diff --git a/nitransforms/nonlinear.py b/nitransforms/nonlinear.py
index 79c3aa45..69c19d35 100644
--- a/nitransforms/nonlinear.py
+++ b/nitransforms/nonlinear.py
@@ -170,17 +170,21 @@ def map(self, x, inverse=False):
             indexes = tuple(tuple(i) for i in indexes.T)
             return self._field[indexes]
 
-        return np.vstack(tuple(
+        new_map = np.vstack(tuple(
             map_coordinates(
                 self._field[..., i],
                 ijk.T,
                 order=3,
                 mode="constant",
-                cval=0,
+                cval=np.nan,
                 prefilter=True,
             ) for i in range(self.reference.ndim)
         )).T
 
+        # Set NaN values back to the original coordinates value = no displacement
+        new_map[np.isnan(new_map)] = x[np.isnan(new_map)]
+        return new_map
+
     def __matmul__(self, b):
         """
         Compose with a transform on the right.