Updated README

Author: tmcgrath325

README.md

@@ -10,6 +10,8 @@ Becaues the runtime of the GOGMA algorithm is O(n^2), and that of the TIV-GOGMA
 ## Construct Isotropic Gaussian Mixture Models (GMMs) for alignment
 
 ```julia
+julia> using GaussianMixtureAlignment; GMA = GaussianMixtureAlignment;
+
 julia> # These are very simple point sets that can be perfectly aligned
 
 julia> xpts = [[0.,0.,0.], [3.,0.,0.,], [0.,4.,0.]];
@@ -32,56 +34,130 @@ julia> overlap(gmmx, gmmy)
 1.1908057504684806
 ```
 
-## Align Isotropic GMMs with TIV-GOGMA
+## Align Isotropic GMMs with GOGMA
 
 ```julia
-julia> res = tiv_gogma_align(gmmx, gmmy);
+julia> res = gogma_align(gmmx, gmmy; nextblockfun=GMA.randomblock, maxsplits=10000);
 
 julia> # upper and lower bounds of alignmnet objective function at search termination
 
 julia> res.upperbound, res.lowerbound
-(-3.2512906351736524, -3.348917632693506)
+(-3.251290635173653, -5.528700740172087)
 
 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
+ -1.38948e-11  -2.5226e-11    1.0
+ -1.0           9.12343e-11  -1.38948e-11
+ -9.12343e-11  -1.0          -2.52261e-11
 
 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
+ 1.0000000000115654
+ 1.000000000557716
+ 0.9999999997059433
 
-julia> # Compute the overlap between the GMMs after alignment
+julia> # Compute the overlap between the GMMs after alignment (equal to res.upperbound)
 
 julia> overlap(res.tform(gmmx), gmmy)
-3.2512906351736524
+3.251290635173653
 ```
 
 ## Plot Isotropic GMMs
 ```julia
+julia> using GLMakie
+
 julia> # Draw the unaligned GMMs
 
-julia> plotdrawing(drawIsotropicGMMs([gmmx,gmmy]))
+julia> gmmdisplay(gmmx,gmmy)
 ```
 <img src="./assets/image/example.png" width="400"/>
 
 ```julia
 julia> # Draw the aligned GMMs
 
-julia> plotdrawing(drawIsotropicGMMs([res.tform(gmmx),gmmy]))
+julia> gmmdisplay(res.tform(gmmx), gmmy)
 ```
 <img src="./assets/image/example_aligned.png" width="400"/>
+
+## Align GMMs with different types of interacting features
+
+```julia
+julia> # four points defining a tetrahedron
+
+julia> tetrahedron = [
+            [0.,0.,1.],
+            [sqrt(8/9), 0., -1/3],
+            [-sqrt(2/9),sqrt(2/3),-1/3],
+            [-sqrt(2/9),-sqrt(2/3),-1/3]
+        ];
+
+julia> # each feature type is given its own IsotropicGMM
+
+julia> chargedGMM = IsotropicGMM([IsotropicGaussian(tetrahedron[1], 1.0, 1.0)]);
+
+julia> stericGMM = IsotropicGMM([IsotropicGaussian(p, 0.5, 1.0) for p in tetrahedron]);
+
+julia> mgmmx = IsotropicMultiGMM(Dict(
+           :positive => chargedGMM,
+           :steric => stericGMM
+       ))
+IsotropicMultiGMM{3, Float64, Symbol} with 2 labeled IsotropicGMM{3, Float64} models made up of a total of 5 IsotropicGMM{3, Float64} distributions.
+
+julia> mgmmy = IsotropicMultiGMM(Dict(
+           :negative => chargedGMM,
+           :steric => stericGMM
+       ))
+IsotropicMultiGMM{3, Float64, Symbol} with 2 labeled IsotropicGMM{3, Float64} models made up of a total of 5 IsotropicGMM{3, Float64} distributions.
+
+julia> # define interactions between each type of feature
+
+julia> interactions = Dict(
+           :positive => Dict(
+               :positive => -1.0,
+               :negative => 1.0,
+           ),
+           :negative => Dict(
+               :positive => 1.0,
+               :negative => -1.0,
+           ),
+           :steric => Dict(
+               :steric => -1.0,
+           ),
+       );
+
+julia> # apply a random rigid transformation to one of the models
+
+julia> using CoordinateTransformations; randtform = AffineMap((rand(), rand(), rand(), rand(), rand(), rand()));
+
+julia> mgmmx = randtform(mgmmx);
+
+julia> # compute alignment between the two MultiIsotropicGMMs
+
+julia> res = gogma_align(mgmmx, mgmmy; interactions=interactions, nextblockfun=GMA.randomblock, maxsplits=10000);
+```
+## Plot MultiGMMs
+
+```
+julia> # Draw the unaligned GMMs
+
+julia> gmmdisplay(mgmmx)
+```
+<img src="./assets/image/multi_example_x.png" width="400"/>
+
+```
+julia> gmmdisplay(mgmmy)
+```
+<img src="./assets/image/multi_example_x.png" width="400"/>
+
+```julia
+julia> # Draw the aligned GMMs. Note that the "steric" features (green) repulse one another.
+
+julia> gmmdisplay(res.tform(mgmmx), mgmmy)
+```
+<img src="./assets/image/multi_example_aligned.png" width="400"/>
+
+## Align GMMs with different types of interacting features