Use compact [1,-2,1] kernel for BendingEnergyLoss second derivatives

[vishnukannaujia]

Fixes #5939.

Summary

BendingEnergyLoss previously computed second-order derivatives by applying the central first-order helper spatial_gradient twice, which yields a wide [1, 0, -2, 0, 1] / 4 stencil for pure second derivatives that spans four voxels per axis. That stencil is less accurate than the standard compact one and forced the validation "all spatial dims > 4".

This PR replaces the second-order computation with compact central stencils evaluated directly on pred:

• Pure d^2/dx_i^2: x[i+1] - 2 * x[i] + x[i-1] (the standard [1, -2, 1] kernel).
• Mixed d^2/(dx_i dx_j): (x[i+1,j+1] - x[i+1,j-1] - x[i-1,j+1] + x[i-1,j-1]) / 4 (compact 4-point central scheme).

Both span three voxels per axis, so the spatial-size validation is relaxed from > 4 to > 2, matching DiffusionLoss. The public API (__init__(normalize, reduction), forward(pred)) and normalize semantics are unchanged. The existing spatial_gradient helper is left untouched because DiffusionLoss still uses it.

Why TEST_CASES expected values do not change

For f(x) = x^2, the analytical second derivative is the constant 2. Both the previous central-of-central stencil (f[i+2] - 2*f[i] + f[i-2]) / 4 and the new compact [1, -2, 1] stencil f[i+1] - 2*f[i] + f[i-1] are exact for quadratics, so both return 2 at every interior voxel. Mixed-partial test inputs are constant in at least one of the two axes, so both stencils return 0 for mixed terms on these cases. Squared and reduced by mean, the existing TEST_CASES expected values (0.0, 4.0, 100.0) are therefore invariant under this change.

What does change in the tests:

• test_ill_shape is updated to trigger on shape 2 (was 4) so it still exercises the spatial-size guard.
• A new TEST_CASES row covers shape (1, 3, 3, 3, 3) of ones (previously rejected by the > 4 guard) → expected 0.0, validating the relaxed guard.

Reference for the compact mixed-partial scheme: Pavel Holoborodko's finite-difference notes cited in the original issue.

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• monai/losses/deform.py

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📝 Walkthrough

Walkthrough

spatial_gradient_squared is added to compute pure second-order spatial derivatives using the compact [1, -2, 1] stencil and mixed partial derivatives via forward-difference cross-term formula, with outputs sliced to [1:-1] on differentiated dimensions. BendingEnergyLoss.forward is updated to call this helper instead of chaining two first-order spatial_gradient calls, accumulating energy from pure and twice-mixed derivative terms with revised normalization. Minimum valid spatial size drops from > 4 to > 2. Tests are adjusted accordingly.

Estimated code review effort

🎯 3 (Moderate) | ⏱️ ~25 minutes

🚥 Pre-merge checks | ✅ 4 | ❌ 1

❌ Failed checks (1 warning)

Check name	Status	Explanation	Resolution
Docstring Coverage	⚠️ Warning	Docstring coverage is 60.00% which is insufficient. The required threshold is 80.00%.	Write docstrings for the functions missing them to satisfy the coverage threshold.

✅ Passed checks (4 passed)

Check name	Status	Explanation
Title check	✅ Passed	Title clearly and concisely summarizes the main change: adopting compact [1,-2,1] kernel for computing second derivatives in BendingEnergyLoss.
Description check	✅ Passed	Description comprehensively covers the rationale, technical implementation, API stability, and test validation. Includes proper issue reference and detailed justification for expected value invariance.
Linked Issues check	✅ Passed	Changes directly address #5939 objectives: replaces numerically unstable double-application of first-order gradient with compact [1,-2,1] kernel for pure derivatives and compact schemes for mixed partials, improving accuracy and stability.
Out of Scope Changes check	✅ Passed	All changes are scoped to BendingEnergyLoss implementation and its tests. New spatial_gradient_squared helper, validation relaxation from >4 to >2, and test adjustments directly support the stated objectives.

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Actionable comments posted: 1

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
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Inline comments:
In `@monai/losses/deform.py`:
- Line 150: The `torch.tensor(0)` initialization at line 150 omits the device
parameter, causing a CPU tensor to be added to GPU tensors in the loop,
inconsistent with line 147's approach using `torch.tensor(pred.shape,
device=pred.device)`. Additionally, torch.tensor(0) defaults to int64 dtype
which may be incorrect. Update the initialization at line 150 to include the
device parameter set to pred.device to match the pattern used earlier in the
function. Also apply the same fix to the DiffusionLoss class around line 237
which has the identical pattern for consistency across the file.

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Configuration used: Path: .coderabbit.yaml

Review profile: CHILL

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📒 Files selected for processing (2)

• monai/losses/deform.py
• tests/losses/deform/test_bending_energy.py