IMAGE-BASED MODELING AND RENDERING 1. HISTOGRAM AND GMM. I-Chen Lin, Dept. of CS, National Chiao Tung University
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1 IMAGE-BASED MODELING AND RENDERING. HISTOGRAM AND GMM I-Che Li, Dept. of CS, Natioal Chiao Tug Uiversity
2 Outlie What s the itesity/color histogram? What s the Gaussia Mixture Model (GMM? Their applicatios ad limitatio. Ref (plety of slides are from : Key A. Hut, The Art of Image Processig. R. C. Gozalez ad R. E. Woods, Digital Image Processig. Tae-Kyu Kim, Machie Learig for Computer Visio. lecture otes. Adrew Roseberg, Machie Learig, lecture otes.
3 Digital Image Samplig ad Quatizatio cotiuous toe scee sampled (partitio space sampled ad quatized (partitioed light levels
4 Digital Image Samplig ad Quatizatio Cotiuous fuctio (upper left is Sampled (above ad the Quatized (upper right to form Digital image (left
5 Image Types Example E.g. RGB (8bits x 3 chael (8bits x chael bits (per pixel
6 Itesity Histogram For istace, a 4 x 4 image (3 bits per pixel itesity (bis
7 Itesity Histogram Examples Fig. from [Gozalez ad Woods]
8 Itesity Histogram Examples Fig. from [Gozalez ad Woods]
9 Histogram Equalizatio Improvig the local cotrast of a image without alterig the global cotrast to a sigificat degree. Creatig a output image with a (early uiform histogram. itesity itesity (ideally
10 Histogram Equalizatio (cot. Estimatig through the Cumulative Distributio Fuctio (CDF. Histogram CDF Fig. from Roger S. Gaborsi, Itro. to Computer Visio
11 Histogram Equalizatio (cot. The goal ow become Histogram CDF (ideally itesity (ideally itesity
12 Numeric Example of Equalizatio
13 Histogram Equalizatio Examples
14 Simple Segmetatio by Histogram itesity itesity
15 Color Histogram 3x5x3 resolutio 3x4x3 resolutio 8x3x3 resolutio Cosider the resolutio of various color histogram biigs i RGB space. The resolutio of each axis may be set idepedetly of the others. Slides from Key A. Hut, The Art of Image Processig.
16 Color Histogram (cot. The segmetatio ow has to rely o a boudig cuboid or thresholdig by plaes i RGB space
17 Color Histogram (cot. How about more complex situatios?
18 Vector Clusterig Data vectors (gree are grouped to homogeous clusters (blue ad red. The cluster ceters are mared x. Parts of the slides are from Tae-Kyu Kim, Machie Learig for Computer Visio. lecture otes.
19 Color Clusterig (Image Quatizatio Image pixels are represeted by 3D vectors of R,G,B values. The vectors are grouped to K=0,3,2 clusters, ad represeted by the mea R values of the respective clusters. ` ` G B Parts of the slides are from Tae-Kyu Kim, Machie Learig for Computer Visio. lecture otes.
20 K-meas Clusterig Fig. from Christopher M. Bishop, Mixture Models ad the EM Algorithm, lecture otes.
21 Gaussia Mixture Models Rather tha idetifyig clusters by earest cetroids Fit a Set of Gaussias to the data Maximum Lielihood over a mixture model Slides are from Adrew Roseberg, Machie Learig, lecture otes.
22 GMM example p ( x 2 2 ( x e : mea : stadard deviatio
23 Multivariate Gaussia distributio For radom variables x, y (scalar vector N N Var( X ( x i x Cov( X, Y ( xi x( yi y N N i i matrix Determiat of Fig. from Christopher M. Bishop, Mixture Models ad the EM Algorithm, lecture otes.
24 Mixture Models Formally a Mixture Model is the weighted sum of a umber of pdfs where the weights are determied by a distributio,
25 Gaussia Mixture Models GMM: the weighted sum of a umber of Gaussias where the weights are determied by a distributio,
26 Expectatio Maximizatio The traiig of GMMs ca be accomplished usig Expectatio Maximizatio Step : Expectatio (E-step Evaluate the resposibilities of each cluster with the curret parameters Step 2: Maximizatio (M-step Re-estimate parameters usig the existig resposibilities Similar to -meas traiig.
27 EM for GMMs (algorithm Iitialize the parameters Evaluate the log lielihood Expectatio-step: Evaluate the resposibilities Maximizatio-step: Re-estimate Parameters Evaluate the log lielihood Chec for covergece
28 EM for GMMs (algorithm E-step: Evaluate the Resposibilities N( x, (2 d 2 2 e ( x 2 T ( x
29 EM for GMMs (algorithm M-Step: Re-estimate Parameters
30 Visual example of EM Slides are from Adrew Roseberg, Machie Learig, lecture otes.
31 Maximum Lielihood over a GMM As usual: Idetify a lielihood fuctio Ad set partials to zero
32 Maximum Lielihood of a GMM Optimizatio of meas. x dx x d l ( ( (2, ( T x x d x e N x x e dx de N N z x z ( (
33 Maximum Lielihood of a GMM Optimizatio of covariace x dx x d l ( ( (2, ( T x x d x e N x x e dx de
34 Maximum Lielihood of a GMM Optimizatio of mixig term x dx x d l ( ( (2, ( T x x d x e N x x e dx de 0, (, ( N K x N x N d df F = 0, (, ( N K x N x N 0 ( N z 0 ( K K N z
35 MLE of a GMM
36 How to Apply the GMMs? Collect traiig data of each category (label. Choose a appropriate feature set. Estimate (Trai the GMM parameters of each category by EM. Evaluate the probability for each category.
37 Practical Issues for Images The computatioal efficiecy. Differet types of covariace matrices. How about these images?! Fig. from W. Matusi, et al., "Image-based visual hulls Fig. from the Grabcut database
38 Appedix: Matrix ad Vector Derivatives
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40 Appedix: Matrix ad Vector Derivatives Ref: J. D. M. Reie. A Simple Exercise o Matrix Derivatives. K. B. Peterse. The Matrix Cooboo
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