Organizations: Shandong University, China · City University of Hong Kong, China · Hong Kong University of Science and Technology, China · The University of Hong Kong, China · Texas A&M University, USA
Implicit neural representations are powerful for geometric modeling, but their practical use is often limited by the high computational cost of network evaluations. We observe that implicit representations require progressively lower accuracy as query points move farther from the target surface, and that even within the same iso-surface, representation difficulty varies spatially with local geometric complexity. However, conventional neural implicit models evaluate all query points with the same network depth and computational cost, ignoring this spatial variation and thereby incurring substantial computational waste. Motivated by this observation, we propose an efficient neural implicit geometry representation framework with spatially adaptive network depth (SAND). SAND leverages a volumetric network-depth map together with a tailed multi-layer perceptron (T-MLP) to model implicit representation. The volumetric depth map records, for each spatial region, the network depth required to achieve sufficient accuracy, while the T-MLP is a modified MLP designed to learn implicit functions such as signed distance functions, where an output branch, referred to as a tail, is attached to each hidden layer. This design allows network evaluation to terminate adaptively without traversing the full network and directs computational resources to geometrically important and complex regions, improving efficiency while preserving high-fidelity representations. Extensive experimental results demonstrate that our approach can significantly improve the inference-time query speed of implicit neural representations.
Figures & tables
Figure 1. Unlike standard implicit neural representations (INRs) that evaluate all queries with a full network depth, SAND performs spatially adaptive evaluation. Each query is evaluated only up to the required depth determined by its spatial importance and representation complexity. This spatially adaptive strategy preserves geometry quality while substantially accelerating inference. Here, SIREN denotes an 8-layer network with 256 hidden units, SIREN* is a larger 8-layer 336-unit variant, and SAND-SIREN applies SAND to the 8-layer 256-unit SIREN backbone. Storage indicates model size, CD denotes Chamfer Distance, and Time measures the inference cost of querying the network during mesh extraction with Marching Cubes at a resolution of 512. The results show that applying SAND to SIREN significantly improves inference efficiency while providing high-fidelity geometric representation.
Figure 2. Overview of SAND. Given a query point x , SAND first performs a lookup in a volumetric network-depth map to determine the required evaluation depth d(x) at that spatial location. The query is then processed by a tailed multi-layer perceptron (T-MLP), which produces intermediate predictions at multiple depths. The network evaluation terminates at depth d(x) , and the accumulated output up to that depth yields the predicted signed distance. By adapting the evaluation depth per query, SAND avoids redundant computation in simple and empty regions without affecting geometric accuracy.
Figure 3. Overview of the T-MLP architecture. Here, the Polynomial Transformation corresponds to Eq. ( 3 ). Built on a standard MLP, the T-MLP attaches an output branch, also called a tail , after each hidden layer. The first tail produces a coarse approximation of the target function. The second tail learns the residual between the target and the first tail’s output. The third tail captures the residual between the target and the cumulative output of the first two tails. In general, the k -th tail models the residual between the target function and the sum of the outputs from the first k−1 tails.
Figure 4. Level of Detail. By varying the maximum network evaluation depth (1, 2, 3, 4), a single learned implicit representation produces surfaces of progressively increasing detail, corresponding to LOD1, LOD2, LOD3, and LOD4.
Method
Stanford
Thingi10K
Storage
Accuracy
Time (s)
Storage
Accuracy
Time (s)
(M)
CD ↓
F-Score ↑
NC ↑
Octree
Network
Total
(M)
CD ↓
F-Score ↑
NC ↑
Octree
Network
Total
FF ( Tancik et al., 2020 )
2.270
2.432
92.13
89.84
0.000
10.81
10.81 (28.6 × )
2.270
3.063
85.92
93.45
0.000
10.83
10.83 (26.9 × )
FF ∗ ( Tancik et al., 2020 )
3.899
1.651
94.69
95.51
0.000
15.02
15.02 (39.7 × )
3.899
1.834
89.75
97.02
0.000
15.18
15.18 (37.7 × )
SIREN ( Sitzmann et al., 2020 )
2.020
1.523
95.16
97.86
0.000
12.14
12.14 (32.1 × )
2.020
1.683
90.44
99.34
0.000
12.24
12.24 (30.4 × )
SIREN ∗ ( Sitzmann et al., 2020 )
3.470
1.513
95.23
97.98
0.000
16.81
16.81 (44.5 × )
3.470
1.680
90.48
99.37
0.000
16.76
16.76 (41.6 × )
Table 1. Quantitative comparisons for 3D shape representation. Total time includes both octree query and network inference. For methods reported with both starred and non-starred versions, the non-starred version uses an 8-layer MLP with 256 hidden units per layer, while the starred version uses an 8-layer MLP with 336 hidden units per layer.
Figure 5. Visual comparisons for 3D shape representation.
Figure 6. Distribution of network depth usage for shapes with different geometric complexity under our SAND framework.
Shape
Depth
Max.
Mean
Med.
Depth
Max.
Mean
Med.
( ×10−3 )
( ×10−4 )
( ×10−4 )
( ×10−3 )
( ×10−4 )
( ×10−4 )
Cube
1
16.78
69.23
69.02
2
5.360
0.815
0.384
Bust
1
21.88
51.10
43.52
2
15.98
15.58
10.32
Statuette
1
24.81
32.04
20.32
2
20.28
31.46
21.36
Cube
3
2.944
0.354
0.225
4
0.326
0.220
0.134
Bust
3
15.62
5.426
2.646
4
10.65
1.617
1.232
Table 2. Statistics of absolute residuals of T-MLP tails at different depths for shapes with varying geometric complexity.
Figure 7. Visual comparisons with Instant-NGP.
Method
Storage (MB)
CD ↓
F-score ↑
NC ↑
Time (s)
FLOPs
Instant-NGP 1
3.823
1.823
88.55
98.23
0.012
1.68×1012
Instant-NGP 2
5.187
1.801
89.02
98.77
0.012
1.73×1012
Instant-NGP 3
7.302
1.736
89.78
99.12
0.013
1.84×1012
SAND-SIREN
3.500
1.671
90.51
99.49
0.344
1.74×1012
SAND-FINER
3.511
1.671
90.52
99.50
0.403
1.76×1012
Table 3. Quantitative comparisons with Instant-NGP.
LoD
Method
Stanford
Thingi10K
CD ↓
F-Score ↑
NC ↑
CD ↓
F-Score ↑
NC ↑
LOD4
NGLOD
1.603
94.62
97.75
1.765
89.61
99.34
BACON ∗
1.618
93.96
96.98
1.709
90.25
99.29
BANF
1.870
89.09
94.82
4.417
74.13
96.19
SAND-SIREN
1.512
95.23
98.29
1.671
90.51
99.49
SAND-FINER
1.498
95.30
98.29
1.671
90.52
99.50
Table 4. Quantitative comparisons for 3D shape representation across multiple levels of detail (LODs).
Figure 8. Visual comparisons for 3D shape LOD representation.
Figure 9. Effect of the error threshold.
Max Depth
6
7
8
9
10
Time (s)
0.886
0.600
0.468
0.378
0.332
Storage (MB)
2.052
2.085
2.230
3.120
7.938
Table 5. Effect of the maximum octree depth.
Network
Storage (MB)
CD ↓
F-score ↑
NC ↑
Time (s)
SAND w/o Octree
2.041
1.521
95.19
97.89
13.67
Full SAND (Ours)
3.120
1.498
95.30
98.29
0.378
Table 6. Ablation study on the octree structure.
Network
CD ↓
F-score ↑
NC ↑
T-MLP w/o Residual Design
1.547
95.02
97.85
T-MLP w/o Mult.
1.509
95.26
98.08
T-MLP w/o Mult. + Scaler
1.505
95.26
98.15
Full T-MLP (Ours)
1.498
95.30
98.29
Table 7. Effect of the residual and multiplicative designs. “Mult.” denotes multiplicative interaction and “Scaler” refers to a learnable scaler modulation.