code: def test_hat_and_lcr():
      hemi = hemi_icosahedron.subdivide(3)
      m, n = sph_harm_ind_list(8)
  
      with warnings.catch_warnings():
          warnings.filterwarnings(
              "ignore",
              message=descoteaux07_legacy_msg,
              category=PendingDeprecationWarning,
          )
  
          B = real_sh_descoteaux_from_index(
              m, n, hemi.theta[:, None], hemi.phi[:, None]
          )
  
      H = hat(B)
      B_hat = np.dot(H, B)
      assert_array_almost_equal(B, B_hat)
  
      R = lcr_matrix(H)
      d = np.arange(len(hemi.theta))
      r = d - np.dot(H, d)
      lev = np.sqrt(1 - H.diagonal())
      r /= lev
      r -= r.mean()
  
      r2 = np.dot(R, d)
      assert_array_almost_equal(r, r2)
  
      r3 = np.dot(d, R.T)
      assert_array_almost_equal(r, r3)
  
imports: import warnings
  import numpy
  import numpy.linalg
  import numpy.testing
  from dipy.testing import assert_true
  from numpy.testing import assert_array_equal, assert_array_almost_equal, assert_equal, assert_raises
  from scipy.special import sph_harm
  from dipy.core.sphere import hemi_icosahedron, Sphere
  from dipy.core.gradients import gradient_table
  from dipy.core.interpolation import NearestNeighborInterpolator
  from dipy.sims.voxel import single_tensor
  from dipy.direction.peaks import peak_directions
  from dipy.reconst.shm import sf_to_sh, sh_to_sf
  from dipy.sims.voxel import multi_tensor_odf
  from dipy.data import mrtrix_spherical_functions
  from dipy.reconst import odf
  from dipy.reconst.shm import real_sh_descoteaux_from_index, real_sym_sh_basis, real_sym_sh_mrtrix, real_sh_descoteaux, real_sh_tournier, sph_harm_ind_list, order_from_ncoef, OpdtModel, normalize_data, hat, lcr_matrix, smooth_pinv, bootstrap_data_array, bootstrap_data_voxel, ResidualBootstrapWrapper, CsaOdfModel, QballModel, SphHarmFit, spherical_harmonics, anisotropic_power, calculate_max_order, sh_to_sf_matrix, gen_dirac, convert_sh_to_full_basis, convert_sh_from_legacy, convert_sh_to_legacy, convert_sh_descoteaux_tournier, descoteaux07_legacy_msg, tournier07_legacy_msg
label_desc: Use image node_id 15 for calling a global function with example usage: test_hat_and_lcr() without return types