Manifold Blurring Mean Shift algorithms for manifold denoising, presentation, 2012

Computer Vision

Manifold Blurring Mean Shift algorithms for manifold denoising

Kevin ADDA, Florent RENUCCI

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

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Manifold Blurring Mean Shift algorithm (MBMS)

Blurring mean-shift update :

Projection on a sub-dimensional space with PCA:

, where K is a Gaussian kernel:

, such that:

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.

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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.

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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.

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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.

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Since we use 2D datasets, we will usually choose L=1, except for GBMS Algorithm (L=0)

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.

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Trade-off between the number of iterations and the other parameters.

Spiral dataset

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Pinwheel.m: generates little two-dimensional datasets that are spirals of noisy data. 

(credit: Harvard intelligent probabilistic systems)

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Initial set: Noisy spiral with uniformely distributed outliers

N = 1250

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 1

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 2

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 3

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 4

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 5

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 6

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 7

Spiral dataset: application

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Parameters : L = 1; k = 15 ; = 1.1

Iteration 8

Number of neighbors effect Initial dataset:

2 sets of parameters: L = 1, k = 10, sigma = 1.1

L = 1, k = 100, sigma = 1.1

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Number of neighbors effect

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K = 10 K = 100

Iteration 1

Number of neighbors effect

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K = 10 K = 100

Iteration 2

Number of neighbors effect

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K = 10 K = 100

Iteration 3

Intrinsic dimension effect Initial dataset:

2 sets of parameters: L = 1, k = 15, sigma = 1.1

L = 0, k = 15, sigma = 1.1

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Number of neighbors effect

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L = 1 L = 0

Iteration 1

Number of neighbors effect

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L = 1 L = 0

Iteration 2

Number of neighbors effect

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L = 1 L = 0

Iteration 3

MNIST Dataset Classification

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Input : 16x8 matrices of 0 and 1 representing the image of a letter.

MNIST Dataset Classification

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Input : 16x8 matrices of 0 and 1 representing the image of a letter.

Parameters :

L = 1; sigma = 1;

k = 4; (must be an even number)

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.

MNIST Dataset Classification

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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

MNIST Dataset Classification

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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%

Conclusion

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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 methodMore congruent image for a human eye

Also more congruent for an automatic classification

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Thank you