1. Outline Multilinear Analysis
M. A. O. Vasilescu and D. Terzopoulos, “Multilinear Independent Components Analysis,” CVPR 2005

2. Motivations
Natural images are generated by the interaction of multiple factors related to scene structure, illumination, and imaging
November 10, 2018
Computer Vision

3. Motivations
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Computer Vision

4. Separating Styles from Content
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Computer Vision

5. Separating Styles from Content
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Computer Vision

6. Separating Styles from Content
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Computer Vision

7. Bilinear Model
Suppose that we want to represent both style s and content c with vectors of parameters
Let ysc denote a K-dimensional observation vector in style s and content class c
November 10, 2018
Computer Vision

11. How to Learn the Bases and Coefficients
Learning is done by minimizing total squared error over the entire training set
Again the solution is given by a SVD decomposition
November 10, 2018
Computer Vision

12. How to Learn the Bases and Coefficients
For symmetric models, we also minimize the total squared error over the training set
This is minimized using an iterative procedure
November 10, 2018
Computer Vision

13. Classification Example
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Computer Vision

14. Classification Example
When the model is trained on 10 faces and tested on the remaining one
1-NN achieves an accuracy of 53.9%±4.3%
The proposed method achieves 73.9%±6.7% when parameters are determined automatically
It achieves 80.6%±7.5% when optimal parameter values are used
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Computer Vision

15. Extrapolation
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Computer Vision

16. Extrapolation
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Computer Vision

17. Translation
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Computer Vision

18. Multilinear Analysis
Tensors are multilinear mappings over a set of vector spaces
An order N tensor is given by
Mode-n vectors are given by
They result in a mode-n flattening
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Computer Vision

19. Multilinear Analysis
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Computer Vision

20. Multilinear Analysis
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Computer Vision

21. Multilinear Analysis Mode-n product of a tensor and matrix is given by
A tensor of n factors
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Computer Vision

22. Mode-n SVD
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Computer Vision

23. TensorFaces Weizmann face image database 28 male subjects
photographed in 5 viewpoints, 4 illuminations, and 3 expressions
Images are aligned to a reference face using a global rigid optical flow and then downsized by a factor of 3 and cropped, yielding a total of 7943 pixels per image within the elliptical cropping window
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Computer Vision

25. Weizmann Face Dataset
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Computer Vision

26. TensorFaces The dataset is then represented by a tensor of order 5
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Computer Vision

27. TensorFaces
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28. TensorFaces
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32. TensorFaces
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33. TensorFaces The bases for TensorFaces are given by November 10, 2018
Computer Vision

34. Face Recognition First experiment
TensorFaces are trained on an ensemble comprising images of 23 people, captured from 3 viewpoints (0,±34 degrees), with 4 illumination conditions (center, left, right, left + right)
It is tested on other images in this 23 person dataset acquired from 2 different viewpoints (±17 degrees) under the same 4 illumination conditions
In this test scenario, the PCA method recognized the person correctly 61% of the time while TensorFaces recognized the person correctly 80% of the time.
November 10, 2018
Computer Vision

35. Face Recognition The second experiment
In a second experiment, TensorFaces is trained on images of 23 people
5 viewpoints (0,±17,±34 degrees), 3 illuminations (center light, left light, right light)
Tested on the 4th illumination (left + right)
PCA yielded a poor recognition rate of 27% while Tensorfaces achieved a recognition rate of 88%
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Computer Vision

36. Dimension Reduction
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38. N-Mode Orthogonal Iteration Algorithm
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Computer Vision

41. Multilinear ICA
Similar to MPCA, MICA is done using an n-mode ICA algorithm
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Computer Vision

42. Different ICA Architectures
Architecture I
ICA computes independent components of DT
Architecture II
ICA computes independent components of D
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Computer Vision

44. MICA
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45. MICA
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46. MICA
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47. Recognition Experiment
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Computer Vision

48. TensorTextures Similar idea can be used for image synthesis
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Computer Vision

49. TensorTextures
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Computer Vision

50. TensorTextures
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Computer Vision

51. TensorTextures
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Computer Vision

52. TensorTextures
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Computer Vision

53. TensorTextures
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Computer Vision

54. TensorTextures
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55. Comparison
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Computer Vision