code: def v_cycle_2d(
      n,
      k,
      delta_field,
      sigma_sq_field,
      gradient_field,
      target,
      lambda_param,
      displacement,
      depth=0,
  ):
      r"""Multi-resolution Gauss-Seidel solver using V-type cycles
  
      Multi-resolution Gauss-Seidel solver: solves the Gauss-Newton linear system
      by first filtering (GS-iterate) the current level, then solves for the
      residual at a coarser resolution and finally refines the solution at the
      current resolution. This scheme corresponds to the V-cycle proposed by
      Bruhn and Weickert[Bruhn05].
  
      Parameters
      ----------
      n : int
          number of levels of the multi-resolution algorithm (it will be called
          recursively until level n == 0)
      k : int
          the number of iterations at each multi-resolution level
      delta_field : array, shape (R, C)
          the difference between the static and moving image (the 'derivative
          w.r.t. time' in the optical flow model)
      sigma_sq_field : array, shape (R, C)
          the variance of the gray level value at each voxel, according to the
          EM model (for SSD, it is 1 for all voxels). Inf and 0 values
          are processed specially to support infinite and zero variance.
      gradient_field : array, shape (R, C, 2)
          the gradient of the moving image
      target : array, shape (R, C, 2)
          right-hand side of the linear system to be solved in the Weickert's
          multi-resolution algorithm
      lambda_param : float
          smoothness parameter, the larger its value the smoother the
          displacement field
      displacement : array, shape (R, C, 2)
          the displacement field to start the optimization from
  
      Returns
      -------
      energy : the energy of the EM (or SSD if sigmafield[...]==1) metric at this
          iteration
  
      References
      ----------
      [Bruhn05] Andres Bruhn and Joachim Weickert, "Towards ultimate motion
                estimation: combining the highest accuracy with real-time
                performance", 10th IEEE International Conference on Computer
                Vision, 2005. ICCV 2005.
      """
      # pre-smoothing
      for i in range(k):
          ssd.iterate_residual_displacement_field_ssd_2d(
              delta_field,
              sigma_sq_field,
              gradient_field,
              target,
              lambda_param,
              displacement,
          )
      if n == 0:
          energy = ssd.compute_energy_ssd_2d(delta_field)
          return energy
  
      # solve at coarser grid
      residual = None
      residual = ssd.compute_residual_displacement_field_ssd_2d(
          delta_field,
          sigma_sq_field,
          gradient_field,
          target,
          lambda_param,
          displacement,
          residual,
      )
      sub_residual = np.array(
          vfu.downsample_displacement_field_2d(residual)
      )
      del residual
      subsigma_sq_field = None
      if sigma_sq_field is not None:
          subsigma_sq_field = vfu.downsample_scalar_field_2d(
              sigma_sq_field
          )
      subdelta_field = vfu.downsample_scalar_field_2d(delta_field)
  
      subgradient_field = np.array(
          vfu.downsample_displacement_field_2d(gradient_field)
      )
  
      shape = np.array(displacement.shape).astype(np.int32)
      half_shape = ((shape[0] + 1) // 2, (shape[1] + 1) // 2, 2)
      sub_displacement = np.zeros(shape=half_shape, dtype=floating)
      sublambda_param = lambda_param * 0.25
      v_cycle_2d(
          n - 1,
          k,
          subdelta_field,
          subsigma_sq_field,
          subgradient_field,
          sub_residual,
          sublambda_param,
          sub_displacement,
          depth + 1,
      )
      # displacement += np.array(
      #    vfu.upsample_displacement_field(sub_displacement, shape))
      displacement += vfu.resample_displacement_field_2d(
          sub_displacement, np.array([0.5, 0.5]), shape
      )
  
      # post-smoothing
      for i in range(k):
          ssd.iterate_residual_displacement_field_ssd_2d(
              delta_field,
              sigma_sq_field,
              gradient_field,
              target,
              lambda_param,
              displacement,
          )
      energy = ssd.compute_energy_ssd_2d(delta_field)
      return energy
  
imports: import abc
  import numpy
  from numpy import gradient
  from scipy import ndimage
  from dipy.align import vector_fields
  from dipy.align import sumsqdiff
  from dipy.align import crosscorr
  from dipy.align import expectmax
  from dipy.align import floating
label_desc: Use image node_id 1 for calling a global function with example usage: v_cycle_2d(n, k, delta_field, sigma_sq_field, gradient_field, target, lambda_param, displacement, depth) and returns: energy, energy