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# Image registration
# image-registration
Find an ND affine transformation that registers an ND test image to a fixed image. Then apply this transformation to
other images.
other images.
`image-registration` is a pure-Rust, dependency-light implementation of intensity-based image registration in the spirit
of [SimpleElastix]([https://elastix.lumc.nl](https://simpleelastix.github.io/)). It finds an affine (rigid + scale + shear + translation) transform
that aligns a *moving* image to a *fixed* image by maximizing
the [Mattes mutual-information](https://doi.org/10.1109/TMI.2003.809072) metric with a B-spline Parzen
window, using a multi-resolution Gaussian image pyramid and a hybrid ASGD + L-BFGS optimization strategy.
Everything operates on [`ndarray`](https://docs.rs/ndarray) arrays, works in arbitrary dimensions (1D, 2D, 3D, …), and
is parallelized with [rayon](https://docs.rs/rayon).
## Features
- **Affine registration** in any number of dimensions — rotation, scale, shear, and translation parameters
(`Transform::register`).
- **B-spline interpolation** of order up to 5 (linear, cubic, …) for resampling/warping images under a transform
(`interpolate`, `interpolate_par`).
- **Mattes mutual-information metric** with B-spline joint-histogram Parzen estimation, analytic gradients, and
configurable sampling (fixed, fixed-at-points, or random) and binning.
- **Multi-resolution registration** with a fixed-smoothing image pyramid with Gaussian smoothing.
- **Two optimizers**: L-BFGS (default, for smooth/consistent gradients) and ASGD (adaptive stochastic gradient descent,
robust to noisy gradients).
- **Flexible parameter control**: fix any subset of transform parameters during registration (`FixedMu`).
- **Transform algebra**: composition, inverse, matrix forms, coordinate transforms, serialization to/from YAML.
- **Parallel** interpolation and metric evaluation via rayon.
- Pure Rust — no BLAS/LAPACK or system libraries required (matrix inversion is implemented internally with LU
factorization).
## Quick start
Register a moving image to a fixed image and obtain the affine transform:
```rust
use image_registration::error::Error;
use image_registration::transform::Transform;
use ndarray::Array2;
fn register(fixed: &Array2<f64>, moving: &Array2<f64>) -> Result<Transform, Error> {
// All 6 parameters of the 2D affine transform are free (None = optimize).
// steps=None uses the default multi-resolution pyramid; initial_guess=None
// initializes by aligning the geometric centers of the two images.
Transform::<ndarray::Ix2>::register(
fixed.view(),
moving.view(),
vec![None; 6],
None,
None,
)
}
```
The returned `Transform` maps coordinates from the *fixed* image to the *moving* image, i.e.
`moving(transform_point(x)) ≈ fixed(x)`. To warp the moving image into the fixed image's frame:
```rust
let transform = register(&fixed, &moving)?;
let registered = transform.interpolate_par::<3, _, _>(moving.view())?; // cubic B-spline, parallel
```
## Concepts
### Parameterization of an affine transform
An ND affine transform is stored as a flattened `N×N` linear part plus `N` translation parameters, for `N*N + N` values
in total:
- 1D: `[scale, translation]`
- 2D: `[m00, m01, m10, m11, tx, ty]` (row-major linear part, then translation)
- 3D: `[m00, m01, m02, m10, m11, m12, m20, m21, m22, tx, ty, tz]`
The transformation is applied about a center saved in ```Transform```:
```text
v = A·(p − c) + t + c
```
where `A` is the linear part, `t` the translation, `c` the center of the transformation, and `p` the input coordinates.
`Transform::new(parameters, shape)` sets the center to `(shape − 1) / 2` automatically; `Transform::new_with_center`
lets you override it.
### Registration convention
The registration minimizes the negative Mattes mutual information between the fixed image and the warped moving image.
For a moving image that was created by applying transform `q` to a fixed image, registration recovers `q⁻¹` (the
transform that maps fixed coordinates back onto the moving image).
## Examples
### 1. Build transforms and transform coordinates
```rust
use image_registration::error::Error;
use image_registration::transform::Transform;
use ndarray::Ix2;
fn main() -> Result<(), Error> {
// Rotation of 45° about the image center of a 200×200 image
let rot = Transform::<Ix2>::from_rotation(std::f64::consts::FRAC_PI_4, &[99.5, 99.5]);
// Compose: translate, then scale, then rotate
let t = rot.with_scaling(&[0.9, 1.1]).with_translation(&[10.0, -5.0]);
// Transform a single point (the center is subtracted and re-added)
let p = t.transform_point(&[200, 300]);
// Transform many points at once (rows of an N×2 array)
let points = ndarray::array![[0.0, 0.0], [199.0, 199.0]];
let pts = t.transform_points(points.view())?;
// Matrix form (3×3 homogeneous) and exact inverse
let m = t.matrix();
let inv = t.inverse()?;
assert!(m.dot(&inv.matrix()).iter().all(|x| (x - 1.0).abs() < 1e-9));
Ok(())
}
```
### 2. Warp an image (resampling)
Interpolate an image under a transform, producing an output image of the same shape. Pixels whose transformed
coordinates fall outside the input image are set to zero.
```rust
use image_registration::error::Error;
use image_registration::transform::Transform;
fn main() -> Result<(), Error> {
// `image` is an Array2<f64> (200×150, say)
let image = ndarray::Array2::<f64>::zeros((200, 150));
let transform = Transform::<ndarray::Ix2>::from_rotation(0.3, &[99.5, 74.5]);
// Order-1 (linear) interpolation, single-threaded
let warped_linear = transform.interpolate::<1, _, _>(image.view())?;
// Order-3 (cubic) B-spline interpolation, parallel
let warped_cubic = transform.interpolate_par::<3, _, _>(image.view())?;
Ok(())
}
```
### 3. Full affine registration with a known ground truth
Create a moving image by applying a known transform to a fixed image, register, and check that the recovered transform
matches the inverse of the ground-truth transform (adapted from the crate's `register2_random_affine` test):
```rust
use image_registration::error::Error;
use image_registration::julia_image;
use image_registration::transform::Transform;
use ndarray::Ix2;
fn main() -> Result<(), Error> {
// Fixed image (a Julia fractal, included for testing)
let shape = [200, 200];
let center = [99.5, 99.5];
let fixed = julia_image(&shape, &[1.0, 0.0, 0.0, 1.0, 0.0, 0.0], &center, &[-0.8, 0.156])
.mapv(|i| i as f64);
// Ground-truth affine: 10° rotation, 5% scale change, (12, -8) px translation
let theta = 10.0f64.to_radians();
let (s, c) = theta.sin_cos();
let params = vec![c * 1.05, -s, s, c * 0.95, 12.0, -8.0];
let ground_truth =
Transform::<Ix2>::new_with_center(params.clone(), center.to_vec(), shape.to_vec());
// Moving image = ground truth applied to the fixed image
let moving = ground_truth.interpolate::<3, _, _>(fixed.view())?;
// Register (all 6 parameters free, default steps)
let t = Transform::<Ix2>::register(fixed.view(), moving.view(), vec![None; 6], None, None)?;
// The registered transform should equal the inverse of the ground truth
let q_inv = ground_truth.inverse()?.parameters;
let sse: f64 = t
.parameters
.iter()
.zip(q_inv.iter())
.map(|(a, b)| (a - b).powi(2))
.sum();
assert!(sse < 1.0);
Ok(())
}
```
### 4. Registration with inspection of every optimization step
```rust
use image_registration::error::Error;
use image_registration::transform::Transform;
fn main() -> Result<(), Error> {
let (t, steps) = Transform::<ndarray::Ix2>::register_debug(
fixed.view(), moving.view(), vec![None; 6], None, None,
)?;
for step in steps {
println!(
"sigma={:?} n_bins={} optimizer_iters={} converged={} -> {:?}",
step.sigma_fixed, step.n_bins, step.iterations, step.converged, step.optimal_point
);
}
Ok(())
}
```
### 5. Custom registration steps (multi-resolution pyramid)
`RegistrationStep` gives you full control over the smoothing, sampling, binning, and optimizer at each level. The
following uses two levels: a coarse level with strong Gaussian smoothing and the robust ASGD optimizer, then a fine
level with L-BFGS for accuracy.
```rust
use image_registration::error::Error;
use image_registration::metric::{SamplingArg, Sigma};
use image_registration::register::{Optimizer, Registration, RegistrationStep};
fn main() -> Result<(), Error> {
let mut coarse = RegistrationStep::new(
Sigma::Absolute(vec![4.0, 4.0]), // Gaussian sigma per dimension
SamplingArg::Fixed(3000), // 3000 sampled points (cached)
32, // histogram bins
1e-4, // tolerance
0.05, // edge fraction
512, // max iterations
1.0, // learning rate
);
coarse.optimizer = Optimizer::ASGD; // robust to noisy gradients
let mut fine = RegistrationStep::new(
Sigma::Absolute(vec![0.5, 0.5]),
SamplingArg::Fixed(3000),
32,
1e-8,
0.05,
2048,
1.0,
); // optimizer stays LBFGS (the default)
let mut reg = Registration::new(vec![None; 6]);
reg.set_steps(vec![coarse, fine]);
let t = reg.register(fixed.view(), moving.view())?;
Ok(())
}
```
> **Tip:** `RegistrationStep::default_steps(ndim, n)` returns a sensible default pyramid: a 5-level schedule that starts
> with ASGD on heavily smoothed images and finishes with L-BFGS on the unsmoothed image.
### 6. Fix (pin) some parameters during registration
Pass a `Vec<Option<f64>>` as `fixed_mu`: `None` optimizes a parameter, `Some(v)` keeps it fixed at `v`. For example,
register for translation only:
```rust
use image_registration::error::Error;
use image_registration::transform::Transform;
fn main() -> Result<(), Error> {
// 2D: fix the linear part to identity, optimize tx and ty
let t = Transform::<ndarray::Ix2>::register(
fixed.view(),
moving.view(),
vec![Some(1.0), Some(0.0), Some(0.0), Some(1.0), None, None],
None,
None,
)?;
Ok(())
}
```
For the common "translation only" case there is also the convenience method
`Transform::register_translation(fixed, moving)`.
### 7. Evaluate the Mattes metric directly
Build a `MattesMetric` and evaluate the (negative) mutual information and its gradient at arbitrary parameters.
`SamplingArg` controls which points are used:
- `Random(n)` — `n` random continuous positions (drawn from a thread-local RNG, cached per metric).
- `Fixed(n)` — `n` random positions, frozen at construction.
- `FixedAt(points)` — an explicit list of sample positions (fully deterministic; recommended for reproducible results).
```rust
use algos::ObjectiveFunction;
use image_registration::bspline::BSpline;
use image_registration::error::Error;
use image_registration::metric::{MattesMetric, SamplingArg, Sigma};
fn main() -> Result<(), Error> {
// `fixed` and `moving` are Array2<f64>
let fixed = ndarray::Array2::<f64>::zeros((64, 64));
let moving = ndarray::Array2::<f64>::zeros((64, 64));
let f = Sigma::Absolute(vec![2.0, 2.0]).smooth(fixed.view())?;
let m = Sigma::Absolute(vec![2.0, 2.0]).smooth(moving.view())?;
let metric = MattesMetric::new(
BSpline::<0, _>::new(f.view()), // fixed image B-spline (nearest-neighbor)
BSpline::<3, _>::new(m.view()), // moving image B-spline (cubic)
SamplingArg::FixedAt((0..100).map(|i| vec![i as f64, i as f64]).collect()),
32,
0.05,
)?
.with_fixed_mu(vec![None; 6]); // which parameters are variable
let id = [1.0, 0.0, 0.0, 1.0, 0.0, 0.0];
let val = metric.evaluate(&id);
let grad = metric.gradient(&id).unwrap();
println!("mi({:?}) = {:.6}, grad = {:?}", id, val, grad);
Ok(())
}
```
### 8. Save and load transforms
Transforms serialize to YAML (via serde):
```rust
use image_registration::error::Error;
use image_registration::transform::Transform;
use std::path::PathBuf;
fn main() -> Result<(), Error> {
let t = Transform::<ndarray::Ix2>::from_rotation(0.2, &[99.5, 99.5]);
t.to_file(PathBuf::from("transform.yaml"))?;
let back = Transform::<ndarray::Ix2>::from_file(PathBuf::from("transform.yaml"))?;
assert_eq!(t, back);
Ok(())
}
```
## Module reference
| Module | Contents |
|----------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| `image_registration::transform` | `Transform`, the affine transform type: construction (`new`, `new_with_center`, `from_translation`, `from_scaling`, `from_rotation`), algebra (`inverse`, `with_translation`, `with_scaling`, `with_rotation`, `with_rotation_around`, `Mul`), coordinate transforms (`transform_point`, `transform_points`, `matrix`, `dmatrix`), resampling (`interpolate`, `interpolate_par`), registration (`register`, `register_debug`, `register_affine`, `register_translation`), serialization (`to_file`, `from_file`), plus the free function `transform_point` and a blas-free `matrix_inverse`. |
| `image_registration::register` | `Registration`, `RegistrationStep`, `RegistrationResult`, `Optimizer` (LBFGS/ASGD). |
| `image_registration::metric` | `MattesMetric` (Mattes mutual information with analytic gradient), `SamplingArg` (Fixed/FixedAt/Random), `Sigma` (None/Absolute/Relative), `FixedMu`. |
| `image_registration::bspline` | `BSpline<N, D>` B-spline interpolation of arbitrary order, `BSplineTrait`, `BSplineMem`, Parzen kernel helpers. |
| `image_registration::filter` | Gaussian smoothing and FFT helpers: `gaussian_smooth`, `gaussian_kernel`, `fft`, `ifft`, `fft_freq`, `fft_shift`. |
| `image_registration::optimize` | Optimizers: `lbfgs_minimize`, `asgd_minimize`, `AsgdConfig`. |
| `image_registration::par_indexed_iter` | Parallel iterators over array indices. |
| `image_registration::error` | `Error` type. |
| `image_registration::julia_image` | Test-image generator (Julia fractal on an `Array2<u8>`). |
## How it works
1. **Image pyramid.** Each registration level smooths the fixed and moving images with a Gaussian of the configured
`Sigma` (and optionally downsamples them). Coarse levels use large sigma so the metric is smooth and far-sighted;
fine levels use small sigma for precision.
2. **Metric.** `MattesMetric` samples points in the fixed image, warps them into the moving image through the current
transform, and builds a joint intensity histogram using cubic B-spline Parzen windows. It minimizes (the negative of)
the mutual information with an analytic gradient computed via the chain rule (image Jacobian × transform Jacobian).
3. **Optimization.** ASGD is used at coarse levels (robust to noisy gradients, with automatic parameter estimation),
L-BFGS at fine levels (quadratic convergence on smooth objectives).
4. **Result.** The final transform maps fixed-image coordinates into moving-image coordinates; its inverse maps moving
coordinates back into the fixed frame.
## Known limitations
- With `SamplingArg::Random` / `SamplingArg::Fixed`, sample positions are drawn from a thread-local RNG, so different
runs can converge to different (local) optima — especially on small or nearly uniform images where the MI signal is
weak. For reproducible results, prefer `SamplingArg::FixedAt` with explicit sample points, and use enough samples to
cover the structure of interest.
- The metric is defined only where the warped sample falls inside the moving image; points near the edge are smoothly
weighted out of the histogram, so registration can be unreliable if the two images overlap only slightly.
## License
Licensed under either of [Apache-2.0](LICENSE-APACHE) or [MIT](LICENSE-MIT), at your option.
src/metric.rs
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@@ -214,8 +214,6 @@ where
/// memory for (point, value, gradient) combination, refcell gives interior mutability
metric: RefCell<IntMut>,
edge: f64,
// mem_f: RefCell<BSplineMem>,
// mem_m: RefCell<BSplineMem>,
}
impl<D> MattesMetric<D>
@@ -231,8 +229,6 @@ where
) -> Result<Self, Error> {
let shape = fixed.shape().iter().map(|&i| i as f64).collect::<Vec<_>>();
let center = shape.iter().map(|i| (i - 1.0) / 2.0).collect();
// let mem_f = fixed.get_mem();
// let mem_m = moving.get_mem();
let (min, max) = fixed
.iter()
.chain(moving.iter())