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Sequences II. Prof. Noah Snavely CS1114 http://cs1114.cs.cornell.edu. Administrivia. Assignments: A5P2 due on Friday Quiz Tuesday, 4/24 Final project proposals due on CMS by this Thursday. Google’s PageRank. H. Start at a random page, take a random walk. Where do we end up?. A. E. I.

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sequences ii

Sequences II

Prof. Noah Snavely

CS1114

http://cs1114.cs.cornell.edu

administrivia
Administrivia
  • Assignments:
    • A5P2 due on Friday
  • Quiz Tuesday, 4/24
  • Final project proposals due on CMS by this Thursday
google s pagerank
Google’s PageRank

H

Start at a random page, take a random walk. Where do we end up?

A

E

I

D

B

C

F

J

G

google s pagerank1
Google’s PageRank

H

Add 15% probability of moving to a random page. Now where do we end up?

A

E

I

D

B

C

F

J

G

google s pagerank2
Google’s PageRank

H

PageRank(P) = Probability that a long random walk ends at node P

A

E

I

D

B

C

F

J

G

slide6

Example

(The ranks are an eigenvector of the transition matrix)

modeling texture
Modeling Texture
  • What is texture?
    • An image obeying some statistical properties
    • Similar structures repeated over and over again
    • Often has some degree of randomness
markov random field

A

D

X

B

C

  • Higher order MRF’s have larger neighborhoods

*

*

*

*

*

*

*

*

X

*

*

*

X

*

*

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*

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Markov Random Field
  • A Markov random field (MRF)
    • generalization of Markov chains to two or more dimensions.
  • First-order MRF:
    • probability that pixel X takes a certain value given the values of neighbors A, B, C, and D:
texture synthesis efros leung iccv 99
Texture Synthesis [Efros & Leung, ICCV 99]
  • Can apply 2D version of text synthesis
more synthesis results
More Synthesis Results
  • Increasing window size
more results
More Results
  • aluminum wire
  • reptile skin
author recognition
Author recognition
  • Simple problem:

Given two Markov chains, say Austen (A) and Dickens (D), and a string s (with n words), how do we decide whether A or D wrote s?

  • Idea: For both A and D, compute the probability that a random walk of length n generates s
probability of a sequence
Probability of a sequence
  • What is the probability of a given n-lengthsequence s?

s = s1 s2 s3 … sn

  • Probability of generating s = the product of transition probabilities:

Probability that a sequence starts with s1

Transition probabilities

likelihood
Likelihood
  • Compute this probability for A and D

Jane Austen wrote s

“likelihood” of A

Charles Dickens wrote s

“likelihood” of D

???

problems with likelihood
Problems with likelihood
  • Most strings of text (of significant length) have probability zero
    • Why?
  • Even if it’s not zero, it’s probably extremely small
    • What’s 0.01 * 0.01 * 0.01 * … (x200) … * 0.01?
    • According to Matlab, zero
  • How can we fix these problems?
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