DSHARP: Deep Incompressible Motion Estimation with Sinusoidal-transformed Harmonic Phase for Tagged MRI

Abstract

Tagged magnetic resonance imaging (tMRI) is a valuable tool for visualizing and quantifying tissue deformation in vivo. Its use is often hampered, however, by tag fading, long computation times, and the challenge of ensuring diffeomorphic, incompressible motion fields. In this paper, we describe a novel integration of the harmonic phase (HARP) approach to tMRI analysis with an unsupervised deep learning-based registration framework to estimate 2D and 3D motion fields that are diffeomorphic and nearly incompressible. The resulting method, called deep sinusoidally transformed HARP, or DSHARP, enables end-to-end network training by implementing a transformation of the harmonic phase to remove phase-wrapping discontinuities. It produces diffeomorphic motion by estimating a stationary velocity field from which motion is computed using the scaling and squaring technique. Finally, it encourages incompressibility using a novel Jacobian determinant loss term during network training. We evaluated DSHARP on 2D and 3D phantom data with simulated incompressible motions, real 3D human tongue data acquired during speech from both healthy and glossectomy subjects, and cardiac tagged MRI from the public STACOM 2011 benchmark. Our approach outperforms HARP, SinMod, SyN, PVIRA, VoxelMorph, and DeepTag in tracking accuracy, computation speed, and preservation of incompressibility.

I. Introduction

ESTIMATING the motion of deforming soft tissues in vivo is an important topic in medical imaging research. Magnetic resonance (MR) tagging [1], [2] has become the gold standard imaging modality for quantifying tissue motion and is employed in a variety of applications, including cardiac motion analysis [3], evaluation of musculoskeletal dynamics [4], investigation of tongue movements during speech and swallowing [5]-[7], quantification of brain deformations resulting from mild head impacts [8], [9], and assessment of liver strain [10]. MR tagging achieves this by temporarily magnetizing tissues with spatially modulated periodic patterns, thereby creating transient tags that move with the tissue and encode motion information for both visual interpretation and automated post-processing.

Several techniques have been developed for acquiring tagged MR images. The spatial modulation of magnetization (SPAMM), first introduced by Axel et al. [1], saturates tissue magnetization through position encoding gradients combined with two non-selective RF pulses. This process creates a periodic modulation of the anatomical signal intensity. An improved variant, complementary SPAMM (CSPAMM) [11], enhances tag visibility by combining two separate SPAMM acquisitions at the cost of doubling the total acquisition time.

Automated tracking techniques for SPAMM and CSPAMM generally fall into two categories: image-domain and Fourier-domain methods. Image-domain methods match voxel intensities based on optical flow assumptions, typically including smoothness constraints. However, the underlying assumption of brightness constancy is compromised by intensity variations due to significant tag fading caused by T1 relaxation and pulse sequence progression to steady state [12], [13]. Consequently, intensity-based registration accuracy is limited in practical scenarios [3], [13], [14].

Fourier-domain methods mitigate tag-fading issues by working in the frequency domain. The harmonic phase (HARP) [15]-[17] and SinMod [18] methods are the most widely adopted Fourier-domain techniques, and both approaches have enjoyed further extensions [19]-[23]. All of these methods first isolate spectral peaks in the Fourier domain of tagged MR images using bandpass filters, then take the inverse Fourier transform, and next extract the phase of the resultant images to produce harmonic phase images. The HARP method tracks motion by assuming that the harmonic phase remains constant for material points throughout motion; in this way, tag fading has less impact on motion estimation [3]. Similarly, the SinMod method estimates the motion by dividing the harmonic phase difference across frames by the estimated local frequency. SinMod then uses a heuristic quality map to locally smooth the estimated motion. Our approach is also based on the underlying initial computation of harmonic phase images.

Instead of tracking points independently, as in HARP and SinMod, image registration-based techniques such as PVIRA [21] use harmonic phase images within an image registration framework (e.g., iLogDemons [24]). Since one of the two phase images must deform to match the other, interpolation is required, and this is a challenge because of discontinuities in phase images caused by phase wrapping. PVIRA handles this by locally unwrapping the phase during interpolation. However, phase interpolation with unwrapping is not differentiable (detailed in Section III-B), making it difficult to leverage modern deep learning techniques in an end-to-end learning framework. Global phase unwrapping [25], [26] is one possible solution to this problem, but unwrapping 3D phase is inherently challenging and further complicated by MR imaging artifacts [27].

Incompressibility is another key property when modeling tissue motion in tMRI, as soft tissues such as the myocardium and tongue undergo only minimal volume change during deformation. Empirical studies have reported that myocardial volume varies by less than 4% throughout a cardiac cycle [28], while the tongue exhibits even smaller volumetric variation during speech and swallowing [29]. Thus, incorporating in-compressibility constraints is highly beneficial for realistic and accurate reflection of tissue motion.

In this work, we demonstrate DSHARP on two clinically important domains: cardiac motion and tongue motion during speech. Cardiac motion estimation from tagged MRI is a well-established application [3], yet challenges remain due to tag fading, through-plane motion, and large deformations. We validate our approach on the public STACOM 2011 cardiac benchmark [30] to enable direct comparison with prior work. As a second application, we focus on tongue motion, which presents several distinctive difficulties beyond those encountered in cardiac imaging. We are concerned with tongue motion as roughly 48,000 individuals in the United States each year are diagnosed with oral or oropharyngeal cancers, and about one-third of these involve the tongue [31]. Characterizing how tongue motion differs between healthy volunteers and post-glossectomy patients provides valuable insight for surgical planning and for developing strategies to restore speech and swallowing. Compared with cardiac motion, estimating tongue motion presents several distinctive difficulties: (1) the tongue exhibits rapid motion relative to the temporal sampling rate of the acquisition; (2) its dynamics are aperiodic and vary considerably across speakers; (3) its intricate interdigitated muscular structure leads to rich deformability during articulation [32]; and (4) the presence of air cavities in the oral region often degrades image quality and introduces artifacts.

In this paper, we introduce the deep sinusoidal-transformed HARP (DSHARP) method, which uses a deep registration framework for motion estimation from tMRI. The use of a registration framework for estimating motion from image sequences has many advantages, particularly since the introduction of modern deep learning registration methods [33]. However, registration of raw tagged images leads to very large motion estimation errors [13], [15] and registration of HARP images must contend with phase wrapping artifacts [21]. DSHARP solves this problem by transforming a HARP phase image into two sinusoidal images that do not have discontinuities. Because it is based on HARP phase images, DSHARP is similarly robust to tag fading, and because the transformed HARP images are free of discontinuities, the gradients required for deep network training can be readily computed. The overall DSHARP framework is an unsupervised, multi-channel, deep learning registration framework for 3D tagged MR images with multiple tag orientations. Our method combines the expressive power and rapid inference of deep learning with the tag-fading robustness of HARP. To our knowledge, this is the first integration of HARP with unsupervised deep learning for 3D motion estimation in tagged MRI.

A preliminary version of this work was previously reported in conference form [34]. In this expanded manuscript, we offer a detailed analysis of the non-differentiability inherent to phase interpolation and present new experiments, including tests on 2D and 3D phantoms with simulated incompressible fields, cardiac tagged MRI from the public STACOM 2011 benchmark, and an extended comparative evaluation against SinMod, HARP, PVIRA, VoxelMorph, DeepTag, and MomentaMorph.

II. Background & Related Work

Acquisition of tagged MRI

Fig. 1 illustrates the acquisition of tagged MR images (using the CSPAMM sequence [11]) for capturing three-dimensional tongue motion during speech. Stacks of image slices are acquired in two orthogonal orientations: sagittal and axial. Sagittal slices with vertical and horizontal tagging are used to capture the anterior-posterior and superior-inferior motions, axial slices with vertical tagging are for the left-right motion. Repetitions of motion are required to cover multiple slices. The duration of the recorded phrase is 1 s, and 26 time frames are acquired over this period.

Fig. 1.

Illustration of tag acquisition for three-dimensional motion. Examples of CSPAMM images acquired at two different time points are shown. Orientation abbreviations: A, anterior; P posterior; L, left; R, right; S, superior; I, inferior.

Deep learning-based image registration

Registering two images acquired at different times yields motion between them. Iterative optimization approaches [24], [35], [36] have been successful in achieving good accuracy in intensity-based deformable image registration. However, these methods optimize for each new image pair and can be slow. Alternatively, deep learning-based registration approaches, once trained, can directly take a pair of images as input and estimate the corresponding deformations, resulting in faster inference speeds and comparable accuracy to iterative methods [33], [37]-[41]. Such frameworks learn the function $g_{\theta }(F,M)$ which gives the transformation that is used to align one fixed image $F$ and one moving image $M$. The function $g$ is parametrized by learnable parameters $\theta$, which are learned by optimizing the generalized objective:

$$\begin{array}{l}\widehat{\theta }=\underset{\theta }{\arg \min }{ℒ}_{\mathrm{sim}}\left(F,M\circ g_{\theta }(F,M)\right)\\ +\lambda {ℒ}_{\text{smooth}}\left(g_{\theta }(F,M)\right).\end{array}$$ (1)

The first term, ${ℒ}_{\mathrm{sim}}$, encourages image similarity between the fixed and warped moving image. The second term, ${ℒ}_{\text{smooth}}$, imposes a smoothness constraint on the transformation. The parameter $\lambda$ determines the trade-off between these two terms. Typically, these approaches register one pair of intensity images—one fixed and one moving. But in tMRI, 3D motion is learned from three harmonic phase images, each one providing motion information primarily associated with one cardinal direction. In the tongue motion imaging, these three tMRI orientations are extracted from two sagittal and one axial acquisitions, as described above.

Several deep learning approaches have been proposed for tagged MRI motion estimation. DeepTag [42] introduces an unsupervised CNN-based framework that directly processes raw cardiac tagged MRI sequences to predict motion. SequenceMorph [43] extends DeepTag by leveraging temporal coherence through sequential composition of pairwise deformation estimates. Both DeepTag and SequenceMorph are 2D methods validated on cardiac short-axis imaging. Unlike these methods, which operate directly on image intensities, our approach leverages harmonic phase information through a sinusoidal transformation, providing inherent robustness to tag fading, and is validated on both 2D and 3D data. MomentaMorph [44] addresses the “tag jumping” problem when tracking large motion across multiple frames by estimating initial momenta and using geodesic shooting to model smooth temporal trajectories, with a correction network to handle residual misalignment. Our present work focuses on input representation and incompressibility, while MomentaMorph focuses on temporal trajectory modeling. The two works are complementary: DSHARP’s sinusoidal-transformed HARP images can serve as input to MomentaMorph, combining robust phase-based features with temporally coherent deformation estimation.

Qin et al. [45] proposed a biomechanics-informed neural network for myocardial motion tracking from cardiac cine MRI, incorporating strain-energy-based regularization that includes near-incompressibility constraints. It operates on cine MRI rather than tagged MRI and relies on intensity-based tracking that is not directly applicable to tagged image sequences where tag fading confounds intensity-based similarity measures.

Phase Interpolation & Unwrapping

Unlike intensity images, harmonic phase images have inherent discontinuities caused by phase wrapping. When using harmonic phase images as input for registration, a standard intensity difference metric fails—a difference of nearly $2\pi$ when comparing a value close to $\pi$ with a value close to $-\pi$, even though these values are almost identical after modulo $2\pi$. Fig. 2 shows a comparison between correct and wrong interpolation. Special care must be taken for interpolating warped phase images.

Fig. 2.

Examples of a correct and a wrong phase interpolation at non-integer location x_θ. Green-star points indicate the correct interpolation procedure, while the blue dot indicates incorrect interpolation.

To solve this problem, both global phase unwrapping and local phase unwrapping [26], [46]-[48] can be considered. Global phase unwrapping detects and removes all $2\pi$ jumps to produce a continuous phase map. In tagged MRI, however, it faces several challenges: (1) Its accuracy depends on correctly detecting true wraps. When Itoh’s condition [49] is violated— e.g., low sampling rate (small tag spacing) or high noise (e.g. air-tissue interfaces in the oral cavity)—spurious wraps are often introduced [50], [51] and phase unwrapping accuracy is affected; (2) Conversely, real abrupt phase changes at tissue boundaries (e.g., air-tongue interface) can be mistaken for wraps; (3) A missed unwrapping will propagate errors throughout the image, due to the global solver’s cumulative nature [49]; (4) Temporal consistency across frames is required for phase-based registration, but tag fading leads to varying image quality over time and may cause inconsistent unwraps; (5) Extending from 2D to 3D compounds these issues: there are exponentially more unwrapping paths, discontinuities occur on surfaces instead of lines, and computational cost increases significantly [50].

Local phase unwrapping, by contrast, resolves discontinuities only within a small neighborhood around each interpolation point, limiting error propagation. For instance, PVIRA [21] applies local unwrapping during the interpolation step of phase registration. However, since unwrapping is inherently non-differentiable, it cannot be embedded in modern backpropagation-based training of deep networks (see Section III-B).

In this work, we introduce a novel sinusoidal transformation for harmonic phase images, which obviates local or global phase unwrapping, and allows the use of standard interpolation operators (e.g., trilinear interpolation) within an end-to-end trainable framework.

Incompressibility

Incompressible flows (also known as divergence-free flows or volume-preserving transformations) have long been studied in fluid dynamics [52] and image registration [53], [54]. In medical image registration, two paradigms prevail. The first comprises iterative solvers that explicitly enforce zero divergence of the velocity field via projection, Hodge-Helmholtz decomposition or stepwise regularization [21], [24], [55], [56]. The second relies on determinant-based regularization, which penalizes deviations of the Jacobian determinant of the transformation from unity [57]–[61]. Our approach belongs to the latter category.

Zhou et al. [62] addressed incompressibility in harmonic phase-based motion estimation by using explicit anatomical models—a volumetric mesh of the left cardiac ventricle. Later, PVIRA [21] simplified this strategy by approximating the anatomy mask with harmonic magnitude images and employed the iLogDemons framework [24], which alternates force-driven updates with divergence-free regularization based on diffeomorphic demons [36]. Lopez et al. [60] combined a neo-Hookean hyperelastic PDE with an ${ℒ}_2$ penalty on the deviation of the Jacobian determinant from one to regularize the deformation field parametrized by a physics-informed neural network [63]. Although accurate, these iterative schemes remain computationally intensive.

Jacobian determinant-based regularization has been employed in both classical and deep-learning-based registration to control local volume changes. Dong et al. [61] proposed penalizing the integral of the Jacobian determinant within a region of interest (ROI) to preserve the overall volume, without constraining the local deformation at every voxel. Ruhaak et al. [64] introduced a Volume Change Control (VCC) regularizer within a variational registration framework for lung CT, solved via interior-point methods that require an initial feasible solution. The VCC penalizes volume changes symmetrically, i.e., expanding by factor $k$ and compressing by $1/k$ incur identical penalties. Kervadec et al. [65] proposed a parametric log-barrier extension that eliminates the need for an initial feasible solution required by interior-point methods. Hering et al. [66] extended this line of work by introducing a parametrized linear folding barrier and were the first to integrate such a constraint into a deep-learning registration architecture.

In this work, we introduce a log-Jacobian-based loss. Unlike previous approaches, our formulation does not rely on anatomical masks or explicit biomechanical models. By penalizing deviations of the logarithm of the Jacobian determinant from zero, our formulation treats expansion and contraction symmet-rically. Our formulation requires no parametrized function, and empirically achieves better tracking accuracy and regularization. Furthermore, our learned model processes a single frame pair in less than one second, compared to the minutes required by traditional approaches.

III. Method

The harmonic phase approach offers several advantages, including computational efficiency and greater robustness than intensity-based registration when handling tag fading, inter-frame geometric distortions, and noise [67], [68]. Integrating HARP with modern deep learning frameworks is a natural extension to leverage the strengths of both.

A. Preprocess Tagged MR Images with HARP

Fig. 3 shows the preprocessing steps. Since tagged images are typically acquired with low through-plane resolution, we first upsample each tagged volume to a finer isotropic grid using cubic B-spline interpolation [21], as illustrated in Fig. 3(A-B). In the HARP algorithm, one of the harmonic peaks in the Fourier domain of a tagged image slice is bandpass filtered to yield a complex-valued image whose phase (HARP phase) encodes motion and whose magnitude (HARP magnitude) encodes anatomy. Here, we use peak-combination HARP [69], which extracts two harmonic peaks using two bandpass filters, as shown in Fig. 3(C), and combines them for a better signal-to-noise ratio.

Fig. 3.

3D HARP pre-processing pipeline: process acquired tagged slices into isotropic 3d harmonic phase volumes. $T$: the total number of time frames.

Specifically, for the vertically-tagged axial volume, denoted by $I_{\text{Av}}(\mathit{x})$, the complex image $J_{\text{Av}}$ resulting from HARP filtering is

$$J_{\text{Av}}(\mathit{x})=D_{\text{Av}}(\mathit{x})e^{j{\text{Ψ}}_{\text{Av}}(\mathit{x})},$$ (2)

where $D_{\text{Av}}$ and ${\text{Ψ}}_{\text{Av}}$ are the magnitude and harmonic phase volumes, respectively. The same notation applies to the horizontally-tagged sagittal $\left(I_{\text{Sh}}\right)$ and vertically-tagged sagittal $\left(I_{\text{Sv}}\right)$ volumes, yielding $D_{\text{Sh}}$, ${\mathrm{\Psi }}_{\text{Sh}}$ and $D_{\text{Sv}}$, ${\mathrm{\Psi }}_{\text{Sv}}$. We average the three magnitude volumes to increase the signal-to-noise ratio and denote the averaged HARP magnitude as $D$ (without any subscript). Fig. 3(D) shows an example of the obtained harmonic phase and averaged HARP magnitude volume. We apply this processing at each time point.

B. Non-Differentiability in Phase Interpolation

In the formulation of Eq. (1), to evaluate image similarity loss between a fixed image $F$ and a warped moving image $M\circ g_{\theta }(F,M)$, we must resample $M$ at the transformed coordinates $g_{\theta }(F,M)$. Specifically, a point $x_{\text{fixed}}$ in the fixed image may be mapped to a non-integer location $x_{\theta }$ in the moving image,$x_{\theta }=g_{\theta }(F,M)\left(x_{\text{fixed}}\right)$. Computing $M\left(x_{\theta }\right)$ therefore requires interpolation. Since $F$ and $M$ are harmonic phase images in our case, the interpolation must account for phase wrapping.

We show here that phase interpolation via local unwrapping is not differentiable. For simplicity, we illustrate the one-dimensional case using linear interpolation, although the argument extends naturally to higher dimensions and higher-order interpolation. We define the phase wrapping operator $𝒲(a)$:

$$𝒲(a)=((a+\pi )\mathrm{mod}2\pi )-\pi$$ (3)

$$=a-2\pi ⌊\frac{a+\pi }{2\pi }⌋,$$ (4)

where mod is the modulo operation, and $⌊\cdot ⌋$ is the floor function. $𝒲(a)$ maps any real $a$ into $[-\pi ,\pi )$.

As shown in Fig. 2, given the phase values at integer locations $x_1$ and $x_2$, the interpolated phase at an arbitrary location $x_{\theta }\in \left[x_1,x_2\right]$ can be computed by locally “unwrapping” the jump between two wrapped samples and then re-wrapping the result:

$$\mathrm{\Phi }\left(x_{\theta }\right)=𝒲\left(\mathrm{\Phi }\left(x_1\right)+\frac{x_{\theta }-x_1}{x_2-x_1}\left[𝒰\left(\mathrm{\Phi }\left(x_2\right)\right)-\mathrm{\Phi }\left(x_1\right)\right]\right).$$ (5)

Here, $\mathrm{\Phi }(x)\in [-\pi ,\pi )$ represents the wrapped phase at position $x$. $𝒰(a)=a+2\pi k$ denotes the unwrapping operator, where k ∈ ℤ is chosen so that $\left[𝒰\left(\mathrm{\Phi }\left(x_2\right)\right)-\mathrm{\Phi }\left(x_1\right)\right]\in [-\pi ,\pi ]$. The interpolated value $\mathrm{\Phi }\left(x_{\theta }\right)$ can then be used to compute the image similarity loss $ℒ$.

To train via backpropagation, we would need

$$\frac{\partial {ℒ}_{\text{sim}}}{\partial \theta }=\frac{\partial {ℒ}_{\text{sim}}}{\partial \mathrm{\Phi }\left(x_{\theta }\right)}\frac{\partial \mathrm{\Phi }\left(x_{\theta }\right)}{\partial x_{\theta }}\frac{\partial x_{\theta }}{\partial \theta }.$$ (6)

From (5), the term $\partial \mathrm{\Phi }\left(x_{\theta }\right)/\partial x_{\theta }$ depends on ${𝒲}^{\prime }$. Since $𝒲(a)$ relies on the floor function that has jump discontinuities at integer values, $𝒲(a)$ is discontinuous whenever $(a+\pi )/2\pi \in \mathbb{Z}$, at which ${𝒲}^{\prime }(a)$ is undefined. These discontinuities make the overall operation non-differentiable. Note that unlike ReLU’s single non-differentiable point at $x=0$ which can be handled by using a sub-gradient, the mod function’s infinite discontinuities make it fundamentally different and less amenable to being approximated by a fixed sub-gradient.

C. Sinusoidal-Transformed HARP

To address the issue, we propose to use a sinusoidally transformed HARP. Given a harmonic phase volume $\mathrm{\Psi }$, we compute two continuous intensity volumes by applying element-wise sine and cosine:

$$I_k^{\sin }(\mathit{x})=\sin \left({\text{Ψ}}_k(\mathit{x})\right)\text{and}{I}_k^{\cos }(\mathit{x})=\cos \left({\text{Ψ}}_k(\mathit{x})\right),$$ (7)

where $k\in \{\text{Av},\text{Sh},\text{Sv}\}$. Note each phase value corresponds uniquely to a (sin, cos) pair and vice versa, i.e., one-to-one mapping. This transformation eliminates discontinuities in the HARP volumes, enabling standard interpolation (e.g., linear) during registration while preserving HARP’s robustness to tag fading and noise. Moreover, the flat (low-contrast) regions and steep (high-contrast) regions in sin and cos are compensated for by each other, as annotated in Fig. 4 (within “sinusoidal transform” module), which help preserve good contrast for the subsequent multi-channel registration. Essentially, we turn a single input feature (the harmonic phase) into two features that represent the same information and become differentiable. Although simple, this strategy is neither obvious nor previously reported in tagged-MRI analysis, yet it has proven to be remarkably effective.

Fig. 4.

Diagram of DSHARP. Extracted 3D HARP phase volumes of the fixed and moving images are sinusoidally transformed and fed into a U-Net-style, multi-channel registration network to estimate motion field.

An alternative way to view sinusoidal-transformed HARP $\left(\left(I_k^{\cos },I_k^{\sin }\right)\right)$ is by writing the complex image in Eq. (2) as

$$J_k(\mathit{x})=D_k(\mathit{x})\left(\cos \left({\text{Ψ}}_k(\mathit{x})\right)+j\sin \left({\text{Ψ}}_k(\mathit{x})\right)\right),$$ (8)

where $k\in \{\text{Av},\text{Sh},\text{Sv}\}$. Thus, $\left(I_k^{\text{cos}},I_k^{\text{sin}}\right)$ can be seen as the real and imaginary parts of the complex image $J_k(\mathit{x})/D_k(\mathit{x})$.

D. Incompressibility Loss Function

Volume preservation, also known as incompressibility, is an important feature for image registration in moving biological tissues. We introduce the following Jacobian determinant-based learning objective to encourage incompressibility within tissue while permitting compressibility in the surrounding background or air gap:

$$\begin{array}{l}{ℒ}_{\text{incompress}}=\sum _x{\mathbb{1}}_{[z\ge 0]}D(\mathit{x})|\log (\max (z,ϵ))|\\ +\sum _x{\mathbb{1}}_{[z<0]}|z|,\end{array}$$ (9)

where $z:=\det \left(J_{\varphi }(\mathit{x})\right)$ is the Jacobian determinant of $\varphi$ at $\mathit{x}$, ${\mathbb{1}}_{[\cdot ]}$ denotes the indicator function, and $|\cdot |$ denotes the absolute value. The small constant ε = 10⁻⁵ prevents taking the logarithm of zero for numerical stability. The first term encourages incompressibility and reaches zero when det(J_ϕ(x)) = 1, i.e., when the deformation is locally volume-preserving. The HARP magnitude $D(\mathit{x})\in [0,1]$ serves as a soft tissue mask, obviating the need for explicit anatomical segmentation, which can be cumbersome to generate for every 3D time frame. The second term penalizes local folding by activating only where the Jacobian determinant is negative.

To highlight the differences between our formulation and prior work, we briefly review alternatives for Jacobian-based volume preservation. Recently, Hering et al. [66] extended the Volume Change Control (VCC) regularizer [64], and introduced a loss parametrized by a linear folding barrier and integrated it with a deep-learning registration architecture. The loss is defined as:

$$𝒱(\varphi ):=\sum _x{\psi }_t\left(\det J_{\varphi }(x)\right)\text{d}x,$$

and

$${\psi }_t(z)=\left\{\begin{array}{ll}\frac{{(z-1)}^2}{z}\hfill & \text{if}z\ge t,\hfill \\ \left(1-\frac{1}{t^2}\right)z+\frac{2(1-t)}{t}\hfill & \text{otherwise,}\hfill \end{array}\right.$$

with $z:=\det J_{\varphi }(x)$. The hyperparameter $t>0$ controls the linear folding barrier for $z<t$. For $z\ge t$, the penalty is symmetric, i.e., ${\psi }_t(z)={\psi }_t(1/z)$.

Figure 5 plots different formulations. Theoretically, our formulation offers four additional advantages: (i) $|\log z|\to \infty$ as $z\to 0^{+}$, providing a natural logarithmic barrier against folding, whereas Hering et al.’s linear barrier for $z<t$ depends on the choice of $t$ and penalizes folding less strongly for larger $t$. (ii) Our formulation is symmetric everywhere when det $(J)>0$, whereas Hering et al.’s symmetry only holds for $z\ge t$; as for $z<t$, the linear branch breaks symmetry $\left({\psi }_t(z)\ne {\psi }_t(1/z)\right)$, which leads to imbalanced penalization of expansion and compression in the folding regime. Practically, (iii) our formulation does not contain parametrized functions, simplifying parameter tuning and searching. (iv) Our formulation spatially weights the incompressibility loss by the HARP magnitude $D(\mathit{x})$, which naturally restricts the constraint to tissue regions without requiring segmentation or additional preprocessing. This is particularly important in tagged MRI, where the region outside the tissue (air, bone) should not be constrained to be incompressible. In comparison, Riihaak et al. [64] and Hering et al. [66] apply the Jacobian penalty uniformly over the entire image domain or require explicit anatomical masks.

Fig. 5.

Comparison of three incompressibility penalty formulations as a function of the Jacobian determinant $\mathbf{det}\left({\mathit{J}}_{\mathit{\varphi }}\right)$. Left: the L2 penalty ${(\mathbf{det}(\mathit{J})-\mathbf{1})}^{\mathbf{2}}$ penalizes expansion and compression asymmetrically, for example, $\mathit{\psi }(\mathbf{2})=\mathbf{1}$ but $\mathit{\psi }(\mathbf{0.5})=\mathbf{0.25}$. Note that symmetry here refers to whether expansion by factor $\mathit{k}$ and compression by $\mathbf{1}/\mathit{k}$ incur equal penalties, not the visual symmetry of the plotted curve. Middle: Hering et al.’s piecewise ${\mathit{\psi }}_{\mathit{t}}$ at three thresholds $(\mathit{t}=\mathbf{0.1},\mathbf{0.2},\mathbf{0.4})$, symmetric for $\mathit{z}\ge \mathit{t}$ with a linear folding barrier for $\mathit{z}<\mathit{t}$. Right: our log-Jacobian formulation with incompressibility $\text{(}|\mathbf{log}\mathbf{det}(\mathit{J})|)$ and folding penalty terms; this is symmetric everywhere for $\mathit{z}>\mathbf{0}$ since $|\mathbf{log}\mathit{z}|=|\mathbf{log}(\mathbf{1}/\mathit{z})|$.

We note that $|\log z|$ is non-smooth at $z=1$; however, it is convex with a well-defined subgradient at that point, so PyTorch’s automatic differentiation handles it seamlessly. The discontinuity at $z=0$ due to the indicator-function partitioning is inconsequential in practice: the logarithmic barrier ($|\log z|\to \infty$ as $z\to 0^{+}$) drives the optimizer away from $z=0$ long before the discontinuity is encountered. We observe no optimization issues during training.

E. Multi-Channel Registration

Harmonic phase is a material property that can be used to solve the aperture problem in optical flow by tracking the three harmonic phase values that come from three linearly independent tag directions. Fig. 4 shows a block diagram of DSHARP. Because of the one-to-one nature of the sinusoidal transformation, we track patterns in the sinusoidally trans-formed image pairs instead of tracking harmonic phase values directly. Rather than directly estimating the deformation field $\varphi$, we estimate the stationary velocity field $v$ from a stack of fixed $I_k^{l,f}$ and moving $I_k^{l,m}$ sinusoidal-transformed HARP images:

$$\mathit{v}=g_{\theta }\left(\left\{I_k^{l,f},I_k^{l,m}\mid k\in \{\text{Av},\text{Sh},\text{Sv}\},l\in \{\sin ,\cos \}\right\}\right).$$ (10)

where $g_{\theta }$ denotes the deep networks with learnable parameters $\theta$. The deformation field $\mathit{\varphi }$ is then computed via exponential mapping,

$$\mathit{\varphi }=\exp (\mathit{v}),$$ (11)

which encourages diffeomorphism. In practice, we approximate the exponential mapping using the scaling-and-squaring method [70].

To jointly align multiple sinusoidal image pairs, we use mean squared error (MSE) as the similarity loss during training:

$${ℒ}_{\text{sim}}=\sum _{k\in \{\text{Av},\text{Sh},\text{Sv}\}}\sum _{l\in \{\text{sin},\text{cos}\}}\mathrm{MSE}(I_k^{l,f},I_k^{l,m}\circ \mathit{\varphi }).$$ (12)

Overall training objective

We encourage the spatial smooth-ness of the displacement $\mathit{u}=\varphi (\mathit{x})-\mathit{x}$, with the smoothness loss ${ℒ}_{\text{smooth}}={\sum }_{\mathit{x}}\Vert \nabla \mathit{u}(\mathit{x}){\Vert }^2$. Thus, the overall loss for training is

$${ℒ}_{\text{total}}={ℒ}_{\text{sim}}+\lambda {ℒ}_{\text{smooth}}+\beta {ℒ}_{\text{incompress}},$$ (13)

where $\lambda$ and $\beta$ are hyperparameters whose values are determined empirically.

Network architecture

We adopt LKUnet [40], a U-Net [71]-based architecture, as our backbone $g_{\theta }$. LKUnet employs larger convolutional kernels, which improves the network’s ability to capture larger motions [40]. To accommodate our multi-channel input, we modify the input channel of the first convolutional layer to accept multi-channel images. For 3D data, the input consists of 12 channels, formed by the combination of the fixed and moving images, the sinusoidally transformed HARP components (i.e., sin and cos terms), and the three orthogonal tag acquisitions, resulting in a total of 2 × 2 × 3 = 12 channels. For 2D data, the input comprises 8 channels, following the same formulation but with two acquisitions (2 × 2 × 2). Note that the design of the sinusoidal-transformed HARP representation and the incompressibility loss is independent of the underlying network architecture. Thus, while LKUnet is used here, the proposed framework can readily be applied to alternative architectures.

IV. Experiments

Simulated incompressible motion

Since true voxel-wise motion of human data is difficult to acquire, we first simulate 2D and 3D incompressible motion to evaluate DSHARP’s performance. These simulations aimed to produce variations in the amount of displacement, motion type, and tag fading.

To generate a 3D incompressible (volume-preserving) deformation field, we introduce a vector potential field $\mathbf{A}$, and by vector calculus, the divergence of its curl vanishes:$\nabla \cdot \mathbf{v}=\nabla \cdot (\nabla \times \mathbf{A})=0$. Rather than generating $\mathbf{A}$ in the spatial domain, we construct it directly in Fourier space. Specifically, we (1) sample random complex values for the non-redundant half of the Fourier spectrum $\widehat{\mathbf{A}}(\mathbf{k})$ (with $k_x\ge 0$); (2) enforce Hermitian symmetry so that $\mathbf{A}$ is real-valued in the spatial domain; (3) set the zero-frequency mode $\widehat{\mathbf{A}}(\mathbf{0})=\mathbf{0}$ to eliminate global shifts; and (4) apply a spectral filter $H(\Vert \mathbf{k}\Vert )=\exp \left[-{\left(\Vert \mathbf{k}\Vert /k_0\right)}^2\right]$ to control smoothness via the cutoff frequency $k_0$.

By the Fourier differentiation property, the velocity follows directly as $\widehat{\mathbf{v}}(\mathbf{k})=i\mathbf{k}\times \widehat{\mathbf{A}}(\mathbf{k})$, and transforming back with an inverse FFT yields a real-valued divergence-free velocity field in the spatial domain. This approach avoids finite difference approximations for computing spatial gradients. We then solve the deformation map $\mathit{\varphi }$ defined by the ordinary differential equation $d\mathit{\varphi }/d\tau =\mathbf{v}(\mathit{\varphi }(\tau ))$, $\mathit{\varphi }(0)=\mathbf{I}$, where $\mathbf{I}$ is the identity mapping. We integrate the velocity field using the fourth-order Runge-Kutta scheme.

To generate a 2D incompressible (area-preserving) motion field, we introduce a scalar stream function $\psi (x,y)$, whose level sets coincide with the flow’s streamlines. We then define the velocity field $\mathbf{v}(x,y)=(\partial \psi /\partial y,-\partial \psi /\partial x)$ which satisfies the zero divergence $\nabla \cdot \mathbf{v}=0$ by construction. Similarly to 3D, we construct $\psi$ directly in Fourier space and use the Fourier differentiation property.

Physical 2D phantom with simulated incompressible motion

To validate performance in 2D, we acquired single-slice 2D tagged MRI data from a static cylindrical gel phantom (made from Sylgard^® 527) using a 1–1 SPAMM tagging pulse, with tag periods of 9 and 12 mm. Both horizontally- and vertically-tagged sequences were acquired. The duration was 1.1s with an 11 ms temporal resolution, yielding 100 time frames. We simulated 50 unique 2D incompressible motion sequences and applied them to each of the acquired tagged sequences, yielding $N=100$ tag sequences each with 100 time frames. The imaging parameters were TR = 3.67 ms, TE = 1.63 ms, and a 128 × 128 voxel matrix with a 2 × 2 mm resolution. The 100 sequences were split into training, validation, and test datasets in a ratio of 4:2:4. Hyper-parameters are tuned based on the performance on the validation set. Results are reported on the test set.

Digital 3D phantom with simulated incompressible motion

To validate the 3D performance, we constructed a 3D cylindrical phantom with an isotropic voxel size of 1 mm and a tag period of 5 mm. Tag fading was simulated using a Bloch-equation-based model [13] with a 1–1 SPAMM sequence and a 64 × 64 × 64 voxel matrix. We generated 200 unique 3D incompressible motion sequences, each comprising three time frames, and applied them to deform the tagged volumes. The dataset was divided into training, validation, and testing sets with a ratio of 6:2:2, respectively.

Human tongue during speech

The study comprised a dataset of 25 distinct tagged MRI sequences, each containing 26 three-dimensional time frames and capturing a participant articulating a short phrase. Eight healthy volunteers and two post-partial glossectomy patients with tongue flaps were enrolled. During data acquisition, participants were instructed to pronounce one or more target phrases—“a thing,” “a souk,” or “a shell”—while tagged MR images were obtained. All participants gave written informed consent, which was approved by the University of Maryland Baltimore Institutional Review Board.

The study used a 3T Prisma MR scanner and a 64-channel head/neck coil to obtain images of the tongue’s motion during speech. Participants repeated a phrase while tagged MR images were acquired with a field of view of 240 × 240 mm², TE = 1.47 ms, and TR = 36 ms. Each data set includes a stack of images covering the entire tongue and surrounding tissues. The recorded phrases had a duration of 1 second, with 26 time frames captured during this period. The in-plane resolution of the MR images is 1.875 × 1.875 mm, the slice thickness is 6 mm, and the tag period is 12 mm. The tagged MR images were collected using the CSPAMM pulse sequence in both sagittal (both vertical and horizontal tags) and axial (vertical tags only) orientations. The two SPAMM datasets collected during each CSPAMM acquisition were subtracted before the preprocessing steps. Data from healthy controls were divided into 12, 4, and 4 tagged MR sequences for training, validation, and testing, respectively, following a 5-fold cross-validation scheme. Additionally, 5 tagged MR sequences from post-partial glossectomy patients were reserved exclusively for testing.

Cardiac tagged MRI (STACOM 2011)

We further evaluated DSHARP on the public STACOM 2011 cardiac motion tracking challenge [30]. Each subject’s data contains three orthogonal tagging orientations (short-axis, long-axis 2-chamber, and long-axis 4-chamber). The number of time frames varies across subjects, ranging from 21–38 time frames. For each subject, 12 myocardial landmarks were independently annotated by two observers throughout the cardiac cycle, yielding 24 reference point trajectories per subject that serve as ground truth. Registration accuracy was assessed by propagating landmarks from the first time frame using the estimated motion fields and computing the Euclidean distance to the ground truth positions. For DSHARP, we adopted a 5-fold cross-validation protocol: the 15 subjects were split into 5 folds, with each model trained on 4 folds and evaluated on the held-out fold, rotating across all folds so that every subject is evaluated exactly once. Importantly, DSHARP is trained in a fully unsupervised manner—no landmark annotations are used during training.

Methods for Comparison

For the 2D physical phantom study, we compared DSHARP with HARP, SinMod, DeepTag [42], and PVIRA. For the 3D digital phantom, human tongue study, and cardiac data, we compared DSHARP with SyN [72] and PVIRA [21]. We additionally evaluated VoxelMorph [37] and MomentaMorph [44] as alternative registration backbones in an architecture ablation study on the 3D digital phantom. We used the HARP, PVIRA, and MomentaMorph software provided by the original authors. The bandpass filter radius was set to half of the tagging frequency for the SPAMM sequence and to the tagging frequency for the CSPAMM sequence. For our SyN implementation, we employed the ANTs library [35], using cross-correlation as the image similarity metric. The source code of SinMod is not publicly available, and thus we implemented it in Python following the original publication. The size of the squared cosine kernel used for smoothing was set to 15 and the exponent for quality measurement was set to 8, as in [18]. We used DeepTag ¹ and VoxelMorph ² implementations from the original authors. Because DeepTag was originally designed for grid-tag patterns, we doubled the number of input channels in the network’s first layer to accommodate separate horizontal and vertical tagged images as a two-channel input. A similar modification was done for VoxelMorph. The training procedure followed the original paper, with the same loss functions, and hyperparameters were grid-searched as the authors suggested. The models with the lowest loss on the validation set were selected for evaluation.

Training details

During training, fixed and moving image pairs were randomly selected from a sequence. We augmented each pair of images with random center-crops during training. No overfitting was observed. The hyperparameters $\lambda$ and $\beta$ were determined by grid search for each model to ensure fair comparison. We empirically set $(\lambda ,\beta )$ as (0.08, 0.4) in the 3D digital phantom and human tongue study, and (10,100) for 2D physical phantom study. We empirically set $\lambda =0.08$ for the “DSHARP w/o inc.” model where the ${ℒ}_{\text{incompress}}$ was not applied. For scaling and squaring, we used 7 as the number of iterations for all model variants. In all experiments, we used the Adam optimizer with a batch size of one and a fixed learning rate of 1 × 10^–4 throughout training. The training procedure took approximately 10, 25, and 30 hours for the 2D physical phantom, 3D digital phantom, and tongue study, respectively, using a single Quadro RTX 5000 GPU. For human tongue data, the images were manually cropped to include the tongue region and then zero-padded to 64 × 64 × 64. For phantom data, we kept the original image size. PVIRA, HARP, SyN, and SinMod were tested on a Linux machine with 20 Intel(R) Xeon(R) Silver 4208 @ 2.10GHz CPUs.

V. Results

A. Evaluation Metrics

Metrics for Registration Accuracy

For the two phantom studies, where we do have ground truth motion, we report the end-point error (EPE), $\mathrm{EPE}(\mathit{x})={‖{\mathbf{u}}_{\text{gt}}(\mathit{x})-{\mathbf{u}}_{\text{est}}(\mathit{x})‖}_2$, the magnitude of the difference between the ground truth and estimated displacement fields. In the tongue study, true 3D dense motion fields are difficult to obtain. Because the harmonic phase moves with tissue, a more accurate field should yield better voxel-wise intensity alignment between fixed and warped phase images. Therefore, we use the root mean squared error (RMSE) between fixed and warped sinusoidally transformed images as a proxy for registration accuracy. Unless otherwise specified, we adopt the Lagrangian convention, using the first time frame as the reference.

Metrics for Incompressibility

A perfectly incompressible flow field has a det $\left(J_{\varphi }(\mathit{x})\right)$ of 1 everywhere. To assess incompressibility, we first compute the histogram of determinant errors (i.e.,$\left|\det \left(J_{\varphi }(\mathit{x})\right)-1\right|$) within the region of interest. We then calculate the area under the cumulative distribution function (CDF) curve (AUC) as a scalar metric between 0 and 1. A higher AUC indicates a more incompressible motion field. Intuitively, a left-leaning CDF curve (with a higher AUC) indicates better preservation of incompressibility, because more determinants are closer to unity. We use the Det_AUC metric instead of mean absolute error because Det_AUC is less sensitive to outliers.

B. Evaluation on 2D Physical Phantom

As shown in Fig. 6, DSHARP achieves the best incompressibility (Det_AUC ≈ 1.0) and low tracking error (mean EPE and 95th EPE), especially in later time frames. DeepTag performs comparably at early time frames but degrades severely as tag fading becomes more pronounced in later time frames, which is likely because it operates directly on raw tagged image intensities that violate the brightness constancy assumption under tag fading. In terms of computational speed, DSHARP (33.33 fps) is comparable to DeepTag (35.70 fps) and substantially faster than PVIRA (0.04 fps) and HARP (0.67 fps). Figure 7 shows a qualitative comparison of the estimated motion fields, displacement magnitude maps, and Jacobian determinant maps at t=1.1s.

Fig. 6.

Quantitative evaluation on the 2D physical phantom across time frames: (a) incompressibility (Det_AUC), (b) mean end-point error (mean EPE), (c) 95th-percentile EPE, and (d) computational speed in frames per second (fps). $\mathit{N}=\mathbf{100}$.

Fig. 7.

Qualitative comparison on the 2D physical phantom at $\mathit{t}=\mathbf{1.1}\mathrm{s}$. Top row: estimated motion fields. Middle row: displacement magnitude maps (colorbar in pixels). Bottom row: Jacobian determinant maps (ideal = 1.0 for incompressible motion). The rightmost column: ground truth.

C. Evaluation on 3D Digital Phantom

Figure 8 demonstrates that DSHARP achieves superior performance in 3D motion tracking under both small and large deformations. In contrast, direct registration of raw tagged images using the SyN algorithm yields the poorest performance, as the tag patterns fade over time. Interestingly, SyN achieves much better performance when applied to our multi-channel sinusoidal-transformed HARP (sHARP) images. Figure 9 provides a visual comparison of the evaluated methods.

Fig. 8.

Quantitative evaluation of 3D digital-phantom experiments: (a) incompressibility, measured by Det_AUC; (b) mean end-point error (mean EPE); (c) 95th-percentile EPE. When comparing DSHaRp with PVIRA, all differences are statistically significant at $\mathit{p}<\mathbf{0.001}$ according to the Wilcoxon signed-rank test; and (d) computational speed in frames per second (fps). N=40.

Fig. 9.

Qualitative results of 3D digital phantom experiments. (A) Illustration of imaging plane. (B) The incompressible motion field, from time frame (TF) 1 to 3. (C) Example of tagged images with tag fading. (D-E) Magnitude and Jacobian determinants of deformation (TF1–3) shown in three orthogonal orientations.

Ablation on registration architecture

To evaluate the generalizability of the proposed sHARP representation and incompressibility loss, we compare three registration backbones: VoxelMorph [37], LKU-Net [40], and MomentaMorph [44]. Each backbone is trained with raw tagged images and with our sHARP input (denoted DSHARP). TABLE I shows the results on the 3D digital phantom. Several observations can be drawn: (i) sHARP input consistently improves registration performance in all architectures. For every backbone, the sHARP-based variant outperforms its raw-tagged counterpart across all metrics. This shows that the sinusoidal-transformed HARP representation provides inherent robustness to tag fading, and is architecture-agnostic. (ii) All DSHARP variants achieve substantially higher Det_AUC than their counterparts, indicating that the proposed log-Jacobian incompressibility loss effectively enforces volume preservation regardless of the registration backbone. (iii) Among architectures, MomentaMorph (MM) and LKU-Net achieve comparable performance, and both are better than VoxelMorph. LKU-Net’s large-kernel design and MM’s incorporation of temporal information are well-suited for capturing large tongue motions, which likely explains their superior performance over VoxelMorph.

TABLE I.

Ablation on registration architectures.

Method	Input	Mean EPE (mm) ↓	95% EPE (mm) ↓	Det_AUC ↑
VM [37]	raw tagged	1.05 ± 0.38	1.78 ± 0.71	0.893 ± 0.024
VM (DSHARP)	sHARP	0.58 ± 0.20	1.12 ± 0.37	0.968 ± 0.012
LKUnet [40]	raw tagged	0.72 ± 0.25	1.55 ± 0.47	0.923 ± 0.017
LKUnet (DSHARP)	sHARP	0.34 ± 0.12	0.49 ± 0.16	0.990 ± 0.005
MM [44]	raw tagged	0.69 ± 0.19	1.48 ± 0.38	0.922 ± 0.014
MM (DSHARP)	sHARP	0.35 ± 0.10	0.47 ± 0.13	0.993 ± 0.004

Results on 3D digital phantom (TF 1→3). VM = VoxelMorph [37], LKUnet = Large Kernel U-Net [40], MM = MomentaMorph [44]. N=40.

Ablation on incompressibility loss

We conducted an ablation study on the 3D digital phantom comparing four incompressibility penalty formulations: no penalty, the simple L2 penalty, Hering et al.’s ${\psi }_t$ (with $t=0.1,0.2,0.4$), and our log-Jacobian loss. Figure 10 summarizes the results. Without any incompressibility penalty, the model achieves the worst incompressibility (measured by Det_AUC), confirming that an explicit volume-preservation loss is necessary for accurate incompressible motion estimation. The L2 penalty ${(\det (J)-1)}^2$ improves incompressibility over the no-penalty baseline. Among the Hering et al. variants, performance is sensitive to the hyperparameter $t$: a smaller threshold $\text{(}t=0.1\text{)}$ yields better incompressibility than larger values $(t=0.2,0.4)$, as the tighter folding barrier more strongly discourages volume change near $\det (J)=0$. However, there is a trade-off between volume preservation and tracking accuracy: the $t=0.1$ variant achieves reasonable incompressibility but incurs a larger EPE than $t=0.2$, suggesting that an overly aggressive folding barrier can distort the deformation field. Our log-Jacobian formulation achieves the lowest EPE and highest Det_AUC simultaneously, which validates the theoretical advantages discussed in the Method section.

Fig. 10.

Ablation study comparing incompressibility penalty variants on the 3D digital phantom (TF 1→3). Box plots show end-point error (EPE, mm) and Det_AUC across six configurations: no penalty, L2 penalty,${(\mathbf{det}(\mathit{J})-\mathbf{1})}^{\mathbf{2}}$ Hering et al.’s ${\psi }_{\mathit{t}}$ at three thresholds $(\mathit{t}=\mathbf{0.1},\mathbf{0.2},\mathbf{0.4})$, and our log-Jacobian formulation.

D. Evaluation on Human Tongue

As shown in Table II, DSHARP significantly outperforms PVIRA in terms of both registration accuracy and incompressibility. When trained without the incompressibility term $\left({ℒ}_{\text{incompress}}\right)$, the model achieves comparable RMSE but often fails to preserve incompressibility and, in some cases, loses diffeomorphic properties. Interestingly, the larger variant (“DSHARP-L”, which doubles the number of channels in each layer) attains the highest registration accuracy while maintaining the best ability to preserve incompressibility. This suggests that increased model expressivity enables a more favorable trade-off between intensity matching and regularity of the deformation field. Fig. 11 shows a visual example: as the subject initiates the phrase “a thing” from a neutral position, the tongue moves back and downward. Our model accurately tracks tags while maintaining incompressibility of the flow fields.

TABLE II.

Quantitative results on human tongue study.

Method	${𝓛}_{inc}$	Registration Acc: RMSE ↓			Incompressibility: Det_AUC ↑			NegDet (%) ↓	Time ↓ (s/pair)
		Mean ± Std	Median	p	Mean ± Std	Median	p	Mean ± Std
PVIRA		0.153 ± 0.053	0.160	†	0.936 ± 0.031	0.935	†	0.000 ± 0.005	49
DSHARP		0.122 ± 0.041	0.126	†	0.862 ± 0.077	0.870	†	0.019 ± 0.039	<1
DSHARP	✓	0.132 ± 0.045	0.137	-	0.950 ± 0.038	0.956	-	0.000 ± 0.000	<1
DSHARP-L	✓	0.122 ± 0.038	0.126	†	0.950 ± 0.039	0.956	†	0.000 ± 0.001	<1

“VM” is VoxelMorph [37] and “LK” is LKU-Net [40]. Wilcoxon signed-rank tests between “DSHARP (with arch LK)” and each other method were performed, with p-values adjusted via Bonferroni correction.

^†The dagger denotes p < 0.001. The best result in each column is in bold.

Fig. 11. (A).

An example of a fixed and moving frame pair is shown, with only the sin pattern displayed (the cos pattern is omitted). The contour of the tongue region is annotated in the fixed image with a red dotted line. (B-D) Results of three different methods are shown, including 3D motion field, the warped moving images, and a sagittal and an axial slice of the 3D Jacobian determinant map. Major differences are highlighted with red circles.

Generalizability to pathological subjects

Our model further shows superior performance on post-glossectomy patient data (Fig. 12). Note that it was trained exclusively on healthy control subjects and then applied directly to the patient cohort, demonstrating its generalization ability. To leverage the unsupervised framework for enhanced accuracy, we employed an instance-specific optimization at test time. Specifically, for each test pair we ran gradient descent for 30 iterations to minimize the loss in Eq. (13). As shown in Fig. 12, the instance-specific optimization (denoted “-inst”) yields higher registration accuracy while preserving comparable Det_AUC.

Fig. 12.

Performance on the pathological cases. The suffix “-inst” indicates the results with instance-specific optimization. Symbols † and ‡ indicate significance at $\mathit{p}<0.001$ under the Wilcoxon signed-rank test; $\oslash$ denotes no significant difference ($\mathit{p}<\mathbf{0.07}$).

E. Evaluation on Cardiac Tagged MRI

We evaluated SyN (with MSE and cross-correlation similarity), PVIRA, SyN+sHARP, and DSHARP on tagged MR sequences of 15 healthy volunteers from the STACOM 2011 cardiac motion tracking challenge [30]. Figure 13 shows the quantitative results. DSHARP achieves a median tracking error of 0.88 mm, approaching the inter-observer variability (IOV: 0.87 mm) and outperforming all other approaches. The improvement of SyN+sHARP (0.93 mm) over SyN with MSE (1.44 mm) confirms the benefit of the HARP-based representation across anatomical domains. Pairwise Wilcoxon signed-rank tests (Bonferroni corrected) were performed on every pair, and every pair reached significance (p < 0.01). In [30], four institutions (MEVIS, IUCL, UPF, and INRIA) analyzed the same volunteer datasets using their own tracking methods, with reported median tracking errors of MEVIS = 1.33 mm, IUCL = 1.52 mm, UPF = 1.09 mm, and INRIA = 1.32 mm. Our approaches, SyN+sHARP and DSHARP, achieve median values of 0.93 mm and 0.88 mm, respectively, outperforming all four institutional methods. In addition, DSHARP requires less than 1 s per frame for inference once trained, compared to tens of minutes of runtime for SyN or PVIRA.

Fig. 13.

Landmark tracking error (mm) on the STACOM 2011 cardiac tagged MRI benchmark (15 healthy volunteers). Violin plots show the distribution of 3D Euclidean tracking error across all landmarks and time frames. The dashed line indicates the inter-observer variability (IOV = 0.87 mm).

Figure 14 shows qualitative tracking results on two subjects. Tag contrast fades and SNR degrades progressively over the cardiac cycle, while rapid dephasing of blood tags and the disappearance of tag lines near myocardial boundaries pose additional challenges, especially in late time frames [3], [15], [73]. Despite these degradations, SyN+sHARP and DSHARP achieve the most accurate landmark tracking throughout the full cardiac cycle.

Fig. 14.

Qualitative 3D landmark tracking results on two STACOM cardiac subjects. Top panels: ground-truth landmark positions at end-diastole. The sequence of tagged MRI in three orientations (SAX, LAX 2-chamber, LAX 4-chamber). Bottom panels: predicted 3D landmark trajectories at end-systole and end-of-cycle for each method, color-coded by cardiac region. SyN (CC) and SyN+sHARP are omitted for space and also because their trajectories are visually similar to the methods shown here at this scale.

VI. Conclusion

We proposed DSHARP, a deep registration framework for estimating 2D and 3D incompressible motion from tagged MRI. At its core, we introduce sinusoidal-transformed HARP, a simple yet effective approach that integrates tag-fading-robust harmonic phase with the expressive power of deep learning. To enforce physical plausibility, we employ a determinant-based objective to learn incompressible motion fields. Compared with competing techniques, DSHARP is faster, more robust to tag fading and large motion, and can generalize well to previously unseen pathological data. We validated DSHARP on both 2D and 3D data spanning simulated phantoms with known ground-truth motion, real 3D human tongue imaging during speech, and cardiac tagged MRI from the public STACOM 2011 benchmark, on which DSHARP achieves tracking accuracy approaching the inter-observer variability. These results demonstrate the generalizability of our approach across anatomical domains and imaging conditions.