In the back of my mind I thought there was some connection ... but put the link anyway jsut in case. Sorry about that, Bob. Gave you another plug anyway
Hope I don't step on toes again cause I believe you basically know what you're talking about with the DPI thing. But this has always been a pet peeve of mine. My past 15 years out of the hobby I spent making 3D meshes and creating 2D images from them. The quality of a 2D image file, like a scan, photo ... a jpeg file ... can be described 2 ways. The one typically used for display on a computer monitor is the resolution in pixels, like your example of 1581x2049, 1581 dots X 2049 dots, if you will.. This completely describes "quality" or resolution.
When printing, the resolution is usually described using 3 variables ... DPI X width X height. The width and height are usually in inches when using Dots Per Inch as the first measure. In the example above for monitors, the 1581x2049 file can be described as 150DPI x 10.54" x 13.66" or, 72 DPI x 21.958" x 28.458". All three of these measurements describe the exact same file. The the only point I was addressing is that you could have a 72 DPI file that has much higher resolution than a 300 DPI file.
Your scannners resolution of 1200 dpi by 1200 dpi does fully describe the resolution because it is based on the maximum area of your scanning bed. But without knowing those dimensions, I can not say what your maximum resolution would be. If it could only scan a 1" by 1" area (i-scan? .. ehhe) you could only get a 1200 x 1200 image from it.
Hope this made sense and it was only meant FYI.