Minimum number of pixels of the image sensor to identify an object?

[eitan77]

Homework Statement

Assume that the camera Field Of View is U, the lens focal length is F and the number of pixels of the detector (image sensor) is N. The camera is observing the objects at a very far distance. Assume that you want to identify an object, whose size is 1% of U. What is the minimum number (approximately) of the pixels of the detector you need for that?

I am new in this field and would appreciate it if you could help me understand how to get to the answer (My solution seems illogical to me)

Note: if it matters this is a theoretical question and not for an actual device.

Relevant Equations

$\theta =2arctan(d/2F)$
$\theta = 2arctan(U/2D)$
$pixels size= (object size)/N$

d- the image sensor size
D- the distance between the object and the lens

d is unknown and D is considered very big.

$d/2F = U/2D$
$d'/2F = 0.01U/2D$
d' - the size of the object's reflection on the image sensor
$pixels size = d/N$

since we want N to be minimal, the pixel size should be maximal: pixels size = d'

hence : $N = d/(pixels size) = d/d' = 100$
My final answer does not depend on U & F which seems strange to me.

 

[DaveC426913]

Doesn't it kind of matter what the object is, and what the context is?

It seems to me that identifying an orange in a desert might require fewer pixels than identifying the "R" variant of an F-4J against the canopy of a jungle.

 

[eitan77]

Doesn't it kind of matter what the object is, and what the context is?

It seems to me that identifying an orange in a desert might require fewer pixels than identifying the "R" variant of an F-4J against the canopy of a jungle.

This is not specified in the task, I guess it can be assumed that this is the simplest case that can be thought of.

 

[Baluncore]

I think this problem is a question of signal detection in noise.

There is only a probability, no certainty, that the object is present, where it may have been detected.

The variability of the target object being searched for, reduces the certainty of the detection.

A probability threshold for detection must be decided based on experience.

 

[hutchphd]

This is very much underspecified. What is the criterion for "recognizing" the object?? All you have written is that the angle subtended by the image and the object (from the lens center) will be the same. You need to specify that minimum required angle somehow. Then the pixels required can be calculated because you know the magnification of the lens when you know
I believe you have calculated N such that the image of the object will fill exactly one pixel....but even that is not clear to me.

 

[eitan77]

I believe you have calculated N such that the image of the object will fill exactly one pixel....but even that is not clear to me.

That's what I did.

This is very much underspecified. What is the criterion for "recognizing" the object?? All you have written is that the angle subtended by the image and the object (from the lens center) will be the same. You need to specify that minimum required angle somehow. Then the pixels required can be calculated because you know the magnification of the lens when you know

the minimum requierd angle is not $arctan (d'/2F)$ ?

 

[hutchphd]

What exactly does "recognized" mean ? If the object is Mickey Mouse, for instance, do we need to be sure it is not Minnie Mouse instead? That changes the criterion.
Also, in the real world, there may be limits imposed by diffraction and lens aberrations. Additionally the visual contrast of Mickey relative to the background will be important.

 

[hutchphd]

Actually the issue of contrast brings along another question: the worst case scenario in your simple case is that the square object images equally onto the corners of four adjacent pixels. The change in response of any one pixel (caused by the object) will then be smaller by a factor of four. Is it good enough? Not enough criteria presented.
So what you did is not wrong, but far too simple to be of practical utility as I hope is clear to you from all the answers. You did warn that it was strictly a theoretical exercise!!

 

[Baluncore]

Shannon says you must sample at twice the spatial frequency of the object you are looking for. The aperture, in square pixels, must then be 4 times the target object area.