File size: 20,171 Bytes
23e73f9
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
import os
import sys
import math
import random
import torch
import numpy as np
import torch.nn.functional as F
from argparse import ArgumentParser

from core.registry import register_method
from core.base_method import BaseMethod

sys.path.insert(0, '/root/autodl-tmp/BAGS_offy')
from utils.loss_utils import l1_loss, ssim
from gaussian_renderer import render as native_render
from scene import Scene, GaussianModel
from arguments import ModelParams, PipelineParams, OptimizationParams

def tv_loss(grids):
    number_of_grids = grids.shape[0]
    h_tv_count = grids[:, :, 1:, :].shape[1] * grids[:, :, 1:, :].shape[2] * grids[:, :, 1:, :].shape[3]
    w_tv_count = grids[:, :, :, 1:].shape[1] * grids[:, :, :, 1:].shape[2] * grids[:, :, :, 1:].shape[3]
    h_tv = torch.pow((grids[:, :, 1:, :] - grids[:, :, :-1, :]), 2).sum()
    w_tv = torch.pow((grids[:, :, :, 1:] - grids[:, :, :, :-1]), 2).sum()
    return 2 * (h_tv / h_tv_count + w_tv / w_tv_count) / number_of_grids

def get_emb(sin_inp):
    emb = torch.stack((sin_inp.sin(), sin_inp.cos()), dim=-1)
    return torch.flatten(emb, -2, -1)

def get_2d_emb(batch_size, x, y, out_ch, device):
    out_ch = int(np.ceil(out_ch / 4) * 2)
    inv_freq = 1.0 / (10000 ** (torch.arange(0, out_ch, 2).float() / out_ch))
    pos_x = torch.arange(x, device=device).type(inv_freq.type()) * 2 * np.pi / x
    pos_y = torch.arange(y, device=device).type(inv_freq.type()) * 2 * np.pi / y
    sin_inp_x = torch.einsum("i,j->ij", pos_x, inv_freq)
    sin_inp_y = torch.einsum("i,j->ij", pos_y, inv_freq)
    emb_x = get_emb(sin_inp_x).unsqueeze(1)
    emb_y = get_emb(sin_inp_y)
    emb = torch.zeros((x, y, out_ch * 2), device=device)
    emb[:, :, : out_ch] = emb_x
    emb[:, :, out_ch : 2 * out_ch] = emb_y
    return emb[None, :, :, :].repeat(batch_size, 1, 1, 1)

@register_method("bags")
class BAGSWrapper(BaseMethod):
    def __init__(self, dataset_config, hyperparams):
        self.parser = ArgumentParser()
        self.lp = ModelParams(self.parser)
        self.op = OptimizationParams(self.parser)
        self.pp = PipelineParams(self.parser)
        self.args = self.parser.parse_args([])

        self.args.source_path = dataset_config["source_path"]
        self.args.model_path = dataset_config["model_path"]
        self.args.eval = True
        self.args.resolution = dataset_config.get("resolution", 1)
        self.track_decoupling = hyperparams.get("track_decoupling", False)

        self.dataset = self.lp.extract(self.args)
        self.opt = self.op.extract(self.args)
        self.pipe = self.pp.extract(self.args)

        self.opt.iterations = 60000
        self.opt.ms_steps = 6000
        self.upsample_iter = [3000, 6000]

        # INJECTED_RES_FIX begin
        import sys as _sys
        _scene, _explicit_res = None, None
        for _i, _a in enumerate(_sys.argv[:-1]):
            _v = _sys.argv[_i + 1]
            if _a == "--scene":          _scene = _v
            elif _a == "--source_path":  _scene = _v.rstrip("/").split("/")[-1]
            elif _a == "--resolution":
                try: _explicit_res = int(_v)
                except: pass
        _OUTDOOR_360 = {"bicycle", "flowers", "garden", "stump", "treehill"}
        if _explicit_res is not None and _explicit_res > 0:
            _res = _explicit_res
        elif _scene is not None:
            _res = 4 if _scene in _OUTDOOR_360 else 2
        else:
            _res = None
        try:
            if _res is not None:
                self.dataset.resolution = _res
                print("[res-fix] scene=%s explicit=%s -> res=%s  (%s)" % (_scene, _explicit_res, _res, __file__))
        except Exception as _e:
            print("[res-fix] FAILED:", _e)
        # INJECTED_RES_FIX end
        self.scene = Scene(self.dataset)

        inp_shape = [len(self.scene.getTrainCameras()), int(np.round(self.scene.orig_h/self.args.resolution)), int(np.round(self.scene.orig_w/self.args.resolution))]

        self.gaussians = GaussianModel(
            self.dataset.sh_degree, inp_shape,
            ks1=self.dataset.kernel_size1, ks2=self.dataset.kernel_size2,
            ks3=self.dataset.kernel_size3, ks_ss=self.dataset.kernel_size_ss,
            not_use_rgbd=self.opt.not_use_rgbd, not_use_pe=self.opt.not_use_pe
        )
        
        self.scene.load_gaussian(self.gaussians)
        self.gaussians.training_setup(self.opt)

        bg_color = [1, 1, 1] if self.dataset.white_background else [0, 0, 0]
        self.background = torch.tensor(bg_color, dtype=torch.float32, device="cuda")

        self.trainCameras_scale4 = self.scene.getTrainCameras(scale=4.0).copy()
        self.trainCameras_scale2 = self.scene.getTrainCameras(scale=2.0).copy()
        self.trainCameras_scale1 = self.scene.getTrainCameras(scale=1.0).copy()
        
        self.viewpoint_stack = self.trainCameras_scale4.copy()
        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale4)

        self.unfold1 = torch.nn.Unfold(kernel_size=(self.dataset.kernel_size1, self.dataset.kernel_size1), padding=self.dataset.kernel_size1 // 2).cuda()
        self.unfold2 = torch.nn.Unfold(kernel_size=(self.dataset.kernel_size2, self.dataset.kernel_size2), padding=self.dataset.kernel_size2 // 2).cuda()
        self.unfold3 = torch.nn.Unfold(kernel_size=(self.dataset.kernel_size3, self.dataset.kernel_size3), padding=self.dataset.kernel_size3 // 2).cuda()
        if self.dataset.kernel_size_ss != self.dataset.kernel_size3:
            self.opt.use_another_mlp = True
            self.unfold_ss = torch.nn.Unfold(kernel_size=(self.dataset.kernel_size_ss, self.dataset.kernel_size_ss), padding=self.dataset.kernel_size_ss // 2).cuda()
        else:
            self.unfold_ss = self.unfold3

        self.last_n_gaussians = self.gaussians.get_xyz.shape[0]

    def train_iteration(self, step):
        ori_iter = step
        adj_iter = step
        
        if ori_iter == self.upsample_iter[0]:
            self.viewpoint_stack = self.trainCameras_scale2.copy()
            self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale2)
        elif ori_iter == self.upsample_iter[1]:
            self.viewpoint_stack = self.trainCameras_scale1.copy()
            self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale1)

        if ori_iter > self.opt.ms_steps:
            adj_iter = step - self.opt.ms_steps

        self.gaussians.update_learning_rate(adj_iter)
        if adj_iter % 1000 == 0:
            self.gaussians.oneupSHdegree()

        if not self.viewpoint_stack:
            if ori_iter >= self.upsample_iter[0] and ori_iter < self.upsample_iter[1]:
                self.viewpoint_stack = self.trainCameras_scale2.copy()
            elif ori_iter >= self.upsample_iter[1]:
                self.viewpoint_stack = self.trainCameras_scale1.copy()
            else:
                self.viewpoint_stack = self.trainCameras_scale4.copy()

        viewpoint_cam = self.viewpoint_stack.pop(random.randint(0, len(self.viewpoint_stack) - 1))

        subpixel_offset = None
        if self.dataset.ray_jitter:
            subpixel_offset = torch.rand((int(viewpoint_cam.image_height), int(viewpoint_cam.image_width), 2), dtype=torch.float32, device="cuda") - 0.5

        render_pkg = native_render(viewpoint_cam, self.gaussians, self.pipe, self.background, kernel_size=self.dataset.kernel_size, subpixel_offset=subpixel_offset)
        image, depth, viewspace_point_tensor, visibility_filter, radii = render_pkg["render"], render_pkg["depth"], render_pkg["viewspace_points"], render_pkg["visibility_filter"], render_pkg["radii"]

        gt_image = viewpoint_cam.original_image.cuda()
        loss_target = torch.tensor(0.0, device="cuda")
        loss_parasitic = torch.tensor(0.0, device="cuda")

        if adj_iter > 250:
            shuffle_rgb = image.unsqueeze(0)
            shuffle_depth = depth.unsqueeze(0) - depth.min()
            shuffle_depth = shuffle_depth / (shuffle_depth.max() + 1e-7)
            pos_enc = get_2d_emb(1, shuffle_rgb.shape[-2], shuffle_rgb.shape[-1], 16, torch.device("cuda"))

            if ori_iter < 3000:
                kernel_weights, mask = self.gaussians.mlp_rgb_ms(0, pos_enc, torch.cat([shuffle_rgb, shuffle_depth], 1).detach(), ori_iter)
                patches = self.unfold1(shuffle_rgb).view(1, 3, self.dataset.kernel_size1 ** 2, shuffle_rgb.shape[-2], shuffle_rgb.shape[-1])
            elif ori_iter >= 3000 and ori_iter < 6000:
                kernel_weights, mask = self.gaussians.mlp_rgb_ms(0, pos_enc, torch.cat([shuffle_rgb, shuffle_depth], 1).detach(), ori_iter)
                patches = self.unfold2(shuffle_rgb).view(1, 3, self.dataset.kernel_size2 ** 2, shuffle_rgb.shape[-2], shuffle_rgb.shape[-1])
            else:
                if (ori_iter > self.opt.ms_steps) and self.opt.use_another_mlp:
                    kernel_weights, mask = self.gaussians.mlp_rgb_ss(0, pos_enc, torch.cat([shuffle_rgb, shuffle_depth], 1).detach(), adj_iter)
                    patches = self.unfold_ss(shuffle_rgb).view(1, 3, self.dataset.kernel_size_ss ** 2, shuffle_rgb.shape[-2], shuffle_rgb.shape[-1])
                else:
                    kernel_weights, mask = self.gaussians.mlp_rgb_ms(0, pos_enc, torch.cat([shuffle_rgb, shuffle_depth], 1).detach(), ori_iter)
                    patches = self.unfold3(shuffle_rgb).view(1, 3, self.dataset.kernel_size3 ** 2, shuffle_rgb.shape[-2], shuffle_rgb.shape[-1])

            kernel_weights = kernel_weights.unsqueeze(1)
            rgb = torch.sum(patches * kernel_weights, 2)[0]
            mask = mask[0]
            blur_image = mask * rgb + (1 - mask) * image

            depthloss = self.opt.depth_loss_alpha * tv_loss(shuffle_depth) if self.opt.use_depth_loss else torch.tensor(0.0, device="cuda")
            maskloss = self.opt.mask_loss_alpha * mask.mean() if self.opt.use_mask_loss else torch.tensor(0.0, device="cuda")
            tvloss = self.opt.rgbtv_loss_alpha * tv_loss(shuffle_rgb) if self.opt.use_rgbtv_loss else torch.tensor(0.0, device="cuda")

            Ll1 = l1_loss(blur_image, gt_image)
            loss_target = (1.0 - self.opt.lambda_dssim) * Ll1
            loss_parasitic = self.opt.lambda_dssim * (1.0 - ssim(blur_image, gt_image)) + tvloss + maskloss + depthloss
        else:
            Ll1 = l1_loss(image, gt_image)
            loss_target = (1.0 - self.opt.lambda_dssim) * Ll1
            loss_parasitic = self.opt.lambda_dssim * (1.0 - ssim(image, gt_image))

        loss = loss_target + loss_parasitic

        grad_cos_sim = 0.0
        parasitic_ratio = 0.0
        metrics = {}

        if self.track_decoupling and step % 100 == 0:
            self.gaussians.optimizer.zero_grad(set_to_none=True)
            loss_target.backward(retain_graph=True)
            grad_target = self.gaussians._xyz.grad.clone() if self.gaussians._xyz.grad is not None else torch.zeros_like(self.gaussians._xyz)
            
            self.gaussians.optimizer.zero_grad(set_to_none=True)
            loss_parasitic.backward(retain_graph=True)
            grad_parasitic = self.gaussians._xyz.grad.clone() if self.gaussians._xyz.grad is not None else torch.zeros_like(self.gaussians._xyz)
            
            valid_mask = (torch.norm(grad_target, dim=1) > 0) & (torch.norm(grad_parasitic, dim=1) > 0)
            if valid_mask.any():
                grad_cos_sim = float(F.cosine_similarity(grad_target[valid_mask], grad_parasitic[valid_mask], dim=1).mean())
            parasitic_ratio = float(torch.norm(grad_parasitic, dim=1).mean() / (torch.norm(grad_target, dim=1).mean() + 1e-7))

            param_groups_map = {
                "spatial": [self.gaussians._xyz],
                "geometry": [self.gaussians._scaling, self.gaussians._rotation],
                "opacity": [self.gaussians._opacity],
                "appearance": [self.gaussians._features_dc, self.gaussians._features_rest],
            }

            self.gaussians.optimizer.zero_grad(set_to_none=True)
            loss_target.backward(retain_graph=True)
            grads_target = {}
            for group_name, params in param_groups_map.items():
                grads_target[group_name] = torch.cat([p.grad.clone().reshape(-1) for p in params if p.grad is not None])

            self.gaussians.optimizer.zero_grad(set_to_none=True)
            loss_parasitic.backward(retain_graph=True)
            grads_parasitic = {}
            for group_name, params in param_groups_map.items():
                grads_parasitic[group_name] = torch.cat([p.grad.clone().reshape(-1) for p in params if p.grad is not None])

            for group_name in param_groups_map:
                gt, gp = grads_target.get(group_name), grads_parasitic.get(group_name)
                if gt is not None and gp is not None and gt.norm() > 0 and gp.norm() > 0:
                    cos = float(F.cosine_similarity(gt.unsqueeze(0), gp.unsqueeze(0)))
                    r = float(gp.norm() / (gt.norm() + gp.norm() + 1e-7))
                    ti = r * max(0.0, -cos)
                else:
                    ti = 0.0
                metrics[f"sti_{group_name}"] = ti

            self.gaussians.optimizer.zero_grad(set_to_none=True)
            loss.backward()
        else:
            loss.backward()

        with torch.no_grad():
            if adj_iter < self.opt.densify_until_iter:
                self.gaussians.max_radii2D[visibility_filter] = torch.max(self.gaussians.max_radii2D[visibility_filter], radii[visibility_filter])
                self.gaussians.add_densification_stats(viewspace_point_tensor, visibility_filter)
                
                if adj_iter > self.opt.densify_from_iter and adj_iter % self.opt.densification_interval == 0:
                    size_threshold = 20 if adj_iter > self.opt.opacity_reset_interval else None
                    if ori_iter <= self.opt.ms_steps:
                        dgt = self.opt.init_dgt
                        min_op = self.opt.init_opacity if self.opt.init_opacity >= 0 else self.opt.min_opacity
                    else:
                        dgt = self.opt.densify_grad_threshold
                        min_op = self.opt.min_opacity
                    self.gaussians.densify_and_prune(dgt, min_op, self.scene.cameras_extent, size_threshold)
                    
                    if ori_iter >= self.upsample_iter[0] and ori_iter < self.upsample_iter[1]:
                        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale2)
                    elif ori_iter >= self.upsample_iter[1]:
                        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale1)
                    else:
                        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale4)

                if adj_iter % self.opt.opacity_reset_interval == 0 or (self.dataset.white_background and adj_iter == self.opt.densify_from_iter):
                    self.gaussians.reset_opacity()

            if adj_iter % 100 == 0 and adj_iter > self.opt.densify_until_iter:
                if adj_iter < (self.opt.iterations - self.opt.ms_steps) - 100:
                    if ori_iter >= self.upsample_iter[0] and ori_iter < self.upsample_iter[1]:
                        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale2)
                    elif ori_iter >= self.upsample_iter[1]:
                        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale1)
                    else:
                        self.gaussians.compute_3D_filter(cameras=self.trainCameras_scale4)

            if ori_iter < self.opt.iterations:
                self.gaussians.optimizer.step()
                self.gaussians.optimizer.zero_grad(set_to_none=True)

        num_gaussians = self.gaussians.get_xyz.shape[0]
        metrics.update({
            "loss": float(loss), "loss_l1": float(loss_target), "loss_ssim": float(loss_parasitic),
            "num_gaussians": int(num_gaussians), "delta_N": int(num_gaussians - self.last_n_gaussians),
            "peak_vram_GB": float(torch.cuda.max_memory_allocated() / (1024 ** 3)),
            "grad_cos_sim": float(grad_cos_sim), "parasitic_ratio": float(parasitic_ratio)
        })
        self.last_n_gaussians = num_gaussians
        
        histograms = {}
        if step % 1000 == 0:
            histograms["opacity"] = torch.sigmoid(self.gaussians._opacity).clone().detach()
            scales = torch.exp(self.gaussians._scaling).clone().detach()
            histograms["scaling"] = scales
            scales_2d = scales[:, :2] if scales.shape[1] >= 2 else scales
            gamma = scales_2d.max(dim=-1)[0] / (scales_2d.min(dim=-1)[0] + 1e-7)
            histograms["anisotropy"] = gamma
            histograms["sh_dc_mag"] = self.gaussians._features_dc.detach().norm(dim=-1)

        return metrics, histograms

    def render(self, camera):
        with torch.no_grad():
            render_pkg = native_render(camera, self.gaussians, self.pipe, self.background)
        return {"image": render_pkg["render"], "depth": render_pkg.get("depth", None)}

    def save(self, save_dir, step):
        self.scene.save(step)

    def load(self, model_path, iteration):
        self.gaussians.load_ply(os.path.join(model_path, 'point_cloud', f'iteration_{iteration}', 'point_cloud.ply'))

    def get_spatial_centers(self):
        return self.gaussians._xyz

    def compute_physical_metrics(self, cameras=None):
        metrics = {}
        with torch.no_grad():
            raw_scales = self.gaussians._scaling
            scales = torch.exp(raw_scales)
            scales_2d = scales[:, :2] if scales.dim() > 1 and scales.shape[1] >= 2 else scales.unsqueeze(-1).expand(-1, 2)
            
            max_S, _ = torch.max(scales_2d, dim=1)
            min_S, _ = torch.min(scales_2d, dim=1)
            gamma = max_S / (min_S + 1e-7)
            
            metrics["gamma_median"] = float(torch.median(gamma))
            metrics["gamma_90th_percentile"] = float(torch.quantile(gamma, 0.90))
            metrics["scale_mean"] = float(torch.mean(scales_2d))
            metrics["alpha_mean"] = float(torch.mean(torch.sigmoid(self.gaussians._opacity)))
            
            dc, rest = self.gaussians._features_dc, self.gaussians._features_rest
            if rest is not None and rest.shape[1] > 0:
                metrics["sh_energy_ratio"] = float(rest.norm(dim=-1).mean() / (dc.norm(dim=-1).mean() + 1e-7))
                
            if cameras is not None and len(cameras) > 0:
                view_dirs = []
                for c in cameras:
                    view_dirs.append(c.world_view_transform[:3, 2].tolist())
                view_dirs = F.normalize(torch.tensor(view_dirs, dtype=torch.float32, device="cuda"), dim=1)
                rots = F.normalize(self.gaussians._rotation.clone(), dim=1)
                w, x, y, z = rots.unbind(dim=-1)
                normals = F.normalize(torch.stack([2*(x*z + w*y), 2*(y*z - w*x), 1-2*(x*x + y*y)], dim=-1), dim=1)
                max_cos, _ = torch.max(torch.abs(torch.matmul(normals, view_dirs.T)), dim=1)
                metrics["billboard_bias_ratio"] = float((max_cos > 0.90).float().mean())
        return metrics

    def evaluate_spatial_field(self, query_points: torch.Tensor, cameras=None) -> torch.Tensor:
        with torch.no_grad():
            V = query_points.shape[0]
            densities = torch.zeros(V, device="cuda")
            xyz, opacities = self.gaussians._xyz, torch.sigmoid(self.gaussians._opacity).squeeze()
            scales = torch.exp(self.gaussians._scaling)
            sigma_sq = (scales[:, :2].max(dim=1)[0].pow(2)) if scales.shape[1] >= 2 else scales.squeeze().pow(2)
            
            N_gaussians = xyz.shape[0]
            chunk_size = max(1, 30_000_000 // (N_gaussians + 1))
            for i in range(0, V, chunk_size):
                end = min(i + chunk_size, V)
                dist_sq = torch.cdist(query_points[i:end], xyz, p=2).pow(2)
                weights = torch.exp(-0.5 * dist_sq / (sigma_sq.unsqueeze(0) + 1e-7))
                densities[i:end] = torch.sum(weights * opacities.unsqueeze(0), dim=1)
            return densities