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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Derivatives of a TPS
Step1:  We start by defining the source and target landmarks. Notice that, in this first example source = target!!!
Step2:  The warp can be effectively computed, although the rendering will not appear to be correct...
Step3:  The next step is to define the set of points at which the derivative of the previous TPS warp must be evaluated. In this case, we use the function meshgrid to generate points inside the convex hull defined by the source landmarks.
Step4:  We evaluate the derivative, reshape the output, and visualize the result.
Step5:  If everything goes as expected, the upper corner of the images defining the derivative of the warp wrt the x and y coordinates of the first of the source landmarks should both contain values close to 1.
Step6:  The sum of all the derivatives wrt the x coordinates should produce an all 1 image
Step7:  and so should the sum of all derivatives wrt the y coordinates.
Step8:  Finally, the derivatives with respect to the x and y coordinates should be in this case exactly the same!!!
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Python Code:
import os
import numpy as np
import scipy.io as sio
import matplotlib.pyplot as plt
from menpo.shape import PointCloud
import menpo.io as mio
from menpofit.transform import DifferentiableThinPlateSplines
Explanation: Derivatives of a TPS
End of explanation
src_landmarks = PointCloud(np.array([[-1, -1],
                                     [-1,  1],
                                     [ 1, -1],
                                     [ 1,  1]]))
tgt_landmarks = PointCloud(np.array([[-1, -1],
                                     [-1,  1],
                                     [ 1, -1],
                                     [ 1,  1]]))
Explanation: We start by defining the source and target landmarks. Notice that, in this first example source = target!!!
End of explanation
tps = DifferentiableThinPlateSplines(src_landmarks, tgt_landmarks)
np.allclose(tps.apply(src_landmarks).points, tgt_landmarks.points)
Explanation: The warp can be effectively computed, although the rendering will not appear to be correct...
End of explanation
x = np.arange(-1, 1, 0.01)
y = np.arange(-1, 1, 0.01)
xx, yy = np.meshgrid(x, y)
points = np.array([xx.flatten(1), yy.flatten(1)]).T
Explanation: The next step is to define the set of points at which the derivative of the previous TPS warp must be evaluated. In this case, we use the function meshgrid to generate points inside the convex hull defined by the source landmarks.
End of explanation
%matplotlib inline
dW_dxy = tps.d_dl(points)
reshaped = dW_dxy.reshape(xx.shape + (4,2))
#dW_dx
plt.subplot(241)
plt.imshow(reshaped[:,:,0,0])
plt.subplot(242)
plt.imshow(reshaped[:,:,1,0])
plt.subplot(243)
plt.imshow(reshaped[:,:,2,0])
plt.subplot(244)
plt.imshow(reshaped[:,:,3,0])
#dW_dy
plt.subplot(245)
plt.imshow(reshaped[:,:,0,1])
plt.subplot(246)
plt.imshow(reshaped[:,:,1,1])
plt.subplot(247)
plt.imshow(reshaped[:,:,2,1])
plt.subplot(248)
plt.imshow(reshaped[:,:,3,1])
Explanation: We evaluate the derivative, reshape the output, and visualize the result.
End of explanation
print(reshaped[1:5,1:5,0,0])
print(reshaped[1:5,1:5,0,1])
Explanation: If everything goes as expected, the upper corner of the images defining the derivative of the warp wrt the x and y coordinates of the first of the source landmarks should both contain values close to 1.
End of explanation
summed_x = np.sum(reshaped[:,:,:,0], axis=-1)
np.allclose(np.ones(xx.shape), summed_x)
plt.imshow(summed_x)
Explanation: The sum of all the derivatives wrt the x coordinates should produce an all 1 image
End of explanation
summed_y = np.sum(reshaped[:,:,:,1], axis=-1)
np.allclose(np.ones(xx.shape), summed_y)
plt.imshow(summed_y)
Explanation: and so should the sum of all derivatives wrt the y coordinates.
End of explanation
np.allclose(reshaped[:,:,:,0], reshaped[:,:,:,1])
Explanation: Finally, the derivatives with respect to the x and y coordinates should be in this case exactly the same!!!
End of explanation
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