![Computing eigenfaces Results References
Eigenvectors and eigenfaces I
Visualizing an eigenface
The eigenvectors are ordered, each one accounting for a different
amount of variation among face images. These vectors can be
thought of as a set of features that characterize the variation
between face images [2].
1 Each image location contributes more or less to each
eigenvector, so that we can display the eigenvector as a sort of
ghostly face which we call an eigenface as shown inf figure 1.
Figure 1: Eigenface](https://image.slidesharecdn.com/project2-150707083050-lva1-app6891/85/Project-2-4-320.jpg)
![Computing eigenfaces Results References
Notes on Results
1 Images of faces, being similar in overall configuration, will not
be randomly distributed in this huge image space R361 and
thus can be described by a relatively low dimensional subspace
[2].
2 The main idea of PCA (or Karhunen-Lo`eve expansion [1]) is
to find the vectors that best account for the distribution of
face images within the entire image space.
3 These vectors define the subspace of the face images which is
referred to as ”face-space” as mentioned in earlier slides.
4 Each vector being vi ∈ R361. All such vectors are eigenvectors
of the covariance matrix corresponding to the original face
images.](https://image.slidesharecdn.com/project2-150707083050-lva1-app6891/85/Project-2-14-320.jpg)
Project 2
- 1. Computing eigenfaces Results References Image processing, retrieval and analysis II Project 2 Raghunandan Palakodety, Himanshu Thakur Universit¨at Bonn June 1, 2015
- 2. Computing eigenfaces Results References Outline 1 Computing eigenfaces 2 Results 3 References
- 3. Computing eigenfaces Results References Eigenface approach In mathematical terms, we wish to find the principal components of the distribution of faces or the eigenvectors of the covariance matrix of the set of face images, treating an image as in a very high dimension space. In our case, the dimension being R361.
- 4. Computing eigenfaces Results References Eigenvectors and eigenfaces I Visualizing an eigenface The eigenvectors are ordered, each one accounting for a different amount of variation among face images. These vectors can be thought of as a set of features that characterize the variation between face images [2]. 1 Each image location contributes more or less to each eigenvector, so that we can display the eigenvector as a sort of ghostly face which we call an eigenface as shown inf figure 1. Figure 1: Eigenface
- 5. Computing eigenfaces Results References Eigenvectors and eigenfaces II 2 Each eigenface deviates from uniform gray where some facial feature differs among the set of training faces; they are a sort of map of the variations between faces. 3 Each individual face can be represented exactly in terms of a linear combination of the eigenfaces. Each face can also be approximated using only the ”best” eigenfaces. 4 That is those eigenfaces that have largest eigenvalues and which therefore account for the most variance within the set of face images. 5 The best k eigenfaces span an k dimensional subspace or ”facespace” of all possible images.
- 6. Computing eigenfaces Results References Results I 1 Set of eigenvalues in descending order is shown in figure 2. Figure 2: Spectrum of Co-variance
- 7. Computing eigenfaces Results References Results II 2 Determine smallest eigenvalue λk such that k i=1 λi d j=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 below a) b) c) d) e) f)
- 8. Computing eigenfaces Results References Results III g) h) i) j) k) l)
- 9. Computing eigenfaces Results References Results IV m) n) o) p) q) r)
- 10. Computing eigenfaces Results References Results V s) t)
- 11. Computing eigenfaces Results References Results VI 6 Randomly select 10 test images, compute their Euclidean distances to all training images, sort (in descending order) and plot the distances. Some of these are shown in figures. u) v) w) x)
- 12. Computing eigenfaces Results References Results VII y) z)
- 13. Computing eigenfaces Results References Results VIII 7 Consider the same 10 test images as above; in the lower dimensional space, compute their Euclidean distances to all the training images, sort the set of distances in descending order and plot them; compare your plots. The comparison of those plots are shown in the previous slide.
- 14. Computing eigenfaces Results References Notes on Results 1 Images of faces, being similar in overall configuration, will not be randomly distributed in this huge image space R361 and thus can be described by a relatively low dimensional subspace [2]. 2 The main idea of PCA (or Karhunen-Lo`eve expansion [1]) is to find the vectors that best account for the distribution of face images within the entire image space. 3 These vectors define the subspace of the face images which is referred to as ”face-space” as mentioned in earlier slides. 4 Each vector being vi ∈ R361. All such vectors are eigenvectors of the covariance matrix corresponding to the original face images.
- 15. Computing eigenfaces Results References References K. Karhunen, ¨Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae scientiarum Fennicae. Series A. 1, Mathematica-physica, 1947. M. A. Turk and A. P. Pentland, Face recognition using eigenfaces, in Computer Vision and Pattern Recognition, 1991. Proceedings CVPR’91., IEEE Computer Society Conference on, IEEE, 1991, pp. 586–591.