We need the expectation here, because again f is realization

of a random field, a two-dimensional random process.

So we observe an image from this collection, or

ensemble of images, but we want the estimation error

to be minimized not just for the image we

observed, but for all the images in this ensemble.

So next time, another image from this ensemble is provided to us to be restored.

We are guaranteed that we can provide the restoration

that will result in the smallest possible restoration error.

The additional requirement imposed by the

Wiener Filter is that this restoration filter

should be, is required, is desired to be a linear, especially in variant filter.

So, in other words, the restored image, f-hat, will be the convolution of the

impulses parts of the restoration field there, r, i, j, with the available data.

So pictorially, here is the degradation restoration system, right?

The original image goes through the degradation system,

noise is added, y is the observed image,.

We want to operate on need with restoration filter with inputs r(i,

j) so that we obtain an estimate of the original image here, so

that the error between f and f hat, our estimate, the absolute

value of this squared in the expected sense is the smallest possible one.