The image editor does not keep information of the file name that has been imported.
However, since the only way to load images through the UI is using the OpenImageCommand, you can implement your OpenImageCommand, similar to the default one, which keeps track of the name of the file that is being opened. You can change the template of the editor to use your command instead of the default one.
This approach has been described in detail about the save command of the image editor in this forum post.
All the best,
the Telerik team
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