Learned Adaptive Multiresolution Diffusion Imaging

Christian Tantardini; Stig Rune Jensen; Roberto Di Remigio Eikås; Joakim Henrik Beck

math.NAOct 6, 2026

Learned Adaptive Multiresolution Diffusion Imaging

Authors: Christian Tantardini, Stig Rune Jensen, Roberto Di Remigio Eikås, Joakim Henrik Beck

Organizations: Center for Integrative Petroleum Research, King Fahd University of Petroleum and Minerals, Dhahran 31261, Kingdom of Saudi Arabia · Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, N-9037 Tromsø, Norway · Algorithmiq S.r.l., c/o STLex, Via della Chiusa 15, 20123, Milano, Italia

Abstract

Adaptive multiresolution methods reduce representation cost by concentrating fine-scale degrees of freedom where needed, but their tree updates are usually governed by fixed local criteria. We introduce Learned Adaptive Multiresolution Diffusion Imaging (Learned AMDI), which preserves the AMDI fixed-tree propagator and hierarchy constraints while replacing the post-propagation selector with a shared local policy trained by proximal policy optimization. Regression tests reproduce deterministic AMDI trajectories to machine precision when identical trees are used. In the Haar implementation studied here, the deterministic one-step selector accepts no refinements in 54 decisions. Across nine held-out cases, Learned AMDI executes 393 refinements and reduces the mean terminal reference discrepancy from $0.17496$ to $0.13657$ , while occupancy rises from $0.13737$ to $0.26660$ . Step-resolved diagnostics reveal occasional small adaptation-energy increases; fixed-tree energy stability therefore does not guarantee monotonicity of the learned outer iteration. At comparable occupancy, a validation-tuned observed-detail threshold reaches a discrepancy of $0.13792$ with slightly better RMSE and SSIM, placing both methods on essentially the same accuracy--occupancy tradeoff. A decision-1-only control reaches $0.13742$ , indicating that most of the improvement on this static benchmark arises from the initial allocation. The shared actor transfers without retraining to $64\times64$ and $128\times128$ images, improving reference discrepancy, RMSE, and SSIM relative to deterministic AMDI, while the frozen threshold rule remains competitive. Learned AMDI thus provides a hierarchy-constrained, resolution-transferable mechanism for adaptive allocation and clarifies the contribution of sequential decisions.

Figures & tables

Method	$E_{\mathrm{ref}}$	$C_{\mathrm{rel}}$	$\overline{N}_{\mathrm{act}}$	RMSE	SSIM	$\overline{\sigma}_{\mathrm{sw}}$	$N_{\mathrm{ref}}/N_{\mathrm{crs}}$
Deterministic AMDI	0.17496	0.13737	140.67	0.05391	0.80851	0.00022	$0/4$
Retain initial tree	0.17494	0.13867	142.00	0.05392	0.80813	0	$0/0$
Top- $K$ , val.-return ( $K=32$ )	0.14442	0.23242	238.00	0.05286	0.80829	0.01563	$288/0$
Top- $K$ , val.-occupancy ( $K=36$ )	0.14175	0.24414	250.00	0.05290	0.80628	0.01758	$324/0$
Threshold, val.-return	0.14617	0.22331	228.67	0.05178	0.81162	0.01411	$260/0$
Threshold, val.-occupancy	0.13792	0.26042	266.67	0.05266	0.80196	0.02029	$374/0$

Table 1: Nine-case holdout comparison.

\overline{N}_{\mathrm{act}}

is the mean terminal number of active degrees of freedom.

N_{\mathrm{ref}}/N_{\mathrm{crs}}

gives the total number of executed refined cells and coarsened parent groups, respectively, over nine cases and six decisions. “Val.-return” and “val.-occupancy” denote parameters selected exclusively on the validation images. All terminal fields are clipped to

[0,1]

before RMSE and SSIM are computed.

Decision	AMDI R/F/C	Proposed R/C	Executed R/C	$\overline{N}_{\mathrm{act}}$	Energy up
1	$8/0/1$	$383/0$	$383/0$	269.67	$9/9$
2	$9/0/0$	$10/0$	$10/0$	273.00	$6/9$
3	$9/0/0$	$0/0$	$0/0$	273.00	$0/9$
4	$9/0/0$	$0/5$	$0/0$	273.00	$0/9$
5	$9/0/0$	$0/10$	$0/0$	273.00	$0/9$
6	$9/0/0$	$0/30$	$0/0$	273.00	$0/9$

Table 2: Nine-case action and energy analysis by adaptation decision. Here R/F/C denotes retain/refine/coarsen for deterministic AMDI, whereas R/C denotes learned refinement/coarsening counts. Learned proposals and executions are totals over the nine cases. “Energy up” reports the number of cases in which the post-adaptation energy exceeds the post-propagation energy.

Difference	Top- $K$	Threshold
$\Delta E_{\mathrm{ref}}$	$+0.00518\pm 0.00958$	$+0.00135\pm 0.00106$
$\Delta C_{\mathrm{rel}}$	$-0.02246\pm 0.03944$	$-0.00618\pm 0.00473$
$\Delta\mathrm{RMSE}$	$-0.00017\pm 0.00148$	$-0.00041\pm 0.00037$
$\Delta\mathrm{SSIM}$	$+0.00728\pm 0.01323$	$+0.00296\pm 0.00297$

Table 3: Paired holdout differences relative to Learned AMDI. Entries are control minus learned and are reported as the mean

\pm

sample standard deviation over nine cases. Both controls use parameters selected by matching the learned validation occupancy.

Figure 1: Representative spatial outcome for the four-region holdout case: (a) clean image; (b) noisy input; (c) deterministic-AMDI reconstruction; (d) Learned-AMDI reconstruction; (e) final deterministic-AMDI refinement levels; and (f) final learned refinement levels. This figure illustrates the terminal allocations; sequential contribution is assessed separately by the decision-1-only control.

Figure 2: Nine-case action and energy analysis. (a) Mean reference error and (b) mean active degrees of freedom for deterministic AMDI, unrestricted Learned AMDI, and the decision-1-only temporal control. (c) Mean learned trajectory energy immediately before propagation, after propagation, and after adaptation. (d) Mean energy change caused by adaptation; labels give the number of cases with a positive change. The near coincidence of the two learned trajectories after decision 1 shows that later decisions make only a small contribution on this benchmark, while panels (c)–(d) show that the fixed-tree safeguard does not extend across the learned tree update.

Figure 3: Accuracy–representation frontiers on the same four test images and noise realizations for every method. Shown are (a) reference error, (b) RMSE, and (c) SSIM versus terminal occupancy. Blue points are learned policies obtained by varying

\lambda_{\mathrm{occ}}

; the

\lambda_{\mathrm{occ}}=0.15

point reuses the selected principal checkpoint. Orange and green curves are the prespecified top-

K

and observed-detail-threshold grids, respectively. The square and triangle denote deterministic AMDI and the retain-tree control.

$N$	$N_{\mathrm{full}}$	Method	$C_{\mathrm{rel}}$	$\overline{N}_{\mathrm{act}}$	$N_{\mathrm{ref}}^{(4)}$	$E_{\mathrm{ref}}$	RMSE	SSIM
32	1024	Deterministic AMDI	0.13648	139.75	0	0.17301	0.05378	0.80623
32	1024	Learned AMDI	0.24927	255.25	154	0.13814	0.05466	0.80512
32	1024	Top- $K$ control	0.24194	247.75	144	0.13977	0.05412	0.81135
32	1024	Threshold control	0.24487	250.75	148	0.13932	0.05398	0.81207
64	4096	Deterministic AMDI	0.03650	149.50	0	0.17534	0.04673	0.72399
64	4096	Learned AMDI	0.04089	167.50	24	0.17104	0.04404	0.73740

Table 4: Same-resolution deployment and resolution transfer on four images at each

N

\overline{N}_{\mathrm{act}}

is the mean absolute terminal basis size, and

N_{\mathrm{ref}}^{(4)}

is the total number of executed refinements over the four trajectories. The top-

K

and threshold parameters are the validation-occupancy choices

K=36

and

\tau_{\mathrm{detail}}=6.12571\times 10^{-3}

Figure 4: Same-resolution deployment at

N=32

and resolution transfer to

N=64

and

N=128

, using four images at each resolution. Shown are (a) RMSE, (b) SSIM, (c) reference error, and (d) relative occupancy for deterministic AMDI, Learned AMDI, the frozen top-

K

control, and the frozen observed-detail-threshold control. The actor and both control parameters are fixed at

N=32

and are not retuned at the finer resolutions. Absolute active basis sizes and executed refinements are reported in Table 4 .

Reward	$E_{\mathrm{ref}}$	$C_{\mathrm{rel}}$	$\overline{\sigma}_{\mathrm{sw}}$	RMSE
Full	0.13769	0.27710	0.02209	0.05414
No occupancy term	0.02080	0.99561	0.14185	0.06483
No switching term	0.15126	0.22363	0.01318	0.05260

Table 5: Reward ablation on the common four-image subset. The Full row reuses the selected checkpoint but is evaluated with the ablation experiment’s noise realizations.

Figure 5: Policy-seed sensitivity and reward ablation. (a) Error–occupancy operating points for three independently trained policy seeds; (b) RMSE and SSIM, normalized by their respective means across the three policy-seed means; and (c) operating points for the full reward and the two reward ablations.

Figure 6: Regional localization for deterministic AMDI, Learned AMDI, and an independent VAMPyR/MRCPP order-five projection of the clean analytic target, computed with precision

10^{-3}

and maximum depth eight. The method-dependent level values are used only as qualitative regional indicators.

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