Those are the slides for the book: Visual Computing: Geometry, Graphics, and Vision. by Frank Nielsen (2005) http://www.sonycsl.co.jp/person/nielsen/visualcomputing/ http://www.amazon.com/Visual-Computing-Geometry-Graphics-Charles/dp/1584504277

1. © Frank Nielsen 2011
INF555
Fundamentals of 3D
Lecture 9:
Laplacian Image pyramids
Expectation-Maximization
+ Overview of computational photography
Frank Nielsen
nielsen@lix.polytechnique.fr
30 Novembre 2011

2. © Frank Nielsen 2011
Télécharger votre projet le 10 Décembre au soir
Examen 13 Décembre
Venez avec une clé USB (Projet, TDs)
→ Utilisez Processing.org+JAMA+JMyron (dans la mesure du possible)

3. © Frank Nielsen 2011
Fourier log power spectrum
Stripes of the hat
Stripes of the hair
Interpreting Fourier spectra
90 degrees
Divide by
blocks

4. © Frank Nielsen 2011
Laplacian image pyramids
Used also in computer graphics for texturing (mipmapping)

5. © Frank Nielsen 2011
Gaussian. Blur and sample, and then
Laplacian. Interpolate and estimate
Laplacian image pyramids
Residual
Reconstruction
→ Precursors of wavelets

6. © Frank Nielsen 2011
Blurring is efficient for sampling as it removes high-frequency
components. (sample at fewer positions.)
Gaussian kernel and resampling at a quarter of the image size.
Blurring and resampling is computed using a single discrete kernel.
●Central limit theorem:
(mean of random variables approach Gaussian distribution)
●Infinitely differentiable functions
●Fourier of Gaussians are Gaussians
●Human brain has neuronal regions doing Gaussian filtering
Laplacian image pyramids
Why Gaussians?

7. © Frank Nielsen 2011
Laplacian image pyramids:
Lossless multi-scale representation of images

8. © Frank Nielsen 2011
Laplacian image pyramids:
Reconstruction process

9. © Frank Nielsen 2011
Laplacian image pyramids: Application to blending
Multiband blending.
Blending two overlapping images using their pyramids
● Compute Laplacian pyramids L(I1) and L(I2) of I1 and I2.
● Generate a hybrid Laplacian pyramid Lr by creating for
each image of the pyramid a 50%/50% mix of images,
obtained by selecting the leftmost half of
L(I1) with the rightmost half of L(I2).
● Reconstruct blended images from the Laplacian pyramid Lr.

10. © Frank Nielsen 2011
Laplacian image pyramids: Application to blending

11. © Frank Nielsen 2011
Laplacian image pyramids: Application to blending

12. © Frank Nielsen 2011
Laplacian image pyramids: Application to blending
Using a region mask
Nowadays, we better use Poisson image editing and
gradient/image reconstruction

13. © Frank Nielsen 2011
Soft clustering using Expectation-Maximization (EM)
http://www.neurosci.aist.go.jp/~akaho/MixtureEM.html
Generative statistical models
GAUSSIAN MIXTURE MODELS (GMMs)

14. © Frank Nielsen 2011
Indicator variables z (also called latent variables)
Multinomially distributed

15. © Frank Nielsen 2011
Generating samples from Gaussian Mixture Models (GMMs)

16. © Frank Nielsen 2011
Maximize the likelihood
(incomplete)
Maximize the likelihood
(complete likelihood)

17. © Frank Nielsen 2011
Expectation-Maximization algorithm: Iteration
Initialize with k-means (or k-means++)
Soft assignment to clusters
Given assignments find best parameters

18. © Frank Nielsen 2011

19. © Frank Nielsen 2011

20. © Frank Nielsen 2011

21. © Frank Nielsen 2011
En Java, http://www.lix.polytechnique.fr/~nielsen/MEF/
Modeling images with
Gaussian mixture models
RGB+XY= Point in 5D
En Python, http://www.lix.polytechnique.fr/~schwander/pyMEF/

22. © Frank Nielsen 2011
Gaussian mixture models for image segmentation
Any smooth density function can be arbitrarily closely approximated
by a Gaussian mixture model