# Source code for dipy.tracking.life

    """
    This is an implementation of the Linear Fascicle Evaluation (LiFE) algorithm
    described in :footcite:p:`Pestilli2014`.

    References
    ----------
    .. footbibliography::
    """

    import numpy as np
    import scipy.linalg as la
    import scipy.sparse as sps

    import dipy.core.optimize as opt
    import dipy.data as dpd
    from dipy.reconst.base import ReconstFit, ReconstModel
    from dipy.testing.decorators import warning_for_keywords
    from dipy.tracking.streamline import transform_streamlines
    from dipy.tracking.utils import unique_rows
    from dipy.tracking.vox2track import _voxel2streamline

    def gradient(f):
        """
        Return the gradient of an N-dimensional array.

        The gradient is computed using central differences in the interior
        and first differences at the boundaries. The returned gradient hence has
        the same shape as the input array.

        Parameters
        ----------
        f : array_like
          An N-dimensional array containing samples of a scalar function.

        Returns
        -------
        gradient : ndarray
          N arrays of the same shape as `f` giving the derivative of `f` with
          respect to each dimension.

        Examples
        --------
        >>> x = np.array([1, 2, 4, 7, 11, 16], dtype=float)
        >>> gradient(x)
        array([ 1. ,  1.5,  2.5,  3.5,  4.5,  5. ])

        >>> gradient(np.array([[1, 2, 6], [3, 4, 5]], dtype=float))
        [array([[ 2.,  2., -1.],
               [ 2.,  2., -1.]]), array([[ 1. ,  2.5,  4. ],
               [ 1. ,  1. ,  1. ]])]

        Notes
        -----
        This is a simplified implementation of gradient that is part of numpy
        1.8. In order to mitigate the effects of changes added to this
        implementation in version 1.9 of numpy, we include this implementation
        here.
        """
        f = np.asanyarray(f)
        N = len(f.shape)  # number of dimensions
        dx = [1.0] * N

        # use central differences on interior and first differences on endpoints
        outvals = []

        # create slice objects --- initially all are [:, :, ..., :]
        slice1 = [slice(None)] * N
        slice2 = [slice(None)] * N
        slice3 = [slice(None)] * N

        for axis in range(N):
            # select out appropriate parts for this dimension
            out = np.empty_like(f)
            slice1[axis] = slice(1, -1)
            slice2[axis] = slice(2, None)
            slice3[axis] = slice(None, -2)

            # 1D equivalent -- out[1:-1] = (f[2:] - f[:-2])/2.0
            out[tuple(slice1)] = (f[tuple(slice2)] - f[tuple(slice3)]) / 2.0
            slice1[axis] = 0
            slice2[axis] = 1
            slice3[axis] = 0
            # 1D equivalent -- out[0] = (f[1] - f[0])
            out[tuple(slice1)] = f[tuple(slice2)] - f[tuple(slice3)]
            slice1[axis] = -1
            slice2[axis] = -1
            slice3[axis] = -2
            # 1D equivalent -- out[-1] = (f[-1] - f[-2])
            out[tuple(slice1)] = f[tuple(slice2)] - f[tuple(slice3)]

            # divide by step size
            outvals.append(out / dx[axis])
            # reset the slice object in this dimension to ":"
            slice1[axis] = slice(None)
            slice2[axis] = slice(None)
            slice3[axis] = slice(None)

        if N == 1:
            return outvals[0]
        else:
            return outvals

    def streamline_gradients(streamline):
        """
        Calculate the gradients of the streamline along the spatial dimension

        Parameters
        ----------
        streamline : array-like of shape (n, 3)
            The 3d coordinates of a single streamline

        Returns
        -------
        Array of shape (3, n): Spatial gradients along the length of the
        streamline.

        """
        return np.array(gradient(np.asarray(streamline))[0])

    def grad_tensor(grad, evals):
        """
        Calculate the 3 by 3 tensor for a given spatial gradient, given a canonical
        tensor shape (also as a 3 by 3), pointing at [1,0,0]

        Parameters
        ----------
        grad : 1d array of shape (3,)
            The spatial gradient (e.g between two nodes of a streamline).

        evals: 1d array of shape (3,)
            The eigenvalues of a canonical tensor to be used as a response
            function.

        """
        # This is the rotation matrix from [1, 0, 0] to this gradient of the sl:
        R = la.svd([grad], overwrite_a=True)[2]
        # This is the 3 by 3 tensor after rotation:
        T = np.linalg.multi_dot([R, np.diag(evals), R.T])
        return T

    @warning_for_keywords()
    def streamline_tensors(streamline, *, evals=(0.001, 0, 0)):
        """
        The tensors generated by this fiber.

        Parameters
        ----------
        streamline : array-like of shape (n, 3)
            The 3d coordinates of a single streamline

        evals : iterable with three entries, optional
            The estimated eigenvalues of a single fiber tensor.

        Returns
        -------
        An n_nodes by 3 by 3 array with the tensor for each node in the fiber.

        Notes
        -----
        Estimates of the radial/axial diffusivities may rely on
        empirical measurements (for example, the AD in the Corpus Callosum), or
        may be based on a biophysical model of some kind.
        """

        grad = streamline_gradients(streamline)

        # Preallocate:
        tensors = np.empty((grad.shape[0], 3, 3))

        for grad_idx, this_grad in enumerate(grad):
            tensors[grad_idx] = grad_tensor(this_grad, evals)
        return tensors

    @warning_for_keywords()
    def streamline_signal(streamline, gtab, *, evals=(0.001, 0, 0)):
        """
        The signal from a single streamline estimate along each of its nodes.

        Parameters
        ----------
        streamline : a single streamline
            Streamline data.

        gtab : GradientTable class instance
            Gradient table.

        evals : array-like of length 3, optional
            The eigenvalues of the canonical tensor used as an estimate of the
            signal generated by each node of the streamline.
        """
        # Gotta have those tensors:
        tensors = streamline_tensors(streamline, evals=evals)
        sig = np.empty((len(streamline), np.sum(~gtab.b0s_mask)))
        # Extract them once:
        bvecs = gtab.bvecs[~gtab.b0s_mask]
        bvals = gtab.bvals[~gtab.b0s_mask]
        for ii, tensor in enumerate(tensors):
            ADC = np.diag(np.linalg.multi_dot([bvecs, tensor, bvecs.T]))
            # Use the Stejskal-Tanner equation with the ADC as input, and S0 = 1:
            sig[ii] = np.exp(-bvals * ADC)
        return sig - np.mean(sig)

    class LifeSignalMaker:
        """
        A class for generating signals from streamlines in an efficient and speedy
        manner.
        """

        @warning_for_keywords()
        def __init__(self, gtab, *, evals=(0.001, 0, 0), sphere=None):
            """
            Initialize a signal maker

            Parameters
            ----------
            gtab : GradientTable class instance
                The gradient table on which the signal is calculated.
            evals : array-like of 3 items, optional
                The eigenvalues of the canonical tensor to use in calculating the
                signal.
            sphere : `dipy.core.Sphere` class instance, optional
                The discrete sphere to use as an approximation for the continuous
                sphere on which the signal is represented. If integer - we will use
                an instance of one of the symmetric spheres cached in
                `dps.get_sphere`. If a 'dipy.core.Sphere' class instance is
                provided, we will use this object. Default: the :mod:`dipy.data`
                symmetric sphere with 724 vertices
            """
            self.sphere = sphere or dpd.default_sphere

            self.gtab = gtab
            self.evals = evals
            # Initialize an empty dict to fill with signals for each of the sphere
            # vertices:
            self.signal = np.empty((self.sphere.vertices.shape[0], np.sum(~gtab.b0s_mask)))
            # We'll need to keep track of what we've already calculated:
            self._calculated = []

        def calc_signal(self, xyz):
            idx = self.sphere.find_closest(xyz)
            if idx not in self._calculated:
                bvecs = self.gtab.bvecs[~self.gtab.b0s_mask]
                bvals = self.gtab.bvals[~self.gtab.b0s_mask]
                tensor = grad_tensor(self.sphere.vertices[idx], self.evals)
                ADC = np.diag(np.linalg.multi_dot([bvecs, tensor, bvecs.T]))
                sig = np.exp(-bvals * ADC)
                sig = sig - np.mean(sig)
                self.signal[idx] = sig
                self._calculated.append(idx)

            return self.signal[idx]

        def streamline_signal(self, streamline):
            """
            Approximate the signal for a given streamline
            """
            grad = streamline_gradients(streamline)
            sig_out = np.zeros((grad.shape[0], self.signal.shape[-1]))
            for ii, g in enumerate(grad):
                sig_out[ii] = self.calc_signal(g)
            return sig_out

    @warning_for_keywords()
    def voxel2streamline(streamline, affine, *, unique_idx=None):
        """
        Maps voxels to streamlines and streamlines to voxels, for setting up
        the LiFE equations matrix

        Parameters
        ----------
        streamline : list
            A collection of streamlines, each n by 3, with n being the number of
            nodes in the fiber.
        affine : array_like (4, 4)
            The mapping from voxel coordinates to streamline points.
            The voxel_to_rasmm matrix, typically from a NIFTI file.
        unique_idx : array, optional.
           The unique indices in the streamlines

        Returns
        -------
        v2f, v2fn : tuple of dicts

        The first dict in the tuple answers the question: Given a voxel (from
        the unique indices in this model), which fibers pass through it?

        The second answers the question: Given a streamline, for each voxel that
        this streamline passes through, which nodes of that streamline are in that
        voxel?
        """
        transformed_streamline = transform_streamlines(streamline, affine)

        if unique_idx is None:
            all_coords = np.concatenate(transformed_streamline)
            unique_idx = unique_rows(np.round(all_coords))

        return _voxel2streamline(transformed_streamline, unique_idx.astype(np.intp))

    class FiberModel(ReconstModel):
        """
        A class for representing and solving predictive models based on
        tractography solutions.

        Notes
        -----
        This is an implementation of the LiFE model described in
        :footcite:p:`Pestilli2014`.

        References
        ----------
        .. footbibliography::
        """

        def __init__(self, gtab):
            """
            Parameters
            ----------
            gtab : a GradientTable class instance

            """
            # Initialize the super-class:
            ReconstModel.__init__(self, gtab)

        @warning_for_keywords()
        def setup(self, streamline, affine, *, evals=(0.001, 0, 0), sphere=None):
            """
            Set up the necessary components for the LiFE model: the matrix of
            fiber-contributions to the DWI signal, and the coordinates of voxels
            for which the equations will be solved

            Parameters
            ----------
            streamline : list
                Streamlines, each is an array of shape (n, 3)
            affine : array_like (4, 4)
                The mapping from voxel coordinates to streamline points.
                The voxel_to_rasmm matrix, typically from a NIFTI file.
            evals : array-like (3 items, optional)
                The eigenvalues of the canonical tensor used as a response
                function. Default:[0.001, 0, 0].
            sphere: `dipy.core.Sphere` instance, optional
                Whether to approximate (and cache) the signal on a discrete
                sphere. This may confer a significant speed-up in setting up the
                problem, but is not as accurate. If `False`, we use the exact
                gradients along the streamlines to calculate the matrix, instead of
                an approximation. Defaults to use the 724-vertex symmetric sphere
                from :mod:`dipy.data`
            """
            if sphere is not False:
                SignalMaker = LifeSignalMaker(self.gtab, evals=evals, sphere=sphere)

            streamline = transform_streamlines(streamline, affine)
            # Assign some local variables, for shorthand:
            all_coords = np.concatenate(streamline)
            vox_coords = unique_rows(np.round(all_coords).astype(np.intp))
            del all_coords
            # We only consider the diffusion-weighted signals:
            n_bvecs = self.gtab.bvals[~self.gtab.b0s_mask].shape[0]
            v2f, v2fn = voxel2streamline(streamline, np.eye(4), unique_idx=vox_coords)
            # How many fibers in each voxel (this will determine how many
            # components are in the matrix):
            n_unique_f = len(np.hstack(list(v2f.values())))
            # Preallocate these, which will be used to generate the sparse
            # matrix:
            f_matrix_sig = np.zeros(n_unique_f * n_bvecs, dtype=float)
            f_matrix_row = np.zeros(n_unique_f * n_bvecs, dtype=np.intp)
            f_matrix_col = np.zeros(n_unique_f * n_bvecs, dtype=np.intp)

            fiber_signal = []
            for _, s in enumerate(streamline):
                if sphere is not False:
                    fiber_signal.append(SignalMaker.streamline_signal(s))
                else:
                    fiber_signal.append(streamline_signal(s, self.gtab, evals=evals))

            del streamline
            if sphere is not False:
                del SignalMaker

            keep_ct = 0
            range_bvecs = np.arange(n_bvecs).astype(int)
            # In each voxel:
            for v_idx in range(vox_coords.shape[0]):
                mat_row_idx = (range_bvecs + v_idx * n_bvecs).astype(np.intp)
                # For each fiber in that voxel:
                for f_idx in v2f[v_idx]:
                    # For each fiber-voxel combination, store the row/column
                    # indices in the pre-allocated linear arrays
                    f_matrix_row[keep_ct : keep_ct + n_bvecs] = mat_row_idx
                    f_matrix_col[keep_ct : keep_ct + n_bvecs] = f_idx

                    vox_fiber_sig = np.zeros(n_bvecs)
                    for node_idx in v2fn[f_idx][v_idx]:
                        # Sum the signal from each node of the fiber in that voxel:
                        vox_fiber_sig += fiber_signal[f_idx][node_idx]
                    # And add the summed thing into the corresponding rows:
                    f_matrix_sig[keep_ct : keep_ct + n_bvecs] += vox_fiber_sig
                    keep_ct = keep_ct + n_bvecs

            del v2f, v2fn
            # Allocate the sparse matrix, using the more memory-efficient 'csr'
            # format:
            life_matrix = sps.csr_array((f_matrix_sig, [f_matrix_row, f_matrix_col]))

            return life_matrix, vox_coords

        def _signals(self, data, vox_coords):
            """
            Helper function to extract and separate all the signals we need to fit
            and evaluate a fit of this model

            Parameters
            ----------
            data : 4D array

            vox_coords: n by 3 array
                The coordinates into the data array of the fiber nodes.
            """
            # Fitting is done on the S0-normalized-and-demeaned diffusion-weighted
            # signal:
            idx_tuple = (vox_coords[:, 0], vox_coords[:, 1], vox_coords[:, 2])
            # We'll look at a 2D array, extracting the data from the voxels:
            vox_data = data[idx_tuple]
            weighted_signal = vox_data[:, ~self.gtab.b0s_mask]
            b0_signal = np.mean(vox_data[:, self.gtab.b0s_mask], -1)
            relative_signal = weighted_signal / b0_signal[:, None]

            # The mean of the relative signal across directions in each voxel:
            mean_sig = np.mean(relative_signal, -1)
            to_fit = (relative_signal - mean_sig[:, None]).ravel()
            to_fit[np.isnan(to_fit)] = 0
            return (to_fit, weighted_signal, b0_signal, relative_signal, mean_sig, vox_data)

        @warning_for_keywords()
        def fit(self, data, streamline, affine, *, evals=(0.001, 0, 0), sphere=None):
            """
            Fit the LiFE FiberModel for data and a set of streamlines associated
            with this data

            Parameters
            ----------
            data : 4D array
                Diffusion-weighted data
            streamline : list
                A bunch of streamlines
            affine : array_like (4, 4)
                The mapping from voxel coordinates to streamline points.
                The voxel_to_rasmm matrix, typically from a NIFTI file.
            evals : array-like, optional
                The eigenvalues of the tensor response function used in
                constructing the model signal. Default: [0.001, 0, 0]
            sphere: `dipy.core.Sphere` instance or False, optional
                Whether to approximate (and cache) the signal on a discrete
                sphere. This may confer a significant speed-up in setting up the
                problem, but is not as accurate. If `False`, we use the exact
                gradients along the streamlines to calculate the matrix, instead of
                an approximation.

            Returns
            -------
            FiberFit class instance
            """
            if affine is None:
                affine = np.eye(4)
            sl_len = np.array([len(s) for s in streamline])
            if np.any(sl_len < 2):
                raise ValueError(
                    "Input contains streamlines with only one node."
                    " The LiFE model cannot be fit with these streamlines included."
                )
            life_matrix, vox_coords = self.setup(
                streamline, affine, evals=evals, sphere=sphere
            )
            (to_fit, weighted_signal, b0_signal, relative_signal, mean_sig, vox_data) = (
                self._signals(data, vox_coords)
            )
            beta = opt.sparse_nnls(to_fit, life_matrix)
            return FiberFit(
                self,
                life_matrix,
                vox_coords,
                to_fit,
                beta,
                weighted_signal,
                b0_signal,
                relative_signal,
                mean_sig,
                vox_data,
                streamline,
                affine,
                evals,
            )

    class FiberFit(ReconstFit):
        """
        A fit of the LiFE model to diffusion data
        """

        def __init__(
            self,
            fiber_model,
            life_matrix,
            vox_coords,
            to_fit,
            beta,
            weighted_signal,
            b0_signal,
            relative_signal,
            mean_sig,
            vox_data,
            streamline,
            affine,
            evals,
        ):
            """
            Parameters
            ----------
            fiber_model : A FiberModel class instance

            params : the parameters derived from a fit of the model to the data.

            """
            ReconstFit.__init__(self, fiber_model, vox_data)

            self.life_matrix = life_matrix
            self.vox_coords = vox_coords
            self.fit_data = to_fit
            self.beta = beta
            self.weighted_signal = weighted_signal
            self.b0_signal = b0_signal
            self.relative_signal = relative_signal
            self.mean_signal = mean_sig
            self.streamline = streamline
            self.affine = affine
            self.evals = evals

        @warning_for_keywords()
        def predict(self, *, gtab=None, S0=None):
            """
            Predict the signal

            Parameters
            ----------
            gtab : GradientTable
                Default: use self.gtab
            S0 : float or array
                The non-diffusion-weighted signal in the voxels for which a
                prediction is made. Default: use self.b0_signal

            Returns
            -------
            prediction : ndarray of shape (voxels, bvecs)
                An array with a prediction of the signal in each voxel/direction
            """
            # We generate the prediction and in each voxel, we add the
            # offset, according to the isotropic part of the signal, which was
            # removed prior to fitting:
            if gtab is None:
                _matrix = self.life_matrix
                gtab = self.model.gtab
            else:
                _model = FiberModel(gtab)
                _matrix, _ = _model.setup(self.streamline, self.affine, evals=self.evals)

            pred_weighted = np.reshape(
                opt.spdot(_matrix, self.beta),
                (self.vox_coords.shape[0], np.sum(~gtab.b0s_mask)),
            )

            pred = np.empty((self.vox_coords.shape[0], gtab.bvals.shape[0]))
            if S0 is None:
                S0 = self.b0_signal

            pred[..., gtab.b0s_mask] = S0[:, None]
            pred[..., ~gtab.b0s_mask] = (pred_weighted + self.mean_signal[:, None]) * S0[
                :, None
            ]

            return pred