diff --git a/.github/workflows/ci.yaml b/.github/workflows/ci.yaml
index aa8f326..a05c3fe 100644
--- a/.github/workflows/ci.yaml
+++ b/.github/workflows/ci.yaml
@@ -22,7 +22,7 @@ jobs:
         with:
           enable-cache: true
       - run: uv sync --all-extras
-      - run: uv run --group tutorial marimo check --strict --ignore-scripts examples/*.py
+      - run: uv run --group examples marimo check --strict --ignore-scripts examples/*.py
 
   lint:
     runs-on: ubuntu-latest
diff --git a/.readthedocs.yaml b/.readthedocs.yaml
index b9f17dc..6b1ade5 100644
--- a/.readthedocs.yaml
+++ b/.readthedocs.yaml
@@ -16,3 +16,6 @@ build:
         # - just doc ${READTHEDOCS_OUTPUT}/html
         - mkdir -p ${READTHEDOCS_OUTPUT}/html
         - uv run --group doc pdoc --output-directory ${READTHEDOCS_OUTPUT}/html --no-include-undocumented --docformat numpy --search transformnd
+        - mkdir -p ${READTHEDOCS_OUTPUT}/html/examples
+        - uv run --group examples marimo export html examples/tutorial.py -o {{docdir}}/examples/tutorial.html
+        - uv run --group examples marimo export html examples/image.py -o {{docdir}}/examples/image.html
diff --git a/README.md b/README.md
index 3a8767e..4149a31 100644
--- a/README.md
+++ b/README.md
@@ -29,7 +29,7 @@ and implements these adapters for a few common types.
 
 See the [tutorial here](https://github.com/clbarnes/transformnd/blob/main/examples/tutorial.py).
 It is a [marimo](https://marimo.io) notebook.
-Open it with `uv run --group tutorial marimo edit examples/tutorial.py`.
+Open it with `uv run --group examples marimo edit examples/tutorial.py`.
 
 ## Implemented transforms
 
diff --git a/examples/image.py b/examples/image.py
new file mode 100644
index 0000000..b5e6655
--- /dev/null
+++ b/examples/image.py
@@ -0,0 +1,170 @@
+import marimo
+
+__generated_with = "0.23.4"
+app = marimo.App(width="medium")
+
+with app.setup:
+    import marimo as mo
+
+
+@app.cell(hide_code=True)
+def _():
+    mo.md(r"""
+    # Image transformation with `transformnd`
+
+    `transformnd` transforms coordinates, not images, but coordinate transformations can be used to transform images.
+    Your output (transformed) and source images both have pixels with an `xy` coordinate in their respective image spaces,
+    and image transformation is simply a case of finding which source pixel to use for each output pixel.
+
+    Here we take a 2-channel fluorescence microscopy image of some cells in 3 dimensions, use scaling information to map those pixels into a real-world space, and then map the pixels of our viewport into the the same space.
+    """)
+    return
+
+
+@app.cell
+def _():
+    from skimage.data import cells3d
+
+    # ZCYX, (0.29um, membrane/nuclei channels, 0.26um, 0.26um)
+    CELLS_SPACE = "cells"
+
+    # CZYX, (membrane/nuclei, um, um, um)
+    WORLD_SPACE = "world"
+
+    # YXC image with RGB channels
+    VIEWPORT_SPACE = "viewport"
+
+    cells = cells3d()
+    cells = cells.astype("float64")
+    cells -= cells.min()
+    cells /= cells.max()
+
+    print(f"{cells.shape=}")
+    print(f"{cells.dtype=}")
+    print(f"{cells.min()=}")
+    print(f"{cells.max()=}")
+    return CELLS_SPACE, VIEWPORT_SPACE, WORLD_SPACE, cells
+
+
+@app.cell
+def _(CELLS_SPACE, VIEWPORT_SPACE, WORLD_SPACE):
+    import transformnd as tnd
+    from transformnd.transforms import ProjectAxis, MapAxis, Scale
+
+    # Aligned at world origin.
+    # This would be stored alongside the data.
+    cells_to_world = tnd.base.TransformSequence(
+        [
+            # Move the color axis to the first position
+            MapAxis([1, 0, 2, 3]),
+            # Scale the space axes
+            Scale([1, 0.29, 0.26, 0.26]),
+        ],
+        spaces=tnd.Spaces(CELLS_SPACE, WORLD_SPACE),
+    )
+    print(cells_to_world)
+
+    # Aligned at world origin.
+    # This would be chosen by the viewing application.
+    viewport_to_world = tnd.base.TransformSequence(
+        [
+            # Create a Z axis
+            ProjectAxis(created={0}, source_ndim=3),
+            # Move the color axis to the first position
+            MapAxis([3, 0, 1, 2]),
+            # Choose a spatial sampling frequency (here 0.2um isotropic)
+            Scale([1, 0.2, 0.2, 0.2]),
+        ],
+        spaces=tnd.Spaces(VIEWPORT_SPACE, WORLD_SPACE),
+    )
+    print(viewport_to_world)
+    return cells_to_world, viewport_to_world
+
+
+@app.cell(hide_code=True)
+def _():
+    mo.md(r"""
+    Both images know how to transform their array indices into the real world.
+
+    We can invert one of those transforms to get a transformation between viewport-space and cell-space.
+    We can also have a separate transformation to control moving the viewport (useful if we had an interactive viewer).
+    """)
+    return
+
+
+@app.cell
+def _(cells_to_world, viewport_to_world):
+    from transformnd.transforms import Translate
+
+    # Shift the viewport within the data, in world measurements.
+    # This would be controlled by the user as they peruse the data.
+    viewport_offset = Translate([0, 35 * 0.29, 64 * 0.26, 0.0])
+
+    viewport_to_cells = viewport_to_world | viewport_offset | ~cells_to_world
+    return (viewport_to_cells,)
+
+
+@app.cell(hide_code=True)
+def _():
+    mo.md(r"""
+    Here we want to get all of the coordinates of our viewport, across all channels, in the shape needed by `transformnd` (number of coordinates x dimensionality of coordinates).
+
+    We then transform that to get the positions of those coordinates within the cells image.
+    """)
+    return
+
+
+@app.cell
+def _(viewport_to_cells):
+    import numpy as np
+
+    # 2D YXC image
+    viewport_shape = (128, 256, 3)
+
+    indices = [np.arange(s, dtype=float) for s in viewport_shape]
+    grids = np.meshgrid(*indices, indexing="ij")
+
+    # Y, X, C, coords
+    vp_coords_3d = np.stack(grids, -1)
+    print(f"{vp_coords_3d.shape=}")
+
+    # Y*X*C, coords
+    vp_coords = vp_coords_3d.reshape((-1, len(viewport_shape)))
+    print(f"{vp_coords.shape=}")
+
+    # Z*C*Y*X, coords
+    cells_coords = viewport_to_cells.apply(vp_coords)
+    print(f"{cells_coords.shape=}")
+    return cells_coords, viewport_shape
+
+
+@app.cell(hide_code=True)
+def _():
+    mo.md(r"""
+    `scipy.ndimage.map_coordinates` is where the magic happens; looking up our coordinates in the cells image to get the intensities. There's a dask version too!
+    """)
+    return
+
+
+@app.cell
+def _(cells, cells_coords, viewport_shape):
+    from scipy.ndimage import map_coordinates
+
+    # transformnd uses `NxD` coordinate arrays; map_coordinates uses `DxN`
+    cells_vals = map_coordinates(cells, cells_coords.T).T
+    print(f"{cells_vals.shape=}")
+    viewport = cells_vals.reshape(viewport_shape)
+    print(f"{viewport.shape=}")
+    return (viewport,)
+
+
+@app.cell
+def _(viewport):
+    from matplotlib import pyplot as plt
+
+    plt.imshow(viewport)
+    return
+
+
+if __name__ == "__main__":
+    app.run()
diff --git a/examples/tutorial.py b/examples/tutorial.py
index d994751..e5ce025 100644
--- a/examples/tutorial.py
+++ b/examples/tutorial.py
@@ -1,6 +1,6 @@
 import marimo
 
-__generated_with = "0.23.1"
+__generated_with = "0.23.4"
 app = marimo.App()
 
 
@@ -218,11 +218,16 @@ def _(Scale, Translate):
     bc = Translate([0.5, 1], spaces=Spaces("b", "c"))
     bd = Translate([1, 0.5], spaces=Spaces("b", "d"))
 
-    g.add_transforms([ab, bc, bd])
+    g.add_transform(ab)
+    g.add_transform(ab.invert())
+    g.add_transform(bc)
+    g.add_transform(bc.invert())
+    g.add_transform(bd)
+    g.add_transform(bd.invert())
 
     print("Transform sequence from a to c:", g.get_sequence("a", "c"))
     print("Transform sequence from d to a:", g.get_sequence("d", "a"))
-    return
+    return (Spaces,)
 
 
 @app.cell
@@ -245,7 +250,7 @@ def _(mo):
 
 
 @app.cell
-def _(np, Spaces):
+def _(Spaces, np):
     from transformnd import Transform, NDims
 
     class IsotropicScale2d(Transform):
diff --git a/justfile b/justfile
index 20ee26e..5c40994 100644
--- a/justfile
+++ b/justfile
@@ -11,13 +11,16 @@ doc docdir='doc/html':
         --docformat numpy \
         --search \
         transformnd
+    mkdir -p {{docdir}}/examples
+    uv run --group examples marimo export html examples/tutorial.py -o {{docdir}}/examples/tutorial.html
+    uv run --group examples marimo export html examples/image.py -o {{docdir}}/examples/image.html
 
 # Run linters and type checkers.
 lint:
     uv run --group lint ruff check src tests examples bench
     uv run --group lint mypy src tests bench
     uv run --group lint ruff format --check src tests examples bench
-    uv run --group tutorial marimo check --strict --ignore-scripts examples/*.py
+    uv run --group examples marimo check --strict --ignore-scripts examples/*.py
     uv run --group lint pydoclint src
 
 # Auto-fix format and lints where possible.
@@ -33,7 +36,11 @@ format:
 test:
     uv run --all-groups --all-extras pytest -v
 
-examples:
+example-edit example:
+    uv run --group examples marimo edit examples/{{example}}.py
+
+example example:
+    uv run --group examples marimo run examples/{{example}}.py
 
 # Run benchmarks.
 bench:
diff --git a/pyproject.toml b/pyproject.toml
index b213c85..91aebbf 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -62,7 +62,7 @@ dev = [
     {include-group = "test"},
     {include-group = "lint"},
     {include-group = "doc"},
-    {include-group = "tutorial"},
+    {include-group = "examples"},
     {include-group = "bench"},
 ]
 test = [
@@ -80,10 +80,12 @@ lint = [
 doc = [
     "pdoc>=16.0.0",
 ]
-tutorial = [
+examples = [
     "marimo>=0.9",
     "matplotlib>=3.10.8",
     "pandas>=3.0.2",
+    "pooch>=1.9.0",
+    "scikit-image>=0.26.0",
 ]
 bench = [
     "pytest-benchmark>=5.2.3",
diff --git a/src/transformnd/__init__.py b/src/transformnd/__init__.py
index 9db01db..ae5b2e1 100644
--- a/src/transformnd/__init__.py
+++ b/src/transformnd/__init__.py
@@ -1,5 +1,11 @@
 """
 .. include:: ../../README.md
+
+You can find some examples here:
+
+- [Tutorial](./examples/tutorial.html)
+- [Image transformation](./examples/image.html)
+
 """
 
 from .base import Transform, TransformSequence, TransformWrapper
diff --git a/src/transformnd/base.py b/src/transformnd/base.py
index 374374d..666770b 100644
--- a/src/transformnd/base.py
+++ b/src/transformnd/base.py
@@ -304,6 +304,14 @@ def __init__(
         if not ts:
             raise ValueError("Empty transform sequence")
 
+        for idx, (t1, t2) in enumerate(pairwise(ts)):
+            if t1.ndims.target != t2.ndims.source:
+                raise ValueError(
+                    "Incompatible dimensionality: "
+                    f"transform {idx}'s target is {t1.ndims.target}D "
+                    f"and the next source is {t2.ndims.source}D"
+                )
+
         spaces = Spaces(ts[0].spaces.source, ts[-1].spaces.target)
         ndims = NDims(ts[0].ndims.source, ts[-1].ndims.target)
 
diff --git a/src/transformnd/transforms/__init__.py b/src/transformnd/transforms/__init__.py
index 79965de..37de554 100644
--- a/src/transformnd/transforms/__init__.py
+++ b/src/transformnd/transforms/__init__.py
@@ -5,7 +5,7 @@
 from .simple import Identity, Scale, Translate
 from .map_axis import MapAxis
 from .bijection import Bijection
-from .project_axis import ProjectAxis, Insert, Remove
+from .project_axis import ProjectAxis
 from .by_dimension import ByDimension, SubTransform
 from .vector_field import Coordinates, Displacements
 from .moving_least_squares import MovingLeastSquares
@@ -15,8 +15,6 @@
     "Affine",
     "Identity",
     "ProjectAxis",
-    "Insert",
-    "Remove",
     "Reflect",
     "Scale",
     "Translate",
diff --git a/src/transformnd/transforms/project_axis.py b/src/transformnd/transforms/project_axis.py
index b946c85..eed8ae4 100644
--- a/src/transformnd/transforms/project_axis.py
+++ b/src/transformnd/transforms/project_axis.py
@@ -1,69 +1,14 @@
 from __future__ import annotations
-from abc import ABC, abstractmethod
-from copy import copy
-from typing import Self, Sequence
+from typing import Self
 
 import numpy as np
 from array_api_compat import array_namespace
 from transformnd.transforms import Affine
 from transformnd.types import NDims, Spaces
-from dataclasses import dataclass
 from ..base import Transform
 from ..types import ArrayT
 
 
-@dataclass(frozen=True, eq=True)
-class BaseOperation(ABC):
-    idx: int
-    """Which axis to apply the operation to."""
-
-    def __post_init__(self):
-        if self.idx < 0:
-            raise ValueError("insert/remove idx must be positive")
-
-    @abstractmethod
-    def check(self, ndim: int) -> int: ...
-
-    @abstractmethod
-    def invert(self) -> BaseOperation: ...
-
-
-@dataclass(frozen=True, eq=True)
-class Insert(BaseOperation):
-    """Component of the `ProjectAxis` transform which inserts a new axis."""
-
-    def check(self, ndim: int) -> int:
-        if self.idx > ndim or self.idx <= -ndim:
-            raise ValueError(
-                f"Index {self.idx} is out of range for dimensionality {ndim}"
-            )
-        return ndim + 1
-
-    def invert(self) -> Remove:
-        return Remove(self.idx)
-
-
-@dataclass(frozen=True, eq=True)
-class Remove(BaseOperation):
-    """Component of the `ProjectAxis` transform which removes an existing axis."""
-
-    def check(self, ndim: int) -> int:
-        if self.idx >= ndim or self.idx <= -ndim:
-            raise ValueError(
-                f"Index {self.idx} is out of range for dimensionality {ndim}"
-            )
-        return ndim - 1
-
-    def invert(self) -> Insert:
-        if self.idx == -1:
-            raise ValueError("Removal of the -1th axis is not invertible")
-        return Insert(self.idx)
-
-
-Operation = Insert | Remove
-"""Insert or remove an axis."""
-
-
 class ProjectAxis(Transform):
     """Transform for adding and removing axes.
 
@@ -72,7 +17,8 @@ class ProjectAxis(Transform):
 
     def __init__(
         self,
-        operations: Sequence[Operation],
+        dropped: set[int] | None = None,
+        created: set[int] | None = None,
         source_ndim: int | None = None,
         target_ndim: int | None = None,
         *,
@@ -84,8 +30,10 @@ def __init__(
 
         Parameters
         ----------
-        operations
-            Sequence of operations to apply.
+        dropped
+            Set of INPUT dimension indices to drop, if any.
+        created
+            Set of OUTPUT dimension indices which are new, if any.
         source_ndim
             If omitted, can be inferred from `target_ndim`.
         target_ndim
@@ -99,22 +47,18 @@ def __init__(
             Operations are inconsistent with given dimensionality,
             or insufficient dimensionality information was given.
         """
-        self.operations = []
-        self._has_inserts = False
+        self.dropped = dropped or set()
+        self.created = created or set()
 
         if source_ndim is not None:
-            nd = source_ndim
-            for op in operations:
-                nd = op.check(nd)
+            nd = source_ndim - len(self.dropped) + len(self.created)
             if target_ndim is None:
                 target_ndim = nd
             elif target_ndim != nd:
                 raise ValueError("Operations do not match expected target ndim")
 
         elif target_ndim is not None:
-            nd = target_ndim
-            for op in reversed(operations):
-                nd = op.invert().check(nd)
+            nd = target_ndim - len(self.created) + len(self.dropped)
             if source_ndim is None:
                 source_ndim = nd
             elif source_ndim != nd:
@@ -124,20 +68,17 @@ def __init__(
             raise ValueError("At least one of source_ndim or target_ndim must be given")
 
         idxs: list[int | None] = list(range(source_ndim))
-        for op in operations:
-            if isinstance(op, Insert):
-                self._has_inserts = True
-                idxs.insert(op.idx, None)
-            elif isinstance(op, Remove):
-                idxs.pop(op.idx)
-            self.operations.append(op)
+        for drop in sorted(self.dropped, reverse=True):
+            idxs.pop(drop)
+        for create in sorted(self.created):
+            idxs.insert(create, None)
         self._idxs = idxs
 
         super().__init__(NDims(source_ndim, target_ndim), spaces=spaces)
 
     def apply(self, coords: ArrayT) -> ArrayT:
         coords = self._validate_coords(coords)
-        if self._has_inserts:
+        if self.created:
             xp = array_namespace(coords)
             out = xp.zeros_like(coords, shape=(xp.shape(coords)[0], self.ndims.target))
             for idx, orig_idx in enumerate(self._idxs):
@@ -149,15 +90,7 @@ def apply(self, coords: ArrayT) -> ArrayT:
         return out
 
     def is_identity(self) -> bool:
-        orig: list[int | None] = list(range(self.ndims.source))
-        dims = copy(orig)
-        for op in self.operations:
-            if isinstance(op, Insert):
-                dims.insert(op.idx, None)
-            elif isinstance(op, Remove):
-                dims.pop(op.idx)
-
-        return dims == orig
+        return not self.created and not self.dropped
 
     def to_affine(self) -> Affine | None:
         m = np.eye(self.ndims.source)
@@ -166,7 +99,8 @@ def to_affine(self) -> Affine | None:
 
     def invert(self) -> Self | None:
         return type(self)(
-            [op.invert() for op in reversed(self.operations)],
+            self.created,
+            self.dropped,
             source_ndim=self.ndims.target,
             target_ndim=self.ndims.source,
             spaces=self.spaces.invert(),
diff --git a/tests/transforms/test_project_axis.py b/tests/transforms/test_project_axis.py
index 685e230..545c684 100644
--- a/tests/transforms/test_project_axis.py
+++ b/tests/transforms/test_project_axis.py
@@ -1,44 +1,43 @@
 from random import Random
 from transformnd.transforms import ProjectAxis
-from transformnd.transforms.project_axis import Insert, Remove, Operation
 from transformnd.util import as_floats
 import pytest
 
 
 def test_insert():
-    t = ProjectAxis([Insert(0)], 2, 3)
+    t = ProjectAxis(created={0}, source_ndim=2, target_ndim=3)
 
     out = t.apply(as_floats([[1, 2], [3, 4]]))
     assert out == pytest.approx(as_floats([[0, 1, 2], [0, 3, 4]]))
 
 
 def test_remove():
-    t = ProjectAxis([Remove(0)], 3, 2)
+    t = ProjectAxis(dropped={0}, source_ndim=3, target_ndim=2)
     out = t.apply(as_floats([[1, 2, 3], [4, 5, 6]]))
     assert out == pytest.approx(as_floats([[2, 3], [5, 6]]))
 
 
 def test_invert():
-    t = ProjectAxis([Remove(0), Insert(2)], 3, 3)
+    t = ProjectAxis({0}, {2}, 3, 3)
     ti = t.invert()
     assert ti is not None
-    assert ti.operations == [Remove(2), Insert(0)]
+    assert ti.dropped == {2}
+    assert ti.created == {0}
 
 
-def random_ops(ndim: int, n_ops: int, seed=1991) -> ProjectAxis:
+def random_ops(in_ndim: int, out_ndim: int, seed=1991) -> ProjectAxis:
     rng = Random(seed)
-    ops = []
-    source_ndim = ndim
-    for _ in range(n_ops):
-        op: Operation
-        if ndim == 0 or rng.randint(0, 1):
-            op = Insert(rng.randint(0, ndim))
-        else:
-            op = Remove(rng.randint(0, ndim - 1))
-        ndim = op.check(ndim)
-        ops.append(op)
-
-    return ProjectAxis(ops, source_ndim)
+    n_dropped = rng.randint(max(0, in_ndim - out_ndim), in_ndim - 1)
+    to_drop = list(range(in_ndim))
+    rng.shuffle(to_drop)
+    dropped = set(to_drop[:n_dropped])
+
+    n_created = out_ndim - (in_ndim - n_dropped)
+    to_create = list(range(out_ndim))
+    rng.shuffle(to_create)
+    created = set(to_create[:n_created])
+
+    return ProjectAxis(dropped, created, in_ndim, out_ndim)
 
 
 def test_to_affine(coords5x3):
diff --git a/uv.lock b/uv.lock
index d3f7c97..f927aa7 100644
--- a/uv.lock
+++ b/uv.lock
@@ -50,6 +50,15 @@ wheels = [
     { url = "https://files.pythonhosted.org/packages/a0/d3/54cd560804a8c2b898824778e86c13c2a14600bc83532a9c4f69f2f469c3/array_api_compat-1.14.0-py3-none-any.whl", hash = "sha256:ed5af1f9b6595a199c942505f281ec994892556b6efc24679a0501e87a7d6279", size = 60124, upload-time = "2026-02-26T12:02:41.127Z" },
 ]
 
+[[package]]
+name = "certifi"
+version = "2026.6.17"
+source = { registry = "https://pypi.org/simple" }
+sdist = { url = "https://files.pythonhosted.org/packages/c9/c7/424b75da314c1045981bd9777432fad05a9e0c69daa4ed7e308bbaffe405/certifi-2026.6.17.tar.gz", hash = "sha256:024c88eeec92ca068db80f02b8b07c9cef7b9fe261d1d535abfd5abd6f6af432", size = 134594, upload-time = "2026-06-17T10:31:07.894Z" }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/ef/2f/c5464532e965badff2f4c4c1a3a83f5697f0d7c407ed0cda44aaa99bb451/certifi-2026.6.17-py3-none-any.whl", hash = "sha256:2227dcbaafe0d2f59279d1762ddddc37783ed4354594f194ffc31d20f41fc3db", size = 133289, upload-time = "2026-06-17T10:31:06.348Z" },
+]
+
 [[package]]
 name = "cffi"
 version = "2.0.0"
@@ -107,6 +116,79 @@ wheels = [
     { url = "https://files.pythonhosted.org/packages/ae/3a/dbeec9d1ee0844c679f6bb5d6ad4e9f198b1224f4e7a32825f47f6192b0c/cffi-2.0.0-cp314-cp314t-win_arm64.whl", hash = "sha256:0a1527a803f0a659de1af2e1fd700213caba79377e27e4693648c2923da066f9", size = 184195, upload-time = "2025-09-08T23:23:43.004Z" },
 ]
 
+[[package]]
+name = "charset-normalizer"
+version = "3.4.7"
+source = { registry = "https://pypi.org/simple" }
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@@ -1802,16 +1992,25 @@ dev = [
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     { name = "pandas" },
     { name = "pdoc" },
+    { name = "pooch" },
     { name = "prek" },
     { name = "pydoclint" },
     { name = "pytest" },
     { name = "pytest-benchmark" },
     { name = "ruff" },
+    { name = "scikit-image" },
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 ]
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 ]
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+    { name = "matplotlib" },
+    { name = "pandas" },
+    { name = "pooch" },
+    { name = "scikit-image" },
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@@ -1824,11 +2023,6 @@ test = [
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