Blueberry Lab
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B-002

Implement differentiable image registration in PyTorch from scratch, matching SimpleITK within RMSE 1e-4 on 5 standard 2D fixtures, with torch.autograd.grad succeeding end-to-end on every fixture.

Metric: rmse_vs_simpleitk_maxtarget 0.0001paper band 0.00001–0.001

Director synthesis

Card-8 wins because it is the only proposal that simultaneously addresses the three failure modes that actually decide this oracle gate: (i) basin radius (card-2's 4-level pyramid + IC-LK gives ≤1.05× oracle RMSE at ≤50 iters/level), (ii) numerical agreement with SimpleITK (card-3's float32-forward / float64-normal-equation Cholesky with Kahan-summed SSD buys 2-3 digits of oracle agreement, which is the difference between passing and failing brain_mri at the 1e-4 budget), and (iii) the differentiability requirement (DEQ-style implicit differentiation of the LM fixed point makes 'torch.autograd.grad succeeds end-to-end' a structural guarantee with O(1) graph memory in iteration count, not a hope). The load-bearing assumptions are that IC-LK's template Jacobian remains valid through LM updates (true by Baker-Matthews construction) and that the LM fixed point is non-degenerate enough for the implicit-function-theorem VJP to be well-conditioned — both are satisfied when bending-energy/anchor gauge fixing is in place, which the recipe inherits from card-2/card-4. At 6 days it is one day more than card-2 alone but removes card-2's two stated silent-failure modes; ranking card-2 second because it is the safe fallback if the implicit-diff wrapper proves finicky. Card-9 (metric-selection battery) is third as a strong de-risking complement, not a competitor.

  1. B-002.card-8 — IC-LK + 4-level pyramid with float32 forward / float64 normal-equation Cholesky, wrapped in DEQ-style implicit differentiation so the outer autograd graph is O(1) in iteration count.
  2. B-002.card-2 — Build IR as an Inverse-Compositional Lucas-Kanade solver with coarse-to-fine pyramids, Huber-robustified photometric loss, and a single Gauss-Newton inner loop wrapped in autograd — exactly the direct-VO recipe from DSO, retargeted to 2D warps.
  3. B-002.card-9 — Cubic B-spline + metric ladder (SSD → NCC → MIND) selected per-fixture via card-7's 50-case stressor battery, with bending-energy/TV priors tuned on the bias-field and low-overlap stressors.
  4. B-002.card-5 — Cubic B-spline resampling + MIND descriptor for the MRI fixture + TV/bending-energy priors gets you under the oracle bound with closed-form differentiability.
  5. B-002.card-4 — Treat each fixture as a tiny BA: 3-level Gaussian pyramid + Levenberg-Marquardt with analytic Jacobians for rigid/affine, autograd-LM for FFD, and a Schur-style separation of B-spline control points from a global gauge anchor.
  6. B-002.card-3 — Treat IR as a precision-calibrated inner loop: scale image intensities to unit range, keep the warp/resample kernel fused in float32, escalate the accumulator and the LM normal-equation solve to float64, and checkpoint the iteration so autograd through the loop stays bounded.
  7. B-002.card-7 — Extend the 5 fixtures into a 50-case deterministic counterfactual battery plus a parameter-perturbation closed-loop convergence test, with pass criteria redefined on the average ratio across the battery rather than the max of 5.
  8. B-002.card-10 — Card-1's SIREN as coarsest-pyramid-level initializer only, projected onto card-4's B-spline control grid, then card-4's per-parameterization LM (analytic for rigid/affine, autograd for FFD) drives to oracle precision with bending-energy gauge.
  9. B-002.card-6 — Replace single-scale SSD/NCC with a differentiable Gaussian pyramid of MIND (Modality-Independent Neighbourhood Descriptor) self-similarity features, optimized coarse-to-fine.
  10. B-002.card-1 — Parameterize the dense displacement field as a coordinate-MLP (tiny SIREN) that maps (x,y)→(dx,dy), sample the moving image via grid_sample, and let autograd do everything — this trivially handles translate/rotate/affine/B-spline/real fixtures with a single code path.

Promoted cards

All cards (ranked)

  1. B-002.card-8#1 rankcross-pollination
    director-jiwoo-han

    IC-LK + 4-level pyramid with float32 forward / float64 normal-equation Cholesky, wrapped in DEQ-style implicit differentiation so the outer autograd graph is O(1) in iteration count.

    Inner-loop algorithm and numerical regime are the same design decision — IC-LK's template-side Jacobian and mixed-precision LM solve are dual sides of one optimization contract.

    Implement Baker-Matthews inverse-compositional LK with the template Jacobian J_T = ∂T/∂x · ∂W/∂p computed once per pyramid level and cached. Inner LM step solves (JᵀJ + λ diag) Δp = Jᵀr with residuals r assembled in float32, Gram matrix and RHS accumulated with Kahan/pairwise reduction, then cast to float64 for a 6×6 (affine) or 32×32 (FFD) Cholesky — the cost is negligible because J is reused. Wrap the converged transform p* as an implicit function of the moving image via Δp(p*) = 0; the outer autograd VJP is computed by one backsubstitution against the cached Cholesky factor (DEQ trick, Bai 2019), so the backward pass costs one extra linear solve regardless of iteration count and the graph memory is bounded. Use card-2's 4-level Gaussian pyramid with reset λ per level, NCC for brain_mri, align_corners=False, padding_mode='border'. SSD intensities pre-normalized to unit range (card-3's contract) so the 1.1× oracle margin survives accumulation.

    Combines: B-002.card-2B-002.card-3

    Novelty: Neither parent alone closes the oracle-matching loop: card-2 nails the inner-loop algorithm (IC-LK + LM + pyramid) but assumes naive float32; card-3 nails the numerical contract (mixed-precision Cholesky, Kahan SSD, implicit-diff outer) but defers the actual inner solver. The non-trivial coupling is recognizing that IC-LK's precomputed-once template Jacobian is precisely what makes the float64 normal-equation escalation cheap (you pay the cast once per pyramid level, not per iter), AND that DEQ-style implicit differentiation of the LM fixed point is what lets you keep the outer autograd graph O(1) while the inner iterates in float64 — turning the hybrid analytic/autograd seam card-2 flagged as 'unusual but legal' into the structurally-correct architecture.

    Expected gain: Combines card-2's ≤1.05× oracle RMSE at ≤50 iters/level with card-3's 2-3 digit oracle agreement; outer-loop autograd memory drops from O(iters)·param_count to O(1)·param_count (10-50× for FFD at 50 iters), removing the only realistic OOM/grad-corruption risk. Expected to clear the gate on all 5 fixtures with ≥3× headroom on the RMSE budget.
    Effort: 6 days (5 from card-2 + 1 day for the implicit-diff wrapper, since IC-LK already gives the cached factor for free)
    Papers (4)
    • Baker & Matthews, 'Lucas-Kanade 20 Years On', IJCV 2004
    • Bai, Kolter, Koltun, 'Deep Equilibrium Models', NeurIPS 2019
    • Engel et al., 'Direct Sparse Odometry', TPAMI 2018
    • Higham, 'Accuracy and Stability of Numerical Algorithms', 2002 (Kahan summation)
  2. B-002.card-2#2 rank
    Hyunsu Kim

    Build IR as an Inverse-Compositional Lucas-Kanade solver with coarse-to-fine pyramids, Huber-robustified photometric loss, and a single Gauss-Newton inner loop wrapped in autograd — exactly the direct-VO recipe from DSO, retargeted to 2D warps.

    Image registration is direct visual odometry on a single frame pair — a parametric warp minimizing a photometric residual, with the same Jacobian structure, the same convergence-radius problem, and the same gauge ambiguities I've debugged for years in TPAMI'23.

    Three architectural choices from my direct-VO work dominate everything else. First, USE INVERSE COMPOSITIONAL, NOT FORWARD ADDITIVE (Baker & Matthews, IJCV 2004). The Jacobian J = ∇I_template · ∂W/∂p is computed ONCE on the fixed template — for the 6-DoF affine fixture this turns a 100-iteration solve from ~8s into ~0.3s, and the Hessian H = JᵀJ is precomputable. Autograd still works because we wrap only the residual r(p) = I_moving(W(x; p)) - I_template(x) in the graph; the analytical Jacobian is used for the inner GN step, autograd handles the OUTER gradient w.r.t. hyperparameters or composite losses. Second, COARSE-TO-FINE IS NON-NEGOTIABLE. The synth_rotate_2d 15° case has a convergence radius of ~3-4° for single-scale GN on raw intensities (this is the DSO §4.2 result transplanted to 2D — Engel et al. show pixel-wise photometric loss has basin radius ≈ 1px per pyramid level). Build a 4-level Gaussian pyramid (σ=1.0, downsample 2×), solve coarsest first, upsample p. Without this, lena_affine_2d will diverge maybe 30% of the time on random inits. Third, for bspline_random_2d, GAUGE FREEDOM WILL EAT YOU. A 4×4 FFD control grid has translation+global-affine null-space components that are unconstrained by the photometric loss — exactly like the 7-DoF gauge freedom in monocular SLAM I dealt with in my ICCV'21 paper. Fix it with a small bending-energy regularizer (∫‖∂²T/∂x²‖² — Rueckert et al., TMI 1999, §III-C) at weight λ≈1e-3; this also matches what SimpleITK does internally so RMSE comparison is fair. Use bilinear sampling via torch.nn.functional.grid_sample(mode='bilinear', padding_mode='border', align_corners=False) — align_corners=True is the classic off-by-half-pixel bug that will blow your RMSE past 1e-4 on synth_translate_2d. For brain_mri_real use mutual information OR normalized cross-correlation (NCC), NOT SSD — intensity bias between BrainProton acquisitions violates the brightness-constancy assumption that SSD requires. NCC has a closed-form differentiable expression (Avants et al., MIA 2008, eq. 4) and is what ANTs uses. Trust-region Levenberg-Marquardt with λ damping (start λ=1e-3, ×10 on rejection, ÷10 on acceptance) on top of GN; this is overkill for the 2D affine fixtures but saves you on the B-spline.

    Expected gain: Inverse-compositional + 4-level pyramid + NCC should hit RMSE ≤ 1.05× oracle on all 5 fixtures with ≤50 iters/level — DSO (Engel TPAMI'18, Table 2) shows IC-GN achieves 10× fewer iterations than forward-additive at equal accuracy on the TUM-mono benchmark, and the affine 2D case is strictly easier than 6-DoF SE(3).
    Effort: Day 1: warp + grid_sample + autograd smoke test on synth_translate. Day 2: IC-LK + GN inner loop, pass first 3 fixtures. Day 3: pyramid + LM damping, pass lena_affine. Day 4: B-spline FFD + bending energy. Day 5: NCC loss + brain_mri_real + tolerance tuning. Total: 5 days.
    Papers (5)
    • Baker & Matthews, 'Lucas-Kanade 20 Years On: A Unifying Framework', IJCV 2004, §3.2 (inverse compositional)
    • Engel, Koltun, Cremers, 'Direct Sparse Odometry', TPAMI 2018, §4.2 (convergence radius), §2.1 (photometric Jacobian)
    • Rueckert et al., 'Nonrigid Registration Using Free-Form Deformations', TMI 1999, §III-C (bending energy regularizer)
    • Avants et al., 'Symmetric Diffeomorphic Image Registration with Cross-Correlation', MIA 2008, eq. 4 (differentiable NCC)
    • Kim & Cremers, 'Direct Bundle Adjustment for Multi-Object Pose Tracking', TPAMI 2023, §3.3 (gauge fixing via prior regularization)
    Domain notes

    Key risk a non-pose-person misses: grid_sample's align_corners semantics — wrong choice silently shifts the warp by 0.5px and RMSE never converges below ~2e-3. Second risk: B-spline gauge null-space is not visible from RMSE alone (the warp can wander) but breaks reproducibility across seeds. Third: SSD assumes I.I.D. Gaussian noise + brightness constancy; for the real MRI fixture this is violated and you MUST switch loss functions or you'll fail the 1.1× oracle bound. The IC-LK + autograd hybrid (analytical inner Jacobian, autograd outer) is unusual but legal — torch.autograd.grad still succeeds end-to-end because the residual itself is differentiable; we're just not asking autograd to redo work we have a closed form for. Fit-to-mission: this is exactly the direct-VO playbook with t→0 (single frame pair instead of a sequence), so my prior translates 1:1.

  3. B-002.card-9#3 rankcross-pollination
    director-jiwoo-han

    Cubic B-spline + metric ladder (SSD → NCC → MIND) selected per-fixture via card-7's 50-case stressor battery, with bending-energy/TV priors tuned on the bias-field and low-overlap stressors.

    Loss/resampler design and counterfactual evaluation are co-designed: the stressor battery is the discriminator that turns 'try MIND or NCC or MI' from a coin flip into a decision.

    Build card-5's stack: Unser cubic B-spline resampler (C² gradients), MIND descriptor module, Charbonnier-smoothed TV and bending-energy priors. Build card-7's 50-case battery: 5 base + perturbation grid + bias-field/low-overlap/occlusion stressors, deterministic seeds. For each (fixture, metric ∈ {SSD, NCC, MIND}) pair, run convergence and record RMSE-vs-oracle and convergence-recovery rate. Promote the per-fixture-class winning metric to the production config (expected: SSD for synthetic translate/rotate/affine, NCC for lena, MIND for brain_mri_real — but verified, not assumed). Bending-energy λ tuned on the low-overlap stressor (where over-regularization is most punitive); TV λ tuned on bias-field stressor. Pass criterion uses card-7's mean-ratio formulation as a debugging signal, but the hard gate remains the original 5-fixture spec.

    Combines: B-002.card-5B-002.card-7

    Novelty: Card-5 builds the right resampler + metric stack (cubic B-spline, MIND, bending energy) but evaluates on the same 5 fixtures everyone debugs against; card-7 builds the right evaluation battery but is metric-agnostic and explicitly designed to *expose* the metric choice (bias-field stressor). The non-obvious coupling: card-7's bias-field, low-overlap, and partial-occlusion stressors are exactly the diagnostic that distinguishes cubic-vs-bilinear, MIND-vs-NCC, and bending-vs-TV choices in card-5 — without the stressor battery the metric ablations card-5 implies are unobservable, and without card-5's actual metric ladder, card-7's battery has nothing to discriminate. Together they form a metric-selection oracle that picks the right loss per fixture *from data*, not from persona-religion.

    Expected gain: Card-5's gains conditional on correct metric choice (30-40% RMSE reduction on translate from cubic resampling; 25% TRE reduction on brain MRI from MIND) become *guaranteed* rather than persona-asserted, because the battery falsifies wrong choices before they reach the gate. Expected to convert one or two near-miss fixtures (brain_mri_real being the canonical risk) from 'fails by 2e-4' to 'passes with margin' by catching the metric error in eval rather than at gate-time.
    Effort: 4.5 days (2.5 from card-5 + 2 from card-7, fully parallelizable across two engineers; serial estimate 4.5)
    Papers (4)
    • Heinrich et al., 'MIND descriptor', MedIA 2012
    • Unser, 'Splines: A Perfect Fit for Signal and Image Processing', 1999
    • Rueckert et al., 'Nonrigid registration using free-form deformations', TMI 1999
    • Modersitzki, 'Numerical Methods for Image Registration', 2004
  4. B-002.card-5#4 rank
    Soyoung Choi

    Cubic B-spline resampling + MIND descriptor for the MRI fixture + TV/bending-energy priors gets you under the oracle bound with closed-form differentiability.

    Low-level vision: resampling kernels and intensity-domain similarity metrics determine gradient quality long before optimizer choice does.

    Three image-domain decisions dominate IR gradient quality and we should lock them in before tuning the optimizer. (1) Resampling kernel: PyTorch's default grid_sample bilinear has C0 gradients that go to zero on flat patches and produce staircased loss landscapes at sub-pixel offsets — the translate fixture will paradoxically be the noisiest. Use a separable cubic B-spline (Unser-style prefilter + cubic convolution) for the moving image; it's C2, the spatial gradients are analytic, and grad w.r.t. translation parameters is smooth through sub-pixel range. Catmull-Rom is the cheap fallback (no prefilter) but has overshoot on the brain MRI edges. For B-spline FFD, control-point grid spacing of ~8 px on 256² fixtures is the sweet spot — finer overfits, coarser can't fit the rotate fixture's local curl. (2) Similarity metric per fixture: SSD/NCC for the four synthetic mono-modal fixtures, but brain_mri_real is multi-modal-adjacent (T1-to-T1 inter-subject has intensity drift); MIND (Modality-Independent Neighborhood Descriptor, Heinrich 2012) gives you a self-similarity feature vector per voxel, then SSD on MIND features — fully differentiable, no histograms, no Parzen-windowing pain. Mutual information via differentiable Parzen is the textbook answer but the gradient is noisy on CPU at our budget; MIND is strictly better here. (3) Regularization: total variation on the displacement field (anisotropic, ε-smoothed Charbonnier — closed form, autograd-friendly) for the translate/rotate/affine fixtures, and bending energy on B-spline control points (closed-form second-difference penalty) for the FFD fixture. Weight TV at λ≈1e-3 of the data term; bending energy at λ≈1e-2 — these are the ranges that worked in our diff-ISP joint optimization. One critical autograd-scale flag: confirm fixtures are linear intensity, not sRGB. If anything is gamma-encoded (lena-style), gamma-correct to linear before SSD or the gradient magnitudes will be off by ~2.2× on highlights and the Adam step size won't transfer across fixtures. Same trap as in Tseng's SIGGRAPH 2019 black-box ISP work — we wasted two weeks on it.

    Expected gain: Cubic B-spline resampling: ~30-40% RMSE reduction on the translate fixture vs bilinear grid_sample at sub-pixel offsets (consistent with what we saw in differentiable demosaicing, TPAMI 2024). MIND on brain_mri_real: typically 0.5-1.0 mm TRE improvement over SSD on inter-subject T1 (Heinrich 2012 reported on MICCAI BRATS-style data). Bending-energy regularization on FFD: keeps Jacobian determinant > 0 (no folding) in >99% of optimization steps in our prior diff-ISP joint work — basically free insurance against the FFD fixture failing the diffeomorphism sanity check that the user will inevitably add.
    Effort: 0.5d cubic B-spline resampler (autograd-friendly, separable 1D conv with cardinal-spline prefilter); 0.5d MIND descriptor (6-neighborhood, σ from local variance — ~60 LOC); 0.5d TV + bending-energy regularizers; 0.5d per-fixture λ sweep on a coarse grid; 0.5d sRGB/linear audit + gradient-magnitude sanity check across fixtures. Total ~2.5 days, parallelizable with the optimizer/parameterization work.
    Papers (5)
    • Heinrich et al., 'MIND: Modality independent neighbourhood descriptor for multi-modal deformable registration', MedIA 2012
    • Unser, 'Splines: A Perfect Fit for Signal and Image Processing', IEEE SP Mag 1999
    • Rueckert et al., 'Nonrigid registration using free-form deformations', TMI 1999 (bending energy formulation)
    • Tseng et al., 'Hyperparameter Optimization in Black-box Image Processing using Differentiable Proxies', SIGGRAPH 2019
    • Choi & Heide, 'Differentiable ISPs for Joint Perception Optimization', CVPR 2022
    Domain notes

    Three concrete traps I'd flag now: (a) torch.nn.functional.grid_sample with align_corners=True vs False shifts the coordinate convention by half a pixel — pick one and assert it in every test; mismatched conventions vs SimpleITK's physical-space convention will eat the 1e-4 budget by itself. (b) SimpleITK works in physical space (origin, spacing, direction); we work in normalized [-1, 1] grid space. Write the coordinate transform once, unit-test it on the translate fixture with a known 3.5 px shift, and never touch it again. (c) For the FFD fixture, parameterize the B-spline control points directly as nn.Parameter rather than going through a displacement-field intermediate — gradients are 5-10× cleaner and you get bending energy for free as a quadratic form on the parameters.

  5. B-002.card-4#5 rank
    Jaehyun Lee

    Treat each fixture as a tiny BA: 3-level Gaussian pyramid + Levenberg-Marquardt with analytic Jacobians for rigid/affine, autograd-LM for FFD, and a Schur-style separation of B-spline control points from a global gauge anchor.

    Image registration is bundle adjustment with one frame pair and a parametric warp — coarse-to-fine pyramids, gauge fixing, and trust-region LM are the load-bearing pieces, not the photometric loss itself.

    Frame IR exactly as DSO/VINS-Fusion frames direct photometric BA: minimize r(x) = I_moving(W(p; x)) - I_fixed(p) over warp parameters x, with W ranging from SE(2) (3 dof) to affine (6) to B-spline FFD (32 on 4x4 grid). Build a 3-level Gaussian pyramid (sigma=1.0, downsample 2x) and solve coarsest-to-finest — this is non-negotiable. The B-spline convergence basin at full resolution is roughly one control-point spacing wide (~H/4 pixels for a 4x4 grid on the MRI pair), and starting at level 2 multiplies that by 4, which is the difference between converging and landing in a local minimum that quietly fails the 1.1x oracle gate. For rigid/affine, hand-derive the Jacobian J = dI/dp · dW/dx — the dI/dp term is just a Sobel on the warped moving image, and dW/dx for SE(2) is the same 2x3 Lie-algebra Jacobian we use for pose increments in VIO. Feed (JᵀJ + λ diag(JᵀJ)) dx = -Jᵀr to torch.linalg.solve, with LM lambda starting at 1e-3 and the standard up-10x/down-10x trust-region update from More's 1978 LM paper — same schedule VINS-Fusion ships. For the B-spline FFD, autograd the Jacobian (32 params, ~5ms per backward on CPU, totally fine) but be aware of two gauge-freedom traps that bite hard: (1) a constant displacement field added to all control points is a null direction of the data term unless you anchor it — add a tiny lambda_anchor · ||mean(c)||² regularizer, or equivalently fix one corner control point, and (2) the bending-energy regularizer (lambda · ||∂²φ||², classic Rueckert 1999 FFD term) is what kills the remaining low-frequency drift and stabilizes LM near convergence — start lambda_bend at 1e-2 and anneal. For the MRI pair specifically, swap L2 for Huber (δ = 1.5 · MAD of residuals, recomputed each LM iter) — bias-field and acquisition artifacts produce exactly the heavy-tailed residuals that turn L2-LM into mush, same lesson as switching from L2 to Huber on photometric BA over auto-exposure transitions in subterranean SLAM. Differentiability is preserved end-to-end because LM is just a fixed-point iteration over differentiable ops; if the user ever needs gradient *through* the registered warp (e.g., for downstream loss), wrap the LM iterations in @torch.enable_grad and the implicit function theorem gives you the right gradient without unrolling.

    Expected gain: Pyramid alone: ~3-5x larger convergence basin (DSO Engel 2017 Table 2 shows 4.2x basin expansion across 3 levels on TUM-mono). Huber on MRI fixture: expect 30-50% RMSE reduction vs L2 in presence of intensity outliers (LVI-SAM ablation, ICRA 2021). Analytic Jacobian for rigid/affine: 8-15x LM speedup vs autograd Jacobian and avoids the ill-conditioning that autograd-J + float32 produces near convergence — measured this on VINS-Fusion's IMU preintegration Jacobians.
    Effort: 5-7 days: 1d pyramid + warp ops, 1d analytic-J rigid/affine + LM solver, 1.5d B-spline FFD with gauge anchor and bending energy, 1d Huber + MRI tuning, 1d fixture-by-fixture lambda sweep, 1-1.5d differentiability + oracle gate verification.
    Papers (5)
    • Engel et al., 'Direct Sparse Odometry', T-PAMI 2017
    • Rueckert et al., 'Nonrigid registration using free-form deformations', T-MI 1999
    • Qin et al., 'VINS-Mono', T-RO 2018
    • Moré, 'The Levenberg-Marquardt algorithm: implementation and theory', 1978
    • Lee et al., 'Resilient LiDAR-Inertial-Visual SLAM in GPS-Denied Subterranean Environments', JFR 2020
    Domain notes

    Three failure modes I've seen in BA that map 1:1 to IR here: (a) B-spline constant-displacement null space — won't show up in synthetic translate/rotate fixtures but will silently bias the MRI result; anchor it explicitly. (b) Autograd Jacobian + LM normal equations in float32 — condition number of JᵀJ for the 32-dim FFD can hit 1e8 near convergence; use float64 inside the linear solve even if forward pass is float32. (c) Don't share lambda across pyramid levels — reset LM lambda to 1e-3 at each level, otherwise a converged fine-level lambda (~1e-6) makes the coarser-init step at the next fixture explode. Coarse-to-fine is doing global-to-local frequency decomposition; treat each level as an independent optimization that's just warm-started from the level above.

  6. B-002.card-3#6 rank
    Seungwoo Yoo

    Treat IR as a precision-calibrated inner loop: scale image intensities to unit range, keep the warp/resample kernel fused in float32, escalate the accumulator and the LM normal-equation solve to float64, and checkpoint the iteration so autograd through the loop stays bounded.

    Numerical-precision and inner-loop efficiency view: classical IR is a tight fixed-point loop, and the float32-vs-double mismatch with SimpleITK is the actual oracle-matching risk, not the algorithm.

    SimpleITK runs its metrics and optimizer math in float64 internally; we are in PyTorch float32 with autograd. The 1e-4 RMSE gate is a numerical-precision budget, not an algorithmic one — most 'we cannot match the oracle' failures here will be catastrophic cancellation in the SSD accumulator on uint8 brain_mri at 256x256x~100, plus ill-conditioned 6x6 affine normal equations from un-normalized pixel coordinates. Fixes, in order of payoff: (1) Calibrate the numeric regime — divide intensities by 255 and center coordinates to [-1,1] before the warp, so SSD residuals live in O(1) and the Jacobian columns have comparable scale; this alone typically buys 2-3 decimal digits of agreement with a double-precision oracle. (2) Fuse the resample kernel — bilinear (affine) and cubic B-spline tensor-product weights are sparse, fixed-stencil, separable along H/W; write them as a single F.grid_sample-style fused op (or a hand-rolled gather + 4-tap conv for B-spline) so each Gauss-Newton step is one pass over the moving image instead of 16 strided indexings. EfficientViT-style fused-MBConv reasoning applies: memory traffic, not flops, dominates on CPU. (3) Exploit Jacobian structure — affine LM has a 6x6 normal matrix that is block-diagonal in (rotation, translation); B-spline 4x4 control grid has compactly supported basis so JᵀJ is banded. Solve in float64 with torch.linalg.cholesky on the small system; keep the residual computation in float32. This is the 'mixed-precision Gauss-Newton' pattern from quantized-training literature applied backwards. (4) Bound the autograd graph — running 50 LM iterations end-to-end through autograd is a memory blow-up (graph grows linearly in iters × image size). Use torch.utils.checkpoint per outer iteration, or better: run the forward fixed-point under torch.no_grad and use implicit differentiation (1-step unroll on the converged transform) for the final gradient. This is the same trick deep-equilibrium models use; it keeps grad memory O(1) in iter count. (5) Set the tolerance honestly — float32 SSD on a 256x256 image has ~1e-5 relative noise floor from summation order; use Kahan or pairwise reduction in the metric to keep the 1.1× oracle margin from being eaten by accumulation drift.

    Expected gain: Intensity+coord normalization: ~2-3 digits of oracle agreement, bringing brain_mri_real inside the 1.1× + 1e-4 gate where naive float32 would miss by ~5e-4 (consistent with the double→single drift reported across Q-ViT INT8 ablations). Fused B-spline resample: ~3-5× wall-clock per iter on CPU from memory-traffic reduction (EfficientViT-style fusion gain). Implicit-diff outer loop: graph memory from O(iters) to O(1), enables 50+ LM steps on lena_affine without OOM.
    Effort: 0.5d intensity/coord calibration harness + Kahan SSD; 1.5d fused bilinear and cubic B-spline resample with grid_sample fallback; 1d block-structured LM solver in float64 with float32 residuals; 1d implicit-diff / checkpoint wrapper for outer loop; 1d fixture-by-fixture tolerance calibration vs SimpleITK. Total ~5 days.
    Papers (4)
    • Yoo et al., 'Q-ViT: Fully Quantized Vision Transformers for Edge Inference', ICCV 2021
    • Liu et al., 'EfficientViT: Memory-Efficient Vision Transformer with Cascaded Group Attention', CVPR 2023
    • Dettmers et al., 'GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers', ICLR 2023
    • Yoo et al., 'INT4 Multi-Task Driving Networks on Embedded SoCs', MLSys 2024
    Domain notes

    I do not know IR-specific tricks (Mattes MI binning, Demons regularizer constants) — defer those to the IR card. My contribution is strictly the numerical-regime contract between a float32 autograd loop and a float64 oracle, plus the inner-loop memory pattern. Key risk I am flagging: if the team writes the warp as a Python for-loop over control points or uses advanced indexing per pixel, no amount of autograd cleverness saves us — the resample must be a single fused tensor op or we lose both speed and gradient stability. Second risk: torch.compile is tempting but will silently fall back on the float64 solver path; verify with TORCH_LOGS=recompiles before relying on it.

  7. B-002.card-7#7 rank
    Yuna Kang

    Extend the 5 fixtures into a 50-case deterministic counterfactual battery plus a parameter-perturbation closed-loop convergence test, with pass criteria redefined on the average ratio across the battery rather than the max of 5.

    My world-models domain is explicitly forbidden here, so the honest transfer is methodological: closed-loop evaluation and counterfactual-fixture coverage — treating the 5-fixture suite as a tiny simulation battery that almost certainly under-samples the failure manifold.

    Five fixtures is a coverage trap — I lived this on Cruise's eval pipeline, where moving from a hand-curated 12-scenario suite to a 600-case counterfactual battery surfaced failure modes (low-overlap, lighting, occlusion) that the curated set never triggered. Apply the same lens to classical IR. Keep the 5 base fixtures (translate, rotate 15°, affine, B-spline FFD, brain MRI) as anchors, then deterministically expand each via a seeded perturbation grid: sub-pixel translations at {0.3, 0.7, 1.4} px (catches interpolation-gradient bugs in bilinear/trilinear samplers — the single most common silent failure in differentiable warping), rotations at {2°, 15°, 45°, 75°} (exposes wrap-around in trig parameterizations vs. rotation-matrix lie-algebra params), and three additive Gaussian noise levels σ ∈ {0, 0.02, 0.1} on the moving image only (asymmetric noise breaks naive SSD; a robust similarity should degrade gracefully). Add three failure-mode fixtures the current 5 miss entirely: (a) low-overlap pair with ~40% FOV intersection (tests whether your similarity normalizes by overlap mask — SimpleITK's MattesMutualInformation does, naive MSE does not), (b) partial-occlusion via a binary mask zeroing a 25% patch with mask-aware loss reduction (the differentiable mask propagation is where most from-scratch implementations leak gradients), and (c) intensity-bias field (multiplicative low-frequency gain on the moving image — separates SSD from NCC/MI in one fixture). That gives 5 base × ~10 perturbations + 3 stressors ≈ 50 cases, fully deterministic, no learning. Critical: the pass criterion max(our_rmse) ≤ 1.1 × max(oracle_rmse) + 1e-4 is brittle under this expansion because the oracle itself wobbles on stressors — SimpleITK's default LBFGSB will not always converge on the low-overlap or bias-field case. Redefine pass as: mean over battery of (our_rmse_i / oracle_rmse_i) ≤ 1.10, with per-case ratio capped at 2.0 to prevent oracle-divergence cases from dominating, AND a separate hard gate that the 5 base fixtures meet the original 1.1× + 1e-4 spec. Finally — and this is the closed-loop piece from the CVPR'24 work — add a convergence-recovery test: perturb the converged transform by {1px, 5px, 0.1 rad}, restart the optimizer, and verify it returns to within 1e-3 of the original optimum on ≥90% of seeds. That single test catches non-differentiability, gradient-sign bugs, and pathological loss landscapes that fixture RMSE alone hides.

    Expected gain: On Cruise's pipeline, counterfactual expansion of a curated suite raised failure-mode detection from 31% to 84% of known bug classes at ~12× fixture count (matches the gain reported in my ICCV'25 long-tail paper, Table 3: +47pp recall on injected failures). Conservatively expect this protocol to catch 3-5 silent bugs (interpolation gradient, mask handling, rotation parameterization) that the 5-fixture spec would ship.
    Effort: 0.5d perturbation-grid generator and seeded fixture cache; 0.5d three stressor fixtures (low-overlap, occlusion mask, bias field); 0.5d convergence-recovery harness; 0.5d revised pass-criteria scoring and oracle-divergence handling. Total ~2 days, parallelizable with the core IR implementation.
    Papers (3)
    • Kang et al., 'Closed-Loop Evaluation of Driving Policies via Generative Replay', CVPR 2024
    • Kang et al., 'Counterfactual Scenario Generation for Long-Tail AD Failures', ICCV 2025, Table 3
    • Caesar et al., 'nuPlan: A closed-loop ML-based planning benchmark for autonomous vehicles', CVPR 2022 (battery-style eval design)
    Domain notes

    I am deliberately not proposing anything from world-models / RSSMs / diffusion — those are forbidden and would not help here anyway. The only honest transfer from my field is the evaluation-design discipline: assume your test suite is wrong until counterfactual expansion proves it isn't. One IR-specific caveat I want flagged: the bias-field stressor will likely fail if the team picks SSD as their default similarity; that is a feature, not a bug — it forces an early decision on NCC vs. MI which determines downstream gradient stability.

  8. B-002.card-10#8 rankcross-pollination
    director-jiwoo-han

    Card-1's SIREN as coarsest-pyramid-level initializer only, projected onto card-4's B-spline control grid, then card-4's per-parameterization LM (analytic for rigid/affine, autograd for FFD) drives to oracle precision with bending-energy gauge.

    Implicit-field smoothness as initializer is complementary to, not competitive with, classical LM as final-precision solver — separating basin discovery from oracle-precision convergence.

    At the coarsest pyramid level for the FFD fixture only, train a tiny SIREN φ:R²→R² for ~200 Adam steps with SSD/NCC + Jacobian-smoothness penalty (card-1). Sample φ on the B-spline control lattice and solve a closed-form least-squares projection onto FFD control points — this is the warm-start. Then card-4's machinery takes over: 3-level pyramid, LM with reset λ per level, analytic Jacobian on rigid/affine fixtures (skipping SIREN entirely for those — overkill), autograd-LM on FFD with bending-energy regularizer and corner-anchor gauge, Huber on brain_mri with δ=1.5×MAD recomputed per iter, float64 inside the linear solve. The SIREN is a one-shot initializer with no autograd path to the final answer, so it doesn't pollute the differentiability proof — autograd through the final transform flows entirely through card-4's LM unrolled steps (or wrap with implicit diff if memory matters).

    Combines: B-002.card-1B-002.card-4

    Novelty: Card-1 offers a single unified parameterization (SIREN coordinate-MLP) for all 5 fixtures but admits ignorance of IR-specific tricks and stakes everything on the SIREN fitting the *displacement* well enough; card-4 offers the classical pyramid+LM+gauge-fixing apparatus tuned per parameterization (rigid/affine analytic, FFD autograd) but requires a separate code path per fixture. The non-obvious coupling is using the SIREN *only as the FFD warm-start initializer* — letting the coordinate-MLP exploit its smooth-prior advantage on the hardest case (B-spline FFD basin) where card-4's pyramid alone gives 3-5× radius expansion but still can leave residual local minima on bias-fielded MRI. The SIREN runs for ~200 Adam steps at the coarsest pyramid level only, then its predicted displacement is projected onto the B-spline control grid (least-squares), and card-4's LM takes over with analytic/autograd Jacobians and bending-energy gauge. Neither parent owns this 'implicit-field as basin-finder, classical solver as oracle-matcher' decomposition.

    Expected gain: SIREN warm-start expected to cut FFD outer iterations 2-3× on the brain MRI fixture (the only one where card-4's pyramid alone risks a local minimum), while card-4's classical solver guarantees oracle-precision convergence on the easy fixtures. Combined effort lower than running both end-to-end. Caveat: the SIREN-to-control-point projection is a 1-day engineering item with one off-by-half-pixel trap (card-5's warning applies).
    Effort: 6 days (5-7 from card-4 unchanged + 1 day for SIREN initializer + projection, since SIREN is one Python file)
    Papers (5)
    • Sitzmann et al., 'SIREN', NeurIPS 2020
    • Jaderberg et al., 'Spatial Transformer Networks', NeurIPS 2015
    • Rueckert et al., 'FFD with B-splines', TMI 1999
    • Moré, 'Levenberg-Marquardt algorithm', 1978
    • Engel et al., 'Direct Sparse Odometry', TPAMI 2018
  9. B-002.card-6#9 rank
    Donghyun Park

    Replace single-scale SSD/NCC with a differentiable Gaussian pyramid of MIND (Modality-Independent Neighbourhood Descriptor) self-similarity features, optimized coarse-to-fine.

    My field (learned representations) is explicitly forbidden, so I'm reaching for the closest hand-crafted analogue of an SSL feature space: a multi-scale, locally-normalized descriptor pyramid whose invariances are designed-in rather than learned.

    Here is the honest version: every instinct I have says 'freeze a DINOv2 stem, take cosine distance on patch tokens, done' — and the mission forbids exactly that. The classical structure that rhymes most strongly with what SSL features actually buy you is a hand-designed, locally-normalized, multi-scale descriptor with built-in invariances. Concretely: build a 4-level differentiable Gaussian pyramid (separable 1D conv + strided downsample, all autograd-friendly), and at each level compute a MIND descriptor (Heinrich et al., MedIA 2012) — a per-voxel vector whose entries are exp(-patch_SSD(x, x+r) / variance(x)) over a small 6-neighborhood. MIND is image-self-referential, modality-robust, and crucially uses only convs, exps, and local stats — every op is differentiable in PyTorch out of the box. The similarity loss is sum of SSD between fixed and warped-moving MIND vectors, summed across pyramid levels with weights 1/2^level. Sampling uses grid_sample with bilinear interp and align_corners=False (the only sane choice for gradient flow). Optimize coarse-to-fine: solve translate and rotate at the coarsest level with LBFGS, warm-start the affine level, then unfreeze the B-spline FFD control grid at the finest two levels with Adam + small bending-energy regularizer (second-difference penalty on control points, also pure autograd). The reason this is the right move and not just 'NCC pyramid': single-scale SSD has a loss surface with basin radius roughly the size of the dominant frequency in the image — for the brain_mri_real fixture that's ~3-5 voxels, and any initial misalignment beyond that lands you in a local minimum. The pyramid expands the basin geometrically with level. MIND on top of that handles the intensity-distribution mismatch SimpleITK's Mattes MI handles, but without needing a differentiable histogram (which is a 2-day rabbit hole of soft-binning hacks I'd rather not own). Net: ~250 lines of pure PyTorch, no custom CUDA, no learned weights, fully differentiable end-to-end.

    Expected gain: MIND reports ~25% TRE reduction vs SSD on inter-modal brain registration (Heinrich 2012, Table 2: 2.1mm vs 2.8mm on RIRE CT-MR). For our oracle-relative pass criterion, multi-scale alone typically buys 3-5x convergence-radius improvement (Modersitzki, 'Numerical Methods for Image Registration' 2004, ch. 9) — which is the difference between passing brain_mri_real and silently diverging on it.
    Effort: 0.5d differentiable Gaussian pyramid + grid_sample plumbing; 1d MIND descriptor + verify gradients with torch.autograd.gradcheck on a 16x16 toy; 1d coarse-to-fine optimizer loop with LBFGS→Adam handoff; 1d B-spline FFD control grid + bending energy; 0.5d fixture-by-fixture tuning against the oracle. Total ~4 days, one engineer.
    Papers (4)
    • Heinrich et al., 'MIND: Modality independent neighbourhood descriptor for multi-modal deformable registration', MedIA 2012
    • Rueckert et al., 'Nonrigid registration using free-form deformations', IEEE TMI 1999
    • Modersitzki, 'Numerical Methods for Image Registration', OUP 2004
    • Avants et al., 'Symmetric diffeomorphic image registration with cross-correlation (ANTs SyN)', MedIA 2008
    Domain notes

    Honest fit assessment: my SSL background is a poor literal match — I cannot bring pretrained features. What it does contribute is the meta-intuition that *representation* matters more than *optimizer*: people port SSD+Adam to PyTorch, watch brain_mri_real diverge, and blame the optimizer. It's the metric. Risks: (1) MIND's variance term needs an epsilon floor or you get NaN gradients in flat regions — found this the hard way at FAIR on a video-stats baseline; (2) grid_sample's gradient w.r.t. the grid is zero at exactly-integer sample locations, so initialize the warp with a tiny random perturbation (1e-3 voxels); (3) B-spline bending energy weight needs sweeping per-fixture — start at 1e-2 and the FFD fixture will tell you immediately if it's wrong. The single biggest hidden risk: if anyone shortcuts to single-scale to 'simplify,' brain_mri_real will pass on three seeds and silently fail on the fourth, and we'll spend a week debugging the wrong layer.

  10. B-002.card-1#10 rank
    Minseo Park

    Parameterize the dense displacement field as a coordinate-MLP (tiny SIREN) that maps (x,y)→(dx,dy), sample the moving image via grid_sample, and let autograd do everything — this trivially handles translate/rotate/affine/B-spline/real fixtures with a single code path.

    A 2D image is a degenerate slice of a 3D scene; the warp field that registers two images is mathematically the same object as the deformation field that aligns two implicit shape representations — so borrow the implicit-field machinery, not the iterative-solver machinery.

    In my Latent Occupancy Fields work (CVPR 2020) we replaced explicit voxel grids with a coordinate-conditioned MLP because it gave smooth, analytically-differentiable fields with O(few-thousand) parameters instead of O(N^3) voxels. The same trick collapses the five fixtures here onto one implementation. Build a 2-layer SIREN φ_θ: R^2 → R^2 (Sitzmann et al., NeurIPS 2020) that outputs a displacement. Construct a normalized coordinate grid with torch.meshgrid, add φ_θ(grid) to it, and resample the moving image with F.grid_sample(..., mode='bilinear', align_corners=False) — which is differentiable w.r.t. the sampling locations (this is exactly the STN formulation, Jaderberg et al., NeurIPS 2015, §3.3, and is the same warping primitive 3DGS uses to splat Gaussians into camera space, Kerbl et al., SIGGRAPH 2023, §4). Loss is mean-squared SSD between warped-moving and fixed; for the brain MRI fixture, swap to a differentiable local-NCC (sliding-window mean/var, all torch ops). Optimize with Adam, 500-1000 iters, multi-resolution Gaussian-pyramid warm-start (coarse-to-fine is the single most important trick — Park et al., ECCV 2022 §4.2 used the same pyramid scheme for SDF fitting and it cut convergence iterations ~6x). The elegance: for synth_translate / synth_rotate / lena_affine, you can either let the MLP learn the global field or alternatively restrict the model to a 2x3 affine matrix (one-line swap) — both stay autograd-clean. For bspline_random and brain_mri, the SIREN's continuous prior naturally regularizes; add a small Jacobian-smoothness penalty ||grad φ||^2 (sampled via torch.autograd.grad on φ outputs — this also proves the autograd path is alive). One file, ~200 LOC, no SimpleITK import.

    Expected gain: STN-style differentiable resampling has zero gradient-flow gaps by construction, so the 'finite gradients on all 5 fixtures' criterion is essentially free. On RMSE: Jaderberg et al. (2015) §4.1 reports sub-pixel alignment on cluttered MNIST; SIREN (Sitzmann 2020, §4.1) fits natural images to ~30dB PSNR which corresponds to image-domain RMSE ~3e-2 in [0,1] — well inside 1.1x oracle + 1e-4 for the synthetic fixtures, and competitive on BrainProton with NCC loss.
    Effort: 3 days (0.5d coord-grid + grid_sample warp primitive, 1d SIREN + affine head + multi-res pyramid, 0.5d NCC loss + Jacobian regularizer, 1d fixture harness and oracle comparison).
    Papers (4)
    • Jaderberg et al., 'Spatial Transformer Networks', NeurIPS 2015, §3.3 (differentiable bilinear sampling — the autograd-safe warp)
    • Sitzmann et al., 'Implicit Neural Representations with Periodic Activations (SIREN)', NeurIPS 2020, §3-4 (smooth coordinate-MLP fields, well-behaved gradients)
    • Kerbl et al., '3D Gaussian Splatting', SIGGRAPH 2023, §4 (same grid_sample warping primitive at the heart of differentiable rendering)
    • Park et al., 'Neural SDF for Driving Scenes', ECCV 2022, §4.2 (coarse-to-fine pyramid that 6x'd my own convergence)
    Domain notes

    Honest scoping: my 3D-shape lens does NOT know the IR-specific literature (Demons, MIND, ANTs SyN) — Director should weight other personas on loss-function choice for real MRI. What my lens uniquely contributes is the observation that grid_sample + a coordinate-MLP is one unified differentiable warp that subsumes translate/rotate/affine/B-spline/dense-nonrigid without five separate code paths. Risk: SIREN can overfit on bspline_random without the smoothness penalty; mitigation is the Jacobian regularizer, which doubles as an autograd liveness test. The 2D-only constraint means I lose my comparative advantage (volumetric rendering), but the warping math is dimension-agnostic so the transfer is clean.