Gaussian Material Fields for Volumetric Multi-Energy CT Decomposition

Jian Lin; Jiancheng Fang; Hongming Shan; Shaoyu Wang; Yang Chen; Qiegen Liu

cs.CVOct 7, 2026

Gaussian Material Fields for Volumetric Multi-Energy CT Decomposition

Authors: Jian Lin, Jiancheng Fang, Hongming Shan, Shaoyu Wang, Yang Chen, Qiegen Liu

Organizations: Laboratory of Image Science and Technology, the School of Computer Science and Engineering, and the Key Laboratory of New Generation Artificial Intelligence Technology and Its Interdisciplinary Applications, Ministry of Education, Southeast University, Nanjing 210096, China

Abstract

Volumetric material decomposition in multi-energy computed tomography requires a representation that organizes multiple three-dimensional material fields in a common spatial domain while retaining differences in composition and local structure. We observe that spatial primitives can be shared across materials without tying their coefficients, but their local capacity must respond to material-specific reconstruction needs. We introduce Gaussian material fields, which represent multiple material distributions with shared anisotropic 3D Gaussian primitives and independent nonnegative material coefficients. The shared geometry defines a continuous spatial basis, while the coefficients determine each primitive's contribution to the individual material fields. To reconstruct this representation from multi-energy projections, a differentiable spectral forward model combines Gaussian material path integrals with a calibrated basis matrix, enabling joint optimization of spatial geometry and material composition. Material-aware adaptive density control retains material-specific refinement evidence before aggregation and adjusts local representation capacity to accommodate both spatially extensive components and sparse details. Experiments use synthesized multi-energy projections generated from pseudo-reference material maps constructed by conventional methods from publicly available CT data. Across 15 cases, our approach improves average PSNR by 4.03 dB and SSIM by 4.96% over the strongest baseline, while reducing NRMSE by 33.45%. Material-wise comparisons and component ablations support improved recovery of localized structures, while runtime and memory measurements show favorable computational scaling. These results establish Gaussian material fields as an explicit, adaptive representation for volumetric multi-material reconstruction.

Figures & tables

Fig. 1: Two organizations of volumetric material reconstruction. (a) A two-stage image-domain pipeline reconstructs energy-dependent attenuation volumes before voxel-wise material decomposition. (b) Gaussian material fields represent material composition on shared spatial primitives and optimize geometry and material-specific coefficients from multi-energy projections.

Fig. 2: (a) Reconstructed energy-dependent attenuation fields $V_{e}$ , corresponding basis-material coefficient fields $C_{m}$ , local error-magnitude visualizations, and quantitative summaries of error transfer across energies and materials. (b) Co-located basis-material coefficient patches and material-specific refinement evidence for a representative Gaussian $G_{i}$ , including naive material averaging and material-wise maximum aggregation.

Fig. 3: Gaussian material fields and their projection-domain optimization. Shared anisotropic primitives carry material-specific coefficients. X-ray projection forms material path integrals, and calibrated spectral mixing produces predicted energy projections. The projection loss updates geometry and coefficients jointly, while the optimized field provides the material outputs.

Fig. 4: Material-aware adaptive density control. Material-specific projection sensitivities are retained before aggregation and combined with Gaussian scale to guide cloning and splitting. Pruning tests material amplitudes across all channels.

Case	$E/M$	Detector	Volume
AAPM	$2/2$	$512^{2}$	$256^{3}$
Lizard head, mouse forelimb	$3/3$	$128^{2}$	$80^{3}$
Powder phantom	$4/4$	$128^{2}$	$80^{3}$
MUSIC 3D	$6/6$	$256^{2}$	$256^{3}$
Walnut (cubic)	$2/2$	$256^{2}$	$256^{3}$
Ore (cubic)	$2/2$	$128^{2}$	$80^{3}$

TABLE I: Simulated cases.

E/M

gives energy/material channels.

Method	L096	L109	L143	L192	L286
FDK+NNLS	23.58/0.3316/0.0705	18.33/0.2113/0.1279	17.10/0.1789/0.1489	19.90/0.2393/0.1065	18.01/0.1943/0.1329
ADJUST	31.15/0.8710/0.0370	26.68/0.8034/0.0586	24.15/0.6455/0.0850	27.20/0.7694/0.0528	25.20/0.6877/0.0692
NeRF+NNLS	31.10/0.7799/0.0294	27.66/0.6679/0.0434	28.29/0.6608/0.0409	29.28/0.7530/0.0359	28.56/0.6764/0.0393
CIL SIRT+NNLS	31.35/0.8079/0.0314	28.10/0.6754/0.0428	27.31/0.5913/0.0495	27.88/0.7211/0.0428	27.52/0.6415/0.0458
R2+NNLS	34.02 /0.8610/ 0.0216	31.32 /0.7987/ 0.0289	29.95/0.7538/ 0.0351	31.76 /0.8331/ 0.0274	31.29 /0.7959/ 0.0296
ASD-POCS+NNLS	33.60/ 0.9011 /0.0243	30.75/ 0.8340 /0.0319	30.17 / 0.8384 /0.0375	30.25/ 0.8698 /0.0328	30.19/ 0.8551 /0.0342

TABLE II: 50-view material-volume PSNR (dB)/SSIM/NRMSE on 15 cases.

Fig. 5: Matched 50-view slices. Rows, top to bottom: AAPM L506 calcification, lizard-head bone, mouse-forelimb background. Columns, left to right: GT, FDK, ADJUST, NeRF, CIL, R2, ASD-POCS, Ours. Main/ROI views are at left/upper right. Auxiliary structural differences are at lower right (red: missing, blue: extra). Window $[0,1]$ . Full-slice 2D PSNR (dB)/SSIM (%).

Fig. 6: 50-view material-slice comparisons. Rows show powder-phantom Al, powder-phantom Fe, and ore Pb, respectively. Enlarged regions highlight narrow powder features and the inner ore-Pb ring.

Fig. 7: 50-view material-slice comparisons. Rows show MUSIC 3D water-like, MUSIC 3D iodine-like, and walnut shell, respectively. Enlarged regions highlight compact material boundaries and the thin walnut-shell rim.

Fig. 8: Material-volume quality at 25/50/75 views: lizard head (left) and ore (right). Rings show PSNR, SSIM, and NRMSE

\times 10^{3}

from outside in. Symbols mark methods. Darker shading is better within each metric. Each setting uses 1 seed.

Lizard head
Setting	PSNR (dB)	SSIM	NRMSE ( $10^{-3}$ )
Full	33.91 $\pm$ 0.03	0.95813 $\pm$ 0.00021	20.368 $\pm$ 0.071
Fixed set	33.33 $\pm$ 0.01	0.95470 $\pm$ 0.00021	21.778 $\pm$ 0.027
Mean agg.	34.06 $\pm$ 0.04	0.96107 $\pm$ 0.00015	20.045 $\pm$ 0.083
Material $\ell_{1}$	34.30 $\pm$ 0.01	0.96396 $\pm$ 0.00002	19.538 $\pm$ 0.027
Energy $\ell_{1}$	29.66 $\pm$ 0.02	0.88741 $\pm$ 0.00031	34.538 $\pm$ 0.095

TABLE III: Material-volume mean

\pm

SD across 3 seeds. NRMSE

\times 10^{-3}

. Bold/underline mark best/second means.

Fig. 9: Matched 25-view slices: ore Pb and lizard-head bone, top to bottom. Columns: GT, R2, ASD-POCS, Ours.

Fig. 10: Per-material quality in 4 scene-specific energy/material configurations. Bars show PSNR (dB), dashed lines case means, shaded strips SSIM, and dots unscaled NRMSE. S/B denote soft tissue/bone, Ce/Zn denote CeO 2 /ZnO, and W/T denote water-like/tungsten-like. MUSIC uses a

256^{3}

grid, the others

80^{3}

Fig. 11: Ore quality versus log time at 50 views on RTX 5090 with method-specific budgets. Bubble area encodes maximum sampled process memory across repeats, not CUDA peak. The lower matrix gives SSIM, unscaled NRMSE, MiB, and repeat count. For

n=3

, time is the median and quality is the mean.

Fig. 12: Paired variant-minus-Full contrasts across 3 cases and seeds. Upper/lower maps show PSNR (dB)/SSIM differences. Blue is negative, ochre positive. Signed-asinh scales are centered at zero, with parameters 0.3 dB and 0.003 SSIM.

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$K$ -NeAS: Scalable Multi-Material CT Reconstruction Using Neural SDFs

Daksh K. Shah, Emmanouil Nikolakakis, Razvan Marinescu

University of California, Santa Cruz

Cone-Beam Computed Tomography Volume Continuous Distance Field Semantic Scene Understanding

Open paper AskPDF arXiv

Computed Tomography (CT) carries significant ionizing radiation risks, driving the need for sparse-view reconstruction. Implicit scene representations (ISRs) address this by recovering continuous volumetric attenuation fields directly from sparse projections, and recent geometry-aware extensions jointly model surface geometry alongside attenuation to improve fidelity and enable clean tissue segmentation without manual thresholding. However, these methods remain limited by manually tuned attenuation bounds and rigid two-material constraints. This paper proposes

K

-NeAS, a unified and scalable architecture for automated, multi-material surface reconstruction. We replace independent material networks with a shared latent backbone and introduce a fully differentiable

K

-material sequential soft selector to model an arbitrary number of overlapping tissues. To eliminate manual tuning, we automate attenuation bounding using a Gaussian Mixture Model (GMM) and implement a scheduled auxiliary floater loss to mitigate geometric hallucinations common under extreme sparsity. Evaluated across four clinical Cone-Beam CT (CBCT) datasets,

K

-NeAS successfully scales to arbitrary material counts, achieving superior 3D volumetric fidelity at

K=3

materials on complex multi-tissue regions such as the Abdomen (

33.28\text{ dB}

3D PSNR vs.

31.40\text{ dB}

single-material NeAS baseline, a

+1.88\text{ dB}

improvement). Furthermore, our model exhibits enhanced robustness under sparse-sampling conditions, outperforming baseline 3D PSNR by up to

1.17\text{ dB}

under 5- and 10-view constraints.

Abstract

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