Prediction error is a discontinuity attribute that removes the predictable image components and reveals the unpredictable. To use prediction error as a discontinuity attribute - the original goal and starting point of my project - one has to devise a prediction-error computation that predicts and removes the plane-wave volumes of sedimentary layers but that is incapable of predicting the discontinuities.

Prediction-error filters Claerbout (1992); Jain (1989)
remove
the image component
that is predictable as a linear combination
of its neighboring values.
Given a two-dimensional image *u*(*m*,*n*) and
the linear prediction estimate

The region *S*_{x} defines
which neighboring values contribute to the linear prediction.
Causal predictions, that involve regions of the shape shown in
Figure 35, lead to white noise driven output images.
The region is called *causal* since given
a hypothetical scan from top to bottom and left to right
all points of *S*_{x} lie to one-side of the predicted value *u*(*m*,*n*).

pefDomain
Prediction-error domain.
The causal domain ensures that the output of the prediction-error filter
tends to be white noise.
Figure 35 |

To compute the prediction error of a given stationary image,
we first find
the prediction coefficients *a*(*k*,*l*)
that minimize the prediction error for all pixels
of the input image.
Once the prediction coefficients are known,
the convolution

To compute the prediction error of a nonstationary image, I, as usual, divide the image into stationary patches, compute the prediction error for each patch, and merge the patches containing the local prediction error into a single quilt. All result images of this section are smoothed along the vertical axis to suppress the prediction error's tendency to enhance high-frequency noise of the original unfiltered image.

- Three-dimensional prediction-error filter
- Residual of three two-dimensional prediction-error filters
- Norm of residual
- Backprojection of residual
- Discussion
- Acknowledgments

3/8/1999