import numpy as np
from ..core.a_matrix import a_matrix
from ..core.h_matrix import h_matrix
from .cost import cost
from .coordinate_system import cart2sph


class Kernel:
    r"""
    Kernel class for variational implicit point set surfaces.

    This class defines the minimization kernel for the implicit domain object used in the construction of variational implicit point set surfaces, as described in Huang et al. [1]_.

    The `Kernel` class sets up the optimization problem for energy minimization of an implicit surface defined by unstructured point cloud data. It computes the necessary matrices and provides methods to evaluate the cost function, its gradient, and Hessian for optimization algorithms.

    Parameters
    ----------
    points : ndarray of shape (N, D)
        The input point cloud data defining the surface. Each row corresponds to a point in D-dimensional space.

    rbf_degree : int, optional
        The degree of the radial basis function used in constructing the interpolation matrix. Default is 3.

    lam : float, optional
        The regularization parameter (lambda) controlling the smoothness of the resulting surface. Default is 0.

    Attributes
    ----------
    n : int
        The number of points in the point cloud (N).

    d : int
        The dimensionality of the embedding space (D).

    a_ : ndarray of shape ((D+1)*N, (D+1)*N)
        The full interpolation matrix \( \mathbf{A} \) constructed using radial basis functions and polynomial terms.

    h_ : ndarray of shape (D*N, D*N)
        The \( \mathbf{H} \) matrix used in the energy minimization problem.

    a_inv : ndarray of shape ((D+1)*N, (D+1)*N)
        The inverse of the interpolation matrix \( \mathbf{A}^{-1} \).

    lam : float
        The regularization parameter controlling the smoothness of the surface.

    x0 : list of ndarray
        Initial values for the optimization variables.

    h0 : list of ndarray
        Corresponding \( \mathbf{H} \) matrices for different regularization parameters.

    Methods
    -------
    set_initial_values(*args, **kwargs)
        Set the initial values and parameters for the optimization problem.

    get_bounds()
        Get the bounds for the optimization variables.

    eval(x)
        Evaluate the cost function at a given point x.

    gradient(x)
        Evaluate the gradient of the cost function at a given point x.

    hessian(x)
        Evaluate the Hessian of the cost function at a given point x.

    Notes
    -----
    **Variational Implicit Point Set Surfaces**

    The variational implicit point set surfaces method aims to reconstruct smooth surfaces from point cloud data by formulating the problem as an energy minimization task. The key idea is to find an implicit function that minimizes an energy functional while interpolating or approximating the given data points. This approach leads to surfaces that are smooth and can handle noisy data gracefully.

    **Optimization Problem Setup**

    The Kernel class prepares the necessary components for setting up the optimization problem:

    - Constructs the interpolation matrix \( \mathbf{A} \) using radial basis functions.
    - Computes the \( \mathbf{H} \) matrix used in the bending energy minimization.
    - Provides methods to set initial values, define variable bounds, and evaluate the cost function and its derivatives.

    References
    ----------
    .. [1] Huang, J., Zhang, C., Han, H., & Zwicker, M. (2019). Variational implicit point set surfaces. *ACM Transactions on Graphics (TOG)*, 38(4), 1-14. https://doi.org/10.1145/3306346.3322994

    Examples
    --------
    **Example:** Setting up and using the Kernel class for surface reconstruction.

    .. code-block:: python

        import numpy as np
        from your_package.kernel import Kernel

        # Generate sample point cloud data
        points = np.random.rand(100, 3)  # 100 points in 3D space

        # Initialize the Kernel object
        kernel = Kernel(points, rbf_degree=3, lam=0.1)

        # Set initial values for optimization
        kernel.set_initial_values()

        # Get the bounds for the optimization variables
        bounds = kernel.get_bounds()

        # Define an optimization routine (e.g., using scipy.optimize)
        from scipy.optimize import minimize

        # Use the first initial guess and cost function
        x0 = kernel.x0[0]
        cost_function = kernel.__costs__[0]

        # Perform optimization
        result = minimize(cost_function, x0, bounds=bounds)

        # Extract optimized parameters
        optimized_params = result.x

        # Evaluate the cost at the optimized parameters
        final_cost = kernel.eval(optimized_params)
        print("Final cost:", final_cost)

    See Also
    --------
    a_matrix : Function to compute the full interpolation matrix \( \mathbf{A} \).
    h_matrix : Function to compute the \( \mathbf{H} \) matrix for bending energy minimization.
    cost : Function to compute the cost function value.
    cart2sph : Function to convert Cartesian coordinates to spherical coordinates.
    sph2cart : Function to convert spherical coordinates to Cartesian coordinates.

    """

    def __init__(self, points, rbf_degree=3, lam=0):
        """
        Initialize the Kernel object.

        Parameters
        ----------
        points : ndarray of shape (N, D)
            The input point cloud data defining the surface.

        rbf_degree : int, optional
            The degree of the radial basis function used in constructing the interpolation matrix. Default is 3.

        lam : float, optional
            The regularization parameter (lambda) controlling the smoothness of the resulting surface. Default is 0.

        """
        self.n = points.shape[0]
        self.d = points.shape[1]
        self.a_ = a_matrix(points, rbf_degree)
        self.h_, self.j00_, self.j01_, self.j11_, self.a_inv = h_matrix(self.a_, self.n, self.d, lam)
        self.__cost__ = lambda x: cost(x, self.h_, self.n, self.d)
        self.__costs__ = None
        self.__grad__ = None
        self.__hess__ = None
        self.x0 = None
        self.lam = lam
        self.h0 = None

    def set_initial_values(self, *args, **kwargs):
        r"""
        Set initial values and parameters for the optimization problem.

        This method prepares the initial guesses for the optimization variables and, if necessary, adjusts the regularization parameters.

        Parameters
        ----------
        *args : optional
            - If provided, the first positional argument should be an ndarray of normals corresponding to the points.

        **kwargs : optional
            - lambda_parameters : list or tuple of shape (3,), optional
                Parameters to linearly vary the thin-plate relaxation parameter (lambda). The list should contain:
                [minimum value, maximum value, number of values to generate].

                Default is [0.001, 1, 5].

        Returns
        -------
        None

        Notes
        -----
        **Usage Scenarios:**

        - **Using Normals as Initial Values:**

          If normals are provided as positional arguments, they are normalized and converted to spherical coordinates to serve as initial guesses.

        - **Varying Lambda Parameters:**

          If no normals are provided, the method will generate initial values by varying the regularization parameter lambda within the specified range.

        Examples
        --------
        **Example 1:** Using normals to set initial values.

        .. code-block:: python

            normals = np.random.rand(100, 3) - 0.5  # Random normals
            kernel.set_initial_values(normals)

        **Example 2:** Specifying lambda parameters.

        .. code-block:: python

            kernel.set_initial_values(lambda_parameters=[0.01, 0.5, 10])

        """
        if len(args) > 0:
            normals = args[0]
            normals = normals / np.linalg.norm(normals, axis=1).reshape(-1, 1)
            spherical_coordinates = cart2sph(normals)
            x0 = spherical_coordinates[:, 1:]
            h0 = [self.h_]
            x0 = [x0.flatten()]
            funcs = [lambda x: cost(x, self.h_, self.n, self.d)]
        else:
            lam_params = kwargs.get("lambda_parameters", [0.001, 1, 5])
            lams = np.linspace(self.lam + lam_params[0], self.lam + lam_params[1], lam_params[2])
            h0 = []
            x0 = []
            funcs = []
            for lam in lams:
                h_, _, _, _, _ = h_matrix(self.a_, self.n, self.d, lam)
                h0.append(h_)
                eigen_values, eigen_vectors = np.linalg.eig(h_)
                eh = np.argmin(eigen_values)
                x0_init = eigen_vectors[:, eh].real
                x0_init = x0_init.reshape(self.n, self.d)
                x0_init = x0_init / np.linalg.norm(x0_init, axis=1).reshape(-1, 1)
                x0_init = cart2sph(x0_init)
                x0_init = x0_init[:, 1:]
                x0_init = x0_init.flatten()
                x0.append(x0_init)
                func = lambda x: cost(x, h_, self.n, self.d)
                funcs.append(func)
        self.x0 = x0
        self.h0 = h0
        self.__costs__ = funcs
        return None

    def get_bounds(self):
        r"""
        Get the bounds for the optimization variables.

        Returns
        -------
        bounds : tuple of two lists
            A tuple containing two lists:
            - Lower bounds for each optimization variable.
            - Upper bounds for each optimization variable.

        Notes
        -----
        The bounds are set based on the ranges of the angular components in spherical coordinates:

        - For angles \( \phi_i \) where \( i < d - 1 \), the bounds are \( [0, \pi] \).
        - For the azimuthal angle \( \phi_{d-1} \), the bounds are \( [0, 2\pi] \).

        Examples
        --------
        .. code-block:: python

            bounds = kernel.get_bounds()
            # Use bounds in an optimization routine
            result = minimize(cost_function, x0, bounds=bounds)

        """
        bounds = []
        lb = []
        ub = []
        for i in range(self.n):
            for j in range(self.d - 1):
                if j < self.d - 2:
                    lb.append(0)
                    ub.append(np.pi)
                else:
                    lb.append(0)
                    ub.append(2 * np.pi)
        bounds.append(lb)
        bounds.append(ub)
        return tuple(bounds)

    def eval(self, x):
        r"""
        Evaluate the cost function at a given point.

        Parameters
        ----------
        x : ndarray
            The optimization variables (angular components) at which to evaluate the cost function.

        Returns
        -------
        cost_value : float
            The value of the cost function at the given point.

        Examples
        --------
        .. code-block:: python

            x = kernel.x0[0]  # Initial guess
            cost_value = kernel.eval(x)
            print("Cost at x:", cost_value)

        """
        return self.__cost__(x)

    def gradient(self, x):
        r"""
        Evaluate the gradient of the cost function at a given point.

        **Note:** This method is currently a placeholder and needs to be implemented.

        Parameters
        ----------
        x : ndarray
            The optimization variables at which to evaluate the gradient.

        Returns
        -------
        grad : ndarray
            The gradient vector of the cost function at the given point.

        """
        if self.__grad__ is not None:
            return self.__grad__(x)
        else:
            raise NotImplementedError("Gradient function is not implemented.")

    def hessian(self, x):
        r"""
        Evaluate the Hessian of the cost function at a given point.

        **Note:** This method is currently a placeholder and needs to be implemented.

        Parameters
        ----------
        x : ndarray
            The optimization variables at which to evaluate the Hessian.

        Returns
        -------
        hess : ndarray
            The Hessian matrix of the cost function at the given point.

        """
        return self.__hess__(x)
