text: """Module containing the functionalities for computing the Fundamental Matrix."""
  
  from typing import Tuple
  
  import torch
  
  import kornia
  
  
  def normalize_points(points: torch.Tensor, eps: float = 1e-8) -> Tuple[torch.Tensor, torch.Tensor]:
      r"""Normalizes points (isotropic).
  
      Computes the transformation matrix such that the two principal moments of the set of points
      are equal to unity, forming an approximately symmetric circular cloud of points of radius 1
      about the origin. Reference: Hartley/Zisserman 4.4.4 pag.107
  
      This operation is an essential step before applying the DLT algorithm in order to consider
      the result as optimal.
  
      Args:
         points: Tensor containing the points to be normalized with shape :math:`(B, N, 2)`.
         eps: epsilon value to avoid numerical instabilities.
  
      Returns:
         tuple containing the normalized points in the shape :math:`(B, N, 2)` and the transformation matrix
         in the shape :math:`(B, 3, 3)`.
  
      """
      if len(points.shape) != 3:
          raise AssertionError(points.shape)
      if points.shape[-1] != 2:
          raise AssertionError(points.shape)
  
      x_mean = torch.mean(points, dim=1, keepdim=True)  # Bx1x2
  
      scale = (points - x_mean).norm(dim=-1).mean(dim=-1)  # B
      scale = torch.sqrt(torch.tensor(2.0)) / (scale + eps)  # B
  
      ones, zeros = torch.ones_like(scale), torch.zeros_like(scale)
  
      transform = torch.stack(
          [scale, zeros, -scale * x_mean[..., 0, 0], zeros, scale, -scale * x_mean[..., 0, 1], zeros, zeros, ones], dim=-1
      )  # Bx9
  
      transform = transform.view(-1, 3, 3)  # Bx3x3
      points_norm = kornia.transform_points(transform, points)  # BxNx2
  
      return (points_norm, transform)
  
  
  def normalize_transformation(M: torch.Tensor, eps: float = 1e-8) -> torch.Tensor:
      r"""Normalize a given transformation matrix.
  
      The function trakes the transformation matrix and normalize so that the value in
      the last row and column is one.
  
      Args:
          M: The transformation to be normalized of any shape with a minimum size of 2x2.
          eps: small value to avoid unstabilities during the backpropagation.
  
      Returns:
          the normalized transformation matrix with same shape as the input.
  
      """
      if len(M.shape) < 2:
          raise AssertionError(M.shape)
      norm_val: torch.Tensor = M[..., -1:, -1:]
      return torch.where(norm_val.abs() > eps, M / (norm_val + eps), M)
  
  
  def find_fundamental(points1: torch.Tensor, points2: torch.Tensor, weights: torch.Tensor) -> torch.Tensor:
      r"""Compute the fundamental matrix using the DLT formulation.
  
      The linear system is solved by using the Weighted Least Squares Solution for the 8 Points algorithm.
  
      Args:
          points1: A set of points in the first image with a tensor shape :math:`(B, N, 2)`.
          points2: A set of points in the second image with a tensor shape :math:`(B, N, 2)`.
          weights: Tensor containing the weights per point correspondence with a shape of :math:`(B, N)`.
  
      Returns:
          the computed fundamental matrix with shape :math:`(B, 3, 3)`.
  
      """
      if points1.shape != points2.shape:
          raise AssertionError(points1.shape, points2.shape)
      if not (len(weights.shape) == 2 and weights.shape[1] == points1.shape[1]):
          raise AssertionError(weights.shape)
  
      points1_norm, transform1 = normalize_points(points1)
      points2_norm, transform2 = normalize_points(points2)
  
      x1, y1 = torch.chunk(points1_norm, dim=-1, chunks=2)  # Bx1xN
      x2, y2 = torch.chunk(points2_norm, dim=-1, chunks=2)  # Bx1xN
  
      ones = torch.ones_like(x1)
  
      # build equations system and solve DLT
      # https://www.cc.gatech.edu/~afb/classes/CS4495-Fall2013/slides/CS4495-09-TwoViews-2.pdf
      # [x * x', x * y', x, y * x', y * y', y, x', y', 1]
  
      X = torch.cat([x2 * x1, x2 * y1, x2, y2 * x1, y2 * y1, y2, x1, y1, ones], dim=-1)  # BxNx9
  
      # apply the weights to the linear system
      w_diag = torch.diag_embed(weights)
      X = X.transpose(-2, -1) @ w_diag @ X
  
      # compute eigevectors and retrieve the one with the smallest eigenvalue
      _, _, V = torch.svd(X)
      F_mat = V[..., -1].view(-1, 3, 3)
  
      # reconstruct and force the matrix to have rank2
      U, S, V = torch.svd(F_mat)
      rank_mask = torch.tensor([1.0, 1.0, 0]).to(F_mat.device)
  
      F_projected = U @ (torch.diag_embed(S * rank_mask) @ V.transpose(-2, -1))
      F_est = transform2.transpose(-2, -1) @ (F_projected @ transform1)
  
      return normalize_transformation(F_est)
  
  
  def compute_correspond_epilines(points: torch.Tensor, F_mat: torch.Tensor) -> torch.Tensor:
      r"""Compute the corresponding epipolar line for a given set of points.
  
      Args:
          points: tensor containing the set of points to project in the shape of :math:`(B, N, 2)`.
          F_mat: the fundamental to use for projection the points in the shape of :math:`(B, 3, 3)`.
  
      Returns:
          a tensor with shape :math:`(B, N, 3)` containing a vector of the epipolar
          lines corresponding to the points to the other image. Each line is described as
          :math:`ax + by + c = 0` and encoding the vectors as :math:`(a, b, c)`.
  
      """
      if not (len(points.shape) == 3 and points.shape[2] == 2):
          raise AssertionError(points.shape)
      if not (len(F_mat.shape) == 3 and F_mat.shape[-2:] == (3, 3)):
          raise AssertionError(F_mat.shape)
  
      points_h: torch.Tensor = kornia.convert_points_to_homogeneous(points)
  
      # project points and retrieve lines components
      a, b, c = torch.chunk(F_mat @ points_h.permute(0, 2, 1), dim=1, chunks=3)
  
      # compute normal and compose equation line
      nu: torch.Tensor = a * a + b * b
      nu = torch.where(nu > 0.0, 1.0 / torch.sqrt(nu), torch.ones_like(nu))
  
      line = torch.cat([a * nu, b * nu, c * nu], dim=1)  # Bx3xN
      return line.permute(0, 2, 1)  # BxNx3
  
  
  def fundamental_from_essential(E_mat: torch.Tensor, K1: torch.Tensor, K2: torch.Tensor) -> torch.Tensor:
      r"""Get the Fundamental matrix from Essential and camera matrices.
  
      Uses the method from Hartley/Zisserman 9.6 pag 257 (formula 9.12).
  
      Args:
          E_mat: The essential matrix with shape of :math:`(*, 3, 3)`.
          K1: The camera matrix from first camera with shape :math:`(*, 3, 3)`.
          K2: The camera matrix from second camera with shape :math:`(*, 3, 3)`.
  
      Returns:
          The fundamental matrix with shape :math:`(*, 3, 3)`.
  
      """
      if not (len(E_mat.shape) >= 2 and E_mat.shape[-2:] == (3, 3)):
          raise AssertionError(E_mat.shape)
      if not (len(K1.shape) >= 2 and K1.shape[-2:] == (3, 3)):
          raise AssertionError(K1.shape)
      if not (len(K2.shape) >= 2 and K2.shape[-2:] == (3, 3)):
          raise AssertionError(K2.shape)
      if not len(E_mat.shape[:-2]) == len(K1.shape[:-2]) == len(K2.shape[:-2]):
          raise AssertionError
  
      return K2.inverse().transpose(-2, -1) @ E_mat @ K1.inverse()
  
  
  # adapted from:
  # https://github.com/opencv/opencv_contrib/blob/master/modules/sfm/src/fundamental.cpp#L109
  # https://github.com/openMVG/openMVG/blob/160643be515007580086650f2ae7f1a42d32e9fb/src/openMVG/multiview/projection.cpp#L134
  
  
  def fundamental_from_projections(P1: torch.Tensor, P2: torch.Tensor) -> torch.Tensor:
      r"""Get the Fundamental matrix from Projection matrices.
  
      Args:
          P1: The projection matrix from first camera with shape :math:`(*, 3, 4)`.
          P2: The projection matrix from second camera with shape :math:`(*, 3, 4)`.
  
      Returns:
           The fundamental matrix with shape :math:`(*, 3, 3)`.
  
      """
      if not (len(P1.shape) >= 2 and P1.shape[-2:] == (3, 4)):
          raise AssertionError(P1.shape)
      if not (len(P2.shape) >= 2 and P2.shape[-2:] == (3, 4)):
          raise AssertionError(P2.shape)
      if P1.shape[:-2] != P2.shape[:-2]:
          raise AssertionError
  
      def vstack(x, y):
          return torch.cat([x, y], dim=-2)
  
      X1 = P1[..., 1:, :]
      X2 = vstack(P1[..., 2:3, :], P1[..., 0:1, :])
      X3 = P1[..., :2, :]
  
      Y1 = P2[..., 1:, :]
      Y2 = vstack(P2[..., 2:3, :], P2[..., 0:1, :])
      Y3 = P2[..., :2, :]
  
      X1Y1, X2Y1, X3Y1 = vstack(X1, Y1), vstack(X2, Y1), vstack(X3, Y1)
      X1Y2, X2Y2, X3Y2 = vstack(X1, Y2), vstack(X2, Y2), vstack(X3, Y2)
      X1Y3, X2Y3, X3Y3 = vstack(X1, Y3), vstack(X2, Y3), vstack(X3, Y3)
  
      F_vec = torch.cat(
          [
              X1Y1.det().reshape(-1, 1),
              X2Y1.det().reshape(-1, 1),
              X3Y1.det().reshape(-1, 1),
              X1Y2.det().reshape(-1, 1),
              X2Y2.det().reshape(-1, 1),
              X3Y2.det().reshape(-1, 1),
              X1Y3.det().reshape(-1, 1),
              X2Y3.det().reshape(-1, 1),
              X3Y3.det().reshape(-1, 1),
          ],
          dim=1,
      )
  
      return F_vec.view(*P1.shape[:-2], 3, 3)
  