Neural KMDS-Net: Dynamic PET Denoising

• The paper introduces Neural KMDS-Net, an end-to-end model unfolding a kernel space-based multidimensional sparse optimization for dynamic PET denoising.
• It integrates a learned kernel representation with tensor sparse coding to preserve both temporal correlations and intra-frame spatial structures.
• Empirical results show superior PSNR and SSIM over traditional methods, particularly improving noise reduction in low-count early PET frames.

Searching arXiv for the cited papers to ground the article in the primary source and adjacent similarly named methods. Neural KMDS-Net is an end-to-end, model-guided neural network for dynamic PET image denoising that is built by unfolding and neuralizing a kernel space-based multidimensional sparse (KMDS) model. It targets 4D dynamic PET, where early short-duration frames are extremely noisy, and combines a kernel-space representation, tensor sparse coding across space and time, and end-to-end learned optimization parameters. The method is designed to exploit temporal or inter-frame spatial correlation together with spatial or intra-frame structural consistency, while avoiding the explicit construction of the large kernel matrix $K$, its pseudo-inverse $K^+$, and the hand-tuned iterative optimization required by the original KMDS formulation (Xiaodong et al., 23 Sep 2025).

1. Clinical setting and problem formulation

Dynamic PET acquires a sequence of PET images over time to capture tracer kinetics. High temporal resolution requires many short frames, especially early after injection, but short frames contain fewer detected counts and therefore have much lower signal-to-noise ratio. Neural KMDS-Net addresses this denoising problem under the additive model

$y = x + n$

where $y \in \mathbb{R}^{M\times N\times Q\times T}$ is the observed noisy dynamic PET image, $x \in \mathbb{R}^{M\times N\times Q\times T}$ is the unknown clean image sequence, and $n$ is additive noise.

The method is motivated by two structural regularities of dynamic PET. First, there is temporal or inter-frame correlation: when no motion occurs, anatomy and spatial support are largely shared across frames even though uptake values change over time. Second, there is spatial or intra-frame structural consistency: each 3D frame contains edges, textures, and object boundaries that should be preserved rather than oversmoothed. The KMDS framework models both regularities explicitly, and Neural KMDS-Net converts that model into a trainable feed-forward architecture (Xiaodong et al., 23 Sep 2025).

This suggests a design philosophy distinct from purely black-box denoisers. Prior model-based methods are described as interpretable but burdened by hand-crafted priors, manual hyperparameter tuning, and high computational cost. Pure deep learning methods can denoise strongly, but may oversmooth, hallucinate, or generalize poorly in difficult clinical conditions, particularly for very low-count early frames. Neural KMDS-Net is positioned between these regimes: it preserves the structure of a sparse kernel model while substituting its parameter-estimation operators with neural modules.

2. Kernel space-based multidimensional sparse model

The original KMDS model begins from a PET kernel representation. The clean dynamic image $x$ is expressed through a kernel matrix $K$ and a coefficient image $\boldsymbol{\alpha}$: $\tau(x)=K\tau(\boldsymbol{\alpha})$ where $K^+$0 reshapes a 4D array of size $K^+$1 into a matrix of size $K^+$2, and $K^+$3 reverses that reshape. In this formulation,

$K^+$4

captures relationships among spatial voxels.

The second ingredient is the multidimensional sparse model. Rather than vectorizing the sequence, the method retains its fourth-order tensor structure and represents $K^+$5 as

$K^+$6

where $K^+$7 is a sparse coefficient tensor, $K^+$8, $K^+$9, $y = x + n$0, $y = x + n$1, and $y = x + n$2 denotes mode-$y = x + n$3 tensor multiplication. The temporal dictionary $y = x + n$4 captures inter-frame correlation, while $y = x + n$5 preserve intra-frame spatial structure.

The full KMDS optimization problem is

$y = x + n$6

Its alternating optimization first estimates $y = x + n$7 using the pseudo-inverse $y = x + n$8: $y = x + n$9 then solves for $y \in \mathbb{R}^{M\times N\times Q\times T}$0 with an $y \in \mathbb{R}^{M\times N\times Q\times T}$1-relaxed tensor iterative shrinkage-thresholding algorithm. The sparse update is written as

$y \in \mathbb{R}^{M\times N\times Q\times T}$2

$y \in \mathbb{R}^{M\times N\times Q\times T}$3

$y \in \mathbb{R}^{M\times N\times Q\times T}$4

with initialization

$y \in \mathbb{R}^{M\times N\times Q\times T}$5

The direct algorithm is described as cumbersome because it requires the huge matrix $y \in \mathbb{R}^{M\times N\times Q\times T}$6, even its pseudo-inverse $y \in \mathbb{R}^{M\times N\times Q\times T}$7, and iterative optimization with hand-tuned $y \in \mathbb{R}^{M\times N\times Q\times T}$8. Neural KMDS-Net is the neuralized substitute for this procedure (Xiaodong et al., 23 Sep 2025).

3. Unfolded neural architecture

Neural KMDS-Net replaces explicit KMDS operators with CNN modules while preserving the structure of the KMDS inference procedure. The network has three stages: a kernel representation module that estimates $y \in \mathbb{R}^{M\times N\times Q\times T}$9 from $x \in \mathbb{R}^{M\times N\times Q\times T}$0, a multidimensional sparse modeling module that iteratively estimates $x \in \mathbb{R}^{M\times N\times Q\times T}$1 from $x \in \mathbb{R}^{M\times N\times Q\times T}$2, and an image estimation module that maps the denoised kernel coefficient $x \in \mathbb{R}^{M\times N\times Q\times T}$3 back to $x \in \mathbb{R}^{M\times N\times Q\times T}$4 (Xiaodong et al., 23 Sep 2025).

The explicit kernel pseudo-inverse operation

$x \in \mathbb{R}^{M\times N\times Q\times T}$5

is approximated by stacked convolutions,

$x \in \mathbb{R}^{M\times N\times Q\times T}$6

and with batch normalization after each convolution,

$x \in \mathbb{R}^{M\times N\times Q\times T}$7

$x \in \mathbb{R}^{M\times N\times Q\times T}$8

The multidimensional sparse module unfolds the tensor ISTA updates into repeated multidimensional sparse blocks. The generalized stage equations are

$x \in \mathbb{R}^{M\times N\times Q\times T}$9

$n$0

$n$1

where $n$2 approximates the synthesis operator, $n$3 approximates the analysis or gradient step together with $n$4, and $n$5 is the learned threshold tensor. Both $n$6 and $n$7 are CNNs with four cascaded convolutional layers and batch normalization after each convolution. The soft-thresholding operator is implemented as

$n$8

The initialization is also learned: $n$9 where $x$0 has the same architecture as $x$1. After the final sparse stage,

$x$2

and the final image estimate is

$x$3

with

$x$4

Component	Neural form	Role
Kernel representation	$x$5	Learned kernel-space-like feature representation
Sparse-code initialization	$x$6	Learned initialization of sparse codes
MDS block	$x$7	Unfolded sparse coding iteration
Coefficient reconstruction	$x$8	Reconstruct denoised kernel coefficient image
Image estimation	$x$9	Map kernel coefficient back to denoised image

Architecturally, the network input and output are tensors of size

$K$0

The implementation uses 3D convolutions and 3D batch normalization, treating time frames as channels or features while convolving over the 3D spatial volume. The convolution kernel size is $K$1, stride is $K$2, padding is $K$3, the depth of $K$4 and $K$5 is $K$6, the number of output filters is $K$7, and the unfolding number is $K$8. The number of input and output channels in $K$9, $\boldsymbol{\alpha}$0, $\boldsymbol{\alpha}$1, and $\boldsymbol{\alpha}$2 is set to $\boldsymbol{\alpha}$3. The reported model size is 10.5M parameters in the simulation study and 14.2M parameters in the real study. The paper does not introduce attention, gating, recurrent cells, or explicit residual skip connections beyond the optimization-inspired residual update in the sparse code iteration.

4. Training methodology and datasets

The network is trained in a fully supervised manner using paired noisy-clean data and the mean squared error loss

$\boldsymbol{\alpha}$4

Implementation details are specified as PyTorch on an NVIDIA Tesla V100 GPU, with Adam, an initial learning rate of $\boldsymbol{\alpha}$5, learning-rate decay by a factor of $\boldsymbol{\alpha}$6 every 80 epochs, batch size $\boldsymbol{\alpha}$7, and 300 epochs. Training images at different frames are divided by their maximum voxel values (Xiaodong et al., 23 Sep 2025).

The simulated dataset uses dynamic $\boldsymbol{\alpha}$8F-FDG scans from a $\boldsymbol{\alpha}$9 Zubal head phantom with 24 time frames: $\tau(x)=K\tau(\boldsymbol{\alpha})$0 Ground-truth TACs are generated from a 3-compartment kinetic model with blood input function, and Poisson noise plus random and scatter events are simulated. The training set contains 2000 noisy-clean pairs, each of size $\tau(x)=K\tau(\boldsymbol{\alpha})$1. The test set contains 600 pairs, described as $\tau(x)=K\tau(\boldsymbol{\alpha})$2, with 10 noisy realizations per true image.

The clinical dataset consists of 65 patients scanned on a DigitMI 930 PET/CT system for 60 minutes after $\tau(x)=K\tau(\boldsymbol{\alpha})$3F-FDG injection. The split is 55 patients for training and 10 for testing. Frame timing is

$\tau(x)=K\tau(\boldsymbol{\alpha})$4

for 28 total frames. The original image size is $\tau(x)=K\tau(\boldsymbol{\alpha})$5 per bed position and is cropped along the third dimension into blocks of $\tau(x)=K\tau(\boldsymbol{\alpha})$6 for GPU feasibility. Full-count reconstruction from total acquired data is used as the label, raw data are downsampled to $\tau(x)=K\tau(\boldsymbol{\alpha})$7 of total counts for low-dose input, and three low-dose realizations per patient are generated for training. For test data, 60 noisy realizations per patient are simulated.

These settings are consequential for interpreting the architecture. The training procedure uses synthetic clean targets in simulation and high-count reconstructions as reference targets in clinical experiments. A plausible implication is that the method’s reported generalization properties are conditioned on the quality and representativeness of those paired targets rather than on self-supervised or physics-only learning.

5. Empirical performance and ablation evidence

The baselines are ML-EM, KEM, ResNet direct denoising, Pix2Pix-based GAN, SMDS-Net, and DDPM-PET. In simulation, qualitative comparisons on frame 2, frame 12, and frame 24 are summarized as follows: ML-EM has the highest noise; KEM improves on ML-EM but remains limited; ResNet suppresses noise well but is not best; GAN performs poorly on early frame 2 because of strong frame-to-frame intensity differences; DDPM-PET is strong on later frames but noisier than ResNet on frame 2; SMDS-Net exhibits strong bias on frame 2 and frame 12; and Neural KMDS-Net gives the best image quality and best detail preservation (Xiaodong et al., 23 Sep 2025).

A key simulation table reports local PSNR for six organs at frame 2 and frame 24, and Neural KMDS-Net is best in every listed ROI. Reported examples are Gray matter $\tau(x)=K\tau(\boldsymbol{\alpha})$8 and $\tau(x)=K\tau(\boldsymbol{\alpha})$9, White matter $K^+$00 and $K^+$01, Caudate $K^+$02 and $K^+$03, Putamen $K^+$04 and $K^+$05, Thalamus $K^+$06 and $K^+$07, and Tumor $K^+$08 and $K^+$09. The ROI analysis further uses

$K^+$10

and

$K^+$11

and the bias-NSD plots show the best tradeoff between low bias and low background noise for both tumor and gray matter, especially in early and mid frames.

In the clinical chest study, the method achieves the best PSNR and SSIM in frames 6, 18, and 28. The reported values are:

Frame	PSNR	SSIM
6	33.081	0.884
18	35.289	0.934
28	40.795	0.963

In a clinical generalization experiment, the model is trained on whole-body PET data from multiple bed positions and tested on brain dynamic PET. It again performs best, with brain PSNR $K^+$12, $K^+$13, and $K^+$14 for frames 6, 18, and 28, and brain SSIM $K^+$15, $K^+$16, and $K^+$17 for the same frames.

The ablation studies isolate several structural choices. Varying the depth $K^+$18 of the kernel representation and image estimation modules over $K^+$19, the best performance is generally at $K^+$20; example PSNR values are $K^+$21 on frame 2 and $K^+$22 on frame 12 at $K^+$23, while on frame 24 the value $K^+$24 at $K^+$25 slightly exceeds $K^+$26 at $K^+$27. Varying the number of unfolding iterations $K^+$28 over $K^+$29, the chosen value is 20 as a good tradeoff, with best PSNR $K^+$30 on frame 2 and $K^+$31 on frame 12 at $K^+$32, and $K^+$33 on frame 24 at $K^+$34, though gains beyond 20 are described as modest. Varying the size of the sparse tensor $K^+$35, larger dimensions generally improve denoising, and the adopted configuration is

$K^+$36

Taken together, these results support the paper’s claim that the optimization-inspired sparse module is more central than simply deepening the front or back convolutional stacks. They also support the stronger claim that the model is particularly effective on short early frames, where standard CNN, GAN, DDPM, and sparse-model baselines are weaker.

6. Interpretation, limitations, and nomenclature

The method is interpreted through three priors aligned with dynamic PET. First, the kernel space acts as a feature representation where spatial correlations among voxels are easier to exploit. Second, the multidimensional sparse prior treats dynamic PET as a 4D object rather than a stack of independent 3D frames, jointly exploiting common anatomy over time, local spatial redundancy within each frame, and temporal redundancy across frames. Third, learned adaptive parameters replace manually chosen $K^+$37, $K^+$38, dictionary matrices, thresholds, and step sizes. The paper explicitly argues that this is why Neural KMDS-Net performs especially well on short early frames, where ordinary CNN, GAN, and DDPM methods often struggle (Xiaodong et al., 23 Sep 2025).

The limitations are also explicit. Performance degrades when the acquisition time or noise level of test data differs substantially from that seen in training, and a refined retraining procedure may be needed. The optimization modules are replaced by CNNs, but more advanced mechanisms such as spatial-temporal attention could improve performance. Only two priors are integrated, namely kernel methods and multidimensional sparse coding, while other explainable priors such as compressed sensing are left open. Parameters such as architecture depth and latent sizes are selected numerically rather than learned automatically. The structural-consistency assumption is strongest when there is little motion across frames, and no explicit motion modeling is introduced.

The nomenclature can be a source of confusion. Neural KMDS-Net is not the same method as KD-Net, which addresses knowledge distillation from a multimodal teacher to a monomodal student for medical image segmentation, nor is it the same method as KM-UNet, which is a KAN-plus-Mamba U-shaped segmentation architecture (Hu et al., 2021, Zhang, 5 Jan 2025). Neural KMDS-Net instead denotes a model-guided unfolding network for dynamic PET denoising derived from a kernel space-based multidimensional sparse model. Its defining elements are therefore kernel-space representation, tensor sparse coding across space and time, and learned replacements for iterative optimization operators, rather than multimodal knowledge distillation or KAN-plus-state-space segmentation design.

In that narrower and more precise sense, Neural KMDS-Net occupies a distinct position in medical imaging research: it is a denoising architecture for dynamic PET whose principal novelty lies not in an arbitrary deep backbone, but in the learned implementation of a KMDS optimizer.