KP-INR: K-Space Implicit Neural Representation for Cardiac MRI

• KP-INR is a scan-specific unsupervised method that uses a dual-branch implicit neural representation to accurately reconstruct undersampled Cartesian cardiac cine MRI in k-space.
• The approach combines a coordinate-based P-INR branch with a feature-driven K-INR branch through cross-branch interaction and iterative refinement to exploit local spatial-temporal context.
• Experimental results demonstrate that KP-INR achieves sharper images and improved aliasing suppression, outperforming traditional methods on cardiac MRI datasets.

KP-INR is a scan-specific, unsupervised implicit neural representation method for reconstructing undersampled Cartesian cardiac cine MRI directly in k-space. It was proposed to address a specific limitation of prior INR-based MRI reconstruction methods: many existing approaches learn coordinate-to-value mappings primarily from positional embeddings, while not explicitly exploiting the local spatial-temporal structure of k-space or the neighboring context around each queried point. KP-INR introduces a dual-branch design that combines a coordinate-driven INR with a feature-driven INR conditioned on local multi-scale k-space embeddings extracted by a complex-valued auto-encoder, with cross-branch interaction and iterative k-space refinement (Lyu et al., 16 Aug 2025).

1. Problem formulation and motivation

The reconstruction target is dynamic multi-coil cardiac cine MRI acquired with undersampled Cartesian k-space. The measured data are represented as

$$\mathbf{y} \in \mathbb{C}^{H \times W \times C \times T},$$

where $H$ and $W$ are k-space spatial dimensions, $C$ is the number of coils, and $T$ is the number of cardiac frames. Each queried k-space location is a spatiotemporal coordinate

$p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$

with $N$ sampled coordinates.

The method is motivated by the standard cine MRI trade-off between acceleration and image quality. In the reported formulation, Cartesian undersampling is especially challenging because missing phase-encoding lines create structured aliasing and, compared with non-Cartesian trajectories, Cartesian sampling has less “direction diversity,” making it harder for a purely coordinate-based model to infer the full k-space distribution. Prior methods such as NIK are described as learning a direct coordinate-to-k-space-value mapping, while other dynamic MRI INR approaches similarly rely heavily on coordinate encodings plus image-domain regularization. KP-INR was introduced to inject a learned k-space contextual prior into this setting (Lyu et al., 16 Aug 2025).

A central point is that KP-INR remains a k-space-domain method throughout. The implicit mapping is learned for k-space values rather than image intensities; the auxiliary branch uses k-space embeddings extracted from a complex-valued U-Net; hard data consistency is imposed in k-space; and images are produced only after inverse FFT and coil combination. This suggests that the method is designed to make the k-space prior, the sampling model, and the reconstruction loop mutually consistent.

2. Dual-branch architecture

KP-INR consists of three components: a complex-valued U-Net auto-encoder, a K-INR branch, and a P-INR branch. The model takes as input the current multi-coil k-space tensor and the sampled coordinate set $P \in \mathbb{R}^{N \times 3}$. The same coordinate set is used in two ways: it is embedded and fed to the positional branch, and it is used to query k-space feature embeddings from the U-Net decoder for the feature branch.

The complex-valued U-Net functions as an auto-encoder over complex-valued k-space. Its role is not direct image reconstruction, but contextual feature extraction. The reported design is a two-level architecture with a fixed channel number of 64. Two complex convolution layers at the ends adjust channel dimensions. Each encoder and decoder level contains a Complex Convolutional Recurrent Block (C-CRNN), recurrent operations are applied in opposite temporal directions, and their outputs are concatenated and fused via a complex convolution to capture bidirectional temporal information and integrate low- and high-level frequency features. A Complex Bidirectional Convolutional Recurrent Block (C-BCRNN) is placed at the bottleneck. No downsampling is applied, so decoder features remain aligned with the original k-space grid. Decoder features queried at sampled coordinates are fused by element-wise addition to form the local multi-scale k-space embedding.

The P-INR branch is the coordinate-based branch. Input coordinates are normalized to $[-1,1]$. Spatial coordinates $P_s$ use NeRF positional encoding,

$H$0

while temporal coordinates $H$1 use random Fourier feature encoding,

$H$2

The spatial embedding length is 480, the temporal embedding length is 96, and the total positional embedding length is 576. The branch uses a 7-layer MLP with complex Gabor wavelet activation (WIRE), and all intermediate hidden layers have dimension 512.

The K-INR branch is the feature-conditioned branch. Its input is

$H$3

where $H$4 is the sampled feature-embedding length. It predicts

$H$5

representing real and imaginary values for all coils at the queried coordinates. It also uses a 7-layer MLP, but with an initial MLP to adjust embedding length and another MLP with leaky ReLU. The output embedding length is initially 64 and, after intermediate cross-branch interaction, expands to 512.

The two branches are explicitly coupled. At the input side, the architecture applies a linear layer to each branch’s embedding and adds the result to the other branch’s input. In the middle of the network, an MLP processes concatenated intermediate features from both branches; the fused output is concatenated back with each branch’s features before subsequent MLP processing. The reported intent is that positional encoding is guided by contextual k-space structure, while contextual features also benefit from coordinate priors.

3. Mathematical formulation and alternating reconstruction

The branch-wise fitting objective is stated as

$H$6

More explicitly, the positional branch is described as

$H$7

and the feature branch as

$H$8

The total loss contains four terms: a reconstruction loss for the K-INR output, a reconstruction loss for the P-INR output, an auto-encoding loss on the acquired k-space region, and an auto-encoding loss on the zero-filled region. The reported weights are

$H$9

The reconstruction losses use the high dynamic range loss from NIK to address the imbalance of k-space magnitudes, while the two auto-encoding losses use MSE on their corresponding masked regions.

Training is scan-specific and unsupervised: each case is optimized directly from its own undersampled measurements. The procedure alternates between an optimization stage and an inference stage. During optimization, the current multi-coil k-space and sampled coordinates are used to train the complex U-Net, K-INR, and P-INR. During inference, all Cartesian-grid coordinates are fed to the frozen KP-INR, the two branch outputs are averaged,

$W$0

and hard data consistency is imposed: $W$1 The refined k-space is then fed back into the next optimization stage.

Final image reconstruction is obtained by inverse FFT on each coil,

$W$2

followed by coil combination using sensitivity maps $W$3,

$W$4

The reported implementation uses PyTorch 2.0.0 on an NVIDIA A100 40GB GPU with AdamW, $W$5, $W$6, $W$7, learning rate $W$8, weight decay $W$9, learning-rate decay by $C$0 every 500 epochs, minimum learning rate $C$1, k-space input update interval every 500 epochs, and total training length of 6000 epochs (Lyu et al., 16 Aug 2025).

4. Experimental protocol and reported results

KP-INR was evaluated on the public CMRxRecon2024 dataset, acquired on a 3T MAGNETOM Vida scanner using dedicated multi-channel cardiac coils. The study focused on long-axis and short-axis cine MRI acquired with a TrueFISP sequence. Reported acquisition characteristics were 3 LAX slices and 8–14 SAX slices per subject, 12–25 cardiac phases, temporal resolution about 50 ms, spatial resolution $C$2, slice thickness 8.0 mm, and slice gap 4.0 mm. The evaluation used 9 randomly selected subjects: 4 LAX and 5 SAX. Two Cartesian trajectories were tested, uniform Cartesian and Gaussian Cartesian, each at 4× and 8× acceleration. The 3D undersampling masks used temporal interleaving, with the central 16 lines fully sampled as the ACS region.

Baselines were L+S, k-t GRAPPA, and P-INR. Metrics were PSNR, SSIM, and DISTS, with DISTS used as a perceptual metric. To avoid inflated scores from large black background areas, the first and last quarters along the height dimension were cropped before metric computation. Statistical significance was assessed with the Wilcoxon signed-rank test. The reported conclusion was that KP-INR achieved the best results across all tested settings, with statistically significant gains over all baselines (Lyu et al., 16 Aug 2025).

Setting	P-INR	KP-INR
Uniform Cartesian, 4×	$C$3 / $C$4 / $C$5	$C$6 / $C$7 / $C$8
Uniform Cartesian, 8×	$C$9 / $T$0 / $T$1	$T$2 / $T$3 / $T$4
Gaussian Cartesian, 4×	$T$5 / $T$6 / $T$7	$T$8 / $T$9 / $p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$0
Gaussian Cartesian, 8×	$p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$1 / $p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$2 / $p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$3	$p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$4 / $p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$5 / $p_i = [p_t^i, p_x^i, p_y^i]^T,\qquad i=1,\dots,N,$6

Qualitatively, KP-INR was reported to better restore fine details and suppress aliasing. P-INR was described as particularly struggling on Cartesian trajectories, producing blurry reconstructions in important cardiac regions. The authors observe that uniform sampling outperforms Gaussian sampling, likely because Gaussian masks are relatively sparse in high-frequency regions.

5. Interpretation of the dual-branch design

The clearest component analysis is the comparison between P-INR and KP-INR. Because P-INR is the coordinate-only INR baseline, the reported improvement of KP-INR isolates the effect of adding the K-INR branch, introducing k-space contextual features, enabling cross-branch interaction, and using alternating refinement. In the reported interpretation, this is the main evidence that local k-space features matter for scan-specific cine MRI reconstruction.

The local-feature branch is conceptually close to classical neighborhood-exploiting reconstruction logic, but its implementation is learned rather than fixed. The discussion explicitly compares this to k-t GRAPPA: both exploit local k-space structure, but KP-INR does so through learned convolutional features enriched with spatial and temporal context. This suggests that the method occupies an intermediate position between classical local k-space interpolation and coordinate-based neural field fitting.

A common misconception is to read KP-INR as a standard coordinate-only INR with a modified positional encoder. The reported design is broader than that. One branch models where a query lies in spatial-temporal k-space; the other models what the local k-space neighborhood around that point looks like. A second misconception is to treat it as an image-domain INR. In the reported formulation, the extra prior, the data-consistency operation, and the iterative refinement loop are all centered on k-space rather than image intensities.

The alternating optimization procedure is also part of the method’s claimed novelty. The reported rationale is that iterative k-space updates progressively refine both the k-space estimate and the latent feature representation extracted by the U-Net, so later iterations give the K-INR branch better inputs than early iterations.

6. Significance, limitations, and open directions

The main strength attributed to KP-INR is that it adds an explicit contextual k-space prior to an INR framework while remaining unsupervised and scan-specific. The formulation is presented as especially useful for Cartesian cine MRI, where simple coordinate-based fitting is weaker than on more diverse non-Cartesian trajectories. The method combines three ideas: INR-based coordinate-to-value fitting, complex-valued spatiotemporal feature extraction from measured k-space, and iterative self-refinement with hard data consistency (Lyu et al., 16 Aug 2025).

Its limitations are also explicit. KP-INR is scan-specific and therefore requires per-scan optimization rather than feed-forward inference from a pretrained model. The reported evaluation uses only 9 subjects. The study does not provide runtime, parameter count, convergence plots, failure cases, or cross-subject generalization analysis. It also does not report explicit ablations for cross-branch interaction alone or for multi-scale feature fusion alone.

The stated future directions are improving k-space feature representations and exploring better interactions between the two branches. A plausible implication is that the reported performance gains derive not only from adding more model capacity, but from structuring that capacity so that coordinate priors and local k-space context are complementary. Within the reported scope, KP-INR therefore represents a k-space-domain INR formulation in which coordinate encoding, contextual feature extraction, and iterative refinement are treated as jointly necessary for robust reconstruction of undersampled Cartesian cardiac cine MRI.