text: import numpy as np
  import scipy as sp
  import nibabel as nib
  import numpy.linalg as npl
  from numpy.testing import (assert_array_equal,
                             assert_array_almost_equal,
                             assert_almost_equal,
                             assert_equal,
                             assert_raises)
  from dipy.core import geometry as geometry
  from dipy.data import get_data
  from dipy.viz import regtools as rt
  from dipy.align import floating
  from dipy.align import vector_fields as vf
  from dipy.align import imaffine
  from dipy.align.imaffine import AffineInversionError
  from dipy.align.transforms import (Transform,
                                     regtransforms)
  from dipy.align.tests.test_parzenhist import (setup_random_transform,
                                                sample_domain_regular)
  
  # For each transform type, select a transform factor (indicating how large the
  # true transform between static and moving images will be), a sampling scheme
  # (either a positive integer less than or equal to 100, or None) indicating
  # the percentage (if int) of voxels to be used for estimating the joint PDFs,
  # or dense sampling (if None), and also specify a starting point (to avoid
  # starting from the identity)
  factors = {('TRANSLATION', 2): (2.0, 0.35, np.array([2.3, 4.5])),
             ('ROTATION', 2): (0.1, None, np.array([0.1])),
             ('RIGID', 2): (0.1, .50, np.array([0.12, 1.8, 2.7])),
             ('SCALING', 2): (0.01, None, np.array([1.05])),
             ('AFFINE', 2): (0.1, .50, np.array([0.99, -0.05, 1.3, 0.05, 0.99,
                                                 2.5])),
             ('TRANSLATION', 3): (2.0, None, np.array([2.3, 4.5, 1.7])),
             ('ROTATION', 3): (0.1, 1.0, np.array([0.1, 0.15, -0.11])),
             ('RIGID', 3): (0.1, None, np.array([0.1, 0.15, -0.11, 2.3, 4.5,
                                                 1.7])),
             ('SCALING', 3): (0.1, .35, np.array([0.95])),
             ('AFFINE', 3): (0.1, None, np.array([0.99, -0.05, 0.03, 1.3,
                                                  0.05, 0.99, -0.10, 2.5,
                                                  -0.07, 0.10, 0.99, -1.4]))}
  
  
  def test_transform_centers_of_mass_3d():
      np.random.seed(1246592)
      shape = (64, 64, 64)
      rm = 8
      sp = vf.create_sphere(shape[0] // 2, shape[1] // 2, shape[2] // 2, rm)
      moving = np.zeros(shape)
      # The center of mass will be (16, 16, 16), in image coordinates
      moving[:shape[0] // 2, :shape[1] // 2, :shape[2] // 2] = sp[...]
  
      rs = 16
      # The center of mass will be (32, 32, 32), in image coordinates
      static = vf.create_sphere(shape[0], shape[1], shape[2], rs)
  
      # Create arbitrary image-to-space transforms
      axis = np.array([.5, 2.0, 1.5])
      t = 0.15  # translation factor
      trans = np.array([[1, 0, 0, -t * shape[0]],
                        [0, 1, 0, -t * shape[1]],
                        [0, 0, 1, -t * shape[2]],
                        [0, 0, 0, 1]])
      trans_inv = npl.inv(trans)
  
      for rotation_angle in [-1 * np.pi / 6.0, 0.0, np.pi / 5.0]:
          for scale_factor in [0.83, 1.3, 2.07]:  # scale
              rot = np.zeros(shape=(4, 4))
              rot[:3, :3] = geometry.rodrigues_axis_rotation(axis,
                                                             rotation_angle)
              rot[3, 3] = 1.0
              scale = np.array([[1 * scale_factor, 0, 0, 0],
                                [0, 1 * scale_factor, 0, 0],
                                [0, 0, 1 * scale_factor, 0],
                                [0, 0, 0, 1]])
  
              static_grid2world = trans_inv.dot(scale.dot(rot.dot(trans)))
              moving_grid2world = npl.inv(static_grid2world)
  
              # Expected translation
              c_static = static_grid2world.dot((32, 32, 32, 1))[:3]
              c_moving = moving_grid2world.dot((16, 16, 16, 1))[:3]
              expected = np.eye(4)
              expected[:3, 3] = c_moving - c_static
  
              # Implementation under test
              actual = imaffine.transform_centers_of_mass(static,
                                                          static_grid2world,
                                                          moving,
                                                          moving_grid2world)
              assert_array_almost_equal(actual.affine, expected)
  
  
  def test_transform_geometric_centers_3d():
      # Create arbitrary image-to-space transforms
      axis = np.array([.5, 2.0, 1.5])
      t = 0.15  # translation factor
  
      for theta in [-1 * np.pi / 6.0, 0.0, np.pi / 5.0]:  # rotation angle
          for s in [0.83, 1.3, 2.07]:  # scale
              m_shapes = [(256, 256, 128), (255, 255, 127), (64, 127, 142)]
              for shape_moving in m_shapes:
                  s_shapes = [(256, 256, 128), (255, 255, 127), (64, 127, 142)]
                  for shape_static in s_shapes:
                      moving = np.ndarray(shape=shape_moving)
                      static = np.ndarray(shape=shape_static)
                      trans = np.array([[1, 0, 0, -t * shape_static[0]],
                                        [0, 1, 0, -t * shape_static[1]],
                                        [0, 0, 1, -t * shape_static[2]],
                                        [0, 0, 0, 1]])
                      trans_inv = npl.inv(trans)
                      rot = np.zeros(shape=(4, 4))
                      rot[:3, :3] = geometry.rodrigues_axis_rotation(axis, theta)
                      rot[3, 3] = 1.0
                      scale = np.array([[1 * s, 0, 0, 0],
                                        [0, 1 * s, 0, 0],
                                        [0, 0, 1 * s, 0],
                                        [0, 0, 0, 1]])
  
                      static_grid2world = trans_inv.dot(
                          scale.dot(rot.dot(trans)))
                      moving_grid2world = npl.inv(static_grid2world)
  
                      # Expected translation
                      c_static = np.array(shape_static, dtype=np.float64) * 0.5
                      c_static = tuple(c_static)
                      c_static = static_grid2world.dot(c_static + (1,))[:3]
                      c_moving = np.array(shape_moving, dtype=np.float64) * 0.5
                      c_moving = tuple(c_moving)
                      c_moving = moving_grid2world.dot(c_moving + (1,))[:3]
                      expected = np.eye(4)
                      expected[:3, 3] = c_moving - c_static
  
                      # Implementation under test
                      actual = imaffine.transform_geometric_centers(
                          static, static_grid2world, moving, moving_grid2world)
                      assert_array_almost_equal(actual.affine, expected)
  
  
  def test_transform_origins_3d():
      # Create arbitrary image-to-space transforms
      axis = np.array([.5, 2.0, 1.5])
      t = 0.15  # translation factor
  
      for theta in [-1 * np.pi / 6.0, 0.0, np.pi / 5.0]:  # rotation angle
          for s in [0.83, 1.3, 2.07]:  # scale
              m_shapes = [(256, 256, 128), (255, 255, 127), (64, 127, 142)]
              for shape_moving in m_shapes:
                  s_shapes = [(256, 256, 128), (255, 255, 127), (64, 127, 142)]
                  for shape_static in s_shapes:
                      moving = np.ndarray(shape=shape_moving)
                      static = np.ndarray(shape=shape_static)
                      trans = np.array([[1, 0, 0, -t * shape_static[0]],
                                        [0, 1, 0, -t * shape_static[1]],
                                        [0, 0, 1, -t * shape_static[2]],
                                        [0, 0, 0, 1]])
                      trans_inv = npl.inv(trans)
                      rot = np.zeros(shape=(4, 4))
                      rot[:3, :3] = geometry.rodrigues_axis_rotation(axis, theta)
                      rot[3, 3] = 1.0
                      scale = np.array([[1 * s, 0, 0, 0],
                                        [0, 1 * s, 0, 0],
                                        [0, 0, 1 * s, 0],
                                        [0, 0, 0, 1]])
  
                      static_grid2world = trans_inv.dot(
                          scale.dot(rot.dot(trans)))
                      moving_grid2world = npl.inv(static_grid2world)
  
                      # Expected translation
                      c_static = static_grid2world[:3, 3]
                      c_moving = moving_grid2world[:3, 3]
                      expected = np.eye(4)
                      expected[:3, 3] = c_moving - c_static
  
                      # Implementation under test
                      actual = imaffine.transform_origins(static,
                                                          static_grid2world,
                                                          moving,
                                                          moving_grid2world)
                      assert_array_almost_equal(actual.affine, expected)
  
  
  def test_affreg_all_transforms():
      # Test affine registration using all transforms with typical settings
  
      # Make sure dictionary entries are processed in the same order regardless
      # of the platform.
      # Otherwise any random numbers drawn within the loop would make
      # the test non-deterministic even if we fix the seed before the loop.
      # Right now, this test does not draw any samples,
      # but we still sort the entries
      # to prevent future related failures.
      for ttype in sorted(factors):
          dim = ttype[1]
          if dim == 2:
              nslices = 1
          else:
              nslices = 45
          factor = factors[ttype][0]
          sampling_pc = factors[ttype][1]
          transform = regtransforms[ttype]
          static, moving, static_grid2world, moving_grid2world, smask, mmask, T = \
              setup_random_transform(transform, factor, nslices, 1.0)
          # Sum of absolute differences
          start_sad = np.abs(static - moving).sum()
          metric = imaffine.MutualInformationMetric(32, sampling_pc)
          affreg = imaffine.AffineRegistration(metric,
                                               [1000, 100, 50],
                                               [3, 1, 0],
                                               [4, 2, 1],
                                               'L-BFGS-B',
                                               None,
                                               options=None)
          x0 = transform.get_identity_parameters()
          affine_map = affreg.optimize(static, moving, transform, x0,
                                       static_grid2world, moving_grid2world)
          transformed = affine_map.transform(moving)
          # Sum of absolute differences
          end_sad = np.abs(static - transformed).sum()
          reduction = 1 - end_sad / start_sad
          print("%s>>%f" % (ttype, reduction))
          assert(reduction > 0.9)
  
      # Verify that exception is raised if level_iters is empty
      metric = imaffine.MutualInformationMetric(32)
      assert_raises(ValueError, imaffine.AffineRegistration, metric, [])
  
  
  def test_affreg_defaults():
      # Test all default arguments with an arbitrary transform
      # Select an arbitrary transform (all of them are already tested
      # in test_affreg_all_transforms)
      transform_name = 'TRANSLATION'
      dim = 2
      ttype = (transform_name, dim)
      aff_options = ['mass', 'voxel-origin', 'centers', None, np.eye(dim + 1)]
  
      for starting_affine in aff_options:
          if dim == 2:
              nslices = 1
          else:
              nslices = 45
          factor = factors[ttype][0]
          sampling_pc = factors[ttype][1]
          transform = regtransforms[ttype]
          id_param = transform.get_identity_parameters()
          static, moving, static_grid2world, moving_grid2world, smask, mmask, T = \
              setup_random_transform(transform, factor, nslices, 1.0)
          # Sum of absolute differences
          start_sad = np.abs(static - moving).sum()
  
          metric = None
          x0 = None
          sigmas = None
          scale_factors = None
          level_iters = None
          static_grid2world = None
          moving_grid2world = None
          for ss_sigma_factor in [1.0, None]:
              affreg = imaffine.AffineRegistration(metric,
                                                   level_iters,
                                                   sigmas,
                                                   scale_factors,
                                                   'L-BFGS-B',
                                                   ss_sigma_factor,
                                                   options=None)
              affine_map = affreg.optimize(static, moving, transform, x0,
                                           static_grid2world, moving_grid2world,
                                           starting_affine)
              transformed = affine_map.transform(moving)
              # Sum of absolute differences
              end_sad = np.abs(static - transformed).sum()
              reduction = 1 - end_sad / start_sad
              print("%s>>%f" % (ttype, reduction))
              assert(reduction > 0.9)
  
              transformed_inv = affine_map.transform_inverse(static)
              # Sum of absolute differences
              end_sad = np.abs(moving - transformed_inv).sum()
              reduction = 1 - end_sad / start_sad
              print("%s>>%f" % (ttype, reduction))
              assert(reduction > 0.9)
  
  
  def test_mi_gradient():
      np.random.seed(2022966)
      # Test the gradient of mutual information
      h = 1e-5
      # Make sure dictionary entries are processed in the same order regardless
      # of the platform. Otherwise any random numbers drawn within the loop would
      # make the test non-deterministic even if we fix the seed before the loop:
      # in this case the samples are drawn with `np.random.randn` below
  
      for ttype in sorted(factors):
          transform = regtransforms[ttype]
          dim = ttype[1]
          if dim == 2:
              nslices = 1
          else:
              nslices = 45
          factor = factors[ttype][0]
          sampling_proportion = factors[ttype][1]
          theta = factors[ttype][2]
          # Start from a small rotation
          start = regtransforms[('ROTATION', dim)]
          nrot = start.get_number_of_parameters()
          starting_affine = start.param_to_matrix(0.25 * np.random.randn(nrot))
          # Get data (pair of images related to each other by an known transform)
          static, moving, static_g2w, moving_g2w, smask, mmask, M = \
              setup_random_transform(transform, factor, nslices, 2.0)
  
          # Prepare a MutualInformationMetric instance
          mi_metric = imaffine.MutualInformationMetric(32, sampling_proportion)
          mi_metric.setup(
              transform,
              static,
              moving,
              starting_affine=starting_affine)
          # Compute the gradient with the implementation under test
          actual = mi_metric.gradient(theta)
  
          # Compute the gradient using finite-diferences
          n = transform.get_number_of_parameters()
          expected = np.empty(n, dtype=np.float64)
  
          val0 = mi_metric.distance(theta)
          for i in range(n):
              dtheta = theta.copy()
              dtheta[i] += h
              val1 = mi_metric.distance(dtheta)
              expected[i] = (val1 - val0) / h
  
          dp = expected.dot(actual)
          enorm = npl.norm(expected)
          anorm = npl.norm(actual)
          nprod = dp / (enorm * anorm)
          assert(nprod >= 0.99)
  
  
  def create_affine_transforms(
          dim, translations, rotations, scales, rot_axis=None):
      r""" Creates a list of affine transforms with all combinations of params
  
      This function is intended to be used for testing only. It generates
      affine transforms for all combinations of the input parameters in the
      following order: let T be a translation, R a rotation and S a scale. The
      generated affine will be:
  
      A = T.dot(S).dot(R).dot(T^{-1})
  
      Translation is handled this way because it is convenient to provide
      the translation parameters in terms of the center of rotation we wish
      to generate.
  
      Parameters
      ----------
      dim: int (either dim=2 or dim=3)
          dimension of the affine transforms
      translations: sequence of dim-tuples
          each dim-tuple represents a translation parameter
      rotations: sequence of floats
          each number represents a rotation angle in radians
      scales: sequence of floats
          each number represents a scale
      rot_axis: rotation axis (used for dim=3 only)
  
      Returns
      -------
      transforms: sequence of (dim + 1)x(dim + 1) matrices
          each matrix correspond to an affine transform with a combination
          of the input parameters
      """
      transforms = []
      for t in translations:
          trans_inv = np.eye(dim + 1)
          trans_inv[:dim, dim] = -t[:dim]
          trans = npl.inv(trans_inv)
          for theta in rotations:  # rotation angle
              if dim == 2:
                  ct = np.cos(theta)
                  st = np.sin(theta)
                  rot = np.array([[ct, -st, 0],
                                  [st, ct, 0],
                                  [0, 0, 1]])
              else:
                  rot = np.eye(dim + 1)
                  rot[:3, :3] = geometry.rodrigues_axis_rotation(rot_axis, theta)
  
              for s in scales:  # scale
                  scale = np.eye(dim + 1) * s
                  scale[dim, dim] = 1
  
              affine = trans.dot(scale.dot(rot.dot(trans_inv)))
              transforms.append(affine)
      return transforms
  
  
  def test_affine_map():
      np.random.seed(2112927)
      dom_shape = np.array([64, 64, 64], dtype=np.int32)
      cod_shape = np.array([80, 80, 80], dtype=np.int32)
      nx = dom_shape[0]
      ny = dom_shape[1]
      nz = dom_shape[2]
      # Radius of the circle/sphere (testing image)
      radius = 16
      # Rotation axis (used for 3D transforms only)
      rot_axis = np.array([.5, 2.0, 1.5])
      # Arbitrary transform parameters
      t = 0.15
      rotations = [-1 * np.pi / 10.0, 0.0, np.pi / 10.0]
      scales = [0.9, 1.0, 1.1]
      for dim in [2, 3]:
          # Setup current dimension
          if dim == 2:
              # Create image of a circle
              img = vf.create_circle(cod_shape[0], cod_shape[1], radius)
              oracle_linear = vf.transform_2d_affine
              oracle_nn = vf.transform_2d_affine_nn
          else:
              # Create image of a sphere
              img = vf.create_sphere(cod_shape[0], cod_shape[1], cod_shape[2],
                                     radius)
              oracle_linear = vf.transform_3d_affine
              oracle_nn = vf.transform_3d_affine_nn
          img = np.array(img)
          # Translation is the only parameter differing for 2D and 3D
          translations = [t * dom_shape[:dim]]
          # Generate affine transforms
          gt_affines = create_affine_transforms(dim, translations, rotations,
                                                scales, rot_axis)
          # Include the None case
          gt_affines.append(None)
  
          for affine in gt_affines:
  
              # make both domain point to the same physical region
              # It's ok to use the same transform, we just want to test
              # that this information is actually being considered
              domain_grid2world = affine
              codomain_grid2world = affine
              grid2grid_transform = affine
  
              # Evaluate the transform with vector_fields module (already tested)
              expected_linear = oracle_linear(img, dom_shape[:dim],
                                              grid2grid_transform)
              expected_nn = oracle_nn(img, dom_shape[:dim], grid2grid_transform)
  
              # Evaluate the transform with the implementation under test
              affine_map = imaffine.AffineMap(affine,
                                              dom_shape[:dim], domain_grid2world,
                                              cod_shape[:dim],
                                              codomain_grid2world)
              actual_linear = affine_map.transform(img, interp='linear')
              actual_nn = affine_map.transform(img, interp='nearest')
              assert_array_almost_equal(actual_linear, expected_linear)
              assert_array_almost_equal(actual_nn, expected_nn)
  
              # Test set_affine with valid matrix
              affine_map.set_affine(affine)
              if affine is None:
                  assert(affine_map.affine is None)
                  assert(affine_map.affine_inv is None)
              else:
                  assert_array_equal(affine, affine_map.affine)
                  actual = affine_map.affine.dot(affine_map.affine_inv)
                  assert_array_almost_equal(actual, np.eye(dim + 1))
  
              # Evaluate via the inverse transform
  
              # AffineMap will use the inverse of the input matrix when we call
              # `transform_inverse`. Since the inverse of the inverse of a matrix
              # is not exactly equal to the original matrix (numerical
              #  limitations) we need to invert the matrix twice to make sure
              # the oracle and the implementation under test apply the same
              # transform
              aff_inv = None if affine is None else npl.inv(affine)
              aff_inv_inv = None if aff_inv is None else npl.inv(aff_inv)
              expected_linear = oracle_linear(img, dom_shape[:dim],
                                              aff_inv_inv)
              expected_nn = oracle_nn(img, dom_shape[:dim], aff_inv_inv)
  
              affine_map = imaffine.AffineMap(aff_inv,
                                              cod_shape[:dim],
                                              codomain_grid2world,
                                              dom_shape[:dim], domain_grid2world)
              actual_linear = affine_map.transform_inverse(img, interp='linear')
              actual_nn = affine_map.transform_inverse(img, interp='nearest')
              assert_array_almost_equal(actual_linear, expected_linear)
              assert_array_almost_equal(actual_nn, expected_nn)
  
          # Verify AffineMap cannot be created with a non-invertible matrix
          invalid_nan = np.zeros((dim + 1, dim + 1), dtype=np.float64)
          invalid_nan[1, 1] = np.nan
          invalid_zeros = np.zeros((dim + 1, dim + 1), dtype=np.float64)
          assert_raises(
              imaffine.AffineInversionError,
              imaffine.AffineMap,
              invalid_nan)
          assert_raises(
              imaffine.AffineInversionError,
              imaffine.AffineMap,
              invalid_zeros)
  
          # Test exception is raised when the affine transform matrix is not
          # valid
          invalid_shape = np.eye(dim)
          affmap_invalid_shape = imaffine.AffineMap(invalid_shape,
                                                    dom_shape[:dim], None,
                                                    cod_shape[:dim], None)
          assert_raises(ValueError, affmap_invalid_shape.transform, img)
          assert_raises(ValueError, affmap_invalid_shape.transform_inverse, img)
  
          # Verify exception is raised when sampling info is not provided
          valid = np.eye(3)
          affmap_invalid_shape = imaffine.AffineMap(valid)
          assert_raises(ValueError, affmap_invalid_shape.transform, img)
          assert_raises(ValueError, affmap_invalid_shape.transform_inverse, img)
  
          # Verify exception is raised when requesting an invalid interpolation
          assert_raises(ValueError, affine_map.transform, img, 'invalid')
          assert_raises(ValueError, affine_map.transform_inverse, img, 'invalid')
  
          # Verify exception is raised when attempting to warp an image of
          # invalid dimension
          for dim in [2, 3]:
              affine_map = imaffine.AffineMap(np.eye(dim),
                                              cod_shape[:dim], None,
                                              dom_shape[:dim], None)
              for sh in [(2,), (2, 2, 2, 2)]:
                  img = np.zeros(sh)
                  assert_raises(ValueError, affine_map.transform, img)
                  assert_raises(ValueError, affine_map.transform_inverse, img)
              aff_sing = np.zeros((dim + 1, dim + 1))
              aff_nan = np.zeros((dim + 1, dim + 1))
              aff_nan[...] = np.nan
              aff_inf = np.zeros((dim + 1, dim + 1))
              aff_inf[...] = np.inf
  
              assert_raises(
                  AffineInversionError,
                  affine_map.set_affine,
                  aff_sing)
              assert_raises(AffineInversionError, affine_map.set_affine, aff_nan)
              assert_raises(AffineInversionError, affine_map.set_affine, aff_inf)
  
  
  def test_MIMetric_invalid_params():
      transform = regtransforms[('AFFINE', 3)]
      static = np.random.rand(20, 20, 20)
      moving = np.random.rand(20, 20, 20)
      n = transform.get_number_of_parameters()
      sampling_proportion = 0.3
      theta_sing = np.zeros(n)
      theta_nan = np.zeros(n)
      theta_nan[...] = np.nan
      theta_inf = np.zeros(n)
      theta_nan[...] = np.inf
  
      mi_metric = imaffine.MutualInformationMetric(32, sampling_proportion)
      mi_metric.setup(transform, static, moving)
      for theta in [theta_sing, theta_nan, theta_inf]:
          # Test metric value at invalid params
          actual_val = mi_metric.distance(theta)
          assert(np.isinf(actual_val))
  
          # Test gradient at invalid params
          expected_grad = np.zeros(n)
          actual_grad = mi_metric.gradient(theta)
          assert_equal(actual_grad, expected_grad)
  
          # Test both
          actual_val, actual_grad = mi_metric.distance_and_gradient(theta)
          assert(np.isinf(actual_val))
          assert_equal(actual_grad, expected_grad)
  