GridTD: Tensor Decomposed Multi-Res Grid Encoding

• GridTD is an unsupervised continuous representation framework that combines tensor decomposition and multi-resolution grid encoding to recover high-dimensional images from compressed measurements.
• It leverages individual one-dimensional multi-resolution encoders fused via a CP-like structure and processes features with a lightweight two-layer MLP for mapping to intensity values.
• Empirical results demonstrate improved PSNR/SSIM and faster runtimes across video SCI, spectral SCI, and dynamic MRI tasks compared to conventional methods.

Searching arXiv for papers on GridTD and closely related tensor-decomposed multi-resolution grid methods. Tensor Decomposed Multi-resolution Grid Encoding (GridTD) is an unsupervised continuous representation framework for compressive imaging reconstruction that combines tensor decomposition, multi-resolution grid encoding, and a lightweight neural network to recover high-dimensional images from compressed measurements. In the formulation introduced for video snapshot compressive imaging (SCI), spectral SCI, and compressive dynamic MRI, GridTD replaces a full $D$-dimensional grid encoder with a tensor-decomposed construction built from one-dimensional multi-resolution encoders, then fuses them through a CP-like structure and a small multilayer perceptron (MLP). The method is presented as retaining the hierarchical modeling ability of multi-resolution grid encoding while inheriting the compactness of tensor decomposition, and the accompanying analysis derives a Lipschitz property, a generalization error bound, and fixed-point convergence results for the reconstruction algorithm (Jin et al., 10 Jul 2025).

1. Problem setting and rationale

GridTD is formulated for linear inverse problems of the form $y = A(x) + \text{noise}$, specialized in the paper to three compressive imaging regimes. In video SCI, a two-dimensional measurement is formed from a three-dimensional video tensor $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$ through

$$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$

In spectral SCI, a hyperspectral cube is encoded by a coded aperture with spectral shift,

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$

with $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$. In compressive dynamic MRI, multi-coil undersampled $k$-space data satisfy

$$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$

These models are treated through a single continuous-representation viewpoint in which the unknown object is modeled as a function of coordinates rather than as a fully stored discrete tensor (Jin et al., 10 Jul 2025).

The paper motivates GridTD by contrasting it with two prevailing families of unsupervised representations. Coordinate MLPs suffer from spectral bias and often require large networks to fit sharp spatial, temporal, or spectral detail. InstantNGP-style multi-resolution grid encoders improve high-frequency modeling, but in high-dimensional settings they require $D$-dimensional grids whose parameter count and encoding cost scale exponentially with the data dimension $D$. GridTD is introduced specifically to preserve the multi-scale behavior of grid encoding while replacing full $y = A(x) + \text{noise}$0-dimensional grids with a tensor-decomposed structure that scales linearly in $y = A(x) + \text{noise}$1 (Jin et al., 10 Jul 2025).

2. Encoding architecture and tensor-decomposed representation

GridTD operates on normalized coordinates $y = A(x) + \text{noise}$2. For each dimension $y = A(x) + \text{noise}$3, it defines a one-dimensional multi-resolution grid encoder

$y = A(x) + \text{noise}$4

where $y = A(x) + \text{noise}$5 is the product of the number of levels $y = A(x) + \text{noise}$6 and the feature dimension $y = A(x) + \text{noise}$7 per level. At level $y = A(x) + \text{noise}$8, a one-dimensional grid tensor $y = A(x) + \text{noise}$9 is queried by linear interpolation: $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$0 with

$$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$1

The per-dimension encoding concatenates these levelwise vectors: $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$2 GridTD then fuses dimensions by element-wise multiplication,

$$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$3

so that each channel is a product of one factor from every dimension. This is the mechanism by which the encoding acquires a CP-decomposition-like structure (Jin et al., 10 Jul 2025).

The paper makes this tensor interpretation explicit under tensor parallelism. If $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$4 denotes the tensor of GridTD features evaluated on a Cartesian grid, then $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$5 is represented as a concatenation of rank-one outer products over dimensions, and summing over the last dimension recovers the standard CP form

$$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$6

In this sense, GridTD is not merely inspired by tensor decomposition; its feature tensor is CP-structured, while the factor vectors themselves are parameterized by one-dimensional multi-resolution grid encoders (Jin et al., 10 Jul 2025).

A small MLP maps the rank-space feature to an intensity value: $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$7 The paper emphasizes that a depth-2 MLP is already sufficient, and ablations show that increasing the depth to 3 or 4 layers yields negligible gain (Jin et al., 10 Jul 2025).

The resulting storage and encoding complexity differ sharply from full $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$8-dimensional multi-resolution grids.

Scheme	Storage complexity	Encoding complexity
InstantNGP-like $$\mathcal{X} \in \mathbb{R}^{n_1 \times n_2 \times n_3}$$9-D grids	$$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$0	$$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$1
GridTD	$$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$2	$$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$3

For the example $$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$4 and $$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$5, the paper reports 3.12M parameters for InstantNGP versus 0.02M for GridTD, and 38.05s versus 1.31s for 300 iterations of inpainting (Jin et al., 10 Jul 2025).

3. Reconstruction algorithm, regularization, and temporal adaptation

GridTD is trained within a plug-and-play ADMM procedure. For video SCI, the optimization is written as

$$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$6

The ADMM loop alternates among an $$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$7-update, a $$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$8-update, and a dual update. The $$\mathbf{Y} = \sum_{t=1}^{n_3} \mathcal{M}_t \odot \mathcal{X}_t + \mathbf{Z}.$$9-subproblem has a closed form for video SCI; the $$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$0-subproblem corresponds to training the GridTD network against the current ADMM target; and the dual variable is updated by

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$1

The same algorithmic pattern is used for spectral SCI and MRI with task-specific forward operators in the fidelity step (Jin et al., 10 Jul 2025).

During the $$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$2-update, the represented tensor is produced by evaluating the GridTD network on normalized coordinates and then minimizing

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$3

Here

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$4

and

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$5

These regularizers are used to exploit spatial and temporal or spectral smoothness and to increase robustness to noise (Jin et al., 10 Jul 2025).

For video SCI, the paper introduces a temporal affine adapter to relax an otherwise strict low-rank temporal structure. A latent low-rank tensor $$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$6 is mapped to $$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$7 by frame-wise affine transforms,

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$8

with

$$\mathbf{Y} = \sum_{t=1}^{n_3} \operatorname{shift}(\mathcal{M}_t \odot \mathcal{X}_t) + \mathbf{Z},$$9

The affine matrix includes per-frame scale $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$0, rotation $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$1, and translations $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$2. Rather than learning independent translations, the paper models $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$3 and $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$4 by small MLPs driven by the temporal GridTD encoder,

$$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$5

This design couples motion modeling to the same temporal feature space used by the main representation (Jin et al., 10 Jul 2025).

4. Theoretical properties

The theoretical analysis in the paper compares the smoothness and generalization of GridTD with those of InstantNGP-style encoders. Assuming a $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$6-Lipschitz activation, bounded grid embeddings, and $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$7, with $$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$8, the paper proves for InstantNGP

$$\mathbf{Y}\in\mathbb{R}^{n_1\times (n_2 + d(n_3-1))}$$9

whereas for GridTD

$k$0

The distinction is that the Lipschitz constant of GridTD scales linearly in $k$1, while the bound for InstantNGP scales as $k$2 (Jin et al., 10 Jul 2025).

Using these constants, the paper derives generalization error bounds for squared loss. With probability $k$3,

$k$4

while the corresponding bound for InstantNGP is

$k$5

The paper interprets this as showing that GridTD maintains stable generalization behavior as dimensionality increases, whereas the high-dimensional grid encoder inherits an exponential dependence on $k$6 (Jin et al., 10 Jul 2025).

A separate analysis addresses the reconstruction algorithm itself. Under a bounded-residual condition for the $k$7-subproblem and standard assumptions on the ADMM penalty update, the sequence $k$8 is proved to converge to a fixed point. The argument is given for video SCI and stated to apply analogously to MRI and spectral SCI variants. This places the method in a rare category among unsupervised continuous representations for inverse problems: the paper provides both representation-level and algorithm-level guarantees (Jin et al., 10 Jul 2025).

5. Empirical behavior across compressive imaging tasks

The main experimental evidence is reported on six $k$9 grayscale video SCI benchmarks, ten KAIST hyperspectral scenes for spectral SCI, and retrospective cardiac cine from OCMR for dynamic MRI. In video SCI, average PSNR/SSIM across Aerial, Crash, Drop, Kobe, Runner, and Traffic are reported as 29.79 dB / 0.889 for InstantNGP and 30.64 dB / 0.903 for GridTD. Other unsupervised baselines in the same table include 26.85 / 0.846 for GAP-TV, 25.47 / 0.735 for PnP-DIP, 25.62 / 0.754 for DVP, 28.21 / 0.845 for Factorized-DVP, and 29.55 / 0.871 for SCI-BDVP. The accompanying qualitative comparison emphasizes sharper edges, improved texture, and fewer artifacts for GridTD (Jin et al., 10 Jul 2025).

On spectral SCI, GridTD is evaluated against both supervised and unsupervised methods. The average PSNR/SSIM reported for GridTD are 36.99 / 0.953, compared with 36.09 / 0.938 for LRSDN, 35.66 / 0.945 for InstantNGP, 35.93 / 0.956 for CST-L, and 36.10 / 0.954 for MST++. The paper highlights both improved spatial detail and better preservation of local spectral signatures, with spectral curves closely matching ground truth in the provided examples (Jin et al., 10 Jul 2025).

On dynamic MRI, the strongest quantitative example is at acceleration factor $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$0 with 21 spokes, where the reported PSNR/SSIM are 27.70 / 0.666 for zero-filled NUFFT, 38.99 / 0.955 for GRASP, 37.84 / 0.935 for an INR with Fourier features, 42.38 / 0.979 for InstantNGP, and 43.98 / 0.983 for GridTD. The paper states that GridTD consistently yields the highest PSNR and SSIM, or tied best SSIM at the highest acceleration factor, while approximately halving runtime relative to InstantNGP in this task (Jin et al., 10 Jul 2025).

Ablations on the Runner video SCI sequence isolate the role of major components. The full model with TV, SSTV, tensor decomposition, temporal affine adapter, and a 2-layer MLP attains 35.22 / 0.957. Removing TV yields 34.79 / 0.950, removing SSTV yields 34.76 / 0.950, removing tensor decomposition—equated in the paper with reverting to InstantNGP—yields 34.50 / 0.955, and removing the affine adapter yields 35.05 / 0.957. Increasing MLP depth to 3 or 4 layers gives negligible improvement. The paper also reports that performance is stable over broad ranges of $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$1, $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$2, and $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$3, with the best values in the reported experiment occurring near $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$4, $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$5, $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$6, and $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$7 for $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$8, $$\mathbf{Y}_c = \mathbf{F}_u \mathbf{S}_c \mathbf{X},\quad 1\le c\le C.$$9, $D$0 (Jin et al., 10 Jul 2025).

6. Relation to neighboring tensor-grid methods and broader design space

GridTD belongs to a broader family of methods that combine structured grids, low-rank tensor models, and lightweight decoders, but it occupies a specific point in that design space. F-Hash defines a full 4D multi-resolution “tesseract” grid over $D$1 with a collision-free linearization and a shallow MLP, but it does not perform tensor decomposition; in the terminology used in that paper, it is better viewed as a full 4D multi-resolution tensor encoding rather than a decomposed one (Sun et al., 4 Jul 2025). C-Pic learns low-rank latent encodings of multidimensional grid data through TT decomposition and differentiable cross-approximation, emphasizing sample-efficient learning from only a fraction of tensor entries; this suggests a different route to GridTD-like systems in which adaptive TT sampling replaces explicit hash-grid parameterization (Usvyatsov et al., 2021).

Other nearby work develops multi-resolution tensor structure more directly. A multi-resolution low-rank tensor decomposition based on sums of CP models over different grouped unfoldings treats “resolution” as a hierarchy of mode groupings rather than spatial downsampling, which is conceptually close to GridTD’s use of low-rank structure across dimensions (Rozada et al., 2024). Multi-resolution low-rank tensor formats represent a tensor as a sum of compressed tensors defined on increasingly coarse grids, using explicit prolongation and averaging operators; that formulation supplies a grid hierarchy with coarse-to-fine residual correction rather than a hash-encoded coordinate network (Mickelin et al., 2019). T4DT, by contrast, compresses time-varying 3D TSDFs with Tucker, TT, QTT, OQTT, and TT–Tucker hybrids, showing that 4D grid data can be made extremely compact through closed-form tensor truncation, without the network-training loop used by GridTD (Usvyatsov et al., 2022). In operator learning, MG-TFNO combines a multi-grid domain decomposition with a global tensor factorization of Fourier weights, which instantiates the same general pattern—multi-resolution grids plus tensorized parameters—in a PDE setting rather than an inverse-imaging one (Kossaifi et al., 2023).

This comparison indicates that GridTD’s distinctive contribution is not merely “tensor decomposition plus grids,” but a particular synthesis: one-dimensional multi-resolution encoders along each axis, CP-like rank fusion by Hadamard products, a very small MLP, and optimization within a physics-informed ADMM loop. The paper suggests several direct extensions of that synthesis, including hyperspectral–multispectral fusion, multi-dimensional super-resolution, more sophisticated motion adapters, and alternative tensor formats such as Tucker or TT within the same encoding idea (Jin et al., 10 Jul 2025).