code: def approximate_conical_frustum_as_gaussians(
      bins: torch.Tensor, radii: torch.Tensor
  ) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
      """
      Approximates a conical frustum as two Gaussian distributions.
  
      The Gaussian distributions are characterized by
      three values:
  
      - rays_dir_mean: mean along the rays direction
          (defined as t in the parametric representation of a cone).
      - rays_dir_variance: the variance of the conical frustum  along the rays direction.
      - radii_variance: variance of the conical frustum with respect to its radius.
  
  
      The computation is stable and follows equation 7
      of `MIP-NeRF <https://arxiv.org/abs/2103.13415>`_.
  
      For more information on how the mean and variances are computed
      refers to the appendix of the paper.
  
      Args:
          bins: A tensor of shape `(..., num_points_per_ray + 1)`
              containing the bins at which the rays are sampled.
              `bin[..., t]` and `bin[..., t+1]` represent respectively
              the left and right coordinates of the interval.
          t0: A tensor of shape `(..., num_points_per_ray)`
              containing the left coordinates of the intervals
              on which the rays are sampled.
          t1: A tensor of shape `(..., num_points_per_ray)`
              containing the rights coordinates of the intervals
              on which the rays are sampled.
          radii: A tensor of shape `(..., 1)`
              base radii of the conical frustums.
  
      Returns:
          rays_dir_mean: A tensor of shape `(..., num_intervals)` representing
              the mean along the rays direction
              (t in the parametric represention of the cone)
          rays_dir_variance: A tensor of shape `(...,  num_intervals)` representing
              the variance of the conical frustum along the rays
              (t in the parametric represention of the cone).
          radii_variance: A tensor of shape `(..., num_intervals)` representing
              the variance of the conical frustum with respect to its radius.
      """
      t_mu = torch.lerp(bins[..., 1:], bins[..., :-1], 0.5)
      t_delta = torch.diff(bins, dim=-1) / 2
  
      t_mu_pow2 = torch.pow(t_mu, 2)
      t_delta_pow2 = torch.pow(t_delta, 2)
      t_delta_pow4 = torch.pow(t_delta, 4)
  
      den = 3 * t_mu_pow2 + t_delta_pow2
  
      # mean along the rays direction
      rays_dir_mean = t_mu + 2 * t_mu * t_delta_pow2 / den
  
      # Variance of the conical frustum with along the rays directions
      rays_dir_variance = t_delta_pow2 / 3 - (4 / 15) * (
          t_delta_pow4
          * (12 * t_mu_pow2 - t_delta_pow2)
          / torch.pow(den, 2)
      )
  
      # Variance of the conical frustum with respect to its radius
      radii_variance = torch.pow(radii, 2) * (
          t_mu_pow2 / 4
          + (5 / 12) * t_delta_pow2
          - 4 / 15 * (t_delta_pow4) / den
      )
      return rays_dir_mean, rays_dir_variance, radii_variance
  
imports: from __future__ import annotations
  from abc import ABC, abstractmethod
  from dataclasses import dataclass, field
  from enum import Enum
  from typing import Any, Dict, List, Optional, Tuple
  import torch
  from pytorch3d.implicitron.tools.config import ReplaceableBase
  from pytorch3d.ops import packed_to_padded
  from pytorch3d.renderer.implicit.utils import ray_bundle_variables_to_ray_points
label_desc: Use image node_id 2 for calling a global function with example usage: approximate_conical_frustum_as_gaussians(bins, radii) and returns: rays_dir_mean,  rays_dir_variance,  radii_variance