Outline Texture modeling - continued Markov Random Field models Fractals

Visual Perception Modeling Some Texture Examples 11/20/2019 Visual Perception Modeling

Visual Perception Modeling Texture Modeling Texture modeling is to find feature statistics that characterize perceptual appearance of textures There are two major computational issues What kinds of feature statistics shall we use? How to verify the sufficiency or goodness of chosen feature statistics? 11/20/2019 Visual Perception Modeling

Texture Modeling – cont. The structures of images The structures in images are due to the inter-pixel relationships The key issue is how to characterize the relationships 11/20/2019 Visual Perception Modeling

Co-occurrence Matrices Gray-level co-occurrence matrix One of the early texture models Was widely used Suppose that there are G different gray values in a texture image I For a given displacement vector (dx, dy), the entry (i, j) of the co-occurrence matrix Pd is 11/20/2019 Visual Perception Modeling

Autocorrelation Features Many textures have repetitive nature of texture elements The autocorrelation function can be used to assess the amount of regularity as well as the fineness/coarseness of the texture present in the image 11/20/2019 Visual Perception Modeling

Visual Perception Modeling Geometrical Models Geometrical models Applies to textures with texture elements First texture elements are extracted Then one can compute the statistics of local elements or extract the placement rule that describes the texture Voronoi tessellation features Structural methods 11/20/2019 Visual Perception Modeling

Visual Perception Modeling Markov Random Fields Markov random fields Have been popular for image modeling, including textures Able to capture the local contextual information in an image 11/20/2019 Visual Perception Modeling

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 11/20/2019 Visual Perception Modeling

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  11/20/2019 Visual Perception Modeling

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 11/20/2019 Visual Perception Modeling

Markov Random Fields – cont. 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: 11/20/2019 Visual Perception Modeling

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) 11/20/2019 Visual Perception Modeling

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 11/20/2019 Visual Perception Modeling

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 ....... 11/20/2019 Visual Perception Modeling

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 11/20/2019 Visual Perception Modeling

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 11/20/2019 Visual Perception Modeling

Markov Random Fields – cont. The -model The energy function is of the form with 11/20/2019 Visual Perception Modeling

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 11/20/2019 Visual Perception Modeling

Markov Random Fields – cont. Pseudo-likelihood Instead of maximizing P(f), the joint probability, we maximize the products of conditional probabilities 11/20/2019 Visual Perception Modeling

Markov Random Fields – cont. Texture synthesis Generate samples from the Gibbs distributions Two sampling techniques Metropolis sampler Gibbs sampler 11/20/2019 Visual Perception Modeling

Markov Random Fields – cont. 11/20/2019 Visual Perception Modeling

Visual Perception Modeling 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 11/20/2019 Visual Perception Modeling

Visual Perception Modeling Fractals – An Example 11/20/2019 Visual Perception Modeling