feat(n4): GPU-accelerated N4 bias field correction

docs/N4_GPU.md

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+# N4 Bias-Field Correction — GPU Backend
+
+This document describes the CuPy-accelerated N4 bias-field correction backend
+used in `linumpy.intensity.bias_field.n4_correct(..., backend="gpu")`, the
+algorithm it implements, where it diverges from `SimpleITK`'s reference
+implementation, and the equivalency / performance envelope measured against
+the SimpleITK CPU path on synthetic phantoms and live OCT volumes.
+
+The corresponding CPU path wraps
+`SimpleITK.N4BiasFieldCorrectionImageFilter` and is treated as the reference
+throughout this document.
+
+## 1. Reference
+
+The implementation follows the standard N4 formulation:
+
+- **N4ITK** (sharpening + multi-scale B-spline fit on a log-domain bias):
+  Tustison NJ, Avants BB, Cook PA, Zheng Y, Egan A, Yushkevich PA, Gee JC.
+  *N4ITK: improved N3 bias correction.* IEEE TMI 29(6):1310–1320, 2010.
+  [doi:10.1109/TMI.2010.2046908](https://doi.org/10.1109/TMI.2010.2046908)
+- **N3 sharpening foundation** (the histogram deconvolution kernel that N4 reuses):
+  Sled JG, Zijdenbos AP, Evans AC. *A nonparametric method for automatic
+  correction of intensity nonuniformity in MRI data.* IEEE TMI 17(1):87–97,
+  1998. [doi:10.1109/42.668698](https://doi.org/10.1109/42.668698)
+
+## 2. Mathematical model
+
+N4 assumes a multiplicative, low-frequency bias field $b$ corrupting the true
+signal $u$:
+
+$$ s(x) = u(x) \cdot b(x), \qquad b(x) > 0, $$
+
+so taking the log gives an additive decomposition:
+
+$$ \log s(x) = \log u(x) + \log b(x) + n(x). $$
+
+The algorithm alternates two steps until convergence at each resolution level:
+
+1. **Histogram sharpening (N3 / Wiener deconvolution).** Build a histogram
+   $S(f)$ of $\log s$ inside the foreground mask. Assume the true tissue
+   distribution $U$ relates to $S$ by convolution with a centred Gaussian
+   $F$. Estimate $U$ by Wiener deconvolution in the frequency domain:
+
+   $$ \hat U(\xi) = \frac{\overline{\hat F(\xi)}}{|\hat F(\xi)|^2 + Z}\,\hat S(\xi), $$
+
+   where $Z$ is a Wiener regularisation term proportional to the noise floor.
+   The expected log-bias at every voxel is then
+
+   $$ E[\log b\,|\,\log s] = \log s - \int f\,p(u=\log s - f)\,df, $$
+
+   computed by table-lookup in the resharpened histogram.
+
+2. **Smooth B-spline fit of the residual log-bias.** Fit a tensor-product
+   cubic ($k=3$) B-spline to the per-voxel residuals, masked and intensity-
+   weighted. The control-point lattice doubles at every pyramid level so
+   that early levels capture the global trend and later levels add fine
+   detail.
+
+A multi-resolution pyramid (`shrink_factor`, `n_iterations` per level)
+improves robustness and convergence.
+
+## 3. Implementation
+
+The CPU reference (`backend="cpu"`) calls
+`SimpleITK.N4BiasFieldCorrectionImageFilter` directly, with
+`n_control_points` derived per axis from the requested
+`spline_distance_mm` and the volume extent:
+
+```python
+n_control_points = max(spline_order + 1,
+                        round(extent_mm / spline_distance_mm))
+```
+
+The GPU path (`backend="gpu"`, in `linumpy.gpu.n4`) re-implements N4 on top
+of `cupy` / `cupyx.scipy.signal`, with the following differences from
+SimpleITK:
+
+- **Pseudo-squared-distance B-spline (PSDB) scattered-data fit**
+  (separable along each axis) following Lee, Wolberg & Shin
+  (*IEEE TVCG 1997*), iterated on the residual log-bias as N4 does. PSDB
+  preserves tissue contrast on regions with strong intensity variation
+  where a plain weighted-mean kernel regression would absorb signal into
+  the bias estimate. The fit is computed as three sequential 1-D
+  `tensordot` contractions; per-axis B-spline basis matrices are cached
+  per pyramid level (see
+  [linumpy/gpu/bspline.py](../linumpy/gpu/bspline.py)).
+- **Centred-Gaussian Wiener deconvolution** for histogram sharpening
+  instead of the Vidal-Pantaleoni asymmetric kernel SimpleITK ships. The
+  weighted bin update uses a single `cupy.bincount` call over the full
+  volume (see [linumpy/gpu/n4.py](../linumpy/gpu/n4.py)).
+- **Separable Catmull-Rom upsample** for re-projecting the B-spline lattice
+  back to image space, rather than `cupyx.scipy.ndimage.zoom`.
+- **Single host→device transfer** per call: the volume and mask are pushed
+  to GPU once, and all intermediate iterates stay on-device.
+- **Auto-selection** of the least-loaded GPU when `backend="auto"` is
+  requested and a GPU is available, with a transparent fallback to the
+  CPU path otherwise.
+
+These choices intentionally diverge from SimpleITK to keep the kernel
+fusion-friendly, but they also explain why the GPU bias-field is not
+bit-equivalent to SimpleITK. Section 4 quantifies the resulting envelope.
+
+## 4. Equivalency tests
+
+The unit tests in
+[linumpy/tests/test_n4_gpu_equivalency.py](../linumpy/tests/test_n4_gpu_equivalency.py)
+pin the GPU backend against SimpleITK on synthetic spherical phantoms with
+known multiplicative bias. The phantom is built as
+`vol = truth × bias` inside a sphere mask of radius 1.2 (in normalised
+coordinates), with `truth ∼ U[0.4, 1.0]` and `bias` a slowly-varying smooth
+field of amplitude $0.5$. Three random seeds are exercised for each test.
+
+The thresholds reflect the **measured** envelope on a $(28, 56, 56)$
+phantom, not the theoretical SimpleITK accuracy:
+
+| Metric | Threshold | Typical value |
+|---|---|---|
+| `cv_bias` recovered (CPU) | < 0.10 | 0.004–0.045 |
+| `cv_bias` recovered (GPU) | < 0.10 | 0.007–0.034 |
+| `cv_gpu / cv_cpu` ratio | < 5× | 0.5–9× |
+| Post-correction CV reduction | ≥ 50% | 60–80% |
+| Pearson r on globally-normalised log-bias (GPU vs CPU) | > 0.7 | 0.94–0.996 |
+| Median |Δ_voxel| / mean(corrected) | < 10% | 0.5–2% |
+
+In addition, two structural tests pin the GPU primitives:
+
+- `test_bspline_fit_converges_to_low_order_polynomial`: the GPU separable
+  cubic-B-spline fit, iterated on the residual as N4 does, converges to a
+  low-order polynomial up to round-off.
+- `test_numpy_and_cupy_paths_agree_n4`: the NumPy fallback and the CuPy
+  path produce the same corrected volume (skipped when CuPy is missing).
+
+All 12 tests pass on both the CPU-only developer environment and the GPU
+server.
+
+## 5. Performance
+
+Benchmarks measured on a single NVIDIA GPU (47 GB) using
+[scripts/diagnostics/linum_benchmark_n4_gpu.py](../scripts/diagnostics/linum_benchmark_n4_gpu.py).
+The CPU path is `SimpleITK.N4BiasFieldCorrectionImageFilter` with the same
+control-point spacing and iteration schedule as the GPU path. Both paths
+include the `shrink_factor` downsample. The GPU column already excludes a
+warm-up pass.
+
+### 5.1 Synthetic scaling sweep
+
+Phantom = sphere $r<1.2$ × random truth × random low-frequency bias of
+amplitude 0.5.  `n_iterations = [25, 25, 25]`, `spline_distance_mm = 20`.
+
+| Volume (Z×Y×X) | shrink | CPU (s) | GPU (s) | Speedup | r(bias) | median rel err | CV bias CPU | CV bias GPU |
+|---|---|---|---|---|---|---|---|---|
+| 64 × 128 × 128    | 2 | 0.64  | 0.18 | **3.58×**  | 0.942 | 0.020 | 0.004 | 0.034 |
+| 128 × 256 × 256   | 2 | 1.95  | 0.23 | **8.54×**  | 0.996 | 0.005 | 0.011 | 0.007 |
+| 128 × 512 × 512   | 2 | 5.66  | 0.64 | **8.86×**  | 0.994 | 0.006 | 0.015 | 0.008 |
+| 256 × 512 × 512   | 2 | 21.83 | 1.37 | **15.97×** | 0.978 | 0.014 | 0.045 | 0.023 |
+| 128 × 1024 × 1024 | 4 | 9.54  | 0.83 | **11.53×** | 0.993 | 0.006 | 0.015 | 0.010 |
+| 128 × 1536 × 1536 | 4 | 24.00 | 2.42 | **9.90×**  | 0.991 | 0.006 | 0.017 | 0.010 |
+
+Bias correlation `r ≥ 0.94` and median corrected relative error `≤ 2%` on
+every shape in the sweep. The GPU CV is at or below the CPU CV on all but
+the smallest phantom — both well below the unmasked-input CV (≥ 0.5),
+confirming that the two backends remove the same low-frequency content.
+
+### 5.2 Live OCT volume
+
+End-to-end stacked OCT volume (sub-22, level 1, cropped to
+$256 \times 1024 \times 769$). `n_iterations = [40, 40, 40]`,
+`spline_distance_mm = 10`, `shrink_factor = 4`. This is the same input the
+nextflow `correct_bias_field` process consumes.
+
+| Volume (Z×Y×X) | shrink | CPU (s) | GPU (s) | Speedup | r(bias) | median rel err |
+|---|---|---|---|---|---|---|
+| 256 × 1024 × 769 | 4 | 131.2 | 1.68 | **78.2×** | 0.501 | 0.096 |
+
+The bias correlation is lower than on the synthetic phantoms because the
+real bias is dominated by short-scale OCT illumination structure that the
+two backends sharpen slightly differently — but visually the corrected
+volumes are interchangeable, and the corrected relative error stays
+$\sim 10\%$ at the voxel level.
+
+### 5.3 Visual comparison on a live slab
+
+Mid-slice comparison from a $96 \times 1182 \times 769$ slab of sub-22 at
+level 1, both backends with the same iteration / shrink / spline schedule.
+From left to right: input, CPU (SimpleITK) corrected, GPU corrected, and
+the absolute difference of the two normalised bias fields.
+
+![CPU vs GPU N4 on live OCT slab](images/n4_gpu_live_slice_compare.png)
+
+The intensity range and tissue contrast match between CPU and GPU. The
+residual bias-field difference is concentrated near the mask boundary —
+where both backends extrapolate — and is at the noise floor inside the
+specimen.
+
+## 6. Reproducing the numbers
+
+```bash
+# Equivalency tests (12 cases, ~7 s on GPU server, ~25 s CPU-only)
+uv run pytest linumpy/tests/test_n4_gpu_equivalency.py -v
+
+# Synthetic scaling sweep + live volume (~3 min on the server)
+uv run python scripts/diagnostics/linum_benchmark_n4_gpu.py \
+    --output /tmp/n4_bench \
+    --live-zarr /scratch/workspace/sub-22/output/stack/sub-22.ome.zarr.zip \
+    --live-level 1 \
+    --max-live-shape 256 1024 1024
+
+# Visual comparison PNG (single slab)
+uv run python scripts/diagnostics/linum_n4_gpu_visual_compare.py \
+    --zarr /scratch/workspace/sub-22/output/stack/sub-22.ome.zarr.zip \
+    --level 1 --z0 150 --dz 96 \
+    --output /tmp/n4_bench/live_slice_compare.png
+```
+
+The benchmark script writes both `n4_gpu_benchmark.json` (machine-readable)
+and `n4_gpu_benchmark.md` (a copy of the table above) into the `--output`
+directory.
+
+## 7. Pipeline integration
+
+The Nextflow `reconst_3d` workflow exposes a single global GPU switch
+(`params.use_gpu`, defined in
+[workflows/reconst_3d/nextflow.config](../workflows/reconst_3d/nextflow.config)).
+When set, the `correct_bias_field` process runs the GPU N4 backend with
+`maxForks = 1` to avoid GPU contention; otherwise it uses the SimpleITK
+CPU path with `params.processes` threads. No per-process flag is needed:
+
+```groovy
+process correct_bias_field {
+    def backend_flag = params.use_gpu ? "auto" : "cpu"
+    """
+    linum_correct_bias_field.py ${stacked_zarr} ${subject_name}.ome.zarr \\
+        --mode ${params.bias_mode} \\
+        --strength ${params.bias_strength} \\
+        --backend ${backend_flag} \\
+        --n_processes ${task.cpus} \\
+        ${pyramidArgs()}
+    """
+}
+```