FractMorph-Light: 3D Registration and Fracture Analysis

• FractMorph-Light is a dual-use term representing both a lightweight 3D deformable registration model in medical imaging and a compact morphological pipeline for archaeological fracture analysis.
• The medical variant leverages a fractional Fourier-based multi-domain transformer encoder–decoder to reduce parameters by over 50% while maintaining registration accuracy on cardiac MRI data.
• The archaeological variant employs Lipschitz-based facet analysis and signed distance transforms to extract multi-scale open and closed fracture surfaces while preserving geometric complementarity.

Searching arXiv for the cited FractMorph papers to ground the article in the relevant literature. FractMorph-Light is a homonymous term used for two distinct lightweight research systems in the arXiv literature. In medical image analysis, it denotes the reduced-capacity variant of FractMorph, a fractional Fourier-based multi-domain transformer for 3D deformable image registration (DIR), designed to preserve most of the full model’s registration accuracy while substantially reducing parameter count and inference memory (Kebriti et al., 17 Aug 2025). In archaeological shape analysis, it denotes a compact implementation of complementarity-preserving fracture morphology, built around morphological scale spaces, Lipschitz fracture facets, and a signed-distance-transform embedding for simultaneous opening and closing of fracture surfaces (ElNaghy et al., 2019). The shared name conceals major differences in objective, representation, and mathematics: one predicts dense displacement fields for paired medical volumes, whereas the other generates simplified fracture-surface geometries across scales.

1. Terminological scope and domain separation

In the medical-imaging usage, FractMorph-Light is a lightweight variant of a 3D dual-parallel transformer encoder–decoder for aligning a fixed volume $I_f$ and a moving volume $I_m$. Its output is a dense deformation field $\phi$ used to warp the moving image by a spatial transformer with trilinear interpolation (Kebriti et al., 17 Aug 2025).

In the archaeological usage, FractMorph-Light is a compact pipeline for hierarchically simplifying fracture surfaces of scanned fragments while preserving geometric complementarity across scales. Its outputs are simplified opened and closed fracture surfaces at target scales $\{\rho_i\}$, rather than voxelwise correspondences or displacement fields (ElNaghy et al., 2019).

A common source of confusion is that both usages involve 3D geometry, “fracture” or deformation structure, and multi-scale processing. However, they belong to different problem classes. The medical system is a learned end-to-end registration model. The archaeological system is a morphology-based geometric analysis pipeline grounded in set operations, Lipschitz assumptions, and distance transforms.

2. FractMorph-Light in deformable image registration

FractMorph-Light inherits the same overall dual-parallel transformer encoder–decoder backbone and the multi-domain fractional Fourier Transform (FrFT)-based Fractional Cross-Attention (FCA) modules from the full FractMorph model (Kebriti et al., 17 Aug 2025). In both models, the fixed and moving volumes are split into non-overlapping 3D patches of size $P_x\times P_y\times P_z$, embedded into $d$-dimensional tokens, and processed in two parallel streams of transformer blocks.

At each level $l$, the FCA module enriches each stream through four parallel FrFT branches applied to layer-normalized features: orders $0^\circ$, $45^\circ$, $90^\circ$, and a log-magnitude branch of the $I_m$0 transform. Each branch is projected back to the spatial domain with an inverse FrFT; the branch outputs and a residual skip connection are concatenated and fused by a $I_m$1 convolution. After this multi-domain enrichment, cross-attention links the fixed and moving streams bidirectionally at every spatial–spectral scale.

The encoder downsamples spatial resolution by 2 while doubling channel width at each level; the decoder mirrors this by upsampling and skip connections. The defining modification in FractMorph-Light is a channel-width reduction inside the FrFT branches by a constant coefficient $I_m$2. If the incoming tensor has $I_m$3 channels, each branch processes $I_m$4 channels rather than $I_m$5. Because the convolutional parameters of the four branches scale as $I_m$6, this reduces the FCA convolutional cost to $I_m$7 of the full version, while keeping the number of levels $I_m$8, attention heads, and depth per level unchanged. The pruning therefore targets the FrFT feature-extractor stage rather than the transformer skeleton itself.

3. Mathematical operators, decoding, and optimization in the medical variant

The FrFT component is defined in the source formulation through the 1D transform of order $I_m$9, with angle $\phi$0,

$\phi$1

with kernel

$\phi$2

In 3D, separability is exploited by successive transforms along $\phi$3, $\phi$4, and $\phi$5, and the inverse uses order $\phi$6 on each axis (Kebriti et al., 17 Aug 2025).

After branch fusion, the FCA block forms learned queries, keys, and values from enriched moving and fixed feature maps. With flattened spatial extent $\phi$7, the moving-to-fixed pass is

$\phi$8

with a symmetric fixed-to-moving pass computed in parallel. Each result is reshaped back to $\phi$9, added residually to the original stream, layer-normalized, and passed through a feed-forward MLP with another residual connection.

Following the final transformer decoder, the two refined streams are concatenated along the channel axis, reverse patch unembedding restores full resolution $\{\rho_i\}$0, and a lightweight 3D U-Net predicts the dense displacement field $\{\rho_i\}$1. This decoder comprises three convolutional downsampling blocks with $\{\rho_i\}$2 convolutions, stride 2, and ReLU; a bottleneck block; three transposed-convolution upsampling blocks with stride 2 and skip connections; and a final $\{\rho_i\}$3 convolution (Kebriti et al., 17 Aug 2025).

Training uses the loss

$\{\rho_i\}$4

where $\{\rho_i\}$5 is local cross-correlation and $\{\rho_i\}$6. The optimizer is Adam with learning rate $\{\rho_i\}$7, batch size 1, and 400 epochs on patches of size $\{\rho_i\}$8. The same $\{\rho_i\}$9, learning rate, and architecture are used for both the full and Light variants, with no scenario-specific tuning.

4. Model size, memory profile, and ACDC performance

For the medical FractMorph family, the parameter and memory reductions are explicit. Full FractMorph has 63.9 M trainable parameters and approximately 594 MB GPU memory at inference, whereas FractMorph-Light has 29.6 M parameters and approximately 320 MB memory, corresponding to reductions of 54% and 46%, respectively (Kebriti et al., 17 Aug 2025).

On the held-out ACDC cardiac MRI test set of 50 cases covering LV cavity, myocardium, and RV cavity, the reported performance of FractMorph-Light is numerically close to that of the full model.

Metric	FractMorph-Light	Full FractMorph
Overall DSC	$P_x\times P_y\times P_z$0	$P_x\times P_y\times P_z$1
Average per-structure DSC	$P_x\times P_y\times P_z$2	$P_x\times P_y\times P_z$3
HD95	$P_x\times P_y\times P_z$4	$P_x\times P_y\times P_z$5
% non-positive Jacobian voxels	$P_x\times P_y\times P_z$6	$P_x\times P_y\times P_z$7
Std of $P_x\times P_y\times P_z$8	$P_x\times P_y\times P_z$9	$d$0

For reference, the cited leading baselines achieve the following values on the same benchmark: VoxelMorph, $d$1 DSC and HD95 $d$2; Fourier-Net, $d$3 DSC and $d$4; TransMorph, $d$5 DSC and $d$6; XMorpher, $d$7 DSC and $d$8; and TransMatch, $d$9 DSC and $l$0 (Kebriti et al., 17 Aug 2025). The accompanying qualitative boxplots for LV, myocardium, and RV are described as showing that FractMorph-Light nearly matches the full model’s means and variances while outperforming all baselines. The paper further characterizes the resulting deformations as smooth and topology-preserving.

5. FractMorph-Light in archaeological fracture morphology

In archaeological reconstruction, FractMorph-Light is defined as a compact pipeline for hierarchically simplifying and comparing archaeological fracture surfaces while preserving geometric complementarity across scales and maintaining robustness to abrasion and chipping (ElNaghy et al., 2019). It builds directly on the ideas of “Complementarity-Preserving Fracture Morphology for Archaeological Fragments.”

Its mathematical basis begins with the Lipschitz property of fracture facets. A fracture facet $l$1 satisfies a Lipschitz condition with constant $l$2 if it can be represented as a Monge patch over some plane with respect to a unit direction $l$3,

$l$4

Equivalently, the normals of $l$5 lie within a cone of opening angle $l$6 about $l$7, with $l$8. This guarantees a single visibility direction and prevents self-overlaps after extrusion.

Complementarity is formulated inside a working mask $l$9. For matching fragments $0^\circ$0 and $0^\circ$1,

$0^\circ$2

which enforces

$0^\circ$3

At scale $0^\circ$4, exact complementarity persists between the opened fracture of one fragment and the closed fracture of its mate, and vice versa.

The scale space is generated with classical morphology using the closed ball $0^\circ$5: $0^\circ$6 Duality relations are given by

$0^\circ$7

Because the task concerns fracture boundaries rather than complete solid interiors, the method adopts a boundary-morphology viewpoint in which boundary dilation and erosion are induced by volumetric dilation and erosion of $0^\circ$8, then restricted back to the boundary (ElNaghy et al., 2019).

6. Embedding, algorithmic pipeline, and complexity in the archaeological variant

The key implementation mechanism is an embedding that permits simultaneous extraction of opening and closing from a single signed distance transform. For each Lipschitz facet $0^\circ$9, the method duplicates the facet with inverted normals, extrudes both copies along the Lipschitz direction $45^\circ$0 by at least $45^\circ$1, voxelizes the resulting thin cylinder $45^\circ$2, and computes a signed distance transform $45^\circ$3 (ElNaghy et al., 2019). In this field, the isosurface $45^\circ$4 is the boundary of the $45^\circ$5-dilation of $45^\circ$6, while $45^\circ$7 gives the $45^\circ$8-erosion, corresponding to opening. A single distance transform therefore provides both boundary-closing and boundary-opening surfaces for arbitrary scales.

The full pipeline takes as input a triangular fragment mesh, target scales $45^\circ$9, grid step $90^\circ$0, and maximum scale $90^\circ$1, and produces simplified open and closed fracture surfaces at each scale. The steps are: preprocessing by facet segmentation; normal computation and clustering; minimal bounding-cone fitting to obtain axis $90^\circ$2, half-angle $90^\circ$3, and Lipschitz constant $90^\circ$4; rotation so that $90^\circ$5 aligns with the $90^\circ$6-axis; extrusion by $90^\circ$7; voxelization of the closed extruded volume; signed Euclidean distance transform; and iso-surface extraction for $90^\circ$8 and $90^\circ$9. The implementation notes specify a sparse or dense 3D grid for $I_m$00, a voxel bit-mask for the extruded mesh, and temporary arrays for the two iso-surface extractions.

The stated asymptotic costs are $I_m$01 for Lipschitz cone fitting in the number of facet vertices, $I_m$02 for voxelization, $I_m$03 for the Euclidean distance transform, and $I_m$04 for iso-surface extraction, with memory $I_m$05 floats for $I_m$06 plus a bit-mask. Practical guidance includes padding the grid by $I_m$07, parallelizing distance-transform and marching-cubes operations over blocks, and choosing $I_m$08 such that the maximum quantization error $I_m$09 remains below the archaeological tolerance; for terracotta, $I_m$10 mm is reported as effective.

The experimental testbed consists of real terracotta sherds with meshes of approximately 5–10 K triangles, typical parameters $I_m$11 mm, $I_m$12 mm, six scales $I_m$13 mm, and grid sizes up to $I_m$14. The listed metrics are complementarity gap or overlap within the eroded mask, stability under simulated abrasion modeled as a small opening of size $I_m$15, and runtime and memory footprint versus naïve volumetric morphology or mesh decimation. Reported results include less than 2 s for Lipschitz analysis plus extrusion per 10 K-triangle facet on an 8-core Xeon, approximately 60 s for a single SDT on a $I_m$16 grid, and approximately 50 s to extract all six closed and opened surfaces. Morphological opening is stated to be provably unaffected for $I_m$17, while closing error under abrasion is bounded by $I_m$18; the example given is $I_m$19 mm and $I_m$20, yielding at most $I_m$21 mm. Compared to standard mesh decimation, the method is described as preserving local complementarity under chipping with less than 1 mm worst-case deviation, whereas decimation errors grow linearly with hole depth (ElNaghy et al., 2019).

7. Conceptual relationship and recurrent misconceptions

The two systems named FractMorph-Light are not variants of a single methodological family. The medical version is a learned registration architecture coupling FrFT-based spectral-spatial attention with a lightweight 3D U-Net decoder. The archaeological version is a non-learning geometric morphology pipeline based on Lipschitz fracture facets, morphological opening and closing, and a single signed distance transform per facet (Kebriti et al., 17 Aug 2025).

Their shared label can obscure this distinction. A plausible implication is that references to “FractMorph-Light” should always be resolved by domain and arXiv identifier, because the same term designates either a 29.6 M-parameter DIR model for ACDC cardiac MRI or a compact complementarity-preserving fracture-surface simplification workflow for terracotta sherd reconstruction (ElNaghy et al., 2019). The overlap is nominal rather than algorithmic.

At a higher level, the two usages nonetheless reflect a common design impulse: both seek lightweight reductions of more expensive procedures while preserving structurally important behavior. In the medical case, the preserved property is registration accuracy with smooth, topology-preserving deformations under reduced memory and parameter budgets. In the archaeological case, it is complementarity-preserving multi-scale simplification under abrasion and discretization constraints. This suggests a methodological resemblance only at the level of engineering objective, not at the level of mathematical formalism or software architecture.