text: # Taken from https://github.com/menpo/menpo/blob/master/menpo/transform/thinplatesplines.py
  
  import numpy as np
  #from .base import Transform, Alignment, Invertible
  from .rbf import R2LogR2RBF
  import scipy.linalg as la
  import logging
  
  
  # Note we inherit from Alignment first to get it's n_dims behavior
  #class ThinPlateSplines(Alignment, Transform, Invertible):
  class ThinPlateSplines(object):
      SVD_RANDOM_POINTS_THRESHOLD = 2000
  
      r"""
      The thin plate splines (TPS) alignment between 2D `source` and `target`
      landmarks.
  
      ``kernel`` can be used to specify an alternative kernel function. If
      ``None`` is supplied, the :class:`R2LogR2RBF` kernel will be used.
  
      Parameters
      ----------
      source : ``(N, 2)`` `ndarray`
          The source points to apply the tps from
      target : ``(N, 2)`` `ndarray`
          The target points to apply the tps to
      kernel : :class:`menpo.transform.rbf.RadialBasisFunction`, optional
          The kernel to apply.
      min_singular_val : `float`, optional
          If the target has points that are nearly coincident, the coefficients
          matrix is rank deficient, and therefore not invertible. Therefore, we
          only take the inverse on the full-rank matrix and drop any singular
          values that are less than this value (close to zero).
  
      Raises
      ------
      ValueError
          TPS is only with on 2-dimensional data
      """
      def __init__(self, source_points, target_points, kernel=None, min_singular_val=1e-4):
          logging.basicConfig(level=logging.DEBUG)
          self.source_points = source_points
          self.target_points = target_points
          self.n_points, self.n_dims = source_points.shape
          #Alignment.__init__(self, source, target)
          if self.n_dims != 2:
              raise ValueError('TPS can only be used on 2D data.')
          if kernel is None:
              kernel = R2LogR2RBF(source_points)
          self.min_singular_val = min_singular_val
          self.kernel = kernel
          logging.log(logging.DEBUG, "TPS c'tor 1 populate done")
          # k[i, j] is the rbf weighting between source i and j
          # (of course, k is thus symmetrical and it's diagonal nil)
          self.k = self.kernel.apply(self.source_points)
          logging.log(logging.DEBUG, "TPS c'tor 2 applied kernel on source points done")
          # p is a homogeneous version of the source points
          self.p = np.concatenate(
              [np.ones([self.n_points, 1]), self.source_points], axis=1)
          o = np.zeros([3, 3])
          top_l = np.concatenate([self.k, self.p], axis=1)
          bot_l = np.concatenate([self.p.T, o], axis=1)
          self.l = np.concatenate([top_l, bot_l], axis=0)
          self.v, self.y, self.coefficients = None, None, None
          logging.log(logging.DEBUG, "TPS c'tor 3 basic concatenation done")
          self._build_coefficients()
  
      def _build_coefficients(self):
          logging.log(logging.DEBUG, "TPS build_coefficents 1 starting")
          self.v = self.target_points.T.copy()
          logging.log(logging.DEBUG, "TPS build_coefficents 2 target_points copy done")
          self.y = np.hstack([self.v, np.zeros([2, 3])])
          logging.log(logging.DEBUG, "TPS build_coefficents 3 hstack done, self.l.shape: {}".format(self.l.shape))
  
          # If two points are coincident, or very close to being so, then the
          # matrix is rank deficient and thus not-invertible. Therefore,
          # only take the inverse on the full-rank set of indices.
          if self.n_points <= ThinPlateSplines.SVD_RANDOM_POINTS_THRESHOLD:
              _u, _s, _v = np.linalg.svd(self.l)
          else:
              logging.log(logging.DEBUG, "TPS too many points, using randomized SVD")
              # Randomized svd uses much less memory and is much faster than the numpy/scipy versions
              from sklearn.utils.extmath import randomized_svd
              _u, _s, _v = randomized_svd(self.l, 4, ThinPlateSplines.SVD_RANDOM_POINTS_THRESHOLD)
              
          logging.log(logging.DEBUG, "TPS build_coefficents 4 svd done")
          keep = _s.shape[0] - sum(_s < self.min_singular_val)
          logging.log(logging.DEBUG, "TPS build_coefficents 5 finding keep done. keep={}, _u.shape: {}, _s.shape: {}, _v.shape: {}".format(keep, _u[:, :keep].shape, _s[:keep, None].shape, _v[:keep, :].shape))
          _s_over = 1.0 / _s[:keep, None]
          logging.log(logging.DEBUG, "TPS build_coefficents 5.1 compute 1.0 / _s done")
          _s_over_mul_v = _s_over * _v[:keep, :]
          logging.log(logging.DEBUG, "TPS build_coefficents 5.2 compute _s_over * _v[:keep, :] done")
          #inv_l = _u[:, :keep].dot(_s_over_mul_v)
          inv_l = la.blas.dgemm(1.0, _u[:, :keep], _s_over_mul_v)
          logging.log(logging.DEBUG, "TPS build_coefficents 5.3 compute _u[:, :keep].dot(_s_over_mul_v) [== inv_l] done")
          #inv_l = _u[:, :keep].dot(1.0 / _s[:keep, None] * _v[:keep, :])
          logging.log(logging.DEBUG, "TPS build_coefficents 6 compute inv_l done, inv_l.shape: {}, self.y.T.shape: {}".format(inv_l.shape, self.y.T.shape))
          #print(inv_l)
          self.coefficients = inv_l.dot(self.y.T)
          logging.log(logging.DEBUG, "TPS build_coefficents 7 compute coefficients done, self.coefficients.shape: {}".format(self.coefficients.shape))
  
      def _sync_state_from_target(self):
          # now the target is updated, we only have to rebuild the
          # coefficients.
          self._build_coefficients()
  
      def _apply(self, points, **kwargs):
          r"""
          Performs a TPS transform on the given points.
  
          Parameters
          ----------
          points : ``(N, D)`` `ndarray`
              The points to transform.
  
          Returns
          -------
          f : ``(N, D)`` `ndarray`
              The transformed points
  
          Raises
          ------
          ValueError
              TPS can only be applied to 2D data.
          """
          if points.shape[1] != self.n_dims:
              raise ValueError('TPS can only be applied to 2D data.')
          x = points[..., 0][:, None]
          y = points[..., 1][:, None]
          # calculate the affine coefficients of the warp
          # (C = Constant component, then X, Y respectively)
          c_affine_c = self.coefficients[-3]
          c_affine_x = self.coefficients[-2]
          c_affine_y = self.coefficients[-1]
          # the affine warp component
          f_affine = c_affine_c + c_affine_x * x + c_affine_y * y
          # calculate a distance matrix (for L2 Norm) between every source
          # and the target
          kernel_dist = self.kernel.apply(points)
          # grab the affine free components of the warp
          c_affine_free = self.coefficients[:-3]
          # build the affine free warp component
          f_affine_free = kernel_dist.dot(c_affine_free)
          return f_affine + f_affine_free
  
      def apply(self, x, batch_size=None, **kwargs):
          r"""
          Applies this transform to ``x``.
          If ``x`` is :map:`Transformable`, ``x`` will be handed this transform
          object to transform itself non-destructively (a transformed copy of the
          object will be returned).
          If not, ``x`` is assumed to be an `ndarray`. The transformation will be
          non-destructive, returning the transformed version.
          Any ``kwargs`` will be passed to the specific transform :meth:`_apply`
          method.
          Parameters
          ----------
          x : :map:`Transformable` or ``(n_points, n_dims)`` `ndarray`
              The array or object to be transformed.
          batch_size : `int`, optional
              If not ``None``, this determines how many items from the numpy
              array will be passed through the transform at a time. This is
              useful for operations that require large intermediate matrices
              to be computed.
          kwargs : `dict`
              Passed through to :meth:`_apply`.
          Returns
          -------
          transformed : ``type(x)``
              The transformed object or array
          """
  
          def transform(x_):
              """
              Local closure which calls the :meth:`_apply` method with the
              `kwargs` attached.
              """
              return self._apply_batched(x_, batch_size, **kwargs)
  
          try:
              return x._transform(transform)
          except AttributeError:
              return self._apply_batched(x, batch_size, **kwargs)
  
      def _apply_batched(self, x, batch_size, **kwargs):
          if batch_size is None:
              return self._apply(x, **kwargs)
          else:
              outputs = []
              n_points = x.shape[0]
              for lo_ind in range(0, n_points, batch_size):
                  hi_ind = lo_ind + batch_size
                  outputs.append(self._apply(x[lo_ind:hi_ind], **kwargs))
              return np.vstack(outputs)
  
  
      @property
      def has_true_inverse(self):
          r"""
          :type: ``False``
          """
          return False
  
      def pseudoinverse(self):
          r"""
          The pseudoinverse of the transform - that is, the transform that
          results from swapping `source` and `target`, or more formally, negating
          the transforms parameters. If the transform has a true inverse this
          is returned instead.
  
          :type: ``type(self)``
          """
          return ThinPlateSplines(self.target, self.source, kernel=self.kernel)
  