Directional Mapping Module in QSMnet-INR

• Directional Mapping Module is a SIREN-based implicit neural representation that learns orientation-specific dipole responses to address the cone-null problem in single-orientation QSM.
• It leverages a multi-orientation dipole loss—combining INR consistency, complementary filling, and data consistency—to enforce analytical kernel behavior and complete missing k-space frequencies.
• Empirical results demonstrate that integrating the module improves structural recovery and reduces artifacts, yielding significant enhancements in SSIM, PSNR, and overall reconstruction quality compared to baseline methods.

Searching arXiv for the specified paper and closely related QSM work to ground the article. arxiv_search query="(Cai et al., 10 Dec 2025) QSMnet-INR Single-Orientation Quantitative Susceptibility Mapping via Implicit Neural Representation in k-Space" max_results=5

Searching for related quantitative susceptibility mapping and implicit neural representation papers for context. arxiv_search query="quantitative susceptibility mapping implicit neural representation k-space single orientation QSM" max_results=10

In the context of single-orientation quantitative susceptibility mapping, the “Directional Mapping Module” (Editor’s term) denotes the component of QSMnet-INR that performs directional, multi-orientation dipole modeling in k-space: specifically, the paper’s INR Dipole Kernel Completion Module together with the frequency-domain residual-weighted Dipole Loss that constrains it (Cai et al., 10 Dec 2025). Its purpose is to mitigate the ill-posedness of single-orientation dipole inversion by learning complementary dipole responses across directions and using them to complete the cone-null region in Fourier space, thereby improving structural recovery and suppressing streaking artifacts without requiring multi-orientation acquisition.

1. Directional mapping in the QSM inverse problem

Quantitative Susceptibility Mapping (QSM) estimates tissue magnetic susceptibility from magnetic-resonance phase data. In Fourier space, the forward model is

$$\mathcal{F}\{\Delta B(\mathbf{r})\} = D(\mathbf{k}) \cdot \chi(\mathbf{k}),$$

with dipole kernel

$$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$

Single-orientation inversion is ill-posed because $D(\mathbf{k}) = 0$ on the cone defined by

$\frac{k_z^2}{|\mathbf{k}|^2} = \frac{1}{3},$

corresponding to the “magic angle” $54.7^\circ$. Around this cone-null region, division by the dipole kernel becomes unstable, producing noise amplification, structural loss, and streaking artifacts (Cai et al., 10 Dec 2025).

Within this setting, the directional character of the module is not a generic spatial-attention mechanism. It is a k-space mechanism that learns how dipole responses vary across directions and orientations, then uses that learned directional structure to compensate for the cone-null. Multi-orientation methods such as COSMOS address the problem by combining measurements from different head orientations; QSMnet-INR seeks part of that multi-directional benefit from single-orientation data by learning a directional k-space completion model.

2. Mathematical structure of the directional dipole mapping

The module is implemented as a SIREN-based implicit neural representation (INR) in k-space. Its input is a continuous coordinate

$\mathbf{r} = (x,y,z) \in [-1,1]^3,$

and for each orientation $i$ it produces a predicted dipole response

$$\hat{D}_i(\mathbf{k}) = f_{\boldsymbol{\theta}}(\mathbf{r}).$$

The SIREN parameterization is

$\begin{cases} \mathbf{h}_0 = \mathbf{r}, \[3pt] \mathbf{h}_l = \sin\!\big(\omega_0(\mathbf{W}_l \mathbf{h}_{l-1} + \mathbf{b}_l)\big), \quad l=1,\dots,L-1, \[3pt] f_{\boldsymbol{\theta}}(\mathbf{r}) = \mathbf{W}_L \mathbf{h}_{L-1} + \mathbf{b}_L, \end{cases}$

with parameters $\boldsymbol{\theta}=\{\mathbf{W}_l,\mathbf{b}_l\}$ and frequency scaling $$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$0 (Cai et al., 10 Dec 2025).

The paper does not write the INR as an explicit function $$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$1 of a direction vector. Instead, the multi-directional behavior enters through orientation-indexed analytical kernels $$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$2, orientation-specific supervision, and multi-orientation aggregation in the loss. This means the module is direction-sensitive at the level of training constraints and predicted responses, even though the core MLP is shared across orientations.

A central quantity is the orientation-averaged magnitude

$$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$3

which expresses whether the learned family of dipole responses remains collectively nonzero at a given k-space location. This is the formal basis for the module’s directional completion behavior: locations that are null or nearly null for one direction are expected to remain supported by other directions.

3. Cone-null completion and the dipole loss

The module is trained with a cone-focused weighting mask computed from a reference analytic kernel:

$$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$4

When $$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$5, $$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$6; away from the cone-null it decays rapidly toward $$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$7. The weighting therefore emphasizes precisely the frequencies at which single-orientation inversion is most unstable (Cai et al., 10 Dec 2025).

The directional mapping is shaped by three loss terms. The INR consistency loss is

$$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$8

This aligns the learned kernel with analytic dipole behavior, especially near the cone-null.

The complementary filling loss is

$$D(\mathbf{k}) = \frac{1}{3} - \frac{k_z^2}{|\mathbf{k}|^2}, \qquad \mathbf{k} = (k_x,k_y,k_z).$$9

This explicitly penalizes locations in which the averaged multi-orientation response is too small. It is the clearest mathematical expression of the module’s directional completion role: within cone-emphasized regions, at least some orientations must retain nonzero response.

The data consistency loss is

$D(\mathbf{k}) = 0$0

Using the analytic $D(\mathbf{k}) = 0$1, this couples the susceptibility estimate $D(\mathbf{k}) = 0$2 to the physical forward model in k-space. The full dipole loss is

$D(\mathbf{k}) = 0$3

A common misconception is that cone-null completion is achieved by simply allowing a flexible MLP to hallucinate missing frequencies. The formulation above indicates otherwise. The completion is restricted by analytic kernels, by orientation-wise complementarity, and by explicit k-space data consistency.

4. Coupling with the QSMnet backbone

QSMnet-INR contains two modules. Module I is a 3D U-Net-based QSMnet that reconstructs susceptibility $D(\mathbf{k}) = 0$4 from the background-field-removed local field map. Module II is the INR Dipole Kernel Completion Module, which predicts $D(\mathbf{k}) = 0$5 from normalized coordinates and is trained through the dipole loss (Cai et al., 10 Dec 2025).

Training uses alternating optimization. In one stage, the INR parameters are fixed and the QSMnet backbone is updated using the supervised reconstruction objective $D(\mathbf{k}) = 0$6. In the other stage, QSMnet is fixed and the total loss

$D(\mathbf{k}) = 0$7

is optimized so that gradients from the dipole loss act mainly on the INR.

This arrangement gives the directional module a dual role. First, it must remain compatible with analytical dipole structure and multi-orientation complementarity. Second, it must learn a completion that is actually useful for susceptibility reconstruction. The directional mapping is therefore not an isolated pretext task in k-space; it is coupled to image-domain susceptibility estimation through end-to-end training.

Another important clarification is that the module is not explicitly conditioned on an orientation vector in the equations. Orientation dependence is introduced by the analytic kernels $D(\mathbf{k}) = 0$8, orientation-specific grids, and multi-orientation sums in the loss. This suggests that the “directional” aspect resides in the supervision geometry rather than in an explicit orientation token or side-channel.

5. Empirical behavior and ablation evidence

The reported experiments include the 2016 QSM Reconstruction Challenge, a multi-orientation GRE dataset, and both in-house and public single-orientation clinical data. Across these settings, QSMnet-INR is reported to outperform conventional and recent deep-learning approaches on multiple quantitative metrics, with particular advantages in structural recovery within cone-null regions and in artifact suppression (Cai et al., 10 Dec 2025).

The ablation study on the 2016 Challenge isolates the contribution of the directional module and the dipole loss:

Variant	HFEN	NRMSE	SSIM
QSMnet baseline	0.591	0.891	0.898
QSMnet + INR (no dipole loss)	0.403	0.644	0.946
Full QSMnet-INR	0.362	0.521	0.967

The corresponding PSNR values are 40.999 dB, 43.821 dB, and 45.669 dB, respectively (Cai et al., 10 Dec 2025).

These numbers indicate two separable effects. Adding the INR alone already improves structural fidelity and reduces streaking, which is consistent with the view that continuous k-space modeling is beneficial. Adding the dipole loss yields a further improvement, implying that explicit cone-focused, multi-directional physical constraints matter beyond the representational flexibility of the INR itself.

A direction-sensitivity experiment on an unseen subject with 18 orientations further shows that QSMnet-INR maintains SSIM $D(\mathbf{k}) = 0$9 for most orientations and PSNR $\frac{k_z^2}{|\mathbf{k}|^2} = \frac{1}{3},$0 dB, with small variations in HFEN and NRMSE and residuals localized mainly to boundaries. The paper interprets this as evidence that the learned directional k-space mapping generalizes across acquisition orientations.

6. Interpretation, scope, and limitations

The paper does not literally use the phrase Directional Mapping Module. In its own terminology, the relevant component is the INR Dipole Kernel Completion Module together with the multi-directional dipole loss (Cai et al., 10 Dec 2025). The shorthand is nevertheless precise in one respect: the module learns a directional family of dipole responses in k-space and uses that family to address the directional null structure of the dipole kernel.

Its significance lies in how it reframes single-orientation QSM inversion. Rather than treating the cone-null solely as a local numerical instability, it treats it as a missing-direction problem in Fourier space and introduces a learned continuous representation whose behavior is constrained by analytic dipole structure, multi-orientation complementarity, and k-space data consistency. This suggests a conceptual proximity to COSMOS, but without requiring multi-orientation acquisition at inference.

The paper also distinguishes this framework from subject-specific INR approaches. The INR module is trained self-supervised through physical losses and does not require per-subject optimization; a plausible implication is improved robustness across datasets and field strengths relative to subject-specific INR-QSM. At the same time, orientation information is introduced implicitly through training losses rather than explicit conditioning, so the learned representation should be understood as a shared continuous model of directional dipole behavior rather than a fully explicit parametric kernel $\frac{k_z^2}{|\mathbf{k}|^2} = \frac{1}{3},$1.

In summary, within QSMnet-INR the directional mapping function is a continuous, SIREN-based k-space representation that predicts orientation-indexed dipole responses, completes the cone-null region through multi-orientation complementarity, and is physically regularized by a cone-weighted dipole loss. Its primary role is to reduce the ill-posedness of single-orientation QSM while preserving structural detail and physical stability.