(General) To retrieve a clean dataset by deleting outliers. (Computer Vision) the recovery of a digital image that has been contaminated by additive white Gaussian noise.

1. Computer Vision
Manifold Blurring Mean Shift algorithms for manifold denoising
Kevin ADDA, Florent RENUCCI

2. Denoising

(General) To retrieve a clean dataset by deleting outliers.

(Computer Vision) the recovery of a digital image that has
been contaminated by additive white Gaussian noise.
Noisy spiral dataset
Handwritten digits recognition
Noisy image

3. Manifold Blurring Mean Shift algorithm
(MBMS)

Blurring mean-shift update :
, where K is a Gaussian kernel:

Projection on a sub-dimensional that: with PCA:
space
, such
Parameters:
the variance of the Gaussian kernel ;
k the number of neighbors to consider ;
L the local instrinsic dimension;
Iteration number for the whole algorithm.

4. Setting the parameters: the kernel variance

related to the level of local noise outside the manifold;

The larger it is, the stronger the denoising effect;

But can distort the manifold shape over iterations.
Trade-off between kernel variance and iteration number.

5. Setting the parameters: the number of
neighbors

k is the number of nearest neighbors that estimates the local
tangent space;

MBMS is quite robust to it. It typically grows sublinearly with
N.

However, it effects strongly the mean-shift blurring effect as
each point is motioned toward the Gaussian kernel mean on
the neighbors.
Trade-off between the number of parameters and kernel variance.

6. Setting the parameters: the intrinsic
dimensionality

If L is too small, it produces more local clustering and can
distort the manifold;

If L is too big, points will move a little : if L is equal to the
dimension of the set, no motion.
Since we use 2D datasets, we will usually choose L=1, except for GBMS Algorithm (L=0)

7. Setting the parameters: the number of
iterations

A few iterations (1 to 5) achieve most of the denoising

More iterations can refine this and produce a better
result, but shrinkage might arise.
Trade-off between the number of iterations and the other parameters.

8. Spiral dataset

Pinwheel.m: generates little two-dimensional datasets
that are spirals of noisy data.
(credit: Harvard intelligent probabilistic systems)

9. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
N = 1250
Initial set: Noisy spiral with uniformely distributed outliers

10. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 1

11. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 2

12. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 3

13. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 4

14. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 5

15. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 6

16. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 7

17. Spiral dataset: application
Parameters : L = 1; k = 15 ;
= 1.1
Iteration 8

18. Number of neighbors effect

Initial dataset:

2 sets of parameters:

L = 1, k = 10, sigma = 1.1

L = 1, k = 100, sigma = 1.1

19. Number of neighbors effect
K = 10
K = 100
Iteration 1

20. Number of neighbors effect
K = 10
K = 100
Iteration 2

21. Number of neighbors effect
K = 10
K = 100
Iteration 3

22. Intrinsic dimension effect

Initial dataset:

2 sets of parameters:

L = 1, k = 15, sigma = 1.1

L = 0, k = 15, sigma = 1.1

23. Number of neighbors effect
L=1
L=0
Iteration 1

24. Number of neighbors effect
L=1
L=0
Iteration 2

25. Number of neighbors effect
L=1
L=0
Iteration 3

26. MNIST Dataset Classification

Input : 16x8 matrices of 0 and 1 representing the image of
a letter.

27. MNIST Dataset Classification

Input : 16x8 matrices of 0 and 1 representing the image of a letter.

Parameters :


k = 4; (must be an even number)


L = 1; sigma = 1;
n_iteration = 1;
Preprocessing algorithm :

Extraction the "1" elements. It means that if m1,3=1 for example, we extract the point 1,3.

coordinates of the white points.

Denoising step.

If the result is not an integer, we round it.

for example if we plan to move a pixel to the coordinates (12,54;14,1), we round it to (13;14).

The vector obtained is transformed in a matrix of 0 and 1.

28. MNIST Dataset Classification

General algorithm :

We learn a neural network that labels the dataset

We compute the good labelling rate

We denoise the images

We learn a new neural network

We compute the good labelling rate

29. MNIST Dataset Classification

Results :
We first run the algorithm on the dataset, and then
separate training set and test set. We compare the good
labelling rates.
Good labelling rates
dataset
Training/test dataset
No blurring
51%
35%
blurring
53%
39%

30. Conclusion

The Manifold Blurring Mean Shift algorithm allows to
blur an image in order to:



Erase some outliers in merging them in the "real" image;
Merge outliers and decreasing their number.
decrease the error rate of a labelling method
 More
 Also
congruent image for a human eye
more congruent for an automatic classification

31. Thank you