SimpleITK Transformation Types

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import SimpleITK as sitk
import numpy as np
      
from __future__ import print_function
   
import matplotlib.pyplot as plt
%matplotlib inline 
from ipywidgets import interact, fixed

OUTPUT_DIR = "Output"

print(sitk.Version())


    

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import numpy as np

def point2str(point, precision=1):
    """
    Format a point for printing, based on specified precision with trailing zeros. Uniform printing for vector-like data 
    (tuple, numpy array, list).
    
    Args:
        point (vector-like): nD point with floating point coordinates.
        precision (int): Number of digits after the decimal point.
    Return:
        String represntation of the given point "xx.xxx yy.yyy zz.zzz...".
    """
    return ' '.join(format(c, '.{0}f'.format(precision)) for c in point)


def uniform_random_points(bounds, num_points):
    """
    Generate random (uniform withing bounds) nD point cloud. Dimension is based on the number of pairs in the bounds input.
    
    Args:
        bounds (list(tuple-like)): list where each tuple defines the coordinate bounds.
        num_points (int): number of points to generate.
    
    Returns:
        list containing num_points numpy arrays whose coordinates are within the given bounds.
    """
    internal_bounds = [sorted(b) for b in bounds]
         # Generate rows for each of the coordinates according to the given bounds, stack into an array, 
         # and split into a list of points.
    mat = np.vstack([np.random.uniform(b[0], b[1], num_points) for b in internal_bounds])
    return list(mat[:len(bounds)].T)


def target_registration_errors(tx, point_list, reference_point_list):
    """
    Distances between points transformed by the given transformation and their
    location in another coordinate system. When the points are only used to evaluate
    registration accuracy (not used in the registration) this is the target registration
    error (TRE).
    """
    return [np.linalg.norm(np.array(tx.TransformPoint(p)) -  np.array(p_ref))
          for p,p_ref in zip(point_list, reference_point_list)]


def print_transformation_differences(tx1, tx2):
    """
    Check whether two transformations are "equivalent" in an arbitrary spatial region 
    either 3D or 2D, [x=(-10,10), y=(-100,100), z=(-1000,1000)]. This is just a sanity check, 
    as we are just looking at the effect of the transformations on a random set of points in
    the region.
    """
    if tx1.GetDimension()==2 and tx2.GetDimension()==2:
        bounds = [(-10,10),(-100,100)]
    elif tx1.GetDimension()==3 and tx2.GetDimension()==3:
        bounds = [(-10,10),(-100,100), (-1000,1000)]
    else:
        raise ValueError('Transformation dimensions mismatch, or unsupported transformation dimensionality')
    num_points = 10
    point_list = uniform_random_points(bounds, num_points)
    tx1_point_list = [ tx1.TransformPoint(p) for p in point_list]
    differences = target_registration_errors(tx2, point_list, tx1_point_list)
    print(tx1.GetName()+ '-' +
          tx2.GetName()+
          ':\tminDifference: {:.2f} maxDifference: {:.2f}'.format(min(differences), max(differences)))


    

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# SimpleITK points represented by vector-like data structures. 
point_tuple = (9.0, 10.531, 11.8341)
point_np_array = np.array([9.0, 10.531, 11.8341])
point_list = [9.0, 10.531, 11.8341]

print(point_tuple)
print(point_np_array)
print(point_list)

# Uniform printing with specified precision.
precision = 2
print(point2str(point_tuple, precision))
print(point2str(point_np_array, precision))
print(point2str(point_list, precision))


    

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# A 3D translation. Note that you need to specify the dimensionality, as the sitk TranslationTransform 
# represents both 2D and 3D translations.
dimension = 3        
offset =(1,2,3) # offset can be any vector-like data  
translation = sitk.TranslationTransform(dimension, offset)
print(translation)


    

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# Transform a point and use the inverse transformation to get the original back.
point = [10, 11, 12]
transformed_point = translation.TransformPoint(point)
translation_inverse = translation.GetInverse()
print('original point: ' + point2str(point) + '\n'
      'transformed point: ' + point2str(transformed_point) + '\n'
      'back to original: ' + point2str(translation_inverse.TransformPoint(transformed_point)))


    

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point = [10, 11]
rotation2D = sitk.Euler2DTransform()
rotation2D.SetTranslation((7.2, 8.4))
rotation2D.SetAngle(np.pi/2)
print('original point: ' + point2str(point) + '\n'
      'transformed point: ' + point2str(rotation2D.TransformPoint(point)))

# Change the center of rotation so that it coincides with the point we want to
# transform, why is this a unique configuration?
rotation2D.SetCenter(point)
print('original point: ' + point2str(point) + '\n'
      'transformed point: ' + point2str(rotation2D.TransformPoint(point)))


    

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# Rotation only, parametrized by Versor (vector part of unit quaternion),
# quaternion defined by rotation of theta around axis n: 
#  q = [n*sin(theta/2), cos(theta/2)]
               
# 180 degree rotation around z axis

# Use a versor:
rotation1 = sitk.VersorTransform([0,0,1,0])

# Use axis-angle:
rotation2 = sitk.VersorTransform((0,0,1), np.pi)

# Use a matrix:
rotation3 = sitk.VersorTransform()
rotation3.SetMatrix([-1, 0, 0, 0, -1, 0, 0, 0, 1]);

point = (10, 100, 1000)

p1 = rotation1.TransformPoint(point)
p2 = rotation2.TransformPoint(point)
p3 = rotation3.TransformPoint(point)

print('Points after transformation:\np1=' + str(p1) + 
      '\np2='+ str(p2) + '\np3='+ str(p3))


    

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dimension = 3        
t =(1,2,3) 
translation = sitk.TranslationTransform(dimension, t)

# Only need to copy the translational component.
rigid_euler = sitk.Euler3DTransform()
rigid_euler.SetTranslation(translation.GetOffset())

rigid_versor = sitk.VersorRigid3DTransform()
rigid_versor.SetTranslation(translation.GetOffset())

# Sanity check to make sure the transformations are equivalent.
bounds = [(-10,10),(-100,100), (-1000,1000)]
num_points = 10
point_list = uniform_random_points(bounds, num_points)
transformed_point_list = [translation.TransformPoint(p) for p in point_list]

# Draw the original and transformed points, include the label so that we 
# can modify the plots without requiring explicit changes to the legend.
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
orig = ax.scatter(list(np.array(point_list).T)[0],
                  list(np.array(point_list).T)[1],
                  list(np.array(point_list).T)[2], 
                  marker='o', 
                  color='blue',
                  label='Original points')
transformed = ax.scatter(list(np.array(transformed_point_list).T)[0],
                         list(np.array(transformed_point_list).T)[1],
                         list(np.array(transformed_point_list).T)[2], 
                         marker='^', 
                         color='red', 
                         label='Transformed points')
plt.legend(loc=(0.0,1.0))

euler_errors = target_registration_errors(rigid_euler, point_list, transformed_point_list)
versor_errors = target_registration_errors(rigid_versor, point_list, transformed_point_list)

print('Euler\tminError: {:.2f} maxError: {:.2f}'.format(min(euler_errors), max(euler_errors)))
print('Versor\tminError: {:.2f} maxError: {:.2f}'.format(min(versor_errors), max(versor_errors)))


    

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rotationCenter = (10, 10, 10)
rotation = sitk.VersorTransform([0,0,1,0], rotationCenter)

rigid_euler = sitk.Euler3DTransform()
rigid_euler.SetMatrix(rotation.GetMatrix())
rigid_euler.SetCenter(rotation.GetCenter())

rigid_versor = sitk.VersorRigid3DTransform()
rigid_versor.SetRotation(rotation.GetVersor())
#rigid_versor.SetCenter(rotation.GetCenter()) #intentional error

# Sanity check to make sure the transformations are equivalent.
bounds = [(-10,10),(-100,100), (-1000,1000)]
num_points = 10
point_list = uniform_random_points(bounds, num_points)
transformed_point_list = [ rotation.TransformPoint(p) for p in point_list]

euler_errors = target_registration_errors(rigid_euler, point_list, transformed_point_list)
versor_errors = target_registration_errors(rigid_versor, point_list, transformed_point_list)

# Draw the points transformed by the original transformation and after transformation
# using the incorrect transformation, illustrate the effect of center of rotation.
from mpl_toolkits.mplot3d import Axes3D
incorrect_transformed_point_list = [ rigid_versor.TransformPoint(p) for p in point_list]
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
orig = ax.scatter(list(np.array(transformed_point_list).T)[0],
                  list(np.array(transformed_point_list).T)[1],
                  list(np.array(transformed_point_list).T)[2], 
                  marker='o', 
                  color='blue',
                  label='Rotation around specific center')
transformed = ax.scatter(list(np.array(incorrect_transformed_point_list).T)[0],
                         list(np.array(incorrect_transformed_point_list).T)[1],
                         list(np.array(incorrect_transformed_point_list).T)[2], 
                         marker='^', 
                         color='red', 
                         label='Rotation around origin')
plt.legend(loc=(0.0,1.0))

print('Euler\tminError: {:.2f} maxError: {:.2f}'.format(min(euler_errors), max(euler_errors)))
print('Versor\tminError: {:.2f} maxError: {:.2f}'.format(min(versor_errors), max(versor_errors)))


    

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def display_center_effect(x, y, tx, point_list, xlim, ylim):
    tx.SetCenter((x,y))
    transformed_point_list = [ tx.TransformPoint(p) for p in point_list]

    plt.scatter(list(np.array(transformed_point_list).T)[0],
                list(np.array(transformed_point_list).T)[1],
                marker='^', 
                color='red', label='transformed points')
    plt.scatter(list(np.array(point_list).T)[0],
                list(np.array(point_list).T)[1],
                marker='o', 
                color='blue', label='original points')
    plt.xlim(xlim)
    plt.ylim(ylim)
    plt.legend(loc=(0.25,1.01))

# 2D square centered on (0,0)
points = [np.array((-1,-1)), np.array((-1,1)), np.array((1,1)), np.array((1,-1))]

# Scale by 2 
similarity = sitk.Similarity2DTransform();
similarity.SetScale(2)

interact(display_center_effect, x=(-10,10), y=(-10,10),tx = fixed(similarity), point_list = fixed(points), 
         xlim = fixed((-10,10)),ylim = fixed((-10,10)));


    

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rotation_center = (100, 100, 100)
theta_x = 0.0
theta_y = 0.0
theta_z = np.pi/2.0
translation = (1,2,3)

rigid_euler = sitk.Euler3DTransform(rotation_center, theta_x, theta_y, theta_z, translation)

similarity = sitk.Similarity3DTransform()
similarity.SetMatrix(rigid_euler.GetMatrix())
similarity.SetTranslation(rigid_euler.GetTranslation())
similarity.SetCenter(rigid_euler.GetCenter())

# Apply the transformations to the same set of random points and compare the results
# (see utility functions at top of notebook).
print_transformation_differences(rigid_euler, similarity)


    

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rotation_center = (100, 100, 100)
axis = (0,0,1)
angle = np.pi/2.0
translation = (1,2,3)
scale_factor = 2.0
similarity = sitk.Similarity3DTransform(scale_factor, axis, angle, translation, rotation_center)

affine = sitk.AffineTransform(3)
affine.SetMatrix(similarity.GetMatrix())
affine.SetTranslation(similarity.GetTranslation())
affine.SetCenter(similarity.GetCenter())

# Apply the transformations to the same set of random points and compare the results
# (see utility functions at top of notebook).
print_transformation_differences(similarity, affine)


    

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# 2D square centered on (0,0).
points = [np.array((-1,-1)), np.array((-1,1)), np.array((1,1)), np.array((1,-1))]

# Scale by half in x and 2 in y.
scale = sitk.ScaleTransform(2, (0.5,2));

# Interactively change the location of the center.
interact(display_center_effect, x=(-10,10), y=(-10,10),tx = fixed(scale), point_list = fixed(points), 
         xlim = fixed((-10,10)),ylim = fixed((-10,10)));


    

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scales = (0.5,0.7,0.9)
translation = (1,2,3)
axis = (0,0,1)
angle = 0.0
scale_versor = sitk.ScaleVersor3DTransform(scales, axis, angle, translation)
print(scale_versor)


    

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scale = (2,2.1,3)
skew = np.linspace(start=0.0, stop=1.0, num=6) #six eqaully spaced values in[0,1], an arbitrary choice
translation = (1,2,3)
versor = (0,0,0,1.0)
scale_skew_versor = sitk.ScaleSkewVersor3DTransform(scale, skew, versor, translation)
print(scale_skew_versor)


    

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#
# This function displays the effects of the deformable transformation on a grid of points by scaling the
# initial displacements (either of control points for bspline or the deformation field itself). It does
# assume that all points are contained in the range(-2.5,-2.5), (2.5,2.5).
#
def display_displacement_scaling_effect(s, original_x_mat, original_y_mat, tx, original_control_point_displacements):
    if tx.GetDimension() !=2:
        raise ValueError('display_displacement_scaling_effect only works in 2D')

    plt.scatter(original_x_mat,
                original_y_mat,
                marker='o', 
                color='blue', label='original points')
    pointsX = []
    pointsY = []
    tx.SetParameters(s*original_control_point_displacements)
  
    for index, value in np.ndenumerate(original_x_mat):
        px,py = tx.TransformPoint((value, original_y_mat[index]))
        pointsX.append(px) 
        pointsY.append(py)
     
    plt.scatter(pointsX,
                pointsY,
                marker='^', 
                color='red', label='transformed points')
    plt.legend(loc=(0.25,1.01))
    plt.xlim((-2.5,2.5))
    plt.ylim((-2.5,2.5))


    

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# Create the transformation (when working with images it is easier to use the BSplineTransformInitializer function
# or its object oriented counterpart BSplineTransformInitializerFilter).
dimension = 2
spline_order = 3
direction_matrix_row_major = [1.0,0.0,0.0,1.0] # identity, mesh is axis aligned
origin = [-1.0,-1.0]  
domain_physical_dimensions = [2,2]

bspline = sitk.BSplineTransform(dimension, spline_order)
bspline.SetTransformDomainOrigin(origin)
bspline.SetTransformDomainDirection(direction_matrix_row_major)
bspline.SetTransformDomainPhysicalDimensions(domain_physical_dimensions)
bspline.SetTransformDomainMeshSize((4,3))

# Random displacement of the control points.
originalControlPointDisplacements = np.random.random(len(bspline.GetParameters()))
bspline.SetParameters(originalControlPointDisplacements)

# Apply the bspline transformation to a grid of points 
# starting the point set exactly at the origin of the bspline mesh is problematic as
# these points are considered outside the transformation's domain,
# remove epsilon below and see what happens.
numSamplesX = 10
numSamplesY = 20
                   
coordsX = np.linspace(origin[0]+np.finfo(float).eps, origin[0] + domain_physical_dimensions[0], numSamplesX)
coordsY = np.linspace(origin[1]+np.finfo(float).eps, origin[1] + domain_physical_dimensions[1], numSamplesY)
XX, YY = np.meshgrid(coordsX, coordsY)

interact(display_displacement_scaling_effect, s= (-1.5,1.5), original_x_mat = fixed(XX), original_y_mat = fixed(YY),
         tx = fixed(bspline), original_control_point_displacements = fixed(originalControlPointDisplacements));


    

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# Create the displacment field. 
    
# When working with images the safer thing to do is use the image based constructor,
# sitk.DisplacementFieldTransform(my_image), all the fixed parameters will be set correctly and the displacement
# field is initialized using the vectors stored in the image. SimpleITK requires that the image's pixel type be 
# sitk.sitkVectorFloat64.
displacement = sitk.DisplacementFieldTransform(2)
field_size = [10,20]
field_origin = [-1.0,-1.0]  
field_spacing = [2.0/9.0,2.0/19.0]   
field_direction = [1,0,0,1] # direction cosine matrix (row major order)     

# Concatenate all the information into a single list
displacement.SetFixedParameters(field_size+field_origin+field_spacing+field_direction)
# Set the interpolater, either sitkLinear which is default or nearest neighbor
displacement.SetInterpolator(sitk.sitkNearestNeighbor)

originalDisplacements = np.random.random(len(displacement.GetParameters()))
displacement.SetParameters(originalDisplacements)

coordsX = np.linspace(field_origin[0], field_origin[0]+(field_size[0]-1)*field_spacing[0], field_size[0])
coordsY = np.linspace(field_origin[1], field_origin[1]+(field_size[1]-1)*field_spacing[1], field_size[1])
XX, YY = np.meshgrid(coordsX, coordsY)

interact(display_displacement_scaling_effect, s= (-1.5,1.5), original_x_mat = fixed(XX), original_y_mat = fixed(YY),
         tx = fixed(displacement), original_control_point_displacements = fixed(originalDisplacements));


    

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displacement_image = sitk.Image([64,64], sitk.sitkVectorFloat64)
# The only point that has any displacement is (0,0)
displacement = (0.5,0.5)
displacement_image[0,0] = displacement

print('Original displacement image size: ' + point2str(displacement_image.GetSize()))

displacement_field_transform = sitk.DisplacementFieldTransform(displacement_image)

print('After using the image to create a transform, displacement image size: ' + point2str(displacement_image.GetSize()))

# Check that the displacement field transform does what we expect.
print('Expected result: {0}\nActual result:{1}'.format(str(displacement), displacement_field_transform.TransformPoint((0,0))))


    

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# Create a composite transformation: T_affine(T_rigid(x)).
rigid_center = (100,100,100)
theta_x = 0.0
theta_y = 0.0
theta_z = np.pi/2.0
rigid_translation = (1,2,3)
rigid_euler = sitk.Euler3DTransform(rigid_center, theta_x, theta_y, theta_z, rigid_translation)

affine_center = (20, 20, 20)
affine_translation = (5,6,7)  

# Matrix is represented as a vector-like data in row major order.
affine_matrix = np.random.random(9)         
affine = sitk.AffineTransform(affine_matrix, affine_translation, affine_center)

# Using the composite transformation we just add them in (stack based, first in - last applied).
composite_transform = sitk.Transform(affine)
composite_transform.AddTransform(rigid_euler)

# Create a single transform manually. this is a recipe for compositing any two global transformations
# into an affine transformation, T_0(T_1(x)):
# A = A=A0*A1
# c = c1
# t = A0*[t1+c1-c0] + t0+c0-c1
A0 = np.asarray(affine.GetMatrix()).reshape(3,3)
c0 = np.asarray(affine.GetCenter())
t0 = np.asarray(affine.GetTranslation())

A1 = np.asarray(rigid_euler.GetMatrix()).reshape(3,3)
c1 = np.asarray(rigid_euler.GetCenter())
t1 = np.asarray(rigid_euler.GetTranslation())

combined_mat = np.dot(A0,A1)
combined_center = c1
combined_translation = np.dot(A0, t1+c1-c0) + t0+c0-c1
combined_affine = sitk.AffineTransform(combined_mat.flatten(), combined_translation, combined_center)

# Check if the two transformations are equivalent.
print('Apply the two transformations to the same point cloud:')
print('\t', end='')
print_transformation_differences(composite_transform, combined_affine)

print('Transform parameters:')
print('\tComposite transform: ' + point2str(composite_transform.GetParameters(),2))
print('\tCombined affine: ' + point2str(combined_affine.GetParameters(),2))

print('Fixed parameters:')
print('\tComposite transform: ' + point2str(composite_transform.GetFixedParameters(),2))
print('\tCombined affine: ' + point2str(combined_affine.GetFixedParameters(),2))


    

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# Global transformation.
translation = sitk.TranslationTransform(2,(1.0,0.0))

# Displacement in region 1.
displacement1 = sitk.DisplacementFieldTransform(2)
field_size = [10,20]
field_origin = [-1.0,-1.0]  
field_spacing = [2.0/9.0,2.0/19.0]   
field_direction = [1,0,0,1] # direction cosine matrix (row major order)     

# Concatenate all the information into  a single list.
displacement1.SetFixedParameters(field_size+field_origin+field_spacing+field_direction)
displacement1.SetParameters(np.ones(len(displacement1.GetParameters())))

# Displacement in region 2.
displacement2 = sitk.DisplacementFieldTransform(2)
field_size = [10,20]
field_origin = [1.0,-3]  
field_spacing = [2.0/9.0,2.0/19.0]   
field_direction = [1,0,0,1] #direction cosine matrix (row major order)     

# Concatenate all the information into a single list.
displacement2.SetFixedParameters(field_size+field_origin+field_spacing+field_direction)
displacement2.SetParameters(-1.0*np.ones(len(displacement2.GetParameters())))

# Composite transform which applies the global and local transformations.
composite = sitk.Transform(translation)
composite.AddTransform(displacement1)
composite.AddTransform(displacement2)

# Apply the composite transformation to points in ([-1,-3],[3,1]) and 
# display the deformation using a quiver plot.
        
# Generate points.
numSamplesX = 10
numSamplesY = 10                   
coordsX = np.linspace(-1.0, 3.0, numSamplesX)
coordsY = np.linspace(-3.0, 1.0, numSamplesY)
XX, YY = np.meshgrid(coordsX, coordsY)

# Transform points and compute deformation vectors.
pointsX = np.zeros(XX.shape)
pointsY = np.zeros(XX.shape)
for index, value in np.ndenumerate(XX):
    px,py = composite.TransformPoint((value, YY[index]))
    pointsX[index]=px - value 
    pointsY[index]=py - YY[index]
    
plt.quiver(XX, YY, pointsX, pointsY);


    

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import os

# Create a 2D rigid transformation, write it to disk and read it back.
basic_transform = sitk.Euler2DTransform()
basic_transform.SetTranslation((1,2))
basic_transform.SetAngle(np.pi/2)

full_file_name = os.path.join(OUTPUT_DIR, 'euler2D.tfm')

sitk.WriteTransform(basic_transform, full_file_name)

# The ReadTransform function returns an sitk.Transform no matter the type of the transform 
# found in the file (global, bounded, composite).
read_result = sitk.ReadTransform(full_file_name)

print('Different types: '+ str(type(read_result) != type(basic_transform)))
print_transformation_differences(basic_transform, read_result)


# Create a composite transform then write and read.
displacement = sitk.DisplacementFieldTransform(2)
field_size = [10,20]
field_origin = [-10.0,-100.0]  
field_spacing = [20.0/(field_size[0]-1),200.0/(field_size[1]-1)]   
field_direction = [1,0,0,1] #direction cosine matrix (row major order)

# Concatenate all the information into a single list.
displacement.SetFixedParameters(field_size+field_origin+field_spacing+field_direction)
displacement.SetParameters(np.random.random(len(displacement.GetParameters())))

composite_transform = sitk.Transform(basic_transform)
composite_transform.AddTransform(displacement)

full_file_name = os.path.join(OUTPUT_DIR, 'composite.tfm')

sitk.WriteTransform(composite_transform, full_file_name)
read_result = sitk.ReadTransform(full_file_name)

print_transformation_differences(composite_transform, read_result)


    

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