ProJo4D

Sparse-view results on the Spring-Gaus Dataset

Metrics

	Method	mean	apple	banana	chess	cream	cross	paste	torus
CD ↓	Spring-Gaus	26.93	12.12	51.35	3.68	2.97	40.30	73.08	15.00
	GIC	16.11	2.13	8.37	7.51	8.16	2.51	81.24	2.81
	MASIV	2.77	1.02	2.23	5.15	2.42	4.75	2.33	1.51
	GIC + ProJo4D	1.60	0.19	0.12	1.37	1.54	0.38	6.93	0.65
EMD ↓	Spring-Gaus	0.178	0.170	0.223	0.097	0.101	0.232	0.248	0.177
	GIC	0.128	0.090	0.106	0.139	0.135	0.084	0.263	0.081
	MASIV	0.082	0.053	0.051	0.124	0.088	0.102	0.110	0.046
	GIC + ProJo4D	0.057	0.054	0.024	0.066	0.052	0.031	0.142	0.031
PSNR ↑	Spring-Gaus	13.78	17.03	15.79	13.85	14.62	11.24	10.94	13.01
	GIC	17.58	20.52	21.84	14.87	13.93	22.51	12.41	17.00
	MASIV	19.13	21.77	24.07	15.05	16.74	21.67	15.94	18.64
	GIC + ProJo4D	22.30	27.10	28.65	17.96	18.76	28.09	15.20	20.35
SSIM ↑	Spring-Gaus	0.799	0.790	0.825	0.792	0.796	0.819	0.737	0.831
	GIC	0.850	0.868	0.910	0.826	0.795	0.889	0.772	0.892
	MASIV	0.879	0.884	0.930	0.826	0.849	0.890	0.852	0.922
	GIC + ProJo4D	0.913	0.930	0.959	0.886	0.885	0.943	0.852	0.933
MAE log E ↓	GIC	0.2311	0.1840	0.4639	0.1807	0.0838	0.3239	0.1380	0.2436
MAE log E ↓	GIC + ProJo4D	0.1043	0.0633	0.1519	0.0326	0.0336	0.0469	0.1705	0.2315
MAE ν ↓	GIC	0.1790	0.1439	0.1049	0.0622	0.1407	0.0955	0.2209	0.4851
MAE ν ↓	GIC + ProJo4D	0.0911	0.0817	0.2237	0.0222	0.0295	0.0307	0.0928	0.1569

Visualizations

All videos are rendered from novel views and present the full 30-frame sequence. Each model was trained on the first 20 frames of each 30-frame sequence.

Ground Truth

ProJo4D (Ours)

GIC

Spring-Gaus

Sparse-view results on the PAC-NeRF Dataset

Metrics

	Method	Elasticity	Newtonian	Non-Newtonian	Plasticine	Sand
CD ↓	Spring-Gaus	64.076	—	—	—	—
	GIC	5.512	0.537	0.689	2.012	20.262
	ProJo4D	0.913	0.339	0.473	1.103	0.264
	Full Joint	1.318	0.346	8.104	17.678	53.564
EMD ↓	Spring-Gaus	0.267	—	—	—	—
	GIC	0.126	0.103	0.040	0.062	0.122
	ProJo4D	0.042	0.039	0.038	0.053	0.045
	Full Joint	0.049	0.040	0.099	0.124	0.223
MAE v₀ ↓	Spring-Gaus	0.292	—	—	—	—
	GIC	0.008	0.009	0.015	0.010	0.007
	ProJo4D	0.007	0.008	0.005	0.024	0.005
	Full Joint	0.020	0.008	0.080	0.092	0.046
MAE log(E) ↓	Spring-Gaus	—	—	—	—	—
	GIC	0.189	—	—	1.597	—
	ProJo4D	0.124	—	—	0.742	—
	Full Joint	0.216	—	—	2.856	—
MAE ν ↓	Spring-Gaus	—	—	—	—	—
	GIC	0.123	—	—	0.134	—
	ProJo4D	0.048	—	—	0.084	—
	Full Joint	0.061	—	—	0.075	—
MAE log(μ) ↓	Spring-Gaus	—	—	—	—	—
	GIC	—	0.103	0.869	—	—
	ProJo4D	—	0.134	0.491	—	—
	Full Joint	—	0.294	2.315	—	—
MAE log(κ) ↓	Spring-Gaus	—	—	—	—	—
	GIC	—	3.180	0.725	—	—
	ProJo4D	—	1.425	0.462	—	—
	Full Joint	—	3.312	1.673	—	—
MAE log(τ_Y) ↓	Spring-Gaus	—	—	—	—	—
	GIC	—	—	0.069	0.327	—
	ProJo4D	—	—	0.144	0.144	—
	Full Joint	—	—	1.839	6.612	—
MAE log(η) ↓	Spring-Gaus	—	—	—	—	—
	GIC	—	—	0.519	—	—
	ProJo4D	—	—	0.463	—	—
	Full Joint	—	—	1.455	—	—
MAE θ_fric ↓	Spring-Gaus	—	—	—	—	—
	GIC	—	—	—	—	6.785
	ProJo4D	—	—	—	—	4.998
	Full Joint	—	—	—	—	67.893

Mean values shown; standard deviations are reported in the paper.

Visualizations

All videos are rendered from novel views and present the full 14-frame sequence. Since this dataset is designed for physical parameter estimation, each model was trained on all 14 frames.

Ground Truth

ProJo4D (Ours)

GIC

Sand

Non-Newtonian

Plasticine

Results on the Spring-Gaus Real-world Dataset

Metrics

	Method	mean	bun	burger	dog	pig	potato
PSNR ↑	Spring-Gaus	30.63	26.79	35.13	30.31	31.95	28.96
	GIC	34.05	32.14	36.89	33.35	32.30	35.02
	GIC + ProJo4D	38.30	37.35	39.01	36.07	38.90	40.18
SSIM ↑	Spring-Gaus	0.991	0.986	0.995	0.993	0.994	0.989
	GIC	0.995	0.994	0.995	0.995	0.996	0.995
	GIC + ProJo4D	0.996	0.997	0.996	0.996	0.997	0.997