As a Roblox developer, it is currently impossible to clip images and have control over what’s visible.
I’m aware of Image Clipping Module by @boatbomber, which might be a solution to cases where you don’t have to move or scale the image (such as player icon clipping).
That method is hacky and can’t be used if you wish to change the display because of the way it works (applies decal over mesh part and shows the mesh part inside a viewport frame).
The reason it doesn’t work with moving images is that Decal doesn’t have properties which allow you to change the scale and positioning of the image itself on the part. It also doesn’t work with surface guis because they don’t clip to the part at all.
Clipping decal on a mesh part:
(Attempt at) clipping surface gui with image label on a mesh part:
What could Roblox do to address this issue:
- Create a way for developers to clip the images
- Add positioning and scaling properties to
Decal - Add a property to SurfaceGuis which makes it clip to the part same way decals do
- Some sort of lens (?) for cameras inside a ViewportFrame
My current use for this would be a custom shape minimap, but I’m sure there are other uses in both building and gui development.
(There doesn’t seem to be a feature request for this, if there is one already, please let me know 
)