CARA: Collision-Aware Resolution Adaptation for Multiresolution Hash Encoding Based Image Fitting

Conference: ECCV 2026
Paper: ECCV Official
Code: See supplementary material of the original paper
Area: Model Compression
Keywords: Implicit Neural Representations (INR), Multiresolution Hash Encoding, Information Equalization, Hash Collisions, Pixel-Shuffle

TL;DR

Addressing the capacity bottlenecks and parameter inefficiencies caused by data-agnostic geometric resolution schedules and hash collisions in Instant-NGP, CARA quantifies band-wise effective information using empirical entropy coupled with a hash collision attenuation factor, solves for a min-max balanced resolution schedule via dynamic programming, and incorporates an invertible pixel-shuffle transform to lower the load factor, matching SOTA fidelity with only 27.76% of the parameters and gaining up to 6.11 dB PSNR under comparable budgets.

Background & Motivation

Implicit neural representations (INRs) have revolutionized signal processing and visual representation by parameterizing discrete signals as continuous functional mappings from coordinates to signal values. To overcome the spectral bias of conventional coordinate MLPs toward low-frequency components and eliminate sluggish training dynamics, hybrid grid-based architectures such as Instant-NGP pair a lightweight MLP decoder with a trainable multiresolution hash encoding. By distributing spatial coordinates across a geometric hierarchy of hash tables, Instant-NGP enables fast \(O(1)\) feature lookups and rapid optimization. However, standard multiresolution hash encodings rely on a static, data-agnostic geometric progression of grid resolutions, allocating identical hash-table capacity across all levels irrespective of the input signal's characteristics.

This data-agnostic design clashes fundamentally with the non-uniform information distribution inherent in natural visual signals. Real-world images exhibit marked diversity in their frequency profiles: some contain rich textures with substantial high-frequency entropy increments, whereas others are predominantly smooth with energy concentrated in lower frequency bands. A rigid, hand-crafted resolution progression assigns excessive parameters to frequency bands carrying sparse information, while under-allocating representational capacity to information-dense bands where high-frequency reconstruction errors dominate perceptual quality.

Compounding this structural inefficiency, existing multiresolution hash frameworks treat resolution allocation and hash collision dynamics as decoupled factors. At finer grid levels where the number of grid vertices vastly exceeds the hash table size (\(N_\ell^2 \gg T\)), severe hash collisions occur, degrading retrievable feature capacity exponentially and forcing the lightweight decoder to absorb severe residual errors. Core idea: formulate multiresolution grid resolution selection as an effective information equalization problem, quantify retrievable capacity by coupling empirical Shannon entropy increments with a uniform hash collision attenuation factor, solve for a min-max balanced resolution schedule via dynamic programming, and employ an invertible pixel-shuffle transform to deflate the spatial load factor without increasing table size.

Method

Overall Architecture

CARA views multiresolution hash encoding through the lens of a spatial sampling-and-interpolation system, where a grid of resolution \(N_f\) per axis induces a Nyquist cutoff frequency \(f_{\text{Nyq}} = N_f / (2W)\). The overall pipeline operates in three coherent stages: first, it computes band information density \(\mathfrak{I}(f)\) across the spectrum by measuring the empirical Shannon entropy of band-pass residual signals derived from anti-aliased low-pass filtering, scaled by the spatial sample count; second, it incorporates a theoretically derived hash collision efficiency factor \(w^{\mathrm{hash}}\) to compute effective retrievable information per band, finding the optimal resolution schedule \(\{N_\ell\}_{\ell=1}^L\) that minimizes the peak effective information load across levels via dynamic programming; third, it applies an invertible pixel-shuffle transform to the coordinate domain, reshaping spatial dimensions into channels to deflate the hash load factor by a factor of \(s^2\), drastically mitigating collisions during encoding and decoding.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Target Image Y"] --> B["Stage 1: Band-wise Information & Collision Modeling<br/>Measure empirical entropy increments + derive hash collision efficiency"]
    B --> C["Stage 2: Min-Max Dynamic Programming Optimization<br/>Partition spectrum to balance effective information load across levels"]
    C --> D["Stage 3: Load-Deflating Invertible Pixel-Shuffle Transform<br/>Reshape spatial dimensions into channels to reduce hash load factor"]
    D --> E["Adaptive Multiresolution Hash Grid & Lightweight MLP Decoder"]
    E --> F["Inverse Pixel-Shuffle for Full-Resolution High-Fidelity Reconstruction"]

Key Designs

1. Band-wise Information & Collision Modeling: Quantifying Spectral Information Demand and Capacity Degradation

To eliminate heuristic resolution choices, CARA first establishes a rigorous metric for the information content each hash level must represent. For an image low-pass filtered to cutoff frequency \(f\), denoted \(\mathbf{Y}_{\le f}\), CARA defines the band information density over a narrow spectral step \(\Delta f\) as:

\[\mathfrak{I}(f) = H\left(\mathbf{Y}_{\le f + \Delta f} - \mathbf{Y}_{\le f}\right) \cdot N_f^2\]

Here, \(H(\cdot)\) denotes the empirical Shannon entropy computed on quantized 9-bit residual samples, reflecting the average uncertainty (bits per sample) introduced by the refinement step, and \(N_f^2\) represents the number of spatial grid samples at resolution \(N_f\). However, because each hash level stores embeddings in a finite table of size \(T\), vertices collide when \(n_\ell = N_\ell^2 > T\). Under the assumption of uniform hashing, CARA derives the expected distinct retrievable entries per inserted key as the collision efficiency factor:

\[w_\ell^{\mathrm{hash}} = \frac{1 - e^{-\alpha_\ell}}{\alpha_\ell}, \quad \text{where } \alpha_\ell = \frac{N_\ell^2}{T}\]

Consequently, the true retrievable information budget allocated to level \(\ell\) is formulated as \(\mathcal{I}_{\text{eff}, \ell} = \sum_{f \in \mathcal{F}_\ell} \mathfrak{I}(f) \cdot w_{k_\ell}^{\mathrm{hash}}\). This design explicitly couples the image's intrinsic spectral information with the physical capacity limitations of the hash table, laying an information-theoretic foundation for resolution selection.

2. Min-Max Dynamic Programming Optimization: Mitigating Single-Band Bottlenecks

Because each hash level specializes in a specific frequency band, overall reconstruction fidelity is bounded by the most overloaded level; if a single band contains far more information than its corresponding table can hold, the residual error spills over to other levels or overburdens the decoder. Conversely, over-allocating capacity to an information-sparse band wastes parameters. CARA formalizes resolution selection over an ordered candidate resolution set \(\{N^{(k)}\}_{k=1}^K\) as a min-max optimization problem seeking cutoff indices \(0 = k_0 < k_1 < \dots < k_L = K\):

\[\min_{0 < k_1 < \dots < k_{L-1} < K} \max_{\ell \in [L]} \mathcal{I}_\ell\]

This objective seeks the tightest achievable uniform upper bound on per-band effective information load. Because the ordered partitioning exhibits optimal substructure, CARA computes the exact global optimum using dynamic programming in \(O(L K^2)\) time. For gigapixel images like Pluto, solving this DP adds merely 0.7 seconds of pre-processing time, introducing negligible overhead relative to training while fundamentally optimizing parameter utility across all levels.

3. Load-Deflating Invertible Pixel-Shuffle Transform: Mitigating Collisions without Enlarging Hash Tables

Even with optimized resolution allocation, extremely high-resolution representations still suffer from high vertex-to-table ratios at the finest levels. Expanding table size \(T\) linearly scales memory and parameter count. To resolve this tension, CARA introduces an invertible pixel-shuffle transform \(\mathcal{S}_s\). Given a shuffle factor \(s\), an input image \(\mathbf{Y} \in \mathbb{R}^{h \times w \times c}\) is rearranged into a lower-resolution, higher-channel tensor \(\tilde{\mathbf{Y}} \in \mathbb{R}^{\frac{h}{s} \times \frac{w}{s} \times (cs^2)}\).

This spatial compaction reduces the spatial coordinates fed into the INR by a factor of \(s\) along each axis, decreasing the distinct grid vertices at level \(\ell\) from \(N_\ell^2\) to \((N_\ell / s)^2\). Under a fixed table size \(T\), the effective load factor is deflated to:

\[\alpha'_\ell = \frac{(N_\ell / s)^2}{T} = \frac{\alpha_\ell}{s^2}\]

This quadratic drop in load factor significantly elevates the collision efficiency factor \(w^{\mathrm{hash}}\), directly reclaiming usable representation capacity at fine levels without modifying the hash table size. Once the INR predicts \(\hat{\tilde{\mathbf{Y}}}\), the lossless inverse transform \(\mathcal{S}_s^{-1}\) restores the full-resolution output \(\hat{\mathbf{Y}}\).

Loss & Training

The network is optimized end-to-end to minimize the standard \(\ell_2\) mean squared error reconstruction loss between predicted and ground-truth values:

\[\mathcal{L}_{\text{recon}} = \frac{1}{M} \sum_{m=1}^M \|\mathfrak{f}_\theta(\mathbf{x}_m) - \mathbf{y}_m\|_2^2\]

Training employs the Adam optimizer with \((\beta_1, \beta_2, \epsilon) = (0.9, 0.99, 10^{-15})\) and a cosine annealing learning-rate schedule. The feature dimension per level is fixed to 2. The candidate resolution pool size \(K\) is set to 30 with uniform frequency spacing for Kodak images, and \(K = 500\) for gigapixel images and WSIs. Anti-aliased resizing serves as the standard low-pass filtering operator.

Key Experimental Results

Main Results

CARA was evaluated across 24 images from the Kodak dataset (\(768 \times 512\)), three gigapixel natural scenes (Pluto, Tokyo, and Girl With a Pearl Earring), and four newly collected raw uncompressed Whole-Slide Images (WSIs).

On the Kodak benchmark (~207K parameter budget), CARA achieves superior reconstruction fidelity compared to both pure implicit MLPs and hybrid grid methods:

Method	Representation	Params (K)	PSNR (dB) ↑	SSIM ↑
WIRE [32]	Implicit	373	37.59	0.9596
SIREN [39]	Implicit	207	38.70	0.9512
SCONE [22]	Implicit	207	39.72	0.9662
Instant-NGP [28]	Hybrid Hash	206	37.06	0.9386
NFFB [48]	Hybrid Grid	208	38.77	0.9537
NeuRBF [6]	Hybrid Grid	207	38.70	0.9488
MetricGrid [46]	Hybrid Grid	207	39.73	0.9568
CARA (Ours)	Hybrid Hash	205	40.03	0.9714

On the gigapixel Girl benchmark (\(20000 \times 23466\)), CARA demonstrates outstanding scalability and Pareto efficiency:

Method	Low Regime Params	PSNR / SSIM	Mid Regime Params	PSNR / SSIM	High Regime Params	PSNR / SSIM
Instant-NGP [28]	8.01 M	21.51 / 0.579	32.01 M	27.30 / 0.896	412.8 M	33.05 / 0.959
MetricGrid [46]	8.02 M	27.03 / 0.817	32.02 M	29.70 / 0.957	412.8 M	37.06 / 0.994
CARA (Ours)	8.02 M	27.53 / 0.897	114.6 M	37.07 / 0.998	418.5 M	43.37 / 0.999

Note: CARA matches MetricGrid's 412.8 M performance using only 114.6 M parameters (27.76% of the budget), and outperforms MetricGrid by 6.11 dB at ~418 M parameters.

Ablation Study

1. Effect of Collision Modeling (\(w^{\mathrm{hash}}\)) on Kodak

Config	Params (K)	PSNR (dB) ↑	SSIM ↑	Note
W.O. Collision Factor (\(w^{\mathrm{hash}} \equiv 1\))	374	40.32	0.9674	Ignores collision; allocates purely by raw entropy
W.O. Collision Factor (\(w^{\mathrm{hash}} \equiv 1\))	403	43.97	0.9973	Shows noticeable parameter inefficiency at mid scale
W.O. Collision Factor (\(w^{\mathrm{hash}} \equiv 1\))	484	47.97	0.9979	Requires more parameters to match high fidelity
W. Collision Factor (Full CARA)	205	41.03	0.9714	Beats 374K unmodeled baseline with nearly half parameters
W. Collision Factor (Full CARA)	341	46.24	0.9906	Outperforms 403K unmodeled baseline by +2.27 dB
W. Collision Factor (Full CARA)	471	48.00	0.9987	Reaches highest fidelity with fewer parameters

2. Alternative Information Proxies for Resolution Scheduling on Kodak

Information Proxy	Low Budget (K)	PSNR / SSIM	Mid Budget (K)	PSNR / SSIM	Note
Instant-NGP Baseline	206	37.06 / 0.9386	300	43.88 / 0.9964	Heuristic geometric schedule
MetricGrid Baseline	207	39.73 / 0.9568	335	44.51 / 0.9978	Nonlinear metric grid approach
CARA + \(\ell_2\) Energy Proxy	214	38.52 / 0.9504	334	43.60 / 0.9913	Energy reflects signal power, insensitive to distribution
CARA + Variance Proxy	211	38.74 / 0.9560	331	44.94 / 0.9893	Variance poorly mirrors coding uncertainty
CARA + Empirical Entropy Proxy	205	41.03 / 0.9714	332	46.63 / 0.9982	Directly aligns with coding rate; yields optimal fidelity

Key Findings

• Collision Modeling is Indispensable: Disabling the collision factor (\(w^{\mathrm{hash}} \equiv 1\)) causes marked performance degradation. Modeling collisions enables the 205K CARA model to surpass a 374K collision-agnostic model, proving that resolution schedules must account for capacity degradation under collisions.
• Shannon Entropy Outperforms Heuristic Statistics: Substituting empirical entropy with \(\ell_2\) energy or variance drops PSNR by 1.7 to 2.5 dB. Entropy directly measures coding uncertainty (bits per sample) under an optimal code, perfectly mirroring the representational demands placed on grid embeddings.
• Amplified Gains at Gigapixel Scale: While CARA yields 0.3 to 3.0 dB improvements on standard Kodak images, its advantages widen dramatically on gigapixel and raw WSI datasets, outperforming MetricGrid by up to 6.11 dB and matching SOTA with 27.76% of parameters. This confirms that higher spatial resolutions amplify the penalties of collision and frequency misallocation.

Highlights & Insights

• The Theoretical Elegance of Coupling Entropy with Hash Degradation: Rather than relying on empirical heuristic tuning, CARA derives a closed-form collision efficiency factor \(w^{\mathrm{hash}} = \frac{1 - e^{-\alpha}}{\alpha}\) from uniform hash collision probabilities. Coupling it with empirical entropy creates a mathematically grounded information budget for resolution scheduling.
• Zero-Parameter Collision Mitigation via Pixel-Shuffle: Rearranging spatial patches into channels compresses the coordinate domain by \(s\) along each axis, cutting the load factor by \(s^2\). This provides an elegant, plug-and-play way to suppress hash collisions without expanding table size or altering MLP decoders.
• Curating the First Uncompressed Raw WSI Benchmark: The authors identify that public WSI datasets suffer from heavy JPEG compression artifacts that distort high-frequency evaluations, contributing the first uncompressed, raw 40× magnification WSI benchmark for academic research.

Limitations & Future Work

• Global Stationarity Assumption: CARA computes empirical entropy globally across the whole image to determine a uniform resolution schedule. For signals with extreme spatial non-stationarity (e.g., regions of vast empty background juxtaposed with ultra-dense focal textures), a spatially uniform resolution schedule may still leave local capacity underutilized or congested.
• Extension to 3D and Volumetric Fields: Currently validated on 2D image fitting and gigapixel slices, applying CARA to 3D NeRFs or dynamic 4D scenes requires computing 3D spatial frequency entropy, where ray marching sampling densities vary dynamically across depth.
• Adaptive Search Overhead in Video Streams: While DP solving takes only ~0.7 seconds on static gigapixel images, recomputing resolution schedules frame-by-frame for long video streams could introduce latency. Developing a lightweight meta-network to predict resolution schedules in a single forward pass represents a promising future direction.

Related Work & Insights

• vs Instant-NGP [28]: Instant-NGP relies on a static geometric progression that is blind to image frequency distributions, causing severe capacity waste in sparse bands and heavy hash collisions in dense bands. CARA replaces this heuristic schedule with an entropy-based, collision-aware schedule that drastically improves parameter efficiency.
• vs MetricGrid [46]: MetricGrid adapts elementary metric grids for nonlinear approximation but incurs massive parameter costs at gigapixel scales. CARA matches MetricGrid's highest fidelity using only 27.76% of its parameters by optimizing information allocation on standard hash grids.
• vs 2D Gaussian Splatting / GaussianImage [58]: Explicit Gaussian primitives suffer from high memory overhead as primitives densify under ultra-high resolutions. CARA demonstrates that properly balanced multiresolution hash grids achieve superior PSNR (48.00 dB vs 43.69 dB) while using only 13% of Gaussian Splatting's parameters.

Rating

• Novelty: ⭐⭐⭐⭐☆ Reformulating hash grid resolution assignment as an information-theoretic equalization problem coupled with collision dynamics is insightful and mathematically sound.
• Experimental Thoroughness: ⭐⭐⭐⭐⭐ Comprehensive benchmarks spanning Kodak, gigapixel scenes, and a newly curated uncompressed WSI dataset, evaluated against implicit, hybrid, and explicit Gaussian baselines.
• Writing Quality: ⭐⭐⭐⭐⭐ Rigorous derivations from Nyquist sampling to Shannon entropy and dynamic programming, supported by compelling quantitative and qualitative figures.
• Value: ⭐⭐⭐⭐☆ Provides a clear theoretical principle and practical toolkit for optimizing hash-grid based neural representations in high-resolution visual processing.