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Visual Perception Modeling using Markov Random Fields

Explore the use of Markov Random Fields in visual perception modeling, including texture modeling and fractal modeling. Understand the structures and relationships within images and learn how to characterize them.

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Visual Perception Modeling using Markov Random Fields

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  1. Outline • Texture modeling - continued • Markov Random Field models - continued • Fractals

  2. Texture Modeling • The structures of images • The structures in images are due to the inter-pixel relationships • The key issue is how to characterize the relationships Visual Perception Modeling

  3. Markov Random Fields • Markov random fields • Have been popular for image modeling, including textures • Able to capture the local contextual information in an image Visual Perception Modeling

  4. Markov Random Fields – cont. • Sites • Let S index a discrete set of m sites S = {1, ...., m} • A site represents a point or a region in the Euclidean space • Such as an image pixel • Labels • A label is an event that may happen to a site • Such as pixel values Visual Perception Modeling

  5. Markov Random Fields – cont. • Labeling problem • Assign a label from the label set L to each of the sites in S • Also a mapping from SL • A labeling is called a configuration • In texture modeling, a configuration is a texture image • The set of all possible configurations is called the configuration space  Visual Perception Modeling

  6. Markov Random Fields – cont. • Neighborhood systems • The sites in S are related to one another via a neighborhood • A neighborhood system for S is defined as • The neighborhood relationship has the following properties • A site is not a neighbor to itself • The neighborhood relationship is mutual Visual Perception Modeling

  7. Markov Random Fields – cont. • Markov random fields • Let F={F1, ...., Fm} be a family of random variables defined on the set S in which each random variable Fi takes a value from L • F is said to be a Markov random field on S with respect to a neighborhood system N if an only if the following two conditions are satisfied: Visual Perception Modeling

  8. Markov Random Fields – cont. • Homogenous MRFs • If P(fi | fNi) is regardless of the relative position of site i in S • How to specify a Markov random field • Conditional probabilities P(fi | fNi) • Joint probability P(f) Visual Perception Modeling

  9. Markov Random Fields – cont. • Gibbs random fields • A set of random variables F is said to be a Gibbs random field on S with respect to N if and only if its configurations obey a Gibbs distribution • and Visual Perception Modeling

  10. Markov Random Fields – cont. • Cliques • A clique c for (S, N) is defined as a subset of sites in S and it consists of • A single site • A pair of neighboring sites • A triple of neighboring sites • ....... Visual Perception Modeling

  11. Markov Random Fields – cont. • Markov-Gibbs equivalence • Hammersley-Clifford theorm • F is an Markov random field on S respect to N if and only if F is a Gibbs random field on S with respect to N • Practical value of the theorem • It provides a simple way to specify the joint probability by specifying the clique potentials Visual Perception Modeling

  12. Markov Random Fields – cont. • Markov random field models for textures • Homogeneity of Markov random fields is assumed • A texture type is characterized by a set of parameters associated with clique types • Texture images can be generated (synthesized) by sampling from the Markov random field model Visual Perception Modeling

  13. Markov Random Fields – cont. • Texture models using Markov random fields • For most existing models, only the single pixel and pair-wise pixel cliques are considered • In other words, all other higher-order cliques are zero • Textures are characterized by a few parameters Visual Perception Modeling

  14. Markov Random Fields – cont. • Auto Models • Only the single pixel and pair-wise pixel cliques can be non-zeros • Auto-logistic models • Ising models • Auto-binomial models • Gaussian Markov random field models Visual Perception Modeling

  15. Markov Random Fields – cont. • The -model • The energy function is of the form • with Visual Perception Modeling

  16. Markov Random Fields – cont. • Parameter estimation • Parameters are generally estimated using Maximum-Likelihood estimator or Maximum-A-Posterior estimator • Computationally, the partition function can not be evaluated • Markov chain Monte Carlo is often used to estimate the partition function by generating typical samples from the distribution Visual Perception Modeling

  17. Markov Random Fields – cont. • Pseudo-likelihood • Instead of maximizing P(f), the joint probability, we maximize the products of conditional probabilities Visual Perception Modeling

  18. Markov Random Fields – cont. • Texture synthesis • Generate samples from the Gibbs distributions • Two sampling techniques • Metropolis sampler • Gibbs sampler Visual Perception Modeling

  19. Markov Random Fields – cont. Visual Perception Modeling

  20. Fractals • Fractals • Many natural surfaces have a statistical quality of roughness and self-similarity at different scales • Fractals are very useful in modeling self-similarity • Texture features based on fractals • Fractal dimension • Lacunarity Visual Perception Modeling

  21. Fractals – An Example Visual Perception Modeling

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