"""
The `core` module provides foundational components for constructing variational implicit point set surfaces, as
described in [1]_.

This module includes functions and submodules for assembling interpolation matrices and solving the variational problem to generate implicit surfaces from unstructured point cloud data.


- **a_matrix**: Constructs the full interpolation matrix :math:`A` used in the variational formulation.
- **h_matrix**: Computes the :math:`H` matrix representing the bending energy minimization problem.
- **n_matrix**: Constructs the :math:`N` matrix incorporating polynomial terms for the interpolation.
- **m_matrix**: Constructs the :math:`M` matrix and its sub-matrices :math:`M_{00}`, :math:`M_{01}`, and :math:`M_{11}`, representing interactions between points and their derivatives.

Overview
--------

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

The `core` module provides the necessary components to set up and solve this variational problem. The matrices
constructed by the functions in this module are used to assemble the linear systems required to compute the
interpolation function.

Detailed Description
--------------------

a_matrix
--------

The :func:`a_matrix` function constructs the full interpolation matrix :math:`A`, which combines the effects of the
radial basis functions (RBFs) and polynomial terms. This matrix is essential in formulating the variational problem
and solving for the interpolation coefficients.

- **Purpose**: Assemble the matrix :math:`A` used in the interpolation equation.

- **Mathematical Formulation**:

  .. math::

        A = \\begin{bmatrix}
                M & N \\\\
                N^T & 0
            \\end{bmatrix}

  where :math:`M` is constructed from RBF interactions, and :math:`N` incorporates polynomial terms.

- **Key Functions Used**:
  - :func:`m_matrix` to compute :math:`M`.
  - :func:`n_matrix` to compute :math:`N`.

h_matrix
--------

The :func:`h_matrix` function computes the :math:`H` matrix, representing the bending energy minimization problem. This
matrix is positive semi-definite and derived from the inverse of the :math:`A` matrix. It incorporates a regularization
parameter :math:`\lambda` to control the smoothness of the resulting surface.

- **Purpose**: Compute the matrix :math:`H` used in minimizing the bending energy of the interpolated surface.

- **Mathematical Formulation**:

  .. math::

      H = J_{11} - \lambda J_{01}^\top (I + \lambda J_{00})^{-1} J_{01}

  where :math:`J` is a partition of :math:`A^{-1}`, and :math:`\lambda` is the regularization parameter.

- **Role of Regularization**:
  - Controls the trade-off between fitting the data points closely and smoothing the surface.
  - A higher :math:`\lambda` results in a smoother surface that may not pass exactly through the data points.

n_matrix
--------

The :func:`n_matrix` function constructs the :math:`N` matrix, incorporating polynomial terms into the interpolation.
This ensures that the interpolation function can represent affine terms and satisfies necessary smoothness conditions.

- **Purpose**: Assemble the matrix :math:`N` that adds polynomial terms to the interpolation.

- **Mathematical Formulation**:

  - For the first :math:`N` rows (function values):

    .. math::

        N_{i, j} = x_i^{(j)}, \quad j = 0, \dots, D-1

        N_{i, D} = 1

  - For the remaining :math:`D \times N` rows (derivative constraints):

    .. math::

        N_{N + (i \cdot D) + k, j} = \delta_{j, k}, \quad k = 0, \dots, D-1

  where :math:`x_i^{(j)}` is the :math:`j`-th coordinate of point :math:`i`, and :math:`\delta_{j, k}` is the
  Kronecker delta.

m_matrix
--------

The :func:`m_matrix` function constructs the :math:`M` matrix by assembling its sub-matrices :math:`M_{00}`,
:math:`M_{01}`, and :math:`M_{11}`. These sub-matrices represent the interactions between points (:math:`M_{00}`),
between points and their derivatives (:math:`M_{01}`), and between derivatives (:math:`M_{11}`).

- **Purpose**: Compute the matrix :math:`M` capturing RBF interactions in the interpolation.

- **Sub-matrices**:
    - :func:`m00`: Computes :math:`M_{00}`, representing point-to-point interactions.
    - :func:`m01`: Computes :math:`M_{01}`, representing point-to-derivative interactions.
    - :func:`m11`: Computes :math:`M_{11}`, representing derivative-to-derivative interactions.

- **Mathematical Formulation**:

  .. math::

        M = \\begin{bmatrix}
                M_{00} & M_{01} \\\\
                M_{01}^T & M_{11} + M_{11}^T
            \\end{bmatrix}

Relevant Literature
-------------------

The methods implemented in the `core` module are based on the variational approach to implicit surface reconstruction described in Huang et al. [1]_. The functions correspond to the mathematical constructs used in the formulation and solution of the variational problem.

- **Interpolation Matrix Construction**: The assembly of the :math:`A` matrix using :math:`M` and :math:`N` aligns with the formulation in the paper, where the interpolation problem is expressed as a linear system.

- **Energy Minimization**: The computation of the :math:`H` matrix facilitates the bending energy minimization, which is central to obtaining smooth surfaces that conform to the input data while maintaining smoothness.

- **Radial Basis Functions and Polynomials**: The use of RBFs in :math:`M` and the inclusion of polynomial terms in :math:`N` are key components in the variational formulation presented in the paper.

By using these components, the module enables the reconstruction of smooth implicit surfaces from point cloud data, leveraging the variational principles and mathematical formulations presented in the paper.

References
----------

.. [1] Huang, J., Zhang, C., Han, H., & Zwicker, M. (2019). Variational implicit point set surfaces. *ACM Transactions on Graphics (TOG)*, 38(4), 1-13. `ACM Digital Library <https://dl.acm.org/doi/10.1145/3306346.3322994>`_

Examples
--------

**Example:** Using the Core Module to Reconstruct a Surface from Point Cloud Data

.. code-block:: python

    import numpy as np
    from core import a_matrix, h_matrix, n_matrix, m_matrix
    from scipy.linalg import solve

    # Load or generate point cloud data
    points = np.random.rand(100, 3)  # Example: 100 points in 3D space

    # Define parameters
    rbf_degree = 3
    lam = 0.1  # Regularization parameter

    # Compute the matrices
    M = m_matrix(points, rbf_degree=rbf_degree)
    N = n_matrix(points)
    A = a_matrix(points, rbf_degree=rbf_degree)
    H, _, _, _, _ = h_matrix(A, points.shape[0], points.shape[1], lam)

    # Set up the right-hand side (e.g., for fitting scalar values at points)
    b = np.zeros(A.shape[0])
    # ... populate b with appropriate values ...

    # Solve for the interpolation coefficients
    coeffs = solve(A, b)

    # Use the coefficients to evaluate the interpolation function at desired locations
    # ... (implementation dependent on specific use case) ...

See Also
--------

  :func:`~svtoolkit.domain.a_matrix.a_matrix` : Constructs the full interpolation matrix :math:`A`.
  :func:`~svtoolkit.domain.h_matrix.h_matrix` : Computes the :math:`H` matrix for bending energy minimization.
  :func:`~svtoolkit.domain.n_matrix.n_matrix` : Constructs the :math:`N` matrix incorporating polynomial terms.
  :func:`~svtoolkit.domain.m_matrix.m_matrix` : Constructs the :math:`M` matrix from its sub-matrices.

"""
