.3825 0.9270

27. 6.462e-027 -2.3118 0.2755

Mean: D=1.708

22. 4.685e+003 -1.2753 0.9418

23. 2.766e+003 -1.2846 0.9456

24. 1.461e+003 -1.2981 0.9485

25. 5.741e+002 -1.3195 0.9494

26. 1.152e+002 -1.3568 0.9441

27. 8.930e+000 -1.4205 0.9232

28. 2.066e-027 -2.3198 0.2843

Mean: D=1.681

22. 4.654e+003 -1.2895 0.9462

23. 2.761e+003 -1.2965 0.9497

24. 1.433e+003 -1.3084 0.9521

25. 6.197e+002 -1.3268 0.9539

26. 1.832e+002 -1.3560 0.9517

27. 2.504e+001 -1.4048 0.9399

Mean: D=1.673