This is a two part question as I think they go hand-in-hand.
Is there a way to determine the minimal set of sensors needed for any given exercise? For example, say one needs to detect specific objects, they start off assuming they might need a camera (more than one?), maybe a lidar, maybe a radar, etc. How does one quantify a minimal set of sensors needed to accomplish this task? Initially I'm thinking that it has something to do with the confidence one has in the detection, but I'm trying to quantify that.
Other question: Once a minimal set of sensors has been determined, how does one quantify the improvement gained in the detection by adding another sensor (could be same type, could be different)?
Can you clarify your question? It is perhaps too general? Could you give an example?
Eliguisz, I will attempt to clarify. Say there's a vehicle that needs to operate in some environment and we wish this vehicle to operate autonomously. And to be more specific, say it's a car and we want it to go down a particular stretch of road all by itself (driverless). Is there a minimal set of sensor inputs needed to accomplish this? My initial thoughts are it would depend on the environment around the road (urban, weather, etc) and the capabilities of the car. I'm wondering if this can be quantified?
The types of sensors in a project depend on the types of parameters that you intend to measure. For example, if speed, weight, accelerate are what you want, at least two types of sensors are needed, pressure sensor and accelerator. Because speed can be calculated from acceleration. However, if your budget allows you to get a specific sensor for speed, you can save computational load. It is very hard to answer your general question in a precious way. But I believe a careful check of your project model will help your decision.
Look for solution in nature. We have two eyes instead of one to see things - well except for some insects and other (they have more than two) . We have two ears instead of one. And the reason is not some fancy term ' stereo' but instead a pair will do a job of three or four sensors. Also position of sensors matters too. For example, if I am building a device to trace a sun (to position a solar panel) all I need is a pair of solar panels separated by a wall in between. Therefore if solar panel on left receives more light than on the right, computer then realizes that the sun has moved to the left. In this example, I do not need a sophistication of cameras or other tracking devices.
Google Driverless Car is being tested now. Please search in Google.The following Videos may give you an idea of number of sensor eyes and placement.
http://www.ted.com/talks/sebastian_thrun_google_s_driverless_car.html
http://www.forbes.com/sites/joannmuller/2013/03/21/no-hands-no-feet-my-unnerving-ride-in-googles-driverless-car/
If it is to be a single sensor(?) then it may have to be like 3 D film camera and probably rotatable over sufficient spherical angle range and sensing a distant object sufficiently in advance to give leeway to take decisions and implement in time to avoid crashes.
P.S.
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