Bi-Temporal Deformable Alignment (BTDA)

• BTDA is a unifying design pattern that learns spatial and temporal deformations to align features across time, addressing misalignment in various domains.
• It leverages mechanisms such as deformable convolutions, gating, and optimal transport, with formulations varying by application context.
• Empirical studies in video super-resolution, remote sensing, and registration show that BTDA improves key metrics like PSNR, F1-score, and overall alignment robustness.

Searching arXiv for BTDA and related deformable alignment papers to ground the article in the cited literature. Bi-Temporal Deformable Alignment (BTDA) denotes a family of temporal alignment mechanisms that learn spatially or temporally varying deformations between observations from different time points. The designation is not uniform across the literature. In remote sensing change detection, DC-Mamba explicitly introduces a Bi-Temporal Deformable Alignment (BTDA) module that “explicitly introduces geometric awareness to correct spatial misalignments at the semantic feature level” (Sun et al., 19 Sep 2025). In video super-resolution, TDAN presents the closely related idea as a Temporal Deformable Alignment Network, and the original work explicitly notes that it “does not introduce a separate ‘Bi-Temporal Deformable Alignment (BTDA)’ block by that name,” even though its core operation is feature-level deformable temporal alignment between a reference and a supporting frame (Tian et al., 2018). In time-series domain adaptation, “Match-And-Deform” also describes a BTDA formulation, but there the alignment variable is not a convolutional offset field; it is the coupling of an optimal transport plan over samples with a dynamic time warping path over timestamps (Painblanc et al., 2023). Across these usages, BTDA is best understood as a general alignment principle: temporal correspondence is modeled by learned, data-dependent deformation rather than by fixed-grid matching alone.

1. Scope, nomenclature, and problem class

BTDA appears in multiple research areas because temporal data are frequently misaligned for reasons that differ in physical origin but are similar algorithmically. In video super-resolution, “the reference frame and each support frame are not aligned” because of “varying motion of cameras or objects,” and temporal alignment is therefore a central bottleneck (Tian et al., 2018). In remote sensing change detection, existing methods “often lack explicit mechanisms to handle geometric misalignments and struggle to distinguish subtle, true changes from noise,” which motivates a module that reduces “pseudo-changes” before classification (Sun et al., 19 Sep 2025). In breast MRI registration, the challenge is “dense tissue” with “highly non-rigid” deformation across time points, so alignment must capture both global structure and local tissue motion (Chen et al., 19 May 2025). In time-series domain adaptation, temporal shifts coexist with domain shift, requiring simultaneous alignment in feature space and in time space (Painblanc et al., 2023).

The term also spans different data types and mathematical objects. Some BTDA formulations align two image feature maps by deformable sampling, as in TDAN, FDAN, DFAR, DC-Mamba, and STDANet (Tian et al., 2018, Lin et al., 2021, Luo et al., 2024, Sun et al., 19 Sep 2025, Zhang et al., 2022). Others align temporally indexed sequences through optimal transport and dynamic time warping, or through time-varying Thin Plate Spline (TPS) fields across short video intervals (Painblanc et al., 2023, Pero et al., 2014). This diversity matters because “bi-temporal” does not uniquely specify the operator: it may mean two image times, two semantic streams, two reference directions, or two temporal domains.

A recurrent source of confusion is nomenclatural rather than technical. TDAN’s paper-level name is “temporal deformable alignment,” FDAN’s description recasts its Flow-guided Deformable Module as a BTDA mechanism, DC-Mamba uses BTDA as an explicit module name, and MAD uses BTDA for an OT+DTW objective (Tian et al., 2018, Lin et al., 2021, Sun et al., 19 Sep 2025, Painblanc et al., 2023). This suggests that BTDA is not a single canonical block, but a recurring design pattern for learned correspondence across time.

2. Core operator: offsets, gates, masks, and resampling

In the convolutional family of BTDA methods, the essential operation is deformable resampling of one temporal feature map with offsets predicted from a pair or set of temporally related features. TDAN is the prototypical one-stage formulation. Given a low-resolution reference feature $F_t^{LR}$ and a supporting feature $F_i^{LR}$, the network predicts per-location offsets for a $3\times 3$ sampling kernel and produces an aligned feature by deformable convolution:

$$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$

Sampling locations are generally fractional, so bilinear interpolation is used (Tian et al., 2018). The key departure from optical-flow warping is that each kernel point may move independently, allowing the operator to “look around” neighborhoods rather than commit to a single source coordinate.

DC-Mamba simplifies the sampling geometry by predicting a two-dimensional offset field and a scalar gate at each spatial location. Its two-layer convolutional head outputs $\Delta_{\mathrm{raw}}^{(l)}$ and $\lambda_{\mathrm{raw}}^{(l)}$, which are post-processed as

$$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$

followed by gated resampling,

$$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$

The gate is intended to “drive gating toward zero in well-aligned zones,” while offset bounding constrains motion to $[-\delta_{max},+\delta_{max}]$ per component (Sun et al., 19 Sep 2025).

Other variants add masks or modulation weights. In learned bi-directional video compression, the contextual decoder predicts, at each scale, both offsets $o_t^s\in\mathbb R^{2K^2\times H_s\times W_s}$ and modulation masks $F_i^{LR}$0, and the aligned feature is

$F_i^{LR}$1

Here the reference is itself the concatenation of past and future features, so the alignment is bi-directional as well as bi-temporal (Yılmaz et al., 2023). DFAR similarly predicts offsets $F_i^{LR}$2 and optional masks $F_i^{LR}$3 for adjacent-frame feature alignment in infrared target detection (Luo et al., 2024).

Flow-guided variants retain the same deformable sampling logic but add a coarse displacement prior. FDAN first estimates a coarse flow, refines it, warps the neighbor feature, and then applies a deformable convolution that learns residual offsets around the warped position (Lin et al., 2021). STDANet uses coarse optical flows as base offsets and adds learned offsets and attention weights in its spatio-temporal deformable attention operator (Zhang et al., 2022). In both cases, BTDA becomes a two-stage sampling process: a global or coarse motion estimate establishes a neighborhood, and deformable refinement resolves residual misalignment.

3. Architectural realizations across domains

The architectural instantiation of BTDA depends strongly on the downstream task, but the alignment block is usually inserted before fusion, enhancement, or classification.

Domain	Aligned entities	BTDA realization
Video super-resolution	Reference and supporting frames	Feature-level deformable alignment without explicit optical flow
Remote sensing change detection	Two temporal semantic feature maps	Two-layer conv head with bounded offsets and gate
Learned video compression	Past/future reference features and current frame	Multi-scale DCN with offsets and modulation masks
Video deblurring	Neighbor features and blurry mid-frame feature	Coarse flows as base offsets plus learned deformable attention
Time-series domain adaptation	Source and target time series	OT plan over samples plus DTW path over timestamps
Registration and correspondence	Moving/fixed volumes or deformable video intervals	Dense displacement fields, DSTN fusion, or time-varying TPS

TDAN organizes BTDA into three stages: a shared feature extractor, a per-supporting-frame temporal deformable alignment module, and a reconstruction network that concatenates aligned frames with the reference frame and upsamples via sub-pixel convolutions (Tian et al., 2018). DFAR adopts a similar pairwise alignment pattern, but precedes offset prediction with four stacked Dilated Convolution Attention Fusion blocks and follows alignment with an Attention-guided Deformable Fusion refinement module (Luo et al., 2024).

DC-Mamba places BTDA “immediately after the two-branch ChangeMamba encoder and before any change-modeling or classification,” so alignment operates on semantic feature maps rather than raw imagery (Sun et al., 19 Sep 2025). In contrast, the flexible-rate bi-directional video compression model performs alignment at three scales, conditioned on a latent $F_i^{LR}$4 produced by a contextual encoder and decoder; BTDA here is tightly coupled to the conditional coding pathway (Yılmaz et al., 2023).

FDAN and STDANet illustrate a hybrid design in which explicit motion estimation and deformable alignment coexist. FDAN’s Matching-based Flow Estimation produces coarse optical flow from global semantic matching, then the Flow-guided Deformable Module performs coarse-to-fine alignment on top of the warped feature (Lin et al., 2021). STDANet’s Spatio-Temporal Deformable Attention module combines motion-estimator flows, learned offsets, and attention weights to extract “sharp pixels” from adjacent blurry frames for deblurring (Zhang et al., 2022).

Outside convolutional image models, BTDA takes distinct forms. MAD formulates alignment as a joint optimization over an optimal-transport plan $F_i^{LR}$5 between source and target series and a DTW warping matrix $F_i^{LR}$6 between timestamps (Painblanc et al., 2023). The deformable object correspondence method of 2014 uses motion-consistent interval pairs and a time-varying TPS sequence $F_i^{LR}$7 with temporally consistent correspondences (Pero et al., 2014). GuidedMorph uses a two-stage registration pipeline with a global displacement field $F_i^{LR}$8, a dense-tissue displacement field $F_i^{LR}$9, and a Dual Spatial Transformer Network fusion rule $3\times 3$0 (Chen et al., 19 May 2025).

4. Supervision, regularization, and optimization

BTDA methods differ sharply in how alignment is supervised. TDAN uses no ground-truth offsets. Instead, it introduces a self-supervised alignment loss on reconstructed aligned low-resolution frames,

$3\times 3$1

together with a super-resolution loss $3\times 3$2, yielding $3\times 3$3 (Tian et al., 2018). FDAN is even more implicit: it is trained end-to-end with a single image reconstruction loss and “no explicit flow-supervision or smoothness regularizer” (Lin et al., 2021).

DC-Mamba regularizes the deformation magnitude directly through an offset amplitude penalty,

$3\times 3$4

while leaving the downstream classification loss “unchanged from the parent ChangeMamba” (Sun et al., 19 Sep 2025). DFAR adds an explicit motion compensation term,

$3\times 3$5

and combines it with YOLOX regression, classification, and objectness losses as $3\times 3$6, with $3\times 3$7 and best $3\times 3$8 in the reported grid search (Luo et al., 2024).

GuidedMorph adopts an unsupervised registration objective that mixes global similarity, local masked similarity, and deformation smoothness:

$3\times 3$9

Its training schedule doubles $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$0 every 100 epochs so that learning shifts from global structure to dense-tissue alignment (Chen et al., 19 May 2025). The earlier spatiotemporal correspondence method likewise optimizes a regularized deformation energy, but with TPS bending energy and soft correspondence matrices that are constrained to remain consistent over time (Pero et al., 2014).

MAD is the most explicit joint-alignment formulation. It minimizes

$$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$1

by block-coordinate descent: fixing $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$2 reduces the problem to optimal transport over $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$3, and fixing $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$4 reduces it to standard DTW over $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$5 (Painblanc et al., 2023). This separates sample correspondence from temporal correspondence while preserving their multiplicative coupling.

5. Empirical findings and reported gains

The empirical literature consistently reports benefits from explicit deformable alignment, but the metrics are task-specific and should be interpreted within each benchmark.

Study	Reported comparison	Observation
TDAN on Vid4, $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$6	$$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$7 dB vs TOFlow’s $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$8 dB	Higher average PSNR
DC-Mamba	F1 $$F_i^{LR'}(x)\;=\;\sum_{k=1}^{K} w_k\,F_i^{LR}\bigl(x + p_k + \Delta p_k(x)\bigr),\qquad K=9.$$9, IoU $\Delta_{\mathrm{raw}}^{(l)}$0	Improvement over ChangeMamba baseline
3-scale DCN compression on UVG	No-align $\Delta_{\mathrm{raw}}^{(l)}$1, single-scale $\Delta_{\mathrm{raw}}^{(l)}$2, full 3-scale $\Delta_{\mathrm{raw}}^{(l)}$3, plus content-adaptive inference $\Delta_{\mathrm{raw}}^{(l)}$4 BD-BR	Alignment improves rate-distortion
FDAN on Vimeo90K-T	$\Delta_{\mathrm{raw}}^{(l)}$5 dB vs baseline $\Delta_{\mathrm{raw}}^{(l)}$6 dB	Gain under $\Delta_{\mathrm{raw}}^{(l)}$7 VSR
DFAR on DAUB / IRDST	Baseline $\Delta_{\mathrm{raw}}^{(l)}$8 mAP$\Delta_{\mathrm{raw}}^{(l)}$9 to full $\lambda_{\mathrm{raw}}^{(l)}$0	Large detection gains
GuidedMorph	Dense-Tissue Dice $\lambda_{\mathrm{raw}}^{(l)}$1, breast Dice $\lambda_{\mathrm{raw}}^{(l)}$2, breast SSIM $\lambda_{\mathrm{raw}}^{(l)}$3	Improved registration quality
TTPS on horses	AP $\lambda_{\mathrm{raw}}^{(l)}$4 vs SIFT+FG $\lambda_{\mathrm{raw}}^{(l)}$5	Better sequence alignment precision-recall

In video super-resolution, TDAN reports that richer offset-prediction capacity improves performance as the number of deformable convolution layers increases from D2 to D5, and that one-stage alignment yields a further $\lambda_{\mathrm{raw}}^{(l)}$6–$\lambda_{\mathrm{raw}}^{(l)}$7 dB gain over naïve multi-frame super-resolution without alignment (Tian et al., 2018). FDAN reports $\lambda_{\mathrm{raw}}^{(l)}$8 dB PSNR on Vimeo90K-T versus $\lambda_{\mathrm{raw}}^{(l)}$9 for its baseline, while remaining “competitive in computation and memory consumption”; it also reports about $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$0 fewer parameters than PCD on UDM-10 and Vid4 (Lin et al., 2021).

In learned bi-directional video compression, the full 3-scale deformable alignment model outperforms both a no-alignment baseline and a single-scale variant on UVG, and on the occlusion-heavy “Honeybee” sequence achieves $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$1 BD-BR versus $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$2 for single-scale and $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$3 for no-align. The same study reports that alignment reduces average pixel-domain motion residual error by $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$4 at scale 3 and $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$5 at scale 1 across all 7 UVG clips (Yılmaz et al., 2023).

For remote sensing change detection, DC-Mamba reports that its “align-then-enhance” strategy raises the F1-score from $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$6 to $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$7 and IoU from $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$8 to $$\tilde \Delta^{(l)}(x)=\delta_{max}^{(l)}\tanh\!\bigl(\Delta_{raw}^{(l)}(x)\bigr),\qquad \lambda^{(l)}(x)=\sigma\!\bigl(\lambda_{raw}^{(l)}(x)\bigr),$$9 relative to ChangeMamba (Sun et al., 19 Sep 2025). For infrared dim-small target detection, DFAR’s ablation study isolates the contribution of TDA, motion-compensation loss, and feature refinement: the full model reaches $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$0 mAP$$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$1/F1 on DAUB and $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$2 on IRDST, compared with $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$3 and $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$4 for the baseline without TDA, $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$5, or refinement (Luo et al., 2024).

In registration and deformable correspondence, GuidedMorph reports improvements “by over $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$6 in dense tissue Dice, $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$7 in breast Dice, and $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$8 in breast SSIM compared to the best learning-based baseline” in the abstract, while its detailed summary gives averaged gains of Dense-Tissue Dice $$\hat F_{t'}^{(l)}(x)=\mathrm{DeformSample}\!\bigl(F_{t'}^{(l)},\;x+\lambda^{(l)}(x)\tilde \Delta^{(l)}(x)\bigr).$$9, Overall-Breast Dice $[-\delta_{max},+\delta_{max}]$0, and Breast SSIM $[-\delta_{max},+\delta_{max}]$1 (Chen et al., 19 May 2025). The time-varying TPS approach reports $[-\delta_{max},+\delta_{max}]$2 alignable consistent-motion pairs for tigers versus $[-\delta_{max},+\delta_{max}]$3 under uniform sampling, and on horses an AP of $[-\delta_{max},+\delta_{max}]$4 versus $[-\delta_{max},+\delta_{max}]$5 for SIFT+FG (Pero et al., 2014).

6. Conceptual interpretation and recurring misconceptions

A common misconception is that BTDA is synonymous with optical-flow warping. The literature does not support that equivalence. TDAN was explicitly proposed to align “at the feature level without computing optical flow,” and its stated motivation is to avoid the artifacts that arise when “image-level wrapping-based models” depend on inaccurate flow estimates (Tian et al., 2018). Yet FDAN and STDANet show that BTDA can also be flow-guided: both use coarse optical flows as base offsets and then learn finer, local deformable corrections (Lin et al., 2021, Zhang et al., 2022). BTDA therefore includes both flow-free and flow-guided regimes.

Another misconception is that BTDA always operates on pixels. In DC-Mamba, alignment is performed on semantic feature maps from two temporal streams, specifically to reduce pseudo-changes before change modeling (Sun et al., 19 Sep 2025). In DFAR, alignment occurs on feature maps used for detection rather than on image intensities (Luo et al., 2024). In MAD, the aligned objects are not image grids at all, but source and target time series coupled through $[-\delta_{max},+\delta_{max}]$6 and $[-\delta_{max},+\delta_{max}]$7 (Painblanc et al., 2023). In GuidedMorph and the TPS-based deformable-object method, the central objects are displacement fields or time-varying spline warps rather than convolutional kernels (Chen et al., 19 May 2025, Pero et al., 2014).

A further misconception is that the term names a fixed module. The sources indicate otherwise. The operator may be a one-stage deformable convolution module, a gated two-layer head, a multi-scale DCN, a deformable attention block, a two-stage registration system, an OT+DTW objective, or a temporally consistent TPS estimator (Tian et al., 2018, Sun et al., 19 Sep 2025, Yılmaz et al., 2023, Zhang et al., 2022, Chen et al., 19 May 2025, Painblanc et al., 2023). A plausible implication is that BTDA is best treated as a unifying descriptor for learned temporal deformation models whose purpose is to establish correspondence before fusion, enhancement, coding, classification, or registration.

Within that broader interpretation, the enduring technical motif is stable: temporal mismatch is handled by learning where to sample, warp, or correspond across time, and the deformation is constrained—by reconstruction, classification, smoothness, offset penalties, motion-compensation losses, OT marginals, DTW admissibility, or TPS bending energy—so that the alignment remains useful for the downstream objective rather than becoming an unconstrained transformation.