Ideas on how to build a line sensor
There are tons of line scan solutions available, all the way from cameras using cameralink interfaces and with ethernet interfaces for $6K to sensors from about 8 different manufacturers.

However, since you are mentioning Arduino, I'll assume that:

1) This is a one off project - there is no consideration for designing something

2) You want to spend as little as possible.

Typically the best (least expensive) result will be to use what is know as a CIS (Contact image sensor) module, which is the lowest cost that you'll be able to get.  The best place to get these is to buy a very inexpensive scanner and tear out the guts and repurposing it, it would have a USB interface to your Arduino then.  This would have the lights and everything you need.

The fundamental problem you have from your drawing is that there is no optics.  Even the scanner CIS module has some lenslets in from of the array but they only image very close to the array itself.  with the drawing as shown your just going to get a blur.  Perhaps it would work to get a scanner that has some lenses in it.  Some of the higher end ones do not use CIS modules but use lenses and CCD's.

The other alternative is to use a smaller section of the CIS array and to put it behind a SLR lens (35 mm - tons of used ones available) . Most machine/industrial imaging solutions tend to use similar lens types as those that are used for SLR, simply because of availability.  If you use a lens solution yo will have to have a lot higher stand off (distance of lens to object).