Papers:Texture:FRAME

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Filters, Random Fields and Maximum Entropy (FRAME): Towards a Unified Theory for Texture Modeling

S-C. Zhu, Y. Wu, and D. Mumford
International Journal on Computer Vision (IJCV), Vol. 27, No.2, pp 107-126 (1998)
<Download from S-C Zhu's home page>
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Summary

The paper presents a probabilistic model of homogeneous textures, which seeks the maximum entropy distribution constrained by filters.

  • It aims at establishing a probabilistic generative model of textures.
  • The main process can be briefly described as follows
    • It select a set of filters to capture the characteristics of the input textures.
    • The histograms of the filtered images will be extracted to serve as constrains in optimizing the probability model
    • The probabilistic distribution is then optimized by maximizing the entropy.
  • Formally, the optimization problem is formulated as follows
    p(\mathbf{I}) = \operatorname{argmax} \left\{- \int p(\mathbf{I}\log p(\mathbf{I}) d\mathbf{I})\right\}
    subject to
    E_p[\delta(z - \mathbf{I}^{k}(x))] = f^{k}(z), \quad \forall z \in R, \ \forall k = 1, \ldots, K, \ \forall x \in D
    and
    \int p(\mathbf{I}) d\mathbf{I} = 1
    here, k indexes the filters, x refers to pixel position in the spatial domain D.
  • With Langrange multiplier and discrete quantization of z values, the form of the constrained maximum entropy distribution is given by
    p(\mathbf{I}; \Lambda_K, S_K) = \frac{1}{Z(\Lambda_k)}\exp\left(-\sum_{k=1}^K \langle \boldsymbol{\lambda}^{(k)}, \mathbf{h}^{(k)} \rangle \right)
    here, \Lambda_K = \{\mathbf{\lambda}_1, \ldots, \mathbf{\lambda}_K\}, and SK is the set of filters.
    In this model, given an image \mathbf{I}, the histograms of filtered images \mathbf{h}_1, \ldots, \mathbf{h}_2 are sufficient statistics. It means that the probability is essentially a function of these histograms.
  • The parameters ΛK are solved by numerical methods. The gradient is given by
    \frac{\partial p}{\partial \boldsymbol{\lambda}^{(k)}} = E_{p(\mathbf{I}; \Lambda_K, S_K)}\mathbf{h}^{(k)} - \mathbf{h}_{input}^{(k)}
    The probability function is convex of the parameters, thus the convergence to the global maxima is guaranteed.
  • The updating step requires Monte Carlo estimation of the expected histogram. The paper employs Gibbs sampler, which flips each pixel at a time. The sampling of the pixel value is based on a local region determined by the filter size.
  • The give a good model, the selection of a compact set of informative filters is very important. The paper adopts a greedy strategy which adds a new filter at a time. The criterion of selecting a new filter is to maximize the margin distribution of the filter response based on current model and that based on input images. The authors prove that such a selection criterion is closely related to the Minimax Entropy Principle.
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My Comments

  • Traditionally the filtering methods lack solid statistical foundation. This paper actually builds up a statistical framework and develops an insight to the relation between filtering the statistical modeling.
  • Since it is a generative model, it can be used in both analysis and synthesis.
  • Though it claimed that the model unifies filters and Markov Random Field (MRF), the relationship between this model and MRF is actually implicit. The Markov property originates from the local support of the filters but not explicit formulation.
  • The model is really beautiful in theoretical view. However, it is computationally too expensive to be used in practice. The main cost comes from the Gibbs sampling process involved in each update of the parameters. In addition, the conditional distribution of each pixel value is not very easy to get. The paper says that it takes one day to synthesize only one 128 x 128 textured image!
  • From the perspective of texture description, it actually uses global histograms of filtered image to characterize the textures, and inner product of two histograms to measure the similarity. In this sense, it is the same as traditionally filter-based methods.
  • The model significantly increase the expressive power of standard MRF, however, it still lacks the capability of modeling regular spatial structures embeded in the textures. This limitation is fundamentally due to the fact that it only makes use of the global histogram of filtering outputs and neglects the inter-region relation that can not be captured by filtering statistics.
  • To sum up, this paper incorporates the statistical formulation with filters. Though it may not be a good choice to directly applied the algorithm introduced in this paper, the insight here is enlightening, and will be useful when we develop more sophisticated statistical models of textures that involve filters.


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