def forward(self, feat_f0: Tensor, feat_f1: Tensor, data: dict[str, Any]) -> None:
        """Run forward.

        Args:
            feat_f0 (torch.Tensor): [M, WW, C]
            feat_f1 (torch.Tensor): [M, WW, C]
            data (dict)
        Update:
            data (dict):{
                'expec_f' (torch.Tensor): [M, 3],
                'mkpts0_f' (torch.Tensor): [M, 2],
                'mkpts1_f' (torch.Tensor): [M, 2]}

        """
        M, WW, C = feat_f0.shape
        W = int(math.sqrt(WW))
        scale = data["hw0_i"][0] / data["hw0_f"][0]
        self.M, self.W, self.WW, self.C, self.scale = M, W, WW, C, scale

        # corner case: if no coarse matches found
        if M == 0:
            if self.training:
                raise ValueError("M >0, when training, see coarse_matching.py")
            # logger.warning('No matches found in coarse-level.')
            data.update(
                {
                    "expec_f": torch.empty(0, 3, device=feat_f0.device, dtype=feat_f0.dtype),
                    "mkpts0_f": data["mkpts0_c"],
                    "mkpts1_f": data["mkpts1_c"],
                }
            )
            return

        feat_f0_picked = feat_f0[:, WW // 2, :]
        sim_matrix = torch.einsum("mc,mrc->mr", feat_f0_picked, feat_f1)
        softmax_temp = 1.0 / C**0.5
        heatmap = torch.softmax(softmax_temp * sim_matrix, dim=1).view(-1, W, W)

        # compute coordinates from heatmap
        coords_normalized = dsnt.spatial_expectation2d(heatmap[None], True)[0]  # [M, 2]
        grid_normalized = create_meshgrid(
            W, W, normalized_coordinates=True, device=heatmap.device, dtype=heatmap.dtype
        ).reshape(1, -1, 2)  # [1, WW, 2]

        # compute std over <x, y>
        var = torch.sum(grid_normalized**2 * heatmap.view(-1, WW, 1), dim=1) - coords_normalized**2  # [M, 2]
        std = torch.sum(torch.sqrt(torch.clamp(var, min=1e-10)), -1)  # [M]  clamp needed for numerical stability

        # for fine-level supervision
        data.update({"expec_f": torch.cat([coords_normalized, std.unsqueeze(1)], -1)})

        # compute absolute kpt coords
        self.get_fine_match(coords_normalized, data)
Run forward.

        Args:
            feat_f0 (torch.Tensor): [M, WW, C]
            feat_f1 (torch.Tensor): [M, WW, C]
            data (dict)
        Update:
            data (dict):{
                'expec_f' (torch.Tensor): [M, 3],
                'mkpts0_f' (torch.Tensor): [M, 2],
                'mkpts1_f' (torch.Tensor): [M, 2]}

        
kornia/feature/loftr/utils/fine_matching.py