Using computer technology... it should be possible to use some kind of HUD for bang-bang gun shooting (real good.)

Maybe this:

The gun has a laser (which only need be visible to the camera) that's affixed to the barrel (real good) and pointing in the right direction.

Maybe "at the factory" it is all bolted on and secured permanently, with a certain tolerance. For example, they attach the laser, go out to the factory shooting range and go through the QA processes. The laser needn't be perfectly aligned, as a computer will be available to clean up a certain amount of fuzz. In any case, a bullet's trajectory is a parabola, a photon's trajectory is a much, much, much, much bigger parabola (a straight line if we leave Einstein out of this) so there should probably just be a tolerance: at the range, for these distances from the from a target, the the bullet passed through the bullseye this many millimeters from the laser. Make it good enough, which should be pretty good.

Then you attach a Pretty Damn Good Camera (with fancy optics to taste) and a computer. IMHO these belong on the gun, especially because computers and components are ever better at taking abuse; but they needn't be. They only need to be close to the muzzle and have the thing in need of a new hole in-frame (centeredish).

The computer (via the camera) can see the view of barrel, pointing forward (of course). It has machine learning technology!

When pointing the gun at anything at all, the camera and computer can judge where the bullet would go (what thing it would strike first), just as a human with one eye could do. And just like a human, the computer will get lots of practice judging this, so no big whoop.

The computer also can keep track of the orientation and location of the gun via accelerometers (6 axis?).

• It knows the history of the guns position and movements over time
• It has video frames whose position in time is also known

So as the barrel sweeps around, since (as an example) the computer knows its angular velocity (2D -- the roll axis does not matter here) and at the same time it can measure the rate at which features in the line of fire move (pixels per second).

With a Pythagorean Parallax Processor, the computer can therefore "know" the distance to any one pixel. And it can also see the laser and knows how the laser points relative to the trajectory of the bullet.

It knows distance to target and it knows how far away (orthogonal to laser direction) the path of the bullet is going to be from the path of the laser. To emphasize, this distance is the distance between the hole the bullet will make in the target to the point where the laser beam falls on the target.

[Note: Why is this even mounted to the gun? Why is there a human finger in here? See below.]

[Note: No, I am not some kinda barrel-grabber.]

[Note: Oh, "glasshole"... I was thinking surely the operator wants a HUD and that would look cool on the head of a gun-pumper.]

EC:

• The laser can make some arbitrarily complex pulse pattern that identifies itself to the computer. Maybe protec from jamming.
• There can be 7..10 frickin lasers. Whatever you like. They can form a sort of cone. This may be useful in the wild. The single laser might hight something a mile away, while the bullet is expected to fall enough to strike something much nearer. At least at this one "frame in time" the computer has no hope of knowing this. However the gun may have swept this nearer object three frames ago, so the system can still be aware of what's going to happen and may have some hope of calculating.
• The trigger: There wants to be a nice, soft-press "fire at will" button; not a trigger. You are indicating to the gun, "You may fire in the next l'il it. I want to shoot that thing."
• Actually, there's no good reason (other than safety) to have any of this directly in human hands. Motors and worm gears and tripods... that's what you want. Firing a gun with your finger is just unfeeling annihilation with extra steps.
• An airsoft would be a perfect (and fun) development platform.
• Maybe the lasers, instead of being fixed, should go through some kind of sweep dance. The computer moves the lasers, but still knows all their angles relative to the direction of the barrel. More information. You needn't sweep the gun around randomly.
• A sub-project, and one that risks less distraction: An intelligent laser range finder. It just figures stuff out in 3D, more or less, with an eye on adding ballistics to the mix, which is not hard. Later, add a thing that maps out wind conditions...