---
title: Multi-Res Hexplane Encoder
url: https://www.emergentmind.com/topics/multi-resolution-hexplane-structure-encoder
type: topic
---

# Multi-Res Hexplane Encoder

A multi-resolution Hexplane structure encoder is a compact and efficient architecture for representing 4D dynamic scenes, based on decomposing space-time features into six orthogonal two-dimensional feature planes, with each plane encoding multi-scale information via wavelet analysis. This methodology is central to recent advances in dynamic novel view synthesis and 3D neural scene representations, where the primary goals are improved training speed, reduced storage, and high-fidelity view synthesis across both static and dynamic scenes [2312.02218, 2301.09632].

## 1. Multi-Resolution Hexplane Feature Architecture

The core principle of a Hexplane encoder is to organize a volumetric 4D function $f(x, y, z, t)$ into six explicit 2D feature planes, corresponding to each unique pair among the four coordinates: $C=\{\textrm{xy}, \textrm{xz}, \textrm{yz}, \textrm{xt}, \textrm{yt}, \textrm{zt}\}$ [2312.02218, 2301.09632]. Each feature plane stores trainable vectors. In multi-resolution variants, these planes are further organized into a resolution pyramid across several levels $r=1,\dots,L$, with increasingly finer spatial and temporal grids.

For each level $r$, the six planes (e.g., $\textrm{Plane}_{\textrm{XY}}^{(r)} \in \mathbb{R}^{S_x^{(r)} \times S_y^{(r)} \times F_r}$) store feature vectors at discretized 2D grids. Coordinate normalization is used so that input points $(x, y, z, t)$ are mapped into $[0, 1]^4$, and then to integer grid coordinates at each resolution via $u_x^{(r)} = \hat{x} \cdot (S_x^{(r)} - 1)$, etc. [2301.09632].

## 2. Wavelet Domain Parameterization and Inverse Synthesis

WavePlanes [2312.02218] innovates by storing each plane not as a dense feature grid, but as a set of multi-scale wavelet coefficient tensors. For each plane $c \in C$, the representation consists of:

- A low-pass ("father") coefficient tensor $\Omega_{c0} \in \mathbb{R}^{B \times \lceil H/2^N \rceil \times \lceil W/2^N \rceil}$
- For $s=1,...,N$, "mother" wavelet detail tensors $\Omega_{cs} \in \mathbb{R}^{B \times 3 \times \lceil H/2^{N-s+1} \rceil \times \lceil W/2^{N-s+1} \rceil}$ (three directions: LH, HL, HH)

The feature plane $P_c(x, y)$ is reconstructed via a multi-scale inverse discrete wavelet transform (IDWT):
$$
P_c(x,y) = \sum_{i,j} a_0 \phi_{i,j}(x,y) + \sum_{s=1}^N \sum_{m \in \{\textrm{LH, HL, HH}\}} \sum_{u,v} d_{sm,u,v} \psi_{m,s;u,v}(x,y)
$$
where $a_0 = \Omega_{c0}[i,j]$, $d_{sm,u,v} = \Omega_{cs}[m,u,v]$.

Trainable per-scale scaling controls gradient flow to each band. Space-time planes ($c \in \{\textrm{xt}, \textrm{yt}, \textrm{zt}\}$) have a +1 bias to encourage a default empty value [2312.02218].

## 3. Sparse Hard-Thresholding and Hash-Map Compression

Since wavelet domains exhibit signal concentration, hard-thresholding is applied after training:
$$
\Omega_{cs}[i,j] \leftarrow \Omega_{cs}[i,j] \cdot 1\left[|\Omega_{cs}[i,j]| > \tau\right]
$$
Typically $\tau \approx 0.1$ effectively eliminates over 90% of coefficients with negligible PSNR loss. Only nonzero coefficients and their linear indices are stored in per-plane, per-scale hash-maps. This achieves $10$–$15\times$ compression (e.g., $<20$ MB for a dynamic scene at $256^2$ resolution compared to $200$ MB dense) [2312.02218].

A final pass using lossless codecs (LZMA superior to BZ2, GZIP, or Pickle) further reduces storage. The precise number of surviving coefficients $K$ is typically $5$–$10\%$ of original, so storage is $O(K)$ floats plus $O(K)$ indices.

## 4. Feature Extraction, Fusion, and NeRF Integration

For each query $(x, y, z, t)$, projections $\pi_c(q)$ map the point onto each 2D plane, yielding six coordinates. Bilinear interpolation samples feature vectors from fine and coarse planes at each projection. Features across resolutions and planes are fused. 

WavePlanes introduces the Zero-Agreement Mask (ZAM) fusion:
$$
f(q) = \text{concat}_{r=0,1}\left[\frac{1}{3} \sum_{ct \in \{\textrm{xt},\textrm{yt},\textrm{zt}\}} f_{cr,ct}(q) \cdot \prod_{c \in C} f_{cr,c}(q)\right]
$$
Only if all three space-time planes agree on zero does the fused representation become zero, preserving sharp motion boundaries [2312.02218].

The fused feature vector $f(q)\in\mathbb{R}^{6B}$ feeds a scene-specific basis-expansion MLP for density $\sigma$ and view-dependent color $c$. Standard NeRF volume rendering $\int T(t)\sigma(q(t))c(q(t), d)dt$ yields rendered colors. The encoder can also drive a Gaussian Splatting renderer; the hexplane logic remains, only the final scene integrator changes [2312.02218].

## 5. Empirical Results and Ablations

Detailed experiments on LLFF, synthetic D-NeRF, and real DyNeRF scenes demonstrate that with only $6 \times 2$ planes (fine and coarse) at $256^2$ resolution and $B=64$, WavePlanes matches or outperforms K-Planes in PSNR/SSIM, while reducing storage requirements by $5$–$15\times$ and incurring no extra training time on a single RTX 3090 GPU [2312.02218]. 

Key ablation findings include:

| Variable            | Options/Ablation         | Storage/Quality Impact                                  |
|---------------------|-------------------------|---------------------------------------------------------|
| Wavelet family      | Haar (db1), db6, coif4  | Coif4 (PSNR ≈ 31.44) maxes regularity/support           |
| Decomposition depth | $N=2$, $N=3$            | $N=2$ gives $\approx$10% less GPU time, no PSNR loss    |
| Threshold $\tau$    | $0.1$, $0.0$            | $>90\%$ coeff. pruned at $\tau=0.1$, PSNR loss $<0.02$  |
| Codec               | LZMA, GZIP, BZ2, Pickle | LZMA yields $<$18MB serializations                      |

## 6. Connections to Prior Encoding Paradigms

Hexplane encoders generalize prior plane-based feature architectures (e.g., K-Planes, HexPlane) by extending explicit 4D spatial-temporal representations to multi-resolution and wavelet-compressed domains [2312.02218, 2301.09632]. The hash-map sparsification leverages wavelet coefficient sparsity (analogous to SBHex/BBHex on hexagonal grids in 2D image coding [2110.02556]) for rate–distortion optimality at low storage.

The mathematical basis of multi-scale decomposition follows bi-orthogonal wavelet theory, e.g., the Cohen–Schlenker bi-orthogonal bases, but is adapted for Cartesian rather than hexagonal lattice filtering—though hexagonal sampling retains advantages in 2D coding contexts [2110.02556].

## 7. Performance Considerations and Applicability

WavePlanes and its multi-resolution hexplane encoder provide dramatic gains in storage efficiency (10–15×), faster training/inference versus classical implicit MLP NeRFs, and robustness for both static and dynamic view synthesis. The encoder preserves scene fidelity across a variety of wavelet families and decomposition depths, and the zero-prioritizing fusion method maintains edge sharpness under motion. These approaches integrate seamlessly with NeRF or Gaussian Splatting, and the design supports further improvements with more advanced coding/quantization, region-of-interest decoding, or lossless lifting extensions, as explored in related image coding work [2312.02218, 2110.02556, 2301.09632].

Source: https://www.emergentmind.com/topics/multi-resolution-hexplane-structure-encoder