Anti-Filter in Image Processing?
What you're describing is call deconvolution. It's a concept frequently used for distorted images and for equalization in communications. As such you should be able to find a number of resources for your specific application. In general, the original image may not be recoverable exactly (it often isn't).

However, you can do a pretty good job depending on how severe the image has been undesirably filtered. In addition, many methods will attempt deconvolution by estimating the filter which it appears the signal has been affected by. If the assumption that the signal was convolved with a linear filter is a poor one, your performance may be severely inhibited.