Appearance Compensation in Volume Rendering

Jianghong Zhou

cs.CVOct 10, 2026

Appearance Compensation in Volume Rendering

Authors: Jianghong Zhou

Organizations: ByteDance

Abstract

Can a volume that reproduces every external RGB view still have an incorrect density? We study this question in emission-absorption rendering with an independently parameterized, physically bounded effective source. Appearance compensates a density change by replacing the radiance that it removes from the rest of each ray. We characterize the surviving density information with a shared spatial appearance field. Our first result is an optimal density-only gain after finite-degree spherical-harmonic source refitting. A coupled tensor gauge and physical localized packets identify the sharp limiting nonlinear inverse coefficient as its reciprocal. Our second result realizes positive gain at every fixed finite degree for a residual set of isotropic spatial sources at one fixed density. At the same references, growing angular capacity hides a fixed nonzero density error while retaining physicality and any prescribed finite set of smoothness bounds. Local identifiability and the absence of a capacity-uniform recovery modulus therefore coexist. Our third result constructs an exact physical SH2 ambiguity: central density doubles and relative density $L^2$ error exceeds $14.52\%$ , yet every external RGB ray is unchanged. We connect these continuum results to shared finite representations, continuous pixel measurements and density-sensitive acquisition. Complete-class certificates separate SH0, SH1 and SH2 fits at one learned density. Finite-camera results expose the distinct role of spatial capacity, and transmission measurements separate otherwise identical RGB fields. Controlled training tests and captured-smoke profiles supply mechanism evidence with explicit observation scopes. The results show why rendering accuracy alone cannot certify density: recovery also depends on appearance capacity, spatial resolution and measurement design.

Figures & tables

Figure 1: Appearance compensation at fixed incorrect density. The density addition is unchanged across the appearance sequence; even $N=0$ includes a compensating source. Error crops share one linear scale and report sampled maxima. This is an affine-reference illustration, separate from the residual source class in Theorem 6.1 . The latter is established analytically by the source-realization argument.

ID	Type	Claim and information supplied	Statement / proof
Core continuum theory
C1	T	The compensated weighted great-circle gain is the optimal frozen density-only lower bound, including in any prescribed nonempty support ball.	Theorems 5.1 and B.1
C2	T	Positive gain gives local nonlinear density recovery with fixed exterior, small support, physical margins and the complete permitted finite-degree source class.	Theorems 5.2 and C.3
C3	T	Physical two-scale packets make $1/\kappa_{L}$ a necessary inverse and asymptotic deterministic minimax coefficient. Shrinking admissible neighborhoods attains it.	Theorems 5.3 , D and E.3
C4	T	At fixed smooth density, a residual class of physical isotropic spatial sources has positive gain at every point in a fixed interior compact set and every finite degree.	Theorems 6.1 and E.2
C5	T	At the same references, a fixed nonzero density addition is hidden in both $Y$ and maximum RGB by physical finite-degree sources as degree grows, with finite smoothness controls.	Theorems 6.1 , E.1 and E.4

Table 2

Figure 2: Same RGB, different response to a background change. Central density doubles while SH2 appearance preserves every black-background ray ( Theorem 7.2 ). White-background composites reveal the changed transmission with each source held fixed. Density and RGB panels share their respective scales.

Figure 3: RGB optimization learns appearance that conceals density error. (a) All $48$ random starts: open/filled markers use $100/400$ training views. (b) One prescribed SH2 checkpoint and unseen camera; density is identical in the three error maps, true-color replacement exposes density compensation, and DC-only color measures angular reliance ( Section R.3 ). The same density is certified in Table 2 . (c) Nonradial density slices at the first seed, with unseen-view PSNR and 3D density error. (d,e) Nonradial trajectories: means and full ranges of two completed seeds per arm. The true-color oracle supplies appearance during training. All image errors use independent test rays.

Figure 4: Color refitting flattens density-scale sensitivity in captured smoke. Left: held-out capture, fixed/refitted SH0 at $1.5\widehat{\sigma}$ , and predicted opacity change. Right: the complete scale sweep at the same time; fixed color is the SH0 fit at $s=1$ . The unit-color reference sets the brightness ceiling responsible for the low-scale decline ( Section S.5 ). Shared crops and intensity scales; $384\times 216$ evaluation rays. All three times and controls: Section S.5 .

Appearance class	Lower	Upper
All physical SH0	$1.1228$	$1.8820$
All physical SH1	$0.4418$	$0.9888$
All physical SH2	$0$	$0.4093$

Table 2: Certified capacity gaps at one learned density. The

32^{3}

density in Figure 3 b stays fixed. Bounds apply to the optimal maximum-RGB error over each complete physical appearance class on all training rays, in linear gray levels (

1/255

Appendix figures & tables38 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: The two limits in the analytic same-reference theorem. All drawings and curves are schematic, with separate scales and no measured rate. The fixed-degree neighborhood limit gives

C_{L}^{\sharp}(x_{0})=1/\kappa_{L}(x_{0})

. Increasing degree then hides a fixed density difference within selected finite derivative bounds. The lower note identifies the quadratic exact SH2 example as a separate reference. The numerical affine-source montage in Figure 1 illustrates R and is outside

\mathcal{G}

Component	$P$	$D$	$C$
Residual upper bound	$.046538$	$.038327$	$.021660$
Exact profile-error upper bound	$.049959$	$.044609$	$.027105$
Exact-source lower bound $j_{2,k}$			$.2062$
Exact absorption margin $\sigma_{2}-j_{2,k}$			$.25$
Nonzero degree-two coefficient $q_{C}(0)$			$.0531$

Appendix

Table 3: Outward bounds for the exact profiles relative to the analytic proxy at

s=1/2

\varepsilon=1

N=Q=32

. Upper values are rounded up and lower values down.

$\nu$	$\varepsilon$	$s$	$\tau_{*}$	$j_{2,k}>$	$\sigma_{2}-j_{2,k}\geq$	$C(0)>$
2	2	.3	4	.4489	.5	.0643
4	4	.2	8	.9749	1	.0435
8	8	.1	16	1.9991	2	.0153

Appendix

Table 4: Continuous certificates at higher reference optical depths.

\tau_{*}=2\nu

is the reference diameter optical thickness. Every row doubles central density; increasing thickness here also changes support radius.

$L$	Point 1	Point 2	Point 3
0	$.029774$	$.025616$	$.026539$
1	$.011857$	$.011341$	$.015706$
2	$.0052801$	$.0056389$	$.0063861$
3	$.0011850$	$.0011672$	$.0026505$
4	$.00038659$	$.00039738$	$.0010602$
5	$.00010679$	$.000080992$	$.00015611$

Appendix

Table 5: Estimated

\widehat{\kappa}_{L}

in the same nonradial scene. The three columns are the three points specified above; these are numerical diagnostics.

Figure 6: Nonradial density slice and the fixed-scene angular ladder. The three curves use the same density and isotropic spatial color at all degrees. The positive plotted values are finite-search estimates, not certified lower bounds.

$L$	Upper bound on $\kappa_{L}(0)$	Necessary inverse lower bound
0	$.029773685$	$33.5867$
1	$.011857206$	$84.3369$
2	$.005280051$	$189.3921$
3	$.001184962$	$843.9092$

Appendix

Table 6: Certified central-point bounds in the fixed nonradial scene. Values are rounded outwards. Only the stated inequality directions are asserted.

Scene	Views	Degree	Lower RMSE	Upper RMSE	Test PSNR
Quadratic	100	1	$0$	$2.721\cdot 10^{-5}$	$48.35$
Quadratic	100	2	$0$	$1.806\cdot 10^{-7}$	$97.47$
Quadratic	400	1	$.000351417764$	$9.262\cdot 10^{-4}$	$57.75$
Quadratic	400	2	$0$	$5.276\cdot 10^{-6}$	$99.28$
Cosine	100	1	$0$	$3.256\cdot 10^{-5}$	$44.52$
Cosine	100	2	$0$	$2.803\cdot 10^{-6}$	$68.94$

Appendix

Table 7: Twofold candidate: certified training bounds and independent evaluation. Lower bounds include exact rational dual rescaling; upper bounds retain the original feasible witnesses.

Scene	Views	SH1 lower	SH2 upper
Quadratic	100	$0$	$1.556610\cdot 10^{-6}$
Quadratic	400	$3.454608\cdot 10^{-4}$	$5.485578\cdot 10^{-6}$
Cosine	100	$0$	$7.063575\cdot 10^{-6}$
Cosine	400	$8.530096\cdot 10^{-4}$	$1.231659\cdot 10^{-4}$

Appendix

Table 8: Continuous integration of the fixed nodal field. SH1 entries are lower bounds, SH2 entries are feasible upper bounds. Displayed bounds are rounded outward.

Scene	Views	SH1 lower	SH2 feasible upper	SH1 lower minus one
Quadratic	100	$1.05081$	$0.00431$	$0.05081$
Cosine	100	$1.79127$	$0.01118$	$0.79127$
Quadratic	400	$1.56965$	$0.02054$	$0.56965$
Cosine	400	$2.53608$	$0.29079$	$1.53608$

Appendix

Table 9: Continuous elevenfold-candidate certificates. The final column is the additional target RMS perturbation that the lower bound can tolerate while keeping a rejection budget of one linear level. Values are rounded outwards.

Candidate	Views	SH1 lower	SH2 feasible upper
Twofold	100	$0.000$	$0.0004$
Twofold	400	$0.088$	$0.0014$
Elevenfold	100	$1.050$	$0.0044$
Elevenfold	400	$1.569$	$0.0206$

Appendix

Table 10: Continuous integration on the same candidate densities and rays. Values are RGB RMS in linear

0

–

255

units, rounded outwards. The declared rejection budget is

1

. A zero lower bound is inconclusive.

Scene	Views	$L$	Density err. (%)	Learned PSNR	True-color PSNR	$Q$	$Q_{\rm DC}$	Depth RMSE
Quadratic	100	0	11.9	58.87	50.86	2.52	1.00	0.0071
Quadratic	100	1	19.2	53.28	39.92	4.65	2.43	0.0175
Quadratic	100	2	30.0	51.98	34.22	7.73	5.31	0.0309
Quadratic	100	3	38.0	49.89	31.68	8.13	3.64	0.0403
Quadratic	400	0	11.3	60.78	51.45	2.93	1.00	0.0066
Quadratic	400	1	17.5	56.95	40.44	6.69	3.63	0.0163

Appendix

Table 11: All retained random-start groups, three seeds per row. PSNR uses only unknown-intersecting heldout rays.

Q

is RMSE after replacing learned color with true color divided by learned RMSE;

Q_{\rm DC}

instead removes all nonconstant SH terms. Depth error is in scene-radius units. Rows average per-run metrics.

Figure 7: Every trained group under field-function interventions. Lines average three seeds; narrow bands give the full seed range. Good unseen-view RGB coexists with incorrect density, and reverting appearance exposes a larger image error. Removing directional terms at fixed learned density also exposes error. The control shares the phenomenon, so this is evidence of appearance compensation in training, not of a quadratic-specific SH2 transition.

Scene	$L$	Full	True	DC	Full	True	DC	$\Delta_{\rm true}$
		True density: color-only fit			Learned density: joint fit
Quadratic	1	0.198	$<.00011$	1.918	0.364	2.455	1.308	2.289
Quadratic	2	0.319	$<.00011$	3.878	0.372	4.991	3.326	4.938
Quadratic	3	0.532	$<.00011$	2.803	0.533	6.561	2.756	6.560
Cosine	1	0.431	$<.00011$	4.851	1.014	10.970	6.404	10.387
Cosine	2	0.480	$<.00011$	7.556	0.561	12.560	8.900	12.479

Appendix

Table 12: All six matched controls. Errors use the same

7{,}936

unknown-intersecting rays from

32

unseen directions. Oracle density is exact; joint density is the original learned checkpoint. True-color oracle RMSE is below

.00011

in every row. Positive

\Delta_{\rm true}

separates the true-color intervention effects.

Appearance	PSNR (dB)	Max. RGB (levels)
True color	$34.17$	$10.9740$
Learned DC only	$37.69$	$9.3612$
Physical SH0 refit	$47.10$	$3.2156$
Physical SH1 refit	$52.90$	$2.0526$
Physical SH2 refit	$60.55$	$1.4589$
Retained trained SH2	$56.73$	$1.9069$

Appendix

Table 13: Independent unseen-view evaluation at the same fixed learned density. Each retained refit was selected solely on training rays. PSNR and maximum error are different objectives.

Figure 8: Predetermined illustration: quadratic scene, $400$ training views, seed $1101$ , first heldout camera; every $L$ is shown. Top: $z=0$ density slice inside the unknown ball, using a common $[0,1.5]$ scale; labels report three-dimensional density error. Remaining rows show per-pixel RGB RMS error after true-color replacement, directional-color removal, and full learned rendering. Labels give PSNR for the displayed view. RGB error uses a common $[0,10]$ range in linear $0$ – $255$ levels; neither color scale clips any displayed value. Density is unchanged between the three lower rows in each column.

Figure 9: Appearance compensation beyond radial density. Top: generating density and final SH1/SH3 density slices at $z=0$ , using the first declared seed and one color scale. Bottom: complete-run means and full seed ranges for unseen RGB, density error, and the image error revealed by restoring true color. The oracle supplies true appearance during training. Finite optimization is nonmonotone; these trajectories do not establish convergence or universal deterioration with SH degree.

Arm	Seed	PSNR	Density err.	Depth	Transmission	$Q$	$Q_{\rm DC}$	MSE ratio
SH1	5101	54.18	19.14	0.0320	0.0570	16.00	11.31	0.406
SH3	5101	45.09	18.70	0.0365	0.0458	4.89	6.43	2.050
Oracle color	5101	74.40	0.26	0.0004	0.0004	1.00	1.00	0.286
SH1	6202	54.26	19.30	0.0323	0.0578	16.32	11.47	0.451
SH3	6202	45.14	19.05	0.0366	0.0462	4.95	6.51	1.980
Oracle color	6202	74.65	0.24	0.0003	0.0003	1.00	1.00	0.277

Appendix

Table 14: Every completed nonradial run at step

6{,}000

. Density error is a percentage; depth is in scene-radius units;

Q

replaces learned color with true color and

Q_{\rm DC}

removes only directional coefficients. The final column is fixed training-monitor MSE at

6{,}000

divided by its value at

3{,}000

, not a convergence certificate.

Figure 10: Longer optimization separates density improvement from appearance compensation. Lines and bands show means and full ranges over all three seeds. SH1 density error decreases while its learned appearance continues to mask the remaining error. SH3 retains a larger density discrepancy. The true-color oracle supplies generating appearance during training. Curves use the prescribed checkpoints, with no endpoint selection.

Arm	Seed	PSNR	Density	Trans.	Depth	$Q$	$Q_{\rm DC}$	MSE ratio
SH1	5101	57.35	7.95	0.0322	0.0138	12.44	15.45	0.915
SH3	5101	46.46	19.34	0.0464	0.0362	4.34	7.17	0.957
Oracle color	5101	76.31	0.20	0.0003	0.0003	1.00	1.00	0.989
SH1	6202	57.14	8.13	0.0327	0.0141	12.37	15.19	0.969
SH3	6202	46.36	19.56	0.0463	0.0361	4.29	7.09	0.780
Oracle color	6202	75.64	0.21	0.0003	0.0003	1.00	1.00	1.131

Appendix

Table 15: Every run at

30{,}000

steps. Density is relative

L^{2}

error in percent; transmission and depth are RMS errors.

Q

replaces learned color by true color, and

Q_{\rm DC}

removes directional terms. The final column is monitor MSE at

30{,}000

divided by its value at

24{,}000

Frame	Density scale	Fixed color	SH0	SH1	SH2	SH0 foreground
40	1.0	29.45	29.92	30.11	30.08	19.70
40	1.5	26.15	31.14	30.82	30.64	20.93
60	1.0	29.14	29.60	29.74	29.72	21.14
60	1.5	25.20	30.29	30.03	29.92	21.72
80	1.0	28.38	28.71	28.87	28.85	21.24
80	1.5	24.49	29.45	29.16	29.15	21.88

Appendix

Table 16: Test camera PSNR (dB) for every prescribed time and density scaling. ROI covers rays through the calibrated volume. The final column restricts SH0 to observed foreground intensity

>.02

, exposing the contribution of dark background to ROI PSNR.

\dagger

denotes an unseen time, interpolated between fitted fields.

Frame	Density scale	Fixed color	SH0	SH1	SH2	SH0 foreground
40	1.0	28.52	29.09	29.19	29.17	19.07
40	1.5	25.67	30.53	30.29	30.05	20.65
60	1.0	27.92	28.51	28.56	28.53	20.26
60	1.5	24.65	29.48	29.23	29.09	21.27
80	1.0	27.44	27.92	28.00	27.99	20.55
80	1.5	24.06	28.87	28.66	28.60	21.50

Appendix

Table 17: Finer evaluation of the same frozen fields: test camera PSNR (dB), at

384\times 216

. The final column uses observed foreground intensity

>.02

\dagger

: time withheld from every fitting objective.

Figure 11: The three prescribed fitting times, all evaluated from held-out camera $3$ , using unchanged fields and finer rays. Columns show capture, unit-color and SH0 predictions, their absolute errors, and opacity change; all rows share the displayed scales. The middle capture retains native video pixels and the outer captures show the $384\times 216$ observations used for evaluation.

Figure 12: Density-scale sensitivity in all three captured-smoke frames. Top: held-out-camera PSNR after the five displayed color treatments. Fixed color is the SH0 field fitted at

s=1

; the reciprocal control is clipped only where it would exceed the physical box. Bottom: predicted opacity changes relative to

s=1

, independent of color. Markers are the six evaluated scales; connecting segments guide the eye.

Frame	$s$	Fixed	Reciprocal	SH0	SH1	SH2	FG
40	0.5	25.05	25.11	25.34	25.36	25.36	14.04
40	0.75	27.59	27.62	27.83	27.88	27.89	17.23
40	1	29.09	29.09	29.09	29.19	29.17	19.07
40	1.5	27.32	28.39	30.53	30.29	30.05	20.65
40	2	24.38	27.46	30.66	30.70	30.46	20.79
40	3	20.78	25.82	30.62	30.59	30.28	20.90

Appendix

Table 18: Every density scale on the held-out camera at finer resolution. Fixed and reciprocal colors use the saved

s=1

SH0 field. FG denotes SH0 PSNR on observed foreground intensity

>.02

. All values are dB; density scales are relative to a fitted reference.

		Test-camera PSNR (dB)					Validation gain (dB)
Frame	Edit	Fixed	Reciprocal	SH0	SH1	SH2	SH1	SH2
40	$1.25$	29.69	30.22	30.66	30.50	30.28	0.32	0.46
40	$x$ ramp	29.57	30.11	30.70	30.53	30.30	0.30	0.43
40	$y$ ramp	30.30	30.44	30.60	30.32	30.08	0.29	0.39
40	$z$ ramp	29.72	30.24	30.68	30.54	30.33	0.30	0.42
60	$1.25$	28.48	29.15	29.59	29.27	29.14	0.09	0.13

Appendix

Table 20: Spatial density edits separate fitting from transfer to a new camera. ROI PSNR on test camera

3

; every prescribed edit and time is shown. Fixed uses the reference SH0 color and Reciprocal divides it nodally by the multiplier. SH0–SH2 refit appearance at the same edited density. The last two columns give the improvement over SH0 on validation camera

1

. These are different held-out viewpoints of the same fitted field.

Frames	SH	Gain (%)	$G>0$ cameras	$255\overline{D}$	$255Q_{D}$
$40/60/80$	1	$+.35$	$3/5$	$.119$	$.391$
$40/60/80$	2	$-.77$	$3/5$	$.010$	$.388$
$100$	1	$+4.59$	$5/5$	$.256$	$.444$
$100$	2	$+4.04$	$5/5$	$.172$	$.440$

Appendix

Table 21: All registered cross-camera endpoints. Relative gain is positive when the angular refit improves mean held-out ROI RMSE. The last two columns give

255\overline{D}_{L}

and its integration-sensitivity threshold

255Q_{D}

. Neither degree passes in either stratum.

Property	Wang–Miller [ 42 ]	This work
Image formation	First-hit surface appearance	Distributed emission and exponential absorption
Geometric variable	Surface/proxy displacement	Shared three-dimensional volume density
Positive conclusion	Angular-capacity conditions separating true and misaligned surfaces	Sharp frozen density signal and local nonlinear stability
Exact ambiguity	Finite-SH billboard in a constructed viewing-circle model	Physical degree-two source compensation on every oriented external line
Quantitative object	Texture/parallax frequency and approximation error	Weighted great-circle residual; necessary and sufficient inverse coefficients
Empirical test	SH-capacity stress tests in 3DGS	Certified color fits and matched voxel optimization

Appendix

Table 22: Different inverse problems linked by constrained appearance fitting. Both works construct exact ambiguities within their stated observation and scene classes.

Figure 14: Layered-model diagnostic. Left: exact planar parallax with $k\Delta=8$ has strongly aperture-dependent tails. Middle: the opacity-weighted best fit tends to the first-hit best fit within the proven sandwich. Right: a hard cutoff occurs in the separate periodic linear-parallax model. These are unconstrained angular projections, hence lower bounds for a globally physical SH fit; they are not measured $\kappa_{L}$ values or training outcomes.

Normal	Frequency	$r=.02$	$r=.1$	$r=.3$
Weak	$12$	$.01287599$	$.01301846$	$.01419802$
Weak	$24$	$.01230651$	$.01246757$	$.01378320$
Axis	$12$	$.01449925$	$.01457301$	$.01523875$
Axis	$24$	$.01468110$	$.01474239$	$.01531347$

Appendix

Table 23: Actual-renderer

Y

gains after SH1 source refitting. The circle gains are

.01206016

(weak normal) and

.01477062

(axis). All primary cases are shown; these are numerical values, not interval certificates.

Figure 15: The retained constant-transmittance packet study. Directional and locality effects are separated from finite-Jacobian/nonlinear agreement. The sampling-refinement audit covers only the two

k=24,r=.02

cases, not the entire grid.

Normal	$T_{\min}$	$T_{\max}$	$T_{\max}/T_{\min}$	$\operatorname{std}(T)/\operatorname{mean}(T)$
$n_{p}$	0.21377	0.39626	1.854	0.210
$n_{x}$	0.21938	0.35897	1.636	0.162
$n_{z}$	0.21313	0.39464	1.852	0.189

Appendix

Table 24: Transport variation at the fixed point. All three normals were prescribed before evaluating new responses.

Normal	$L$	$g_{w}$	$g_{TU}$	$g_{\rm rms}$	Measured	Error (%)
$n_{p}$	0	0.038129	0.067975	0.039571	0.038116	0.03
$n_{p}$	1	0.013190	0.015472	0.012909	0.013504	2.33
$n_{p}$	2	0.005798	0.008147	0.005651	0.006445	10.04
$n_{p}$	3	0.001815	0.003949	0.001810	0.002630	31.00
$n_{x}$	0	0.035760	0.057131	0.035763	0.035874	0.32
$n_{x}$	1	0.013359	0.012648	0.013053	0.013576	1.60

Appendix

Table 25: Every primary small-support case,

r=.02

, in the

Y

norm. “Measured” is the actual finite-renderer derivative after refitting the prescribed source envelope. Errors are for

g_{w}

. No normal or degree is selected from these outcomes.

Figure 16: Transport weighting changes the prediction. Nonradial packets at

r=.02,k=24

. (a) The first prescribed normal, with all degrees retained. (b) SH1 relative errors against the actual renderer at all three normals. Colors match (a). The comparison uses

Y

; the refinement audit covers only the prior-normal and

x

-axis cases.

Figure 17: Two stages of compensation at one fixed density error. Top: unchanged source (gray), isotropic compensation (

N=0

), and angular refinement, including

N=3

. Left shows

Y

discrepancy; right shows numerical maximum RGB in linear gray levels. Bottom: sampled spatial, angular and mixed derivatives divided by their analytic all-degree bounds. All panels use the same affine reference and density addition ( Table 26 ).

Source	$Y$	Max. RGB	Gray levels	$\max\|\partial_{x}j\|$	$\max\|\Omega j\|$	$\max\|\partial_{x}\Omega j\|$
Unchanged $j$	$0.048601719$	$0.136901595$	$34.910$	$.15$	$0$	$0$
$N=0$	$0.003630594$	$0.014027883$	$3.577$	$24.02625$	$0.000000$	$0.00000$
$N=1$	$0.002452099$	$0.009697377$	$2.473$	$24.64066$	$0.067386$	$1.10920$
$N=2$	$0.001854743$	$0.007433374$	$1.896$	$25.21150$	$0.102371$	$1.67977$
$N=3$	$0.001492635$	$0.006033923$	$1.539$	$25.56058$	$0.123867$	$2.02840$
$N=4$	$0.001249327$	$0.005080907$	$1.296$	$25.79645$	$0.138435$	$2.26378$

Appendix

Table 26: Same reference and altered density, including the unchanged-source baseline. Maxima are numerical, including multistart ray refinement for RGB. Derivative entries are component maxima; complete mixed seminorms include lower orders.

Scene	Views	$L=1$ lower	$L=2$ upper
Quadratic	100	.004430018493	.000001185647
Quadratic	400	.006126135873	.000068440310
Control	100	.007309708730	.000022267172
Control	400	.009951989652	.001128158093

Appendix

Table 27: Integer-certified training RMSE bounds for the fixed alternative density. All displayed lower bounds are rounded down and upper bounds up. The control’s 100-view separation shows that this effect is not exclusive to the quadratic reference.

Scene	Views	$L$	PSNR (dB)	Density error	Depth error	$D$
Quadratic	100	0	$58.87\pm 0.04$	$0.119\pm 0.001$	$0.0071\pm 0.0000$	0.082
Quadratic	100	1	$53.28\pm 0.08$	$0.192\pm 0.001$	$0.0175\pm 0.0001$	0.085
Quadratic	100	2	$51.98\pm 0.02$	$0.300\pm 0.002$	$0.0309\pm 0.0001$	0.099
Quadratic	100	3	$49.88\pm 0.05$	$0.380\pm 0.002$	$0.0403\pm 0.0001$	0.097
Quadratic	400	0	$60.78\pm 0.05$	$0.113\pm 0.001$	$0.0066\pm 0.0000$	0.077
Quadratic	400	1	$56.95\pm 0.07$	$0.175\pm 0.001$	$0.0163\pm 0.0001$	0.074

Appendix

Table 28: All random-start cells. Depth is conditional absorption-depth RMSE in scene units. No seed is omitted.

Reference	Density edit	R	D	H
Trained NeRF Lego	Scale	1.216	1.189	1.106
	Bias	1.319	1.411	1.643
Released NeRF Fern	Scale	1.025	1.000	1.000
	Bias	1.433	1.469	1.472
Released NeRF Lego	Scale	.999	.942	1.000
	Bias	1.126	1.185	1.275

Appendix

Table 29: All finite native-head refits. Entries give frozen/refitted test RGB RMSE ratios in R/D/H order; values above one mean improvement. No arm meets the joint criterion of

5\times

reduction, reference agreement

\geq 35

dB, density change

\geq 20\%

and sampling consistency.

Inverse Rendering

Inverse rendering estimates scene geometry, materials, lighting, and other causes of images from observed visual or sensor data.

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Sep 4, 2026Florian Hahlbohm, Jorge Condor, Linus Franke +23D Scene Representation 3D Gaussian Splatting
TU Braunschweig · Universit`a della Svizzera italiana · University of Würzburg +1
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Dec 10, 2025Patrick Noras, Jun Myeong Choi, Didier Stricker +2Inverse Rendering 3D Gaussian Splatting
University of Kaiserslautern-Landau, Kaiserslautern, Germany · German Research Center for Artificial Intelligence, Kaiserslautern, Germany · University of North Carolina at Chapel Hill, NC, USA +1
Beyond Spherical Harmonics: Rethinking Appearance Models for Radiance Reconstruction
Jun 8, 2026Ewa Miazga, Jorge Condor, Piotr DidykNovel View Synthesis 3D Scene Representation
1École Polytechnique Fédérale de Lausanne, Switzerland · 2Università della Svizzera Italiana, Lugano, Switzerland

Sep 4, 2026J/K move · Enter open · S save

Compact Neural Appearance Models for Efficient Gaussian Splatting

Florian Hahlbohm, Jorge Condor, Linus Franke, Martin Eisemann+1

TU Braunschweig · Universit`a della Svizzera italiana · University of Würzburg+1

3D Scene Representation 3D Gaussian Splatting 3DGS Compression Spherical Harmonics

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Explicit primitive-based radiance fields such as 3D Gaussian Splatting typically model view-dependent appearance using low-order spherical harmonics (SH). Although efficient to evaluate, SH coefficients dominate per-primitive storage and memory traffic, while their band-limited basis restricts angular detail. We present a thorough, end-to-end comparison of SH and recent spherical appearance models and introduce an implicit alternative that decodes compact per-primitive latent codes using a tiny shared MLP. We integrate all models into the same optimized pipeline, fusing their forward and backward passes into a differentiable CUDA rasterizer and provide a portable WebGL viewer for laptop and mobile GPUs. Our evaluation across reconstruction quality, memory use, and optimization and rendering performance shows that recent spherical models offer the strongest overall quality-efficiency trade-off. Our neural representation is the most compact model evaluated and, compared to third-degree SH, reduces the per-primitive appearance footprint from 192 to 28 bytes, accelerates optimization by 1.3

\times

, while improving reconstruction quality. We further analyze how appearance parametrization shapes optimization, identifying differences in recovered geometry and the tendency of expressive models to absorb non-static scene content. Together, our framework and analysis provide practical guidance for replacing SH beyond what image metrics alone can capture.

Abstract

Inverse Rendering

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