# GaussianMixtureAlignment.jl

A Julia implementation of the Globally-Optimal Gaussian Mixture Alignment (GOGMA) algorithm (Campbell, 2016), with modifications inspired by Li et. al. (2018).

The GOGMA algorithm uses a branch-and-bound procedure to return a globally optimal alignment of point sets via rigid transformation. In order to improve speed for small point sets, the alignment problem can be split to separately optimize rotational and translational alignments, while still guaranteeing global optimality, through the using of translation invariant vectors (TIVs).

Becaues the runtime of the GOGMA algorithm is O(n^2), and that of the TIV-GOGMA algorithm is O(n^4), they may be unsuitable for use with large point sets without downsampling.

## Construct Isotropic Gaussian Mixture Models (GMMs) for alignment

```
julia> # These are very simple point sets that can be perfectly aligned
julia> xpts = [[0.,0.,0.], [3.,0.,0.,], [0.,4.,0.]];
julia> ypts = [[1.,1.,1.], [1.,-2.,1.], [1.,1.,-3.]];
julia> σ = ϕ = 1.;
julia> gmmx = IsotropicGMM([IsotropicGaussian(x, σ, ϕ) for x in xpts])
IsotropicGMM{3, Float64} with 3 IsotropicGaussian{3, Float64} distributions.
julia> gmmy = IsotropicGMM([IsotropicGaussian(y, σ, ϕ) for y in ypts])
IsotropicGMM{3, Float64} with 3 IsotropicGaussian{3, Float64} distributions.
julia> # Compute the overlap between the two GMMs
julia> overlap(gmmx, gmmy)
1.1908057504684806
```

## Align Isotropic GMMs with TIV-GOGMA

```
julia> res = tiv_gogma_align(gmmx, gmmy);
julia> # upper and lower bounds of alignmnet objective function at search termination
julia> res.upperbound, res.lowerbound
(-3.2512906351736524, -3.348917632693506)
julia> # rotation component of the best transformation
julia> res.tform.linear
3×3 AngleAxis{Float64} with indices SOneTo(3)×SOneTo(3)(2.0944, -0.57735, 0.57735, -0.57735):
1.63136e-10 1.90925e-10 1.0
-1.0 2.24997e-10 1.63136e-10
-2.24997e-10 -1.0 1.90925e-10
julia> # translation component of the best transformation
julia> res.tform.translation
3-element StaticArrays.SVector{3, Float64} with indices SOneTo(3):
1.0
1.0
1.0
julia> # repeat alignment with stricter tolerance
julia> res.upperbound, res.lowerbound
(-3.2512906351736524, -3.2522904816062654)
julia> # The result has not changed, but there is a tighter lower bound
julia> # Compute the overlap between the GMMs after alignment
julia> overlap(res.tform(gmmx), gmmy)
3.2512906351736524
```

## Plot Isotropic GMMs

```
julia> # Draw the unaligned GMMs
julia> plotdrawing(drawIsotropicGMMs([gmmx,gmmy]))
```

```
julia> # Draw the aligned GMMs
julia> plotdrawing(drawIsotropicGMMs([res.tform(gmmx),gmmy]))
```