Continuous models of anatomy

Deformable registration asks a simple question with a hard answer. Given two scans of different people, what is the smooth spatial warp that carries one onto the other? Get it right and you can compare anatomy across a population, build an atlas, follow a disease over years, or move a hand drawn segmentation onto an unlabelled scan.

Almost every deep learning method answers it by predicting a displacement vector at every voxel of a fixed lattice. That choice is so standard it is rarely defended, and it costs three things quietly. The field only exists where the grid has nodes, so asking what the deformation is between two voxels means interpolating something you never modelled. Its spatial derivatives are piecewise constant, which matters because the Jacobian determinant of the warp is precisely how you judge whether a deformation is physically possible. And the discretisation itself can produce warps that fold, meaning tissue passes through itself.

I wanted to know what happens if the lattice simply goes away.

QueryPhi predicts a stationary velocity field as a continuous function of coordinates. A shared 3D CNN encoder builds a feature pyramid at strides 1, 2, 4 and 8. A three level local cost volume, in the style of PWC-Net and RAFT, produces a coarse to fine velocity estimate with bounded heads and deep supervision. That bound is not a stylistic choice: unbounded heads diverge, taking Dice from 0.41 to 0.02.

The residual detail comes from a modulated SIREN. Each coordinate receives its own FiLM modulation, predicted from a 222 dimensional conditioning vector sampled at the corresponded location rather than the query location. The decoder is zero initialised so training begins exactly at the identity warp, which matters more than it sounds: a model that starts by scrambling the image never recovers.

Scaling and squaring, with ten squarings, turns the velocity into the deformation through its exponential map. Because the field is a function rather than an array, its Jacobian is available analytically through automatic differentiation at any continuous coordinate, not merely at grid nodes.

Training is fully unsupervised. Similarity comes from MIND-SSC, multi window local normalised cross correlation, and an edge based normalised gradient field. Above them sits a confidence weighted correspondence objective: a softmax matching distribution over the MIND cost volume with a Charbonnier penalty and a learned precision that discounts coordinates where the image evidence is ambiguous. Regularisation is a log barrier on the Jacobian determinant, which is the term actually preventing folds, plus diffusion, bending and masked kinetic energy. Segmentation labels are used only to score the result; the training Dice weight is exactly zero.

On IXI atlas to subject registration it attains the highest mean Dice of all ten evaluated methods at 0.7726, with a 95 percent confidence interval of 0.769 to 0.776, and it is best on every anatomy group: subcortical 0.794, cortical 0.710, ventricular 0.814. It also leads on boundary distance at 1.653 mm.

The margin is consistent case by case rather than an average effect. Two sided paired Wilcoxon tests place it above every baseline, including the strongest near fold free one, TransMorph-bspl at 0.7534 with p = 1.5e-8, and the flagship TransMorph at 0.7419 with p = 5.9e-13. All nine comparisons hold at p below 3e-8 after Holm correction.

A Friedman test with Nemenyi post hoc analysis over all ten methods and 115 cases gives it the best average rank at 1.77, against 1.81 for TransMorph-bspl, with both well separated from the rest.

It does this at 5.51 million parameters against 46.8 million for the transformer baselines, which is 8.5 times smaller. The very small methods in the comparison, VoxelMorph and MIDIR at around 0.27 million, are smaller still but far less accurate.

The accuracy comes with a near fold free deformation, 0.0001 percent of voxels. Two baselines record no folds at all, and one of them, TransMorph-bspl, is also the closest on accuracy at 0.7534, so the claim is that QueryPhi leads a field that keeps topology rather than that it is alone in keeping it. The transformers that come nearest on accuracy fold 1.5 to 1.6 percent of voxels, four orders of magnitude more. Holding both at once is the result.

Transferred zero shot to OASIS, a different cohort with different acquisition and a different label protocol, with no retraining at all, it gives the best boundary accuracy of all nine methods in that comparison at 1.502 mm, and retains 91 percent of the published in domain ceiling while staying near fold free.

Because the field is genuinely a function, the deformation can be re-queried at any output resolution after training. I built instruments to verify that rather than assert it: field continuity divergence, super Nyquist spectral energy, and a finite difference to automatic differentiation Jacobian convergence certificate whose relative error falls from 0.26 to 0.023 as the step shrinks.

A control that shares everything except the one thing you are claiming credit for is worth more than another baseline comparison.

Continuity is a property people assert about implicit representations and almost nobody measures. Building the instrument turned out to be worth as much as the model.

Parameter count is not a vanity metric in medical imaging. A model eight times smaller is one that fits alongside a clinical pipeline instead of replacing it.

Reporting the confidence interval and the per structure breakdown, rather than a single mean, is what makes a result survive contact with a careful reader.