Complex Harmonic Cardiac Coordinates

• Complex harmonic cardiac coordinates are a latent representation that unifies modeling of cardiac phase (cycle timing) and amplitude (beat-to-beat variability) in MRI.
• The method integrates a scan-specific CVAE with Hilbert transform and polar representation to robustly capture arrhythmic and variable contraction patterns.
• This framework enhances quantitative T1/T2 mapping by nearly doubling SNR and reducing coefficient of variation compared to conventional approaches.

Complex harmonic cardiac coordinates are a latent representation introduced to unify the modeling of periodic cardiac motion and beat-to-beat functional variability in cardiac magnetic resonance imaging (CMR). Developed within the Generative Multitasking reconstruction framework, they provide a mathematically interpretable mechanism to bridge gated ("cine") and real-time cardiac MRI using a scan-specific conditional variational autoencoder (CVAE). This representation encapsulates both cardiac phase (timing within the heartbeat cycle) and amplitude (capturing variations such as contraction strength or arrhythmic events), thereby supporting phase-resolved, time-resolved, and intermediate reconstructions from a single free-breathing, non-ECG-gated MR acquisition (Fang et al., 22 Nov 2025).

1. Mathematical Formulation of Complex Harmonic Cardiac Coordinates

Let $u_{\mathrm{card}}(t)$ denote the latent cardiac signal extracted by band-pass filtering the encoder’s output at physiological cardiac frequencies (0.67–2.5 Hz). The corresponding analytic signal is computed by obtaining the Hilbert transform, $\mathcal{H}\{\cdot\}$, yielding the imaginary component:

$$v_{\mathrm{card}}(t) = \mathcal{H}\{u_{\mathrm{card}}(t)\}$$

These combine into a complex-valued latent coordinate:

$$z_{\mathrm{card}}(t) = u_{\mathrm{card}}(t) + i \, v_{\mathrm{card}}(t)$$

This admits the polar form:

$$z_{\mathrm{card}}(t) = |z_{\mathrm{card}}(t)|\,e^{i\,\varphi(t)}$$

where

• $$\varphi(t) = \arg z_{\mathrm{card}}(t) = \mathrm{atan2}\left(v_{\mathrm{card}}(t), u_{\mathrm{card}}(t)\right)$$ encodes the instantaneous cardiac phase (location within the cycle),
• $$|z_{\mathrm{card}}(t)| = \sqrt{u_{\mathrm{card}}(t)^2 + v_{\mathrm{card}}(t)^2}$$ represents the latent amplitude that tracks beat-to-beat functional variability.

In this formulation, $\varphi(t)$ produces the cyclic phase structure familiar from ECG-gated cine MRI, while $|z_{\mathrm{card}}(t)|$ captures cycle-to-cycle modulation, enabling representation of phenomena such as arrhythmias or variable contraction strengths observed in real-time imaging (Fang et al., 22 Nov 2025).

2. Integration into Conditional Variational Autoencoders

Within the Generative Multitasking framework, the complex harmonic coordinates are embedded in the latent space of a scan-specific CVAE comprising:

• Encoder $E$: Receives at each time $t$ a feature vector $Q_t$ (from low-rank reconstruction) and a conditional vector $$c_t = [T_{\mathrm{prep}}[t], \mathrm{TI}[t], \mathrm{FA}[t], k[t-\mathrm{TR}]]^T$$ encoding MR sequence timings and the previous k-space angle.
• Latent Output: Two parameter pairs are produced:
  • $$u_{\mathrm{resp}}(t),\ \log \sigma^2_{\mathrm{resp}}(t)$$ (respiratory mean & variance)
  • $$u_{\mathrm{card}}(t),\ \log \sigma^2_{\mathrm{card}}(t)$$ (cardiac mean & variance)
• Filter Bank: Applies low-pass filtering to respiratory and band-pass plus Hilbert transform to cardiac parameters to isolate mode-specific dynamics.
• Sampling: Latents $$z_{\mathrm{resp}}(t)\sim\mathcal{N}\big(u'_{\mathrm{resp}}(t),\sigma^2_{\mathrm{resp}}(t)\big)$$ and $$z_{\mathrm{card}}(t)\sim\mathcal{N}\big(u'_{\mathrm{card}}(t)+i v'_{\mathrm{card}}(t), \sigma^2_{\mathrm{card}}(t)\big)$$ form the input to the decoder.
• Decoder $D$: Combines $$[z_{\mathrm{resp}}, \Re(z_{\mathrm{card}}), \Im(z_{\mathrm{card}})]$$ and $c_t$ to reconstruct $Q_t$.

The training objective jointly minimizes reconstruction error and KL divergence, yielding:

$$\sum_t \|Q_t - D(...)\|^2 + \beta\,\mathrm{KL}\big[E(Q,C)\,\|\,\mathcal{N}(0,I)\big]$$

The result is a latent space with explicit, physiologically interpretable axes for cardiac and respiratory dynamics (Fang et al., 22 Nov 2025).

3. Unified Reconstruction of Cardiac MRI

The decoder’s structure enables three principal reconstruction modes:

• Time-resolved ("real-time-like"): By traversing the empirical (scan-specific) trajectory $$\{z_{\mathrm{resp}}(t), \varphi(t) = \arg z_{\mathrm{card}}(t), |z_{\mathrm{card}}(t)|\}$$, the decoder generates a sequence that faithfully tracks observed beat-to-beat variability.
• Phase-resolved ("gated-like" cine): By fixing respiratory state and specifying a grid of cardiac phases $\varphi_k=2\pi k/N$, setting $z_{\mathrm{card}} = A_0 e^{i\varphi_k}$ with $A_0$ as average or unit amplitude, the decoder produces archetypal cines equivalent to ECG-gated segmented imaging but with access to latent amplitude information.
• Intermediate or customized modes: Arbitrary queries along the latent axes facilitate respiratory-frozen, mean-cycle, or other bespoke reconstructions.

By deferring the gating decision to the reconstruction stage, the complex harmonic framework provides high spatial resolution and maintains robustness to arrhythmic or irregular physiological motion, all from a single acquisition (Fang et al., 22 Nov 2025).

4. Artifact Suppression via Conditional Manipulation

Modeling eddy-current and gradient-memory artifacts is accomplished by including the preceding k-space angle $k[t-\mathrm{TR}]$ in the conditional input $c_t$. During inference, artifact suppression is achieved by replacing this angle with a fixed self-gating angle $K_{\mathrm{SG}}$, thereby producing temporal factors free from trajectory-synchronized peaks (such as golden-angle harmonics), without global smoothing of genuine high-frequency motion:

$$\phi_{\mathrm{eddy-corr}}(t) = D(z_{\mathrm{resp}}(t), \Re z_{\mathrm{card}}(t), \Im z_{\mathrm{card}}(t); [T_{\mathrm{prep}}[t], \mathrm{TI}[t], \mathrm{FA}[t], K_{\mathrm{SG}}])$$

This improves the clarity of high-frequency content in the reconstructed images (Fang et al., 22 Nov 2025).

5. Impact on Quantitative T1 and T2 Mapping

The adoption of complex-harmonic cardiac coordinates in Generative Multitasking leads to substantial improvements in the signal-to-noise ratio (SNR) of quantitative maps. In dual-flip-angle $T_1/T_2$ mapping, measured on seven subjects, the intraseptal coefficients of variation (CoV = $\sigma/\mu$) exhibited significant reductions:

Metric	Conventional Multitasking	Generative Multitasking	p-value
$T_1$ CoV	$0.31 \pm 0.06$	$0.13 \pm 0.04$	$<0.001$
$T_2$ CoV	$0.32 \pm 0.13$	$0.12 \pm 0.05$	$<0.001$

These results indicate that replacing low-rank tensor subspaces with the complex-harmonic CVAE approximately doubles the effective SNR for $T_1$ and $T_2$ maps, without requiring any additional spatial regularization. The mechanism is directly attributed to the latent amplitude component of $z_{\mathrm{card}}(t)$, which preserves beat-to-beat variability crucial for robust mapping (Fang et al., 22 Nov 2025).

6. Significance and Broader Implications

Complex harmonic cardiac coordinates provide a rigorous mathematical and algorithmic framework for describing cardiac motion within a neural generative model. By simultaneously encoding phase and amplitude in a two-dimensional latent space, they unify phase-resolved (gated) and time-resolved (real-time) CMR, enable flexible artifact correction, and enhance quantitative mapping accuracy. The system is scan-specific and does not require ECG gating or separate acquisitions for different reconstruction targets. A plausible implication is that this formalism offers a generalizable approach to multidimensional, motion-encoded CMR in settings with arrhythmic or nonstationary motion, supporting both research and translational clinical scenarios (Fang et al., 22 Nov 2025).