Generalized Procrustes Analysis

Generalized Procrustes Analysis is an iterative technique for optimally rigidly aligning a set of shapes. The contents of this notebook are:

Simple example
Example with faces

First, let's import everything we need.



In [1]:

    
%matplotlib inline
from pathlib import Path

from menpo.transform import GeneralizedProcrustesAnalysis
from menpo.shape import PointCloud

import numpy as np
import matplotlib.pyplot as plt

1. Simple example

Let's make three PointCloud instances



In [2]:

    
src_1 = PointCloud(np.array([[-1.0, -1.0],
                             [1.0, -1.0],
                             [1.0, 1.0],
                             [-1.0, 1.0]]))

src_2 = PointCloud(np.array([[0.0, 0.0],
                             [1.5, 0.1],
                             [1.5, 1.5],
                             [0.1, 1.5]]))

src_3 = PointCloud(np.array([[-2.0, 0.0],
                             [0.0, 3.0],
                             [2.0, 0.0],
                             [0.0, -3.0]]))



In [3]:

    
viewer = src_1.view(marker_face_colour='r', marker_size=80);
src_2.view(figure_id=viewer.figure_id, marker_face_colour='g', marker_size=80);
src_3.view(figure_id=viewer.figure_id, marker_face_colour='b', marker_size=80);
plt.legend(['src_1', 'src_2', 'src_3']);

GeneralizedProcrustesAnalysis accepts a list of Shape instances upon construction. This causes it to run it's iterative alignment process.



In [4]:

    
gpa = GeneralizedProcrustesAnalysis([src_1, src_2, src_3])

Printing the object gives you a summary of what happened



In [5]:

    
print(gpa)









    



Converged after 35 iterations with av. error 1.297239

GeneralizedProcrustesAnalysis provides a property called .transforms, which gives one transform per source that aligns it to the target frame. The target frame is also accessable at .target



In [6]:

    
print('There are {} transforms'.format(len(gpa.transforms)))
print('Each transform is a {}'.format(type(gpa.transforms[0])))
print('The target: {}'.format(gpa.target))









    



There are 3 transforms
Each transform is a <class 'menpo.transform.homogeneous.similarity.AlignmentSimilarity'>
The target: PointCloud: n_points: 4, n_dims: 2

The optimal transform found to align src_1 is:



In [7]:

    
src_1_similarity = gpa.transforms[0]
print(src_1_similarity)









    



Similarity decomposing into:  
  CCW Rotation of 90.0 degrees about [0 0 1]
  UniformScale by 0.7765218033713611
  CCW Rotation of 124.1 degrees about [0 0 1]
  Translation by [ 0.25833333  0.25833333]
[[ 0.64328493 -0.43493747  0.25833333]
 [ 0.43493747  0.64328493  0.25833333]
 [ 0.          0.          1.        ]]

and for src_2 we used:



In [8]:

    
src_2_similarity = gpa.transforms[1]
print(src_2_similarity)









    



Similarity decomposing into:  
  CCW Rotation of 180.0 degrees about [0 0 1]
  UniformScale by 1.069793262684277
  CCW Rotation of 145.9 degrees about [0 0 1]
  Translation by [ 0.03588139 -0.89288118]
[[ 0.88623642 -0.59920166  0.03588139]
 [ 0.59920166  0.88623642 -0.89288118]
 [ 0.          0.          1.        ]]

src_1_similarity has a source which is the same as src_1:



In [9]:

    
print(src_1_similarity.source)
print(np.all(src_1 == src_1_similarity.source))









    



PointCloud: n_points: 4, n_dims: 2
True

The target of this object was set in the first iteration of the algorithm to the mean of all the sources:



In [10]:

    
print(src_1_similarity.target)









    



PointCloud: n_points: 4, n_dims: 2

Its simple to check how the alignment did



In [11]:

    
colours_list = ['r' ,'g', 'b']
for i, (t, a_c) in enumerate(zip(gpa.transforms, gpa.sources)):
    aligned_src = t.apply(a_c)
    aligned_src.view(marker_face_colour=colours_list[i], marker_size=10);
plt.legend(['src_1', 'src_2', 'src_3']);

2. Example with faces

In this example, we will use one of the most popular and widely used annotated facial databases, the Labeled Faces Parts in the Wild (LFPW) database. Both images and corresponding facial landmark annotations are publicly available and can be downloaded from the following link:

http://ibug.doc.ic.ac.uk/download/annotations/lfpw.zip/

In order to continue with this notebook, the user is required to simply:

Click on the previous link.
Fill out the form with his/her details.
Proceed to download the LFPW database.
Unzip and save the LFPW database to a location of his/her like.
Paste the path to the local copy of the LFPW database on the next cell.

Note that the .zip file containing the whole annotated database is of approximately 350MB.



In [12]:

    
path_to_lfpw = Path('/vol/atlas/databases/lfpw')
# path_to_lfpw = Path('/Users/pts08/data/lfpw')

Let's load the shapes



In [ ]:

    
import menpo.io as mio
shapes = [s['all'] for s in mio.import_landmark_files(path_to_lfpw / 'trainset' / '*.pts', 
                                                      as_generator=True, verbose=True)]

We can visualize them using the visualize_pointclouds widget.



In [14]:

    
from menpowidgets import visualize_pointclouds

visualize_pointclouds(shapes)

It is very easy to align the shapes



In [15]:

    
from menpo.transform import Translation

# centralize shapes
centered_shapes = [Translation(-s.centre()).apply(s) for s in shapes]

# align centralized shape using Procrustes Analysis
gpa = GeneralizedProcrustesAnalysis(centered_shapes)

print(gpa)

# get the aligned shapes
aligned_shapes = [s.aligned_source() for s in gpa.transforms]









    



Converged after 4 iterations with av. error 80.790883

Let's see the result!



In [16]:

    
n_shapes = 101

plt.subplot(121)
plt.title('Before GPA')
for s in shapes[:n_shapes]:
    s.view(axes_x_limits=10, axes_y_limits=10)
    
plt.subplot(122)
plt.title('After GPA')
for s in aligned_shapes[:n_shapes]:
    s.view()