We have a number of line-art charts unfortunately saved as JPEGs. They are now riddled with distracting compression artifacts or "speckles". Is there any way of removing these? I do not have the original files and it will be very difficult to recreate them.

I am running Windows 7 and tried Paint.Net; none of the filters help. Posterize washed out all the colors and leaves the speckles. Blur makes text unreadable. Noise Reduction wrecks antialiasing of curved lines, and perversely enhances the speckles, making them look like checkerboards.

Yes, I have Googled for software to do this; there are many programs that advertise despeckling but, after my experience with Paint.Net, do not want to experiment with applications that show no before and after images. The only example I have seen that does what I want is from a Photoshop tutorial. I have dozens of files and the tutorial requires considerable manual fine-tuning. I would prefer to automate or batch-process this task. Commercial apps are fine, but I do not want to spend over $600 and learning a complex program for a single task.