Linear prediction is the problem of finding the minimum mean square estimate of using a linear combination of the past signal values from to The most commonly used forward one step Finite Impulse Response (FIR) linear predictor of order is given by

where are the coefficients of the prediction filter. The solution is given by the Wiener-Hopf [18] equation

where is an autocorrelation matrix, is predictor coefficient vector, and is autocorrelation vector. The autocorrelation is defined as

is defined as for to It is assumed that signal values are real. Consider the set and let be substituted by which is a prediction from or all previous elements. Let so that is the new substitute pixel for Now, let be substituted by the prediction . Again, let . We substitute for and so on. The new set is now wherein are substitution pixels for noisy pixels by linear prediction from noise-free pixels. Rewriting as we have This is the substitution set introduced in Section 4.

The substitution concept proposed in this section requires a recursive-type prediction. One ideal approach is to start from a causal Infinite Impulse Response (IIR) linear predictor [18]. Suppose that the image can be modeled as an Auto Regressive Moving Average (ARMA) process with a known power spectrum such that where is the minimum phase spectral factor and is the variance of the white noise driving the model. The causal Infinite Impulse Response (IIR) predictor is given by which, in time domain, becomes

In image processing with a short finite data, assumption of a power spectrum with known characteristics is generally not possible. The predictor coefficients can be determined from autocorrelation of the available data where signal model is not available. This is a reasonable approach in realistic situations [18].

Let be a prediction from one or more noise-free pixels. An outlier (a salt or pepper noise pixel) is substituted by . This is acceptable because has some correlation with previous data and, therefore, is a better candidate than an impulse. After substitution, let be treated as an image pixel-free of impulse noise corruption. Let be Define

Let a first-order recursive linear predictor be defined as . The error due to prediction is . Minimization of the square of the error leads to where The above procedure is repeated for all impulse corrupted pixels. All of the substitute pixels are obtained by this procedure. The resulting set is a substitute set for in this new scheme and not an estimate. We have proved in Section 4 that a subsequent optimization by median filtering of the substitute set takes the current noisy pixel closer to original noise-free image pixel. One of the computationally simplest optimizations that preserve edges is median filtering and, therefore, the resulting substitute pixel set *Z* is filtered using median operation, which is an L1 optimization in Maximum Likelihood sense. Figure 1 shows the flow chart of the proposed scheme.

There are several advantages of the proposed scheme. In DBA the current noisy pixel under processing is replaced with the median of the processing window. If the median itself is corrupted, then the median is replaced by a previously processed neighborhood pixel. At higher noise densities most of the pixels will be corrupted necessitating repeated replacement. This repeated replacement produces streaking. The proposed method avoids this.

In robust statistics estimation filter [19–21], the current noisy pixel under processing is replaced by an image data estimated using an estimation algorithm. But the computation time is much longer. It will be demonstrated in Section 7 that the linear prediction substitution followed by median filtering as introduced by this paper can overcome the problem of streaking and blur while the computational complexity is reduced in comparison with robust statistics estimation filter.