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I have been studying about the Yule-Walker equations for prediction of a time series data from knowledge of past values of the series. Is there any way I can use the same in an image to exploit the correlation between image pixels in order to perform image interpolation,, or to increase the resolution of the image?

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well, the term interpolation may have different interpretations but in the classical sense, interpolation is a deterministic calculation process of computing intermediate values from the existing ones. It's not a prediction.

Yet, one can devise a probabilistic interpolation scheme where requested unknown values are predicted (instead of a deterministic computation) from available data based on probabilitistic laws. Whether this is usefel or not depends on the context of application.

A typical natural image signal is composed of deterministic (yet unknown) patterns of textures. Hence the most suitable interpolation form is the classical deterministic ones.

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The main problem is Auto Regressive (AR) / IIR models are based on the idea of something before and after.
While for time series it is well defined, how will you address that in image for a pixel in the middle of the image?

I addressed some ideas on that in my answer to the question Denoising an Image Using Kalman Filter.

Another idea is to work across rows and then across columns.
Then you can build an AR model based on the samples with unit "Time Parameter" and then ask what should happen for $ n / 2 $ or so.
You can even take it farther, and in order to use the 2D information take $ k $ rows / columns at a time and build a vectorized AR model.
Then it means that for each pixel you'll get $ k $ estimations which you could average or chose any other idea to fuse.

Actually it could be nice idea to have a try...

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