def sample_pdf(self, bins, weights, N_importance, det=False, eps=1e-5):
        """
        Sample @N_importance samples from @bins with distribution defined by @weights.
        Inputs:
            bins: (N_rays, N_samples_+1) where N_samples_ is "the number of coarse samples per ray - 2"
            weights: (N_rays, N_samples_)
            N_importance: the number of samples to draw from the distribution
            det: deterministic or not
            eps: a small number to prevent division by zero
        Outputs:
            samples: the sampled samples
        """
        N_rays, N_samples_ = weights.shape
        weights = weights + eps # prevent division by zero (don't do inplace op!)
        pdf = weights / torch.sum(weights, -1, keepdim=True) # (N_rays, N_samples_)
        cdf = torch.cumsum(pdf, -1) # (N_rays, N_samples), cumulative distribution function
        cdf = torch.cat([torch.zeros_like(cdf[: ,:1]), cdf], -1)  # (N_rays, N_samples_+1)
                                                                   # padded to 0~1 inclusive

        if det:
            u = torch.linspace(0, 1, N_importance, device=bins.device)
            u = u.expand(N_rays, N_importance)
        else:
            u = torch.rand(N_rays, N_importance, device=bins.device)
        u = u.contiguous()

        inds = torch.searchsorted(cdf, u, right=True)
        below = torch.clamp_min(inds-1, 0)
        above = torch.clamp_max(inds, N_samples_)

        inds_sampled = torch.stack([below, above], -1).view(N_rays, 2*N_importance)
        cdf_g = torch.gather(cdf, 1, inds_sampled).view(N_rays, N_importance, 2)
        bins_g = torch.gather(bins, 1, inds_sampled).view(N_rays, N_importance, 2)

        denom = cdf_g[...,1]-cdf_g[...,0]
        denom[denom<eps] = 1 # denom equals 0 means a bin has weight 0, in which case it will not be sampled
                             # anyway, therefore any value for it is fine (set to 1 here)

        samples = bins_g[...,0] + (u-cdf_g[...,0])/denom * (bins_g[...,1]-bins_g[...,0])
        return samples

        Sample @N_importance samples from @bins with distribution defined by @weights.
        Inputs:
            bins: (N_rays, N_samples_+1) where N_samples_ is "the number of coarse samples per ray - 2"
            weights: (N_rays, N_samples_)
            N_importance: the number of samples to draw from the distribution
            det: deterministic or not
            eps: a small number to prevent division by zero
        Outputs:
            samples: the sampled samples
        
training/volumetric_rendering/renderer.py