[core] Fix spatial derivatives when using mode='bspline' or mode='gaussian'

src/deepali/core/image.py

@@ -1486,7 +1486,10 @@ def spatial_derivatives(
             If ``None``, ``forward_central_backward`` is used as default mode.
         sigma: Standard deviation of Gaussian kernel in grid units. If ``None`` or zero,
             no Gaussian smoothing is used for calculation of finite differences, and a
-            default standard deviation of 0.4 is used when ``mode="gaussian"``.
+            default standard deviation of 0.7355 is used when ``mode="gaussian"``. With a smaller
+            standard deviation, the magnitude of the derivative values starts to deviate between
+            ``mode="gaussian"`` and finite differences of a Gaussian smoothed input. This is likely
+            due to a too small discretized Gaussian filter and its derivative.
         spacing: Physical spacing between image grid points, e.g., ``(sx, sy, sz)``.
             When a scalar is given, the same spacing is used for each image and spatial dimension.
             If a sequence is given, it must be of length equal to the number of spatial dimensions ``D``,
@@ -1556,7 +1559,7 @@ def spatial_derivatives(
     if mode in ("forward", "backward", "central", "forward_central_backward", "prewitt", "sobel"):
         if sigma and sigma > 0:
             blur = gaussian1d(sigma, dtype=torch.float, device=data.device)
-            data = conv(data, blur, padding=PaddingMode.ZEROS)
+            data = conv(data, blur, padding=PaddingMode.REPLICATE)
         if mode in ("prewitt", "sobel"):
             avg_kernel = torch.tensor([1, 1 if mode == "prewitt" else 2, 1], dtype=data.dtype)
             avg_kernel /= avg_kernel.sum()
@@ -1589,7 +1592,7 @@ def spatial_derivatives(
 
         if sigma and sigma > 0:
             blur = gaussian1d(sigma, dtype=torch.float, device=data.device)
-            data = conv(data, blur, padding=PaddingMode.ZEROS)
+            data = conv(data, blur, padding=PaddingMode.REPLICATE)
 
         if stride is None:
             stride = 1
@@ -1616,27 +1619,41 @@ def bspline1d(s: int, d: int) -> Tensor:
             for spatial_dim in SpatialDerivativeKeys.split(code):
                 order[spatial_dim] += 1
             kernel = [bspline1d(s, d) for s, d in zip(stride, order)]
-            derivs[code] = evaluate_cubic_bspline(data, kernel=kernel)
+            deriv = evaluate_cubic_bspline(data, kernel=kernel)
+            if sum(order) > 0:
+                denom = torch.ones(N, dtype=spacing.dtype, device=spacing.device)
+                for delta, d in zip(spacing.transpose(0, 1), order):
+                    if d > 0:
+                        denom.mul_(delta.pow(d))
+                denom = denom.reshape((N,) + (1,) * (deriv.ndim - 1))
+                deriv = deriv.div_(denom.to(deriv))
+            derivs[code] = deriv
 
     elif mode == "gaussian":
+
+        def pad_spatial_dim(data: Tensor, sdim: int, padding: int) -> Tensor:
+            pad = [(padding, padding) if d == sdim else (0, 0) for d in range(data.ndim - 2)]
+            pad = [n for v in pad for n in v]
+            return F.pad(data, pad, mode="replicate")
+
         if not sigma:
-            sigma = 0.4
-        kernel_0 = gaussian1d(sigma, normalize=False, dtype=torch.float)
-        kernel_1 = gaussian1d_I(sigma, normalize=False, dtype=torch.float)
-        norm = kernel_0.sum()
-        kernel_0 = kernel_0.div_(norm).to(data.device)
-        kernel_1 = kernel_1.div_(norm).to(data.device)
+            sigma = 0.7355  # same default value as used in downsample()
+        kernel_0 = gaussian1d(sigma, normalize=False, dtype=torch.float, device=data.device)
+        kernel_1 = gaussian1d_I(sigma, normalize=False, dtype=torch.float, device=data.device)
         for i in range(max_order):
             for code in unique_keys:
                 key = code[: i + 1]
                 if i < len(code) and key not in derivs:
                     sdim = SpatialDim.from_arg(code[i])
-                    result = data if i == 0 else derivs[code[:i]]
+                    deriv = data if i == 0 else derivs[code[:i]]
                     for d in range(D):
-                        dim = SpatialDim(d).tensor_dim(result.ndim)
+                        dim = SpatialDim(d).tensor_dim(deriv.ndim)
                         kernel = kernel_1 if sdim == d else kernel_0
-                        result = conv1d(result, kernel, dim=dim, padding=len(kernel) // 2)
-                    derivs[key] = result
+                        deriv = pad_spatial_dim(deriv, d, len(kernel) // 2)
+                        deriv = conv1d(deriv, kernel, dim=dim, padding=0)
+                    denom = spacing.narrow(1, sdim, 1).reshape((N,) + (1,) * (deriv.ndim - 1))
+                    deriv = deriv.div_(denom.to(deriv))
+                    derivs[key] = deriv
         derivs = {key: derivs[SpatialDerivativeKeys.sorted(key)] for key in which}
 
     else:

tests/_test_core_flow_deriv.py

@@ -0,0 +1,205 @@
+r"""Interactive test and visualization of vector flow derivatives."""
+
+# %%
+# Imports
+from typing import Dict, Optional, Sequence
+
+import matplotlib.pyplot as plt
+
+import torch
+from torch import Tensor
+from torch.random import Generator
+
+from deepali.core import Axes, Grid
+import deepali.core.bspline as B
+import deepali.core.functional as U
+
+
+# %%
+# Auxiliary functions
+def change_axes(flow: Tensor, grid: Grid, axes: Axes, to_axes: Axes) -> Tensor:
+    if axes != to_axes:
+        flow = U.move_dim(flow, 1, -1)
+        flow = grid.transform_vectors(flow, axes=axes, to_axes=to_axes)
+        flow = U.move_dim(flow, -1, 1)
+    return flow
+
+
+def flow_derivatives(
+    flow: Tensor, grid: Grid, axes: Axes, to_axes: Optional[Axes] = None, **kwargs
+) -> Dict[str, Tensor]:
+    if to_axes is None:
+        to_axes = axes
+    flow = change_axes(flow, grid, axes, to_axes)
+    axes = to_axes
+    if "spacing" not in kwargs:
+        if axes == Axes.CUBE:
+            spacing = tuple(2 / n for n in grid.size())
+        elif axes == Axes.CUBE_CORNERS:
+            spacing = tuple(2 / (n - 1) for n in grid.size())
+        elif axes == Axes.GRID:
+            spacing = 1
+        elif axes == Axes.WORLD:
+            spacing = grid.spacing()
+        else:
+            spacing = None
+        kwargs["spacing"] = spacing
+    return U.flow_derivatives(flow, **kwargs)
+
+
+def random_svf(
+    size: Sequence[int],
+    stride: int = 1,
+    generator: Optional[Generator] = None,
+) -> Tensor:
+    cp_grid_size = B.cubic_bspline_control_point_grid_size(size, stride=stride)
+    cp_grid_size = tuple(reversed(cp_grid_size))
+    data = torch.randn((1, 3) + cp_grid_size, generator=generator)
+    data = U.fill_border(data, margin=3, value=0, inplace=True)
+    return B.evaluate_cubic_bspline(data, size=size, stride=stride)
+
+
+def visualize_flow(
+    ax: plt.Axes,
+    flow: Tensor,
+    grid: Optional[Grid] = None,
+    axes: Optional[Axes] = None,
+    label: Optional[str] = None,
+) -> None:
+    if grid is None:
+        grid = Grid(shape=flow.shape[2:])
+    if axes is None:
+        axes = grid.axes()
+    flow = change_axes(flow, grid, axes, grid.axes())
+    x = grid.coords(channels_last=False, dtype=flow.dtype, device=flow.device)
+    x = U.move_dim(x.unsqueeze_(0).add_(flow), 1, -1)
+    target_grid = U.grid_image(shape=flow.shape[2:], inverted=True, stride=(5, 5))
+    warped_grid = U.warp_image(target_grid, x, align_corners=grid.align_corners())
+    ax.imshow(warped_grid[0, 0, flow.shape[2] // 2], cmap="gray")
+    if label:
+        ax.set_title(label, fontsize=24)
+
+
+# %%
+# Random velocity fields
+generator = torch.Generator().manual_seed(42)
+grid = Grid(size=(128, 128, 64), spacing=(0.5, 0.5, 1.0))
+flow = random_svf(grid.size(), stride=8, generator=generator).mul_(0.1)
+
+fig, axes = plt.subplots(1, 1, figsize=(4, 4))
+
+ax = axes
+ax.set_title("v", fontsize=24, pad=20)
+visualize_flow(ax, flow, grid=grid, axes=grid.axes())
+
+
+# %%
+# Visualise first order derivatives for different modes
+configs = [
+    dict(mode="forward_central_backward"),
+    dict(mode="bspline"),
+    dict(mode="gaussian", sigma=0.7355),
+]
+
+fig, axes = plt.subplots(len(configs), 4, figsize=(16, 4 * len(configs)))
+
+for i, config in enumerate(configs):
+    derivs = flow_derivatives(
+        flow,
+        grid=grid,
+        axes=grid.axes(),
+        to_axes=Axes.GRID,
+        which=["du/dx", "du/dy", "dv/dx", "dv/dy"],
+        **config,
+    )
+    for ax, (key, deriv) in zip(axes[i], derivs.items()):
+        if i == 0:
+            ax.set_title(key, fontsize=24, pad=20)
+        ax.imshow(deriv[0, 0, deriv.shape[2] // 2], vmin=-1, vmax=1)
+
+
+# %%
+# Compare magnitudes of first order derivatives for different modes
+flow_axes = [Axes.GRID, Axes.WORLD, Axes.CUBE_CORNERS]
+
+sigma = 0.7355
+
+configs = [
+    dict(mode="bspline"),
+    dict(mode="gaussian", sigma=sigma),
+    dict(mode="forward_central_backward", sigma=sigma),
+    dict(mode="forward_central_backward"),
+]
+
+for to_axes in flow_axes:
+    for config in configs:
+        print(f"axes={to_axes}, " + ", ".join(f"{k}={v!r}" for k, v in config.items()))
+        derivs = flow_derivatives(
+            flow,
+            grid=grid,
+            axes=grid.axes(),
+            to_axes=to_axes,
+            which=["du/dx", "du/dy", "dv/dx", "dv/dy"],
+            **config,
+        )
+        for key, deriv in derivs.items():
+            print(f"- max(abs({key})): {deriv.abs().max().item():.5f}")
+        print()
+    print("\n")
+
+
+# %%
+# Visualise second order derivatives for different modes
+configs = [
+    dict(mode="forward_central_backward"),
+    dict(mode="bspline"),
+    dict(mode="gaussian", sigma=0.7355),
+]
+
+fig, axes = plt.subplots(len(configs), 4, figsize=(16, 4 * len(configs)))
+
+for i, config in enumerate(configs):
+    derivs = flow_derivatives(
+        flow,
+        grid=grid,
+        axes=grid.axes(),
+        to_axes=Axes.GRID,
+        which=["du/dxx", "du/dxy", "dv/dxy", "dv/dyy"],
+        **config,
+    )
+    for ax, (key, deriv) in zip(axes[i], derivs.items()):
+        if i == 0:
+            ax.set_title(key, fontsize=24, pad=20)
+        ax.imshow(deriv[0, 0, deriv.shape[2] // 2], vmin=-0.4, vmax=0.4)
+
+
+# %%
+# Compare magnitudes of second order derivatives for different modes
+flow_axes = [Axes.GRID, Axes.WORLD, Axes.CUBE_CORNERS]
+
+sigma = 0.7355
+
+configs = [
+    dict(mode="bspline"),
+    dict(mode="gaussian", sigma=sigma),
+    dict(mode="forward_central_backward", sigma=sigma),
+    dict(mode="forward_central_backward"),
+]
+
+for to_axes in flow_axes:
+    for config in configs:
+        print(f"axes={to_axes}, " + ", ".join(f"{k}={v!r}" for k, v in config.items()))
+        derivs = flow_derivatives(
+            flow,
+            grid=grid,
+            axes=grid.axes(),
+            to_axes=to_axes,
+            which=["du/dxx", "du/dxy", "dv/dxy", "dv/dyy"],
+            **config,
+        )
+        for key, deriv in derivs.items():
+            print(f"- max(abs({key})): {deriv.abs().max().item():.5f}")
+        print()
+    print("\n")
+
+# %%