No Cones, No Duals: A Fast and Scalable Solver for NRSfM
Authors: Suryansh Kumar
Organizations: Visual and Spatial AI Lab, VCCM Section, College of PVFA, Department of ECEN, Department of CSCE, Texas A&M University, College Station, Texas, USA
Non-rigid structure from motion (NRSfM) aims to recover camera motion and time-varying 3D shape from tracked 2D image measurements. Recent high-performing methods show that shape-space factorization remains highly effective for this task. These methods often use costly solvers for two key steps. First, a semidefinite programming (SDP) cone solver to perform Euclidean correction. Next, an alternating direction method of multipliers (ADMM) solver to recover the low-rank shape. These steps often scale poorly to long sequences and dense point tracks. We propose a fast, scalable NRSfM solver that avoids these costly steps. For rotation estimation, we introduce a differentiable Euclidean correction that directly optimizes small factors for each sequence. A row-polar projection then maps each estimated camera block to the closest row-orthonormal matrix. For shape estimation, we introduce an unrolled spectral-proximal solver. It combines reprojection-gradient steps with weighted singular-value shrinkage in the reshuffled shape domain. This solver further uses FISTA-style restart and continuation. The spectral shrinkage operator itself is classical, but we embed it in a fixed differentiable solver, while preserving NRSfM's geometric constraints. The proposed approach is category-agnostic, requires no external training data, and maps naturally to modern accelerator-friendly linear algebra primitives. Across sparse, dense, noisy, missing-data, and long-sequence benchmarks, our solver achieves substantial speedups with competitive reconstruction accuracy, including 7x-1000x runtime improvements on dense sequences with 28,887 tracked points.
Figures & tables
Figure 1: Accuracy-runtime and dense-sequence behavior. (a) Accuracy-runtime Pareto comparison on CMU MoCap [ 5 ] . Smaller, faded markers show individual sequences, while larger markers show method averages (lower left is better). The proposed solver achieves the fastest runtime while retaining competitive reconstruction accuracy relative to recent shape-space baselines. (b) On dense sequences [ 18 ] with 28,887 tracks, the proposed solver achieves large runtime gains over recent shape-space baselines such as TSM [ 57 ] and OPM [ 33 ] , while retaining competitive accuracy.
\Block [tikz=top color=orange!25]*-1 Stage
\Block [tikz=top color=lightgray!25]*-1 Current Solvers
\Block [tikz=top color=OliveGreen!25]*-1 Ours
Rotation
SDP/cone solver for Gram recovery
Factor-side Yk + row-polar projection
Shape
ADMM with auxiliary/dual variables
Unrolled layer, WSVT shrinkage, restart
Camera
Orthographic camera constraints
row-polar camera projection
Computation
Cone, ADMM subproblem machinery
Accelerator-friendly solver
Table 1: No-cone, no-dual positioning relative to recent high-performing shape-space NRS f M factorization pipelines.
Data
(F,P,K)
BMM [ 15 ]
R-BMM [ 32 ]
OPM [ 33 ]
TSM [ 57 ] (Rinit/Rref)
Ours
Drink
(1102, 41) K=11
0.0086 (10.01)
0.0240 (108.91)
0.0096 (143.54)
0.0086/ 1.3950 (88.94)
0.0060 (8.51)
Pickup
(357, 41) K=12
0.1210 (09.68)
0.1271 (96.24)
0.1144 (98.87)
0.1210/ 1.3197 (14.79)
0.1082 (7.79)
Yoga
(307, 41) K=11
0.0945 (08.08)
0.0944 (70.08)
0.0972 (76.88)
0.0945/ 0.1114 (10.48)
0.0996 (7.73)
Stretch
(370, 41) K=13
0.0682 (08.45)
0.0682 (73.32)
0.0688 (115.06)
0.0682/ 1.4116 (19.40)
0.0647 (7.83)
Table 2: eR and rotation-stage runtime comparison with baselines. All methods are re-simulated for given K . TSM uses two rotation estimates, i.e., BMM initialization Rinit and the refined rotation Rref after its refinement module . Our approach is considerably faster while achieving competitive or better rotation accuracy. The number in parentheses shows compute time in seconds.
Figure 2: Stable convergence of the proposed solver on CMU MoCap sequences. The data-fit values f(S) for R-BMM [ 32 ] , AOW [ 24 ] , and OPM [ 33 ] are normalized for visualization. Qualitative 3D reconstruction results obtained using our solver are also shown for representative sequences. The 4 curves correspond to data-fit values on 4 standard CMU sequences.
Data
BMM [ 15 ]
R-BMM [ 32 ]
OPM [ 33 ]
TSM [ 57 ]
Ours
Drink
0.0266 (61.22)
0.0119 (38.75)
0.0071 (36.08)
0.0033 (142.96)
0.0166 (06.45)
Pickup
0.1731 (26.95)
0.0198 (11.44)
0.0152 (10.34)
0.0126 (12.31)
0.0267 (04.88)
Yoga
0.1150 (12.87)
0.0129 (07.60)
0.0122 (07.51)
0.0118 (09.62)
0.0148 (04.19)
Stretch
0.1034 (15.28)
0.0144 (09.32)
0.0124 (08.73)
0.0131 (14.24)
0.0128 (04.21)
Dance
0.1864 (68.68)
0.1499 (12.82)
0.1209 (8.98)
0.0921 (11.39)
0.1218 (6.39)
Face
0.0303 (18.68)
0.0179 (7.97)
0.0145 (6.140)
0.0144 (14.09)
0.0179 (4.66)
Table 3: e3d and shape-stage runtime comparison with high-performance methods after rotations are fixed . Best K from the previous methods is used to compare e3d and time. Our approach is faster and its e3d compares favorably with competing methods. Numbers in parentheses denote computation time in seconds.
Figure 3: Qualitative results on NRS f M challenge dataset.
Data
CSF2 [ 22 ]
BMM [ 15 ]
R-BMM [ 32 ]
AOW [ 24 ]
BP [ 50 ]
OPM [ 33 ]
TSM [ 57 ]
Ours
Articulated
11.52
18.49
16.00
15.03
16.10
12.18
10.69
10.58
Balloon
10.14
10.39
7.84
8.05
8.29
6.29
7.28
5.86
Paper
9.72
8.94
10.69
10.45
6.70
8.86
7.91
10.42
Stretch
8.65
10.02
7.53
9.01
7.66
6.36
5.43
7.02
Tearing
12.04
14.23
16.34
16.20
11.26
10.91
10.77
10.61
Table 4: 3D reconstruction error comparison with state-of-the-art on [ 26 ] dataset.
Data
SMSR [ 7 ]
DS [ 16 ]
GM [ 37 ]
ND [ 58 ]
DV [ 18 ]
OPM [ 33 ]
TSM [ 57 ]
Ours
Seq.1
0.1893
0.0636
0.0443
-
0.0531
0.0624 (55.60)
0.2223 (10964.22)
0.0774 (8.29)
Seq.2
0.2133
0.0569
0.0381
-
0.0457
0.0451 (53.87)
0.1271 (10500.35)
0.0442 (8.18)
Seq.3
0.1345
0.0374
0.0294
0.0450 (3h+ train)
0.0346
0.0279 (100.72)
0.0393 (11624.89)
0.0382 (15.01)
Seq.4
0.0984
0.0428
0.0309
0.0490 (3h+ train)
0.0379
0.0419 (102.19)
0.0615 (11609.17)
0.0522 (15.16)
Table 5: e3d and overall run-time comparison with high-performing approaches on dense sequences [ 18 ] . Timings for other methods are unavailable due to the lack of open-source code. Observation: Our solver is faster than baselines: OPM and TSM.
Figure 4: (a) Overall compute time comparison with current SOTA approaches [ 57 , 33 ] on [ 26 ] dataset. Observation : Our approach compute time is significantly better as the number of frames and points increases (for e.g., Tearing seq. with F=405 , P=132 ). (b) Our approach compares favorably with other methods on missing dataset sequence.
Data
CSF2 [ 22 ]
BMM [ 15 ]
PND [ 42 ]
CNS [ 43 ]
R-BMM [ 32 ]
TSM [ 57 ]
Ours
Eating2
0.0923
0.0665
0.0701
0.0823
0.0633
0.0520
0.0651
Smoking1
0.1230
0.0844
0.1049
0.1073
0.0746
0.0738
0.0726
Directions
0.0787
0.0648
0.0536
0.0621
0.0595
0.0529
0.0643
Smoking
0.0702
0.0492
0.0622
0.0578
0.0363
0.0352
0.0337
Waiting2
0.1029
0.1081
0.1114
0.1142
0.1117
0.1035
0.1012
Table 6: e3d on One-circle-type camera motion dataset [ 70 ] .
Figure 5: Importance of each loss term included in Eq.( 8 ). Variation in eR as we add the three loss terms.
Figure 6: e3d variation w.r.t different noise levels in W .
Data
CSF1 [ 20 ]
CSF2 [ 22 ]
KSTA [ 21 ]
PTA [ 5 ]
NP [ 63 ]
PBezier [ 2 ]
PPTA [ 4 ]
Ours
Drink
0.0223
0.0223
0.0156
0.0250
0.0354
0.009
0.011
0.0166
Pickup
0.2301
0.2277
0.2322
0.2369
0.0284
0.138
0.235
0.0267
Yoga
0.1467
0.1464
0.1476
0.1624
0.0210
0.111
0.158
0.0148
Stretch
0.0710
0.0685
0.0674
0.1088
0.0151
0.062
0.084
0.0128
Dance
0.2705
0.1983
0.2504
0.2960
0.0980
0.143
0.229
0.1218
Face
0.0363
0.0314
0.0339
0.0436
0.0352
–
–
0.0179
Table 7: e3d comparison on MoCap datasets [ 5 , 61 ] with trajectory-space and trajectory-prior NRSfM methods. Piecewise Bézier [ 2 ] and PPTA [ 4 ] are included as efficient trajectory-prior alternatives. Best performance for each sequence is highlighted in bold. PBezier and PPTA results are reported from their published papers because public code is unavailable; “–” denotes unavailable results.
Data
CSF1 [ 20 ]
CSF2 [ 22 ]
PTA [ 5 ]
NP [ 63 ]
Ours
Face Seq.1
0.5325
0.4677
0.1559
-
0.0774
Face Seq.2
0.9266
0.7909
0.1503
-
0.0442
Face Seq.3
0.5274
0.5474
0.1252
0.0394
0.0382
Face Seq.4
0.5392
0.5292
0.1348
0.0574
0.0522
Table 8: e3d comparison on Dense Face Dataset with popular trajectory space [ 18 ] . Best performance for each sequence is highlighted in black.
Fixed-type camera motion
Data
CSF2 [ 22 ]
BMM [ 15 ]
PND [ 42 ]
CNS [ 43 ]
R-BMM [ 32 ]
TSM [ 57 ]
Ours
Eating2
0.2662
0.2084
0.2309
0.2250
0.1779
0.1767
0.1671
Smoking1
0.3534
0.2287
0.3397
0.2982
0.2290
0.2040
0.2181
Directions
0.2953
0.2872
0.3002
0.2674
0.2964
0.2688
0.2149
Smoking
0.1980
0.1759
0.4349
0.1968
0.1760
0.1693
0.1664
Waiting2
0.2125
0.1712
0.3306
0.1848
0.1478
0.1097
0.1161
Table 9: e3d comparison on H3WB [ 70 ] with Fixed-type camera motion. 2nd best result is underlined.
Multi-circle-type camera motion
Data
CSF2 [ 22 ]
BMM [ 15 ]
PND [ 42 ]
CNS [ 43 ]
R-BMM [ 32 ]
TSM [ 57 ]
Ours
Eating2
0.0658
0.0384
0.0395
0.0605
0.0335
0.0276
0.0333
Smoking1
0.0507
0.0486
0.0358
0.0662
0.0332
0.0294
0.0325
Directions
0.0456
0.0569
0.0322
0.0535
0.0518
0.0315
0.0438
Smoking
0.0708
0.0308
0.0292
0.0346
0.0328
0.0282
0.0273
Waiting2
0.0892
0.0742
0.0450
0.0873
0.0637
0.0390
0.0424
Table 10: e3d comparison on H3WB [ 70 ] with Multi-circle-type camera motion. 2nd best result is underlined.
Figure 7: Constraint-satisfaction analysis on CMU MoCap sequences without ground-truth rotations. For each frame, we report the row-orthonormality residual ∥RiRiT−I2∥F of the recovered 2×3 camera block after row-polar projection using Newton-Schulz inverse-square-root iterations. The plots show frame-wise residuals together with the sequence-level mean and maximum values. Across both the sequence, the residuals remain on the order of 10−7, confirming that the recovered camera blocks satisfy the orthographic Stiefel constraint to high numerical precision.
Methods
Subj. 07
Subj. 20
Subj. 23
Subj. 33
Subj. 34
Subj. 38
Subj. 39
Subj. 43
Subj. 93
C3DPO [ 49 ]
0.286
0.361
0.413
0.421
0.401
0.263
0.330
0.491
0.325
DNRSFM [ 29 ]
0.097
0.219
0.264
0.219
0.209
0.137
0.127
0.223
0.164
PR-PRN [ 66 ]
0.061
0.167
0.249
0.254
0.265
0.108
0.028
0.080
0.242
MHR-Net [ 67 ]
0.398
0.467
0.511
0.474
0.471
0.435
0.428
0.506
0.476
DNRSFM-R [ 17 ]
0.069
0.188
0.288
0.251
0.213
0.096
0.084
0.121
0.120
Table 11: e3d on large CMU MOCAP (UNSEEN categories). Brown shows 2nd best.
Data
K-value
BMM [ 15 ]
R-BMM [ 32 ]
OPM [ 33 ]
TSM [ 57 ]
Drink
11
0.0164 0.0266
0.0084 0.0119
0.0083 0.0071
0.0041 0.0033
Pickup
12
0.0321 0.1731
0.0164 0.0198
0.0155 0.0152
0.0128 0.0126
Yoga
11
0.0284 0.1150
0.0122 0.0129
0.0140 0.0122
0.0114 0.0118
Stretch
13
0.0294 0.1034
0.0126 0.0144
0.0116 0.0124
0.0139 0.0131
Table 12: Suitability of our rotation solver when used in other high-performance methods. Bold indicate improved e3d using our rotation estimates. For each method, Top row shows result using our rotation and Bottom row shows results reported by these methods using their rotation estimates.
Figure 8: e3d variation w.r.t different noise levels in W .
Figure 9: Detailed computational pipeline of the proposed no-cone, no-dual approach. The rotation stage performs factor-side Euclidean correction using learnable per-sequence factors Yk and row-polar projection to recover valid camera blocks. The shape stage applies a fixed unrolled spectral-proximal loop consisting of reprojection-gradient updates, reshaping to S♯ , weighted singular-value shrinkage, FISTA-style restart, and continuation. The main computations are matrix multiplications, SVD, shrinkage, and small row-polar projections.
Stage
Classical (SDP/ADMM)
Ours
Rotation estimation
O(K6) per sequence (SDP)
O(FK2)
Shape estimation (per iter.)
O(F(3P)2)+(3F)3P
O(FP2)ifF>3P,andO(F2P)if3P>F
Linear solves / duals
required
none
Hardware acceleration
NA (CPU-bound)
excellent
Scales with frames F
superlinear
Regime-dependent: linear in F , if F>3P else quadratic
Scales with points P
quadratic
Regime-dependent: linear in P , if 3P>F else quadratic
Table 13: Asymptotic complexity comparison between SOTA SDP and ADMM-based NRS f M and the proposed approach.
Dataset
Rotation time
Shape time
Total time
CMU MoCap Sequence
Drink
8.51
6.45
14.96
Pickup
7.79
4.88
12.67
Yoga
7.73
4.19
11.92
Stretch
7.83
4.21
12.04
Dance
8.57
6.39
14.96
Table 14: Stage-wise runtime of our solver on sparse MoCap and dense face benchmarks. Rotation time includes factor-side Euclidean correction and row-polar projection. Shape time includes the fixed-depth no-dual spectral-proximal solver after rotations are fixed. Total time is the combined solver runtime. All times are reported in seconds.
Sequence
K=10
K=11
K=12
K=13
Drink
0.0161
0.0166
0.0164
0.0153
Pickup
0.0272
0.0239
0.0267
0.0256
Yoga
0.0257
0.0148
0.0236
0.0207
Stretch
0.0122
0.0118
0.0119
0.0128
Table 15: Sensitivity of e3d to the shape-space rank K on CMU MoCap sequences. The underlined entries indicate the K values used in the main-paper comparison following prior shape-space baselines, while bold entries indicate the best result in the sweep. The results show that the proposed solver is stable across nearby ranks and is not dependent on aggressive per-sequence rank tuning.
η↓ /cap →
0.90
0.95
0.99
1.02
(0.0660, 0.0211)
(0.0108, 0.0200)
(0.0112, 0.0177)
1.05
(0.1194, 0.0271)
(0.0640, 0.0265)
(0.0168, 0.0267)
1.10
(0.1369, 0.0577)
(0.1437, 0.0592)
(0.1248, 0.0591)
Table 16: (e3dDrink,e3dPickup) with Npres=0
η↓ /cap →
0.90
0.95
0.99
1.02
(0.0510, 0.0233)
( 0.0108 , 0.0204)
(0.0111, 0.0176 )
1.05
(0.1219, 0.0280)
(0.0618, 0.0264)
( 0.0166 , 0.0267 )
1.10
(0.1360, 0.0698)
(0.1411, 0.0621)
(0.1234, 0.0590)
Table 17: (e3dDrink,e3dPickup) with Npres=1
η↓ /cap →
0.90
0.95
0.99
1.02
(0.0976, 0.0238)
(0.0108, 0.0224)
(0.0113, 0.0186)
1.05
(0.1496, 0.0261)
(0.1205, 0.0274)
(0.0249, 0.0268)
1.10
(0.1690, 0.0880)
(0.1658, 0.0820)
(0.1570, 0.0826)
Table 18: (e3dDrink,e3dPickup) with Npres=2
Sequence
Ours R
GT R
Gap
Drink
0.0166
0.0158
0.0008
Pickup
0.0267
0.0237
0.0030
Yoga
0.0148
0.0148
0.0000
Stretch
0.0127
0.0128
0.0001
Table 19: Pose-impact diagnostic on MoCap sequences with ground-truth rotations. We compare the 3d reconstruction results using estimated rotation and ground-truth rotation, respectively. The gap is computed as Gap=∣e3d(OursR)−e3d(GTR)∣ .
Figure 10: Qualitative results on real dense dataset sequence [ 18 ] .
Figure 11: Qualitative results on different categories present in H3WB dataset [ 57 ] .
Figure 12: Qualitative results on different categories in NRS f M challenge dataset [ 26 ] .
Novel view synthesis (NVS) approaches such as NeRFs or 3DGS can produce photo-realistic 3D scene representation from a set of images with known extrinsic and intrinsic parameters. The necessary camera poses and calibrations are typically obtained from the images via Structure-from-Motion (SfM). Classical SfM approaches rely on local feature matches between the images to estimate both the poses and a sparse 3D model of the scene, using bundle adjustment to refine initial pose, intrinsics, and geometry estimates. In order to increase run-time efficiency, recent SfM systems forgo optimization via bundle adjustment. Instead, they train feed-forward (transformer-based) neural networks to directly regress camera parameters and the 3D structure. While orders of magnitude more efficient, such recent works produce significantly less accurate estimates. To stimulate research on developing SfM approaches that are both efficient and effective, this paper develops a benchmark focused on SfM for novel view synthesis. Using existing datasets and two simple strategies for making the reconstruction process more efficient, we show that: (1) simply using fewer features already significantly accelerates classical SfM methods while maintaining high pose accuracy. (2) using feed-forward networks to obtain initial estimates and refining them using classical SfM techniques leads to the best efficiency-effectiveness trade-off. Our benchmark and code are publicly available at https://github.com/jhacsonmeza/sfm-benchmark-nvs.
Jhacson Meza, Martin R. Oswald, Torsten Sattler
Faculty of Electrical Engineering, Czech Technical University in Prague · Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague · University of Amsterdam
Recent 3D geometric foundation models, such as VGGT, provide robust feed-forward 3D reconstruction by directly predicting camera poses and 3D scene points from input images. However, their results remain inaccurate, and scaling them to long sequences or large unordered image sets typically requires chunk-wise processing, which can introduce drift and inconsistency. We present Glob3R, a global SfM-style reconstruction built on 3D foundation models. Our key idea is to explicitly optimize feed-forward geometric predictions. To this end, we augment a frozen Pi3X backbone with a lightweight dense matching head that predicts image warps between selected reference frames and neighboring views. These dense warps are converted into sparse but reliable multi-view feature tracks, which provide correspondence constraints for global optimization. We further introduce a keyframe-based sliding-window association strategy that propagates tracks and relative poses across overlapping windows, enabling scalable reconstruction. Finally, we perform global motion averaging and bundle adjustment to refine camera poses, reduce scale inconsistencies, and recover dense scene geometry. Extensive experiments on indoor, outdoor, large-scale driving, and unordered SfM benchmarks demonstrate that Glob3R achieves robust and accurate reconstruction. It consistently improves over feed-forward foundation-model baselines and recent scalable reconstruction methods, while being more robust than classical SfM pipelines. The refined poses also lead to higher-quality neural rendering, validating the benefit of combining foundation-model priors with global geometric optimization. Project page: https://junyuandeng.github.io/Glob3r
Junyuan Deng, Heng Li, Kejie Qiu +7
The Hong Kong University of Science and Technology · Tongyi Lab, Alibaba Group · Nanjing University +1
Recent monocular 3D face reconstruction methods achieve high fidelity by fitting a 3D Morphable Model (3DMM) to dense priors predicted by networks, but the optimization stage is computationally expensive, often taking tens of seconds per image. We present RealDenseFace, a real-time optimization-based 3D face reconstruction method with dense UV-space network predictions. Our key idea is to formulate 3DMM fitting as a nonlinear least-squares problem and solve it with a tailored Gauss-Newton solver that converges in only a few iterations. The reconstruction is conducted in two stages. In the first stage, the network predicts two dense UV-space maps from a single RGB image: a correspondence map for UV-to-image alignment, and a relative-depth map for geometric constraints along the viewing direction. In the second stage, the solver fits per-vertex targets sampled from these maps at the vertex UV coordinates. The solver supports all three reconstruction settings: single-image fitting, offline sequence reconstruction, and online tracking. Our method achieves state-of-the-art accuracy on the NeRSemble SVFR benchmark. The online tracker runs at 80+ FPS, and the offline sequence reconstruction is over 20 times faster than previous optimization-based baselines.