Using 4/8 Neighboring Algorithm To Remove Undesired Pixels

Hello Guys. I am working on a project in which after the conversion of image into binary, I have to remove the undesired white spots,patches, technically saying Pixels. Actually i want to left some certain places having white pixels and the remaining one's should be wipe out.

I have also uploaded an image to make things more obvious. The bottom having yellow bounded rectangular region should remain as it is,while all other white spots,pixels should be removed; turned into black. My supervisor suggest me to go through neighboring algorithm. I'll give some specific position to the image,and apply some neighboring technique to remove the undesired white pixels around it and so on. I have also go through IXNEIGHBOUR in Matlab but it didn't help me.I am hoping you guys will help me to get through this

Thank you in Advance,waiting for your response. Cheers.