S. Hongo et I. Yoroizawa, STOCHASTIC RELAXATION FOR CONTINUOUS VALUES - STANDARD REGULARIZATIONBASED ON GAUSSIAN MRF, IEICE transactions on information and systems, E77D(4), 1994, pp. 425-432
We propose a fast computation method of stochastic relaxation for the
continuous-valued Markov random field (MRF) whose energy function is r
epresented in the quadratic form. In the case of regularization in vis
ual information processing, the probability density function of a stat
e transition can be transformed to a Gaussian function, therefore, the
probablistic state transition is realized with Gaussian random number
s whose mean value and variance are calculated based on the condition
of the input data and the neighborhood. Early visual information proce
ssing can be represented with a coupled MRF model which consists of co
ntinuity and discontinuity processes. Each of the continuity or discon
tinuity processes represents a visual property, which is like an inten
sity pattern, or a discontinuity of the continuity process. Since most
of the energy function for early visual information processing can be
represented by the quadratic form in the continuity process, the prob
ability density of local computation variables in the continuity proce
ss is equivalent to the Gaussian function. If we use this characterist
ic, it is not necessary for the discrimination function computation to
calculate the summation of the probabilities corresponding to all pos
sible states, therefore, the computation load for the state transition
is drastically decreased. Furthermore, if the continuous-valued disco
ntinuity process is introduced, the MRF model can directly represent t
he strength of discontinuity. Moreover, the discrimination function of
this energy function in the discontinuity process, which is linear, c
an also be calculated without probability summation. In this paper, a
fast method for calculating the state transition probability for the c
ontinuous-valued MRF on the visual information processing is theoretic
ally explained. Next, initial condition dependency, computation time a
nd dependency on the statistical estimation of the condition are inves
tigated in comparison with conventional methods using the examples of
the data restoration for a corrupted square wave and a corrupted one-d
imensional slice of a natural image.