# image-registration Find an ND affine transformation that registers an ND test image to a fixed image. Then apply this transformation to 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, moving: &Array2) -> Result { // 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::::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::::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 (200×150, say) let image = ndarray::Array2::::zeros((200, 150)); let transform = Transform::::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], ¢er, &[-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::::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::::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::::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>` 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::::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 let fixed = ndarray::Array2::::zeros((64, 64)); let moving = ndarray::Array2::::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::::from_rotation(0.2, &[99.5, 99.5]); t.to_file(PathBuf::from("transform.yaml"))?; let back = Transform::::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` 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`). | ## 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.