project 2

Computing eigenfaces Results References

Image processing, retrieval and analysis IIProject 2

Raghunandan Palakodety, Himanshu Thakur

Universitat Bonn

June 1, 2015


Outline

1 Computing eigenfaces

2 Results

3 References


Eigenface approach

In mathematical terms, we wish to find the principal componentsof the distribution of faces or the eigenvectors of the covariancematrix of the set of face images, treating an image as in a veryhigh dimension space. In our case, the dimension being R361.


Eigenvectors and eigenfaces I

Visualizing an eigenface

The eigenvectors are ordered, each one accounting for a differentamount of variation among face images. These vectors can bethought of as a set of features that characterize the variationbetween face images [2].

1 Each image location contributes more or less to eacheigenvector, so that we can display the eigenvector as a sort ofghostly face which we call an eigenface as shown inf figure 1.

Figure 1: Eigenface


Eigenvectors and eigenfaces II

2 Each eigenface deviates from uniform gray where some facialfeature differs among the set of training faces; they are a sortof map of the variations between faces.

3 Each individual face can be represented exactly in terms of alinear combination of the eigenfaces. Each face can also beapproximated using only the ”best” eigenfaces.

4 That is those eigenfaces that have largest eigenvalues andwhich therefore account for the most variance within the setof face images.

5 The best k eigenfaces span an k dimensional subspace or”facespace” of all possible images.


Results I

1 Set of eigenvalues in descending order is shown in figure 2.

Figure 2: Spectrum of Co-variance


Results II

2 Determine smallest eigenvalue λk such that∑k

i=1 λi∑dj=1 λj

≥ 0.9.

3 In our case we determined k = 20.

4 Visualize the first k eigen vectors vi ∈ R361 as 19x19 images.

5 As shown belowa) b)

c) d)

e) f)


Results III

g) h)

i) j)

k) l)


Results IV

m) n)

o) p)

q) r)


Results V

s) t)


Results VI

6 Randomly select 10 test images, compute their Euclideandistances to all training images, sort (in descending order) andplot the distances. Some of these are shown in figures.

u) v)

w) x)


Results VII

y) z)


Results VIII

7 Consider the same 10 test images as above; in the lowerdimensional space, compute their Euclidean distances to allthe training images, sort the set of distances in descendingorder and plot them; compare your plots. The comparison ofthose plots are shown in the previous slide.


Notes on Results

1 Images of faces, being similar in overall configuration, will notbe randomly distributed in this huge image space R361 andthus can be described by a relatively low dimensional subspace[2].

2 The main idea of PCA (or Karhunen-Loeve expansion [1]) isto find the vectors that best account for the distribution offace images within the entire image space.

3 These vectors define the subspace of the face images which isreferred to as ”face-space” as mentioned in earlier slides.

4 Each vector being vi ∈ R361. All such vectors are eigenvectorsof the covariance matrix corresponding to the original faceimages.


References

K. Karhunen, Uber lineare Methoden in derWahrscheinlichkeitsrechnung, Annales Academiae scientiarumFennicae. Series A. 1, Mathematica-physica, 1947.

M. A. Turk and A. P. Pentland, Face recognition usingeigenfaces, in Computer Vision and Pattern Recognition, 1991.Proceedings CVPR’91., IEEE Computer Society Conferenceon, IEEE, 1991, pp. 586–591.

project 2

Documents

best eigenfaces

set of face images

images of faces

training images

test images

19x19 images

possible images

set of distances