Note

Go to the end to download the full example code.

Why we should use semi-Lagrangian scheme?

In this example I will try to demonstrate to you why we should use the semi-Lagrangian scheme during geodesic integration rather than Eulerian. Obviously, we could use even more developed schemes like Runge-Kutta, but semi-Lagrangian is a good compromise between speed and accuracy.

Importing the necessary libraries

import matplotlib.pyplot as plt

import demeter.metamorphosis as mt
import demeter.utils.reproducing_kernels as rk
import demeter.utils.torchbox as tb
from demeter.constants import DLT_KW_IMAGE

Open images

size = (200,200)
S = tb.reg_open('08',size=size)
T = tb.reg_open('m0',size=size)

fig,ax = plt.subplots(1,3)
ax[0].imshow(S[0,0].cpu(),**DLT_KW_IMAGE)
ax[0].set_title('source')
ax[1].imshow(T[0,0].cpu(),**DLT_KW_IMAGE)
ax[1].set_title('target')
ax[2].imshow(tb.imCmp(S,T,'seg'),origin='lower')
ax[2].set_title('superposition of S and T')
source, target, superposition of S and T
Text(0.5, 1.0, 'superposition of S and T')

Now we try to register the source to the target using the Eulerian scheme.

kernel_op = rk.GaussianRKHS(sigma=(6, 6))

momentum_ini = 0
mr = mt.lddmm(S, T, momentum_ini,
      kernelOperator = kernel_op,
      cost_cst=.0001,
      integration_steps=100,
      n_iter=15,
      grad_coef=1,
      integration_method="Eulerian",
      dx_convention='pixel',
)
mr.plot()
mr.plot_deform()
mr.mp.plot()
plt.show()
  • Lambda = 0.0001 rho = 1.0
  • source, target, Integrated source image, comparaison deformed image with target
  • diffeo = tensor(True)
  • t = 0.0, diffeo = tensor(True), t = 0.2, diffeo = tensor(True), t = 0.5, diffeo = tensor(True), t = 0.7, diffeo = tensor(True), t = 1.0, diffeo = tensor(True)
ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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Progress: [##--------]  20.00%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [###-------]  26.67%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [###-------]  33.33%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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Progress: [####------]  40.00%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [#####-----]  46.67%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [#####-----]  53.33%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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Progress: [######----]  60.00%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [#######---]  66.67%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [#######---]  73.33%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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Progress: [########--]  80.00%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [#########-]  86.67%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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Progress: [#########-]  93.33%  (Ssd : ,4710.3496).ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
    self.rho: 1
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ic| classic.py:85 in _step_fullEulerian()
    self.rho: 1
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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ic| classic.py:85 in _step_fullEulerian()
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Total running time of the script: (0 minutes 20.709 seconds)

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