diff --git a/nitransforms/nonlinear.py b/nitransforms/nonlinear.py
index 4f91c24c..ef92eb6c 100644
--- a/nitransforms/nonlinear.py
+++ b/nitransforms/nonlinear.py
@@ -21,6 +21,7 @@
     SpatialReference,
     _as_homogeneous,
 )
+from scipy.ndimage import map_coordinates
 
 
 class DenseFieldTransform(TransformBase):
@@ -132,6 +133,13 @@ def map(self, x, inverse=False):
         >>> xfm.map([[-6.5, -36., -19.5], [-1., -41.5, -11.25]]).tolist()
         [[0.0, -0.47516798973083496, 0.0], [0.0, -0.538356602191925, 0.0]]
 
+        >>> np.array_str(
+        ...     xfm.map([[-6.7, -36.3, -19.2], [-1., -41.5, -11.25]]),
+        ...     precision=3,
+        ...     suppress_small=True,
+        ... )
+        '[[ 0.    -0.482  0.   ]\n [ 0.    -0.538  0.   ]]'
+
         >>> xfm = DenseFieldTransform(
         ...     test_dir / "someones_displacement_field.nii.gz",
         ...     is_deltas=True,
@@ -139,16 +147,34 @@ def map(self, x, inverse=False):
         >>> xfm.map([[-6.5, -36., -19.5], [-1., -41.5, -11.25]]).tolist()
         [[-6.5, -36.47516632080078, -19.5], [-1.0, -42.03835678100586, -11.25]]
 
+        >>> np.array_str(
+        ...     xfm.map([[-6.7, -36.3, -19.2], [-1., -41.5, -11.25]]),
+        ...     precision=3,
+        ...     suppress_small=True,
+        ... )
+        '[[ -6.7   -36.782 -19.2  ]\n [ -1.    -42.038 -11.25 ]]'
+
         """
 
         if inverse is True:
             raise NotImplementedError
         ijk = self.reference.index(x)
         indexes = np.round(ijk).astype("int")
-        if np.any(np.abs(ijk - indexes) > 0.05):
-            warnings.warn("Some coordinates are off-grid of the field.")
-        indexes = tuple(tuple(i) for i in indexes.T)
-        return self._field[indexes]
+
+        if np.all(np.abs(ijk - indexes) < 1e-3):
+            indexes = tuple(tuple(i) for i in indexes.T)
+            return self._field[indexes]
+
+        return np.vstack((
+            map_coordinates(
+                self._field[..., i],
+                ijk.T,
+                order=3,
+                mode="constant",
+                cval=0,
+                prefilter=True,
+            ) for i in range(self.reference.ndim)
+        )).T
 
     def __matmul__(self, b):
         """