---
title: 'Coil2Coil (C2C): Stellarator & MRI Applications'
url: https://www.emergentmind.com/topics/coil2coil-c2c-method
type: topic
---

# Coil2Coil (C2C): Stellarator & MRI Applications

Coil2Coil (C2C) refers to two distinct, independently developed methodologies: one for the discretization of surface currents into practical coil geometries in stellarator optimization (fusion plasma physics context) and another for self-supervised denoising in multi-coil Magnetic Resonance Imaging (MRI). The following article presents the technical foundations, methodologies, and implications of the C2C method in both domains, referencing [2508.09321] for stellarator optimization and [2208.07552] for MRI denoising.

## 1. Formulations and Motivations

The term "Coil2Coil" (abbreviated "C2C"; *Editor's term*) originated independently in the fields of stellarator coil optimization and multi-coil MRI. In stellarator design, C2C formalizes the conversion of an idealized continuous surface current solution into discrete, manufacturable coil sets that preserve desired magnetic properties. In MRI, C2C enables self-supervised training of deep denoisers by constructing synthetic noisy image pairs from redundant coil measurements, circumventing the need for clean ground-truth images or explicit noise realizations.

In both cases, C2C addresses core limitations in traditional approaches: the lack of detailed methodology for discretizing optimized surface currents into coils in the stellarator setting [2508.09321], and data scarcity/lack of independence in self-supervised MRI denoising [2208.07552].

## 2. C2C in Stellarator Coil Optimization

### 2.1. Surface-Current Formalism

Given a fixed plasma boundary (parametrized by $(\theta, \phi)$), the surface-current sheet is described by the ansatz:
$$
K(\theta, \phi) = \hat{n} \times \nabla \psi(\theta, \phi)
$$
where $\psi$ is the scalar current potential, $\hat{n}$ is the normal to the winding surface, and $\nabla$ is the surface gradient. $\psi$ is expanded as:
$$
\psi(\theta, \phi) = \sum_{m, n} \left[ \psi^c_{m,n} \cos(m\theta-n\phi) + \psi^s_{m,n} \sin(m\theta-n\phi) \right] + I \frac{\theta}{2\pi} + G \frac{\phi}{2\pi}
$$
Here, $I$ and $G$ denote net toroidal and poloidal linking currents, respectively.

### 2.2. Optimization Objective

Optimization selects the Fourier coefficients of $\psi$ to minimize the normal component of the total magnetic field on the plasma surface:
$$
\chi^2_B = \int_{S_{\text{plasma}}} (B_n)^2\, dA
$$
with an optional current regularization term:
$$
\chi^2_K = \int_{S_{\text{winding}}} |K|^2\, dA'
$$
Combined, the regularized minimization problem is:
$$
C[\psi_{\text{SV}}] = \chi^2_B + \lambda \chi^2_K
$$
where $\lambda$ is a tunable regularization parameter.

## 3. The Coil2Coil ("Coil-cutting") Discretization Algorithm

### 3.1. Workflow

Once the optimal continuous current potential $\psi(\theta, \phi)$ is computed, the C2C algorithm discretizes the sheet current as follows:

1. **Grid evaluation**: Evaluate $\psi$ over a uniform $(\theta, \phi)$ grid on the winding surface.
2. **Stream-function partitioning**: Select $N_{\text{coils}}$ equally spaced target values within each field period:
   $$
   \psi_\text{target,\,k} = \psi_{\min} + (k + 0.5) \Delta\psi,\quad \Delta\psi = \frac{\psi_{\max} - \psi_{\min}}{N_{\text{coils}}}
   $$
3. **Contour extraction**: For each $\psi_\text{target,\,k}$, extract the closed contour $\{\theta_i, \phi_i\}$ such that $\psi(\theta_i, \phi_i) = \psi_\text{target,\,k}$ using a 2D contouring (e.g., marching squares, isocontour ODE).
4. **Real-space mapping**: Map contour points to real-space coordinates using the Fourier surface representation $(R(\theta, \phi), Z(\theta, \phi), \phi)$.
5. **Curve fitting**: Fit each real-space contour with a smooth parameterization (cubic spline or Fourier series).
6. **Current assignment**: Assign to each coil a current $I_k = \Delta\psi$.
7. **Symmetry replication**: Repeat each coil configuration for all $N_{\text{FP}}$ field periods via rigid toroidal rotation.

### 3.2. Error Analysis and Post-processing

After discretization, the resulting filamentary coil set is evaluated by recomputing the normal-field error:
$$
\chi^2_B = \int_{S_{\text{plasma}}} [B_{\text{discrete}}(\theta, \phi) \cdot n(\theta, \phi)]^2 dA
$$
If the error exceeds acceptable bounds, one can increase $N_{\text{coils}}$, adjust $\lambda$, or refine coil geometries with local optimization [2508.09321].

#### Example Results
In a demonstration with a quasi-axisymmetric configuration ($N_{\text{FP}}=1$, offset winding surface of 0.2 m), the C2C procedure produced:
- Modular coilset (24 coils): RMS normal-field error $\sim 10^{-4}$ T
- Helical coilset (16 coils): RMS error $<10^{-3}$ T

### 3.3. DESC Code Implementation

The DESC code implements the C2C workflow:
- `regcoil_surfcurr`: Solves for $\psi(\theta, \phi)$
- `cut_coils_from_ψ`: Performs the coil-cutting and produces filamentary coils as `FourierXYZCoil` or `SplineXYZCoil` objects

User parameters include regularization, mode truncation, coil count, field periods, helicity, and grid/contour algorithm choices [2508.09321].

## 4. C2C in Multi-Coil MRI Denoising

### 4.1. Motivation and Data Model

Supervised MRI denoising requires paired clean/noisy data, which are often unavailable. Traditional self-supervised approaches (Noise2Void, Noise2Self) either demand statistical independence unachievable in k-space splitting or result in inferior denoising.

C2C exploits phased-array coil redundancy, using the acquisition model:
$$
y_i = S_i x + n_i,\quad n_i \sim \mathcal{CN}(0, \sigma_i^2 I)
$$
where $S_i$ are coil sensitivities and $n_i$ is complex Gaussian noise.

### 4.2. Paired-Noisy Image Construction

Coil channels are split into disjoint sets $\mathcal{J}, \mathcal{K}$ and coil-combined images are formed:
$$
I_{\text{input}} = \sum_{i \in \mathcal{J}} S_i^H y_i,\qquad I_{\text{label}} = \sum_{i \in \mathcal{K}} S_i^H y_i
$$
This produces pairs with identical noise-free content and independent, zero-mean noise (under ideal uncorrelated noise).

### 4.3. Noise Decorrelation and Sensitivity Matching

Because coil noise is in practice correlated, C2C applies a whitening transform:
$$
\tilde{I}_{\text{label}} = \alpha I_{\text{input}} + \beta I_{\text{label}}
$$
with coefficients derived from the voxelwise covariance matrix to enforce uncorrelated, unit variance noise:
$$
\alpha = -\frac{\sigma_k^2}{\sigma_j \sigma_k - \sigma_{jk}^2}, \quad \beta = \frac{\sigma_{jk}}{\sigma_j \sigma_k - \sigma_{jk}^2}
$$
$\tilde{I}_{\text{label}}$ is further normalized to ensure both images match the underlying signal $x$:
$$
\hat{I}_{\text{label}} = \frac{\sum_{i \in \mathcal{J}} S_i^H}{\sum_{i \in \mathcal{K}} S_i^H} \tilde{I}_{\text{label}}
$$

### 4.4. Denoiser Training

The denoising network $f_\theta$ (18-layer DnCNN variant) is trained via a mean squared error loss:
$$
\mathcal{L}(\theta) = \mathbb{E}_{\text{voxels}}\left\| f_\theta(I_{\text{input}}) - \hat{I}_{\text{label}} \right\|_2^2
$$
Random coil partitions are used at each epoch to enhance generalization [2208.07552].

#### Algorithmic Steps

| Step | Operation                                                         | Purpose                                 |
|------|-------------------------------------------------------------------|-----------------------------------------|
| 1    | Random coil split                                                | Ensures statistical independence        |
| 2    | Coil-combine to $I_{\text{input}}$, $I_{\text{label}}$           | Synthesizes noise pairs                 |
| 3    | Compute covariance, perform whitening                            | Decorrelates noise                      |
| 4    | Sensitivity normalization                                        | Matches underlying signals $x$          |
| 5    | Network forward, loss computation, backprop                      | Parameter update                        |

### 4.5. Experimental Performance

On the NYU fastMRI dataset (brain, 4–20 coils), C2C matched or exceeded all prior self-supervised methods in PSNR/SSIM, achieving parity with fully supervised approaches:
- C2C: PSNR $= 38.97 \pm 2.91$, SSIM $= 0.968 \pm 0.017$ (for $\sigma=1.0$)
- Supervised (Noise2Noise): PSNR $= 39.18 \pm 2.96$, SSIM $= 0.968 \pm 0.017$
Other self-supervised results were lower by statistically significant margins ($p<0.001$).

Ablation studies confirmed that omitting noise decorrelation or sensitivity normalization degraded denoising efficacy or prevented convergence [2208.07552].

In real-world DICOM experiments, C2C removed noise robustly with minimal bias, outperforming alternative training frameworks.

## 5. Algorithmic Distinctions and Limitations

The C2C methodology in stellarator coil design is characterized by its robust mapping from an optimized sheet current to a practical set of filamentary coils with minimized field error, implemented in the DESC code and enabling immediate quantitative and graphical error assessment. In MRI, C2C uniquely circumvents the need for available clean/noisy ground-truth pairs by engineering a statistical context in which the N2N objective becomes theoretically and empirically optimal [2208.07552].

Potential limitations in the stellarator application include the dependence on contour fidelity, coil smoothness, and potential loss of optimality after discretization, necessitating further filamentary-optimizer step if high-precision field matching is required. In MRI, reliance on accurate noise covariance estimation and sensitivity maps is critical, and strong noise coupling between channels can degrade, but not eliminate, performance.

## 6. Software and User Accessibility

In fusion, C2C is accessible via the DESC software suite:
- Python API for description and manipulation of winding surfaces, current potentials, and coil discretization
- User-specifiable parameters for geometric, regularization, and algorithmic choices
- Downstream compatibility with further coil optimizers and validation tools [2508.09321]

For MRI, C2C is implemented in PyTorch and evaluated using standard datasets (NYU fastMRI) with established image normalization, masking, and deep denoising architectures [2208.07552].

## 7. Significance and Future Directions

C2C provides a repeatable, mathematically principled framework bridging theoretical optimization and practical implementation in both stellarator coil engineering and MR image denoising. For plasma physics, it addresses an otherwise underspecified yet pivotal translation from continuous to discrete coil representations, enabling high-fidelity realization of MHD-optimized configurations. In MRI, C2C enables high-performance denoising from routine multi-coil acquisitions, potentially narrowing the gap between self-supervised and supervised methodologies and broadening applicability to real-world clinic data.

A plausible implication is that the C2C paradigm—leveraging redundancy and noise decorrelation for transferring solutions between theoretical and practical domains—may inform future algorithm development in other areas involving measurement redundancy and optimization under noisy, incomplete, or impractical ground-truth conditions.

---

*Primary sources: [2508.09321], [2208.07552]*

Source: https://www.emergentmind.com/topics/coil2coil-c2c-method