(slides 9) visual computing: geometry, graphics, and vision
DESCRIPTION
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/1584504277TRANSCRIPT
© Frank Nielsen 2011
INF555
Fundamentals of 3D
Lecture 9:Laplacian Image pyramidsExpectation-Maximization+ Overview of computational photography
Frank Nielsen
30 Novembre 2011
© 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)
© Frank Nielsen 2011
Fourier log power spectrum
Stripes of the hat
Stripes of the hair
Interpreting Fourier spectra
90 degrees
Divide by blocks
© Frank Nielsen 2011
Laplacian image pyramids
Used also in computer graphics for texturing (mipmapping)
© Frank Nielsen 2011
Gaussian. Blur and sample, and thenLaplacian. Interpolate and estimate
Laplacian image pyramids
Residual
Reconstruction
→ Precursors of wavelets
© 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?
© Frank Nielsen 2011
Laplacian image pyramids:Lossless multi-scale representation of images
© Frank Nielsen 2011
Laplacian image pyramids:Reconstruction process
© 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 ofL(I1) with the rightmost half of L(I2).
● Reconstruct blended images from the Laplacian pyramid Lr.
© Frank Nielsen 2011
Laplacian image pyramids: Application to blending
© Frank Nielsen 2011
Laplacian image pyramids: Application to blending
© Frank Nielsen 2011
Laplacian image pyramids: Application to blending
Using a region mask
Nowadays, we better use Poisson image editing and gradient/image reconstruction
© 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)
© Frank Nielsen 2011
Indicator variables z (also called latent variables)
Multinomially distributed
© Frank Nielsen 2011
Generating samples from Gaussian Mixture Models (GMMs)
© Frank Nielsen 2011
Maximize the likelihood(incomplete)
Maximize the likelihood(complete likelihood)
© Frank Nielsen 2011
Expectation-Maximization algorithm: Iteration
Initialize with k-means (or k-means++)
Soft assignment to clusters
Given assignments find best parameters
© Frank Nielsen 2011
© Frank Nielsen 2011
© Frank Nielsen 2011
© Frank Nielsen 2011
En Java, http://www.lix.polytechnique.fr/~nielsen/MEF/
Modeling images withGaussian mixture models
RGB+XY= Point in 5D
En Python, http://www.lix.polytechnique.fr/~schwander/pyMEF/
© Frank Nielsen 2011
Gaussian mixture models for image segmentation
Any smooth density function can be arbitrarily closely approximatedby a Gaussian mixture model