Resolution and depth of field

When specifying an imaging based system, one has to deal with resolution requirement : what is the smallest detail we need to acquire ?

Spatial resolution is defined in term of resolving power : the spatial resolution is the minimum distance between distinguishable objects. This resolution is limited by aberration and by diffraction. This last one is due to the fact that the light interferes with itself when crossing a lens.

Airy pattern and Point Spread Function

Let's consider perfect lenses without aberration : the spatial resolution is only limited by the diffraction. This one produces Airy pattern illustrated bellow.

Airy pattern depends on a Bessel function J1 depending on the aperture diameter d.
I(\theta)=I_{0}(\frac{2J_{1}(x)}{x})^2
x=\frac{\pi}{\lambda}d\sin{\theta}
This function has zeros when x ≈ 0,3.8,7.0,10.2
The maximum of the first ring is found for x ≈ 5.1

So the first dark ring occurs for small angles when
\sin{\theta}=1.22\frac{\lambda}{d}
And the first bright ring occurs for small angles when
\sin{\theta}=1.63\frac{\lambda}{d}

Using approximation of sin(θ) as x/f and the f-number f# defined as f# = f/d, the first dark ring is found for

x=1.22 \lambda f\#

For a wavelength of 500nm, f/2 gives 1.22 µm when f/8 gives x = 4.9 µm.

Resolutions in relation with f# and lamda
f# 1.4 2 2.8 4 5.6 8 11 16
blue 400nm 0.7 1.0 1.4 1.9 2.8 3.9 5.5 7.8
green 500nm 0.9 1.2 1.7 2.4 3.4 4.9 6.7 9.8
orange 600nm 1.0 1.5 2.0 2.9 4.1 5.9 8.3 11.7
red 700nm 1.2 1.7 2.4 3.4 4.8 6.8 9.7 13.7

So, Airy partern spreads out when increasing f#.

Point Spread Function

When the lens is not perfect, the Airy pattern becomes the Point Spread Function (PSF) describing the response of an imaging system to a point source.

A point source as imaged by a system with negative (top), zero (center), and positive (bottom) spherical aberration. Images to the left are defocused toward the inside, images on the right toward the outside. (wiki)

Modulation Transfert Function

The PSF gives the system resolution as the Airy disc gives the perfect lens resolution. In practice, the system resolution is measured with the ability to distinguish series of line pair. This is known as the Modulation Transfert Function (MTF) that plots the line pair contrast versus the spatial frequency.

The upper plot displays a sine pattern and a square pattern without and with lens blur. The middle plot displays the luminance of the square pattern. The lower plot displays the contrast versus the line frequencies. (link)

The 10% and 50% MTF give a good idea of the system resolution (sharpness): here MTF is 50% at 61 lp/mm and 10% at 183 lp/mm.

It is possible to measure the "vanishing resolution" corresponding to 5-10% MTF by looking the highest spatial frequency where USAF 1951 bars are visibly distinct.

In this USAF 1951 we the vanishing resolution is reach for the group 8,2.

With the following table we find the corresponding resolution of 287 lp/mm

Number of Line Pairs / mm in USAF Resolving Power Test Target 1951

Group Number

Element -2 -1 0 1 2 3 4 5 6 7 8 9
1 0.250 0.500 1.00 2.00 4.00 8.00 16.00 32.0 64.0 128.0 256.0 512.0
2 0.280 0.561 1.12 2.24 4.49 8.98 17.95 36.0 71.8 144.0 287.0 575.0
3 0.315 0.630 1.26 2.52 5.04 10.10 20.16 40.3 80.6 161.0 323.0 645.0
4 0.353 0.707 1.41 2.83 5.66 11.30 22.62 45.3 90.5 181.0 362.0 —–
5 0.397 0.793 1.59 3.17 6.35 12.70 25.39 50.8 102.0 203.0 406.0 —–
6 0.445 0.891 1.78 3.56 7.13 14.30 28.50 57.0 114.0 228.0 456.0 —–

Depth of field

When considering resolution, a majority of the emphasis is placed on point-to-point lateral resolution in the plane perpendicular to the optical axis. Another important aspect to resolution is the axial (or longitudinal) resolving power of an objective, which is measured parallel to the optical axis and is most often referred to as depth of field.

Depth of field is computed in terms of angular blur as :

DOF=\delta min + \delta plus

δmin and δmax are illustrated bellow and solved using similar triangle:

\omega=\frac{Bf}{D}=\frac{Bp}{D+\delta plus}=\frac{Bm}{D-\delta min} \frac{A}{D}=\frac{Bp}{\delta plus}=\frac{Bm}{\delta min} \frac{A}{D}=\frac{(D+\delta plus).\omega}{\delta plus}=\frac{(D-\delta min).\omega}{\delta min} \delta min=\frac{D^2\omega}{A+D\omega}; \delta plus=\frac{D^2\omega}{A-D\omega}

it gives : (see edmund optics)

\delta min + \delta plus = \frac{D^2\omega.2A}{A^2-D^2\omega^2}\approx \frac{2.\omega.D^2}{A} = 2.f\#_{obj}.Bf

Choosing the resolution to be one pixel width (Wpix)

DOF \approx 2.\frac{f\#}{magn}.Bf= 2.\frac{f\#}{magn^2}.W_{pix} DOF \approx \frac{2}{1.2 \lambda}.\frac{W_{pix}^2}{magn^2}

Thus for a given pixel size and magnification, the DOF is only related to the f#.

For instance, Xavier who studies carbonates has a magnification of 1 with MCT pixels of 30 µm.

What is his f*cking DOF for wavelengths of 1000 and 2500 nanometers :

(2 * 30 X 30) / (1.2 x 1 x 1) = 1.5 mm and (2 * 30 X 30) / (1.2 x 2.5 x 1) = 0.6 mm

References :

Wikipedia : Airy diskPoint spread function, MTF, USAF1951, DOFMaster

 

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