167.

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Page 248

Once the prior energy function formulation is established, the second step is to compute the likelihood energy. For an image contaminated by independent identical distribution (i.i.d) Gaussian noise e_r with zero mean and variance σ², the observation d_r can be expressed as d_r=w_r+e_r, where w_r is the true pixel intensity (grey) value, and e_r is the i.i.d. Gaussian noise. The probability P(d_r|w_r) and likelihood energy U(d_r|w_r) can be expressed as:

(6.27)

and the posterior energy U(w|d) can be derived as:

(6.28)

or, alternatively, as:

(6.29)

where q is a weighting parameter such that the larger the value of q, the more the solution will be affected by the second term, i.e. the prior energy.

If the smoothness prior energy U(w) takes a quadratic form as shown above, U(w) is also called a regulariser. Without this regulariser, one or more of the following conditions may apply (Tikhonov and Arsenin, 1977): (1) a solution does not exist; (2) the computed solution is not unique; or (3) the computed solution does not depend on the initial state. In such circumstances, the problem is ill-posed. However, the use of this regulariser can convert an ill-posed problem into well-posed problem, i.e. guarantee that a unique and initial-state-dependent solution is obtained.

The simplest way to minimise the energy function in Equation (6.29) is to use a gradient descent minimisation algorithm (some alternative minimisation algorithms are discussed in Section 6.4). Given an initial configuration w⁽⁰⁾, then, at iteration i, the new value for w_r⁽ⁱ⁺¹⁾ at site r is obtained from:

(6.30)

or equivalently (by using Equation (6.24)):

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