code: def csdeconv(dwsignal, X, B_reg, tau=0.1, convergence=50, P=None):
      r"""Constrained-regularized spherical deconvolution (CSD) [1]_
  
      Deconvolves the axially symmetric single fiber response function `r_rh` in
      rotational harmonics coefficients from the diffusion weighted signal in
      `dwsignal`.
  
      Parameters
      ----------
      dwsignal : array
          Diffusion weighted signals to be deconvolved.
      X : array
          Prediction matrix which estimates diffusion weighted signals from FOD
          coefficients.
      B_reg : array (N, B)
          SH basis matrix which maps FOD coefficients to FOD values on the
          surface of the sphere. B_reg should be scaled to account for lambda.
      tau : float
          Threshold controlling the amplitude below which the corresponding fODF
          is assumed to be zero.  Ideally, tau should be set to zero. However, to
          improve the stability of the algorithm, tau is set to tau*100 % of the
          max fODF amplitude (here, 10% by default). This is similar to peak
          detection where peaks below 0.1 amplitude are usually considered noise
          peaks. Because SDT is based on a q-ball ODF deconvolution, and not
          signal deconvolution, using the max instead of mean (as in CSD), is
          more stable.
      convergence : int
          Maximum number of iterations to allow the deconvolution to converge.
      P : ndarray
          This is an optimization to avoid computing ``dot(X.T, X)`` many times.
          If the same ``X`` is used many times, ``P`` can be precomputed and
          passed to this function.
  
      Returns
      -------
      fodf_sh : ndarray (``(sh_order + 1)*(sh_order + 2)/2``,)
           Spherical harmonics coefficients of the constrained-regularized fiber
           ODF.
      num_it : int
           Number of iterations in the constrained-regularization used for
           convergence.
  
      Notes
      -----
      This section describes how the fitting of the SH coefficients is done.
      Problem is to minimise per iteration:
  
      $F(f_n) = ||Xf_n - S||^2 + \lambda^2 ||H_{n-1} f_n||^2$
  
      Where $X$ maps current FOD SH coefficients $f_n$ to DW signals $s$ and
      $H_{n-1}$ maps FOD SH coefficients $f_n$ to amplitudes along set of
      negative directions identified in previous iteration, i.e. the matrix
      formed by the rows of $B_{reg}$ for which $Hf_{n-1}<0$ where $B_{reg}$
      maps $f_n$ to FOD amplitude on a sphere.
  
      Solve by differentiating and setting to zero:
  
      $\Rightarrow \frac{\delta F}{\delta f_n} = 2X^T(Xf_n - S) + 2 \lambda^2
      H_{n-1}^TH_{n-1}f_n=0$
  
      Or:
  
      $(X^TX + \lambda^2 H_{n-1}^TH_{n-1})f_n = X^Ts$
  
      Define $Q = X^TX + \lambda^2 H_{n-1}^TH_{n-1}$ , which by construction is a
      square positive definite symmetric matrix of size $n_{SH} by n_{SH}$. If
      needed, positive definiteness can be enforced with a small minimum norm
      regulariser (helps a lot with poorly conditioned direction sets and/or
      superresolution):
  
      $Q = X^TX + (\lambda H_{n-1}^T) (\lambda H_{n-1}) + \mu I$
  
      Solve $Qf_n = X^Ts$ using Cholesky decomposition:
  
      $Q = LL^T$
  
      where $L$ is lower triangular. Then problem can be solved by
      back-substitution:
  
      $L_y = X^Ts$
  
      $L^Tf_n = y$
  
      To speeds things up further, form $P = X^TX + \mu I$, and update to form
      $Q$ by rankn update with $H_{n-1}$. The dipy implementation looks like:
  
          form initially $P = X^T X + \mu I$ and $\lambda B_{reg}$
  
          for each voxel: form $z = X^Ts$
  
              estimate $f_0$ by solving $Pf_0=z$. We use a simplified $l_{max}=4$
              solution here, but it might not make a big difference.
  
              Then iterate until no change in rows of $H$ used in $H_n$
  
                  form $H_{n}$ given $f_{n-1}$
  
                  form $Q = P + (\lambda H_{n-1}^T) (\lambda H_{n-1}$) (this can
                  be done by rankn update, but we currently do not use rankn
                  update).
  
                  solve $Qf_n = z$ using Cholesky decomposition
  
      We'd like to thanks Donald Tournier for his help with describing and
      implementing this algorithm.
  
      References
      ----------
      .. [1] Tournier, J.D., et al. NeuroImage 2007. Robust determination of the
             fibre orientation distribution in diffusion MRI: Non-negativity
             constrained super-resolved spherical deconvolution.
  
      """
      mu = 1e-5
      if P is None:
          P = np.dot(X.T, X)
      z = np.dot(X.T, dwsignal)
  
      try:
          fodf_sh = _solve_cholesky(P, z)
      except la.LinAlgError:
          P = P + mu * np.eye(P.shape[0])
          fodf_sh = _solve_cholesky(P, z)
      # For the first iteration we use a smooth FOD that only uses SH orders up
      # to 4 (the first 15 coefficients).
      fodf = np.dot(B_reg[:, :15], fodf_sh[:15])
      # The mean of an fodf can be computed by taking $Y_{0,0} * coeff_{0,0}$
      threshold = B_reg[0, 0] * fodf_sh[0] * tau
      where_fodf_small = (fodf < threshold).nonzero()[0]
  
      # If the low-order fodf does not have any values less than threshold, the
      # full-order fodf is used.
      if len(where_fodf_small) == 0:
          fodf = np.dot(B_reg, fodf_sh)
          where_fodf_small = (fodf < threshold).nonzero()[0]
          # If the fodf still has no values less than threshold, return the fodf.
          if len(where_fodf_small) == 0:
              return fodf_sh, 0
  
      for num_it in range(1, convergence + 1):
          # This is the super-resolved trick.  Wherever there is a negative
          # amplitude value on the fODF, it concatenates a value to the S vector
          # so that the estimation can focus on trying to eliminate it. In a
          # sense, this "adds" a measurement, which can help to better estimate
          # the fodf_sh, even if you have more SH coefficients to estimate than
          # actual S measurements.
          H = B_reg.take(where_fodf_small, axis=0)
  
          # We use the Cholesky decomposition to solve for the SH coefficients.
          Q = P + np.dot(H.T, H)
          fodf_sh = _solve_cholesky(Q, z)
  
          # Sample the FOD using the regularization sphere and compute k.
          fodf = np.dot(B_reg, fodf_sh)
          where_fodf_small_last = where_fodf_small
          where_fodf_small = (fodf < threshold).nonzero()[0]
  
          if (
              len(where_fodf_small) == len(where_fodf_small_last)
              and (where_fodf_small == where_fodf_small_last).all()
          ):
              break
      else:
          msg = "maximum number of iterations exceeded - failed to converge"
          warnings.warn(msg)
  
      return fodf_sh, num_it
  
imports: import warnings
  import numpy
  import numbers
  from scipy.integrate import quad
  from scipy.special import lpn, gamma
  import scipy.linalg
  import scipy.linalg.lapack
  from dipy.data import small_sphere, get_sphere, default_sphere
  from dipy.core.geometry import cart2sphere
  from dipy.core.ndindex import ndindex
  from dipy.sims.voxel import single_tensor
  from dipy.reconst.multi_voxel import multi_voxel_fit
  from dipy.reconst.dti import TensorModel, fractional_anisotropy
  from dipy.reconst.shm import sph_harm_ind_list, real_sh_descoteaux_from_index, sph_harm_lookup, lazy_index, SphHarmFit, real_sh_descoteaux, sh_to_rh, forward_sdeconv_mat, SphHarmModel
  from dipy.reconst.utils import _roi_in_volume, _mask_from_roi
  from dipy.direction.peaks import peaks_from_model
  from dipy.core.geometry import vec2vec_rotmat
  from dipy.utils.deprecator import deprecate_with_version, deprecated_params
label_desc: Use image node_id 6 for calling a global function with example usage: csdeconv(dwsignal, X, B_reg, tau, convergence, P) and returns: fodf_sh,  num_it, fodf_sh,  int