Understanding Spatial Resolution

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Basic Concepts

Spatial Resolution

Spatial resolution refers to the minimum distance at which two nearby points can still be distinguished as separate points.

For example, if the spatial resolution is:

• 10 μm, two points at least about 10 μm apart can be distinguished.

• 1 μm, two points at least about 1 μm apart can be distinguished.

In other words, the smaller the spatial resolution value, the closer two points can be while still being distinguishable — and this generally indicates better performance.

Spatial Frequency

Just as a signal that repeats over time is expressed as a frequency (Hz),
a pattern that repeats over position in space is expressed as spatial frequency.

For example, consider the following black-and-white pattern.

████░░░░████░░░░

Bright and dark regions repeat.

Spatial frequency describes how many times this repetition occurs within a unit length.

The unit is lp/mm (Line Pair per Millimeter), where a line pair is one bright line plus one dark line.

The higher the spatial frequency, the finer the structure it represents.

For example, 100 lp/mm means that one line pair has a period of 10 μm.

So if an optical system can resolve 100 lp/mm, it can be interpreted as being able to distinguish a repeating pattern with a period of about 10 μm.

MTF (Modulation Transfer Function)

MTF is a metric that indicates how well an optical system transfers the contrast of each spatial frequency.

For example, suppose the original pattern is
████░░░░████░░░░
a sharp black-and-white structure like this.
Its contrast is 100%.

However, after passing through an optical system, effects such as diffraction, aberration, and defocus can
███▓▒░░░███▓▒░░░

blur it like this.
In this case, the difference between the bright and dark regions decreases.
In other words, the structure still exists, but the contrast is reduced, making it harder to distinguish.

MTF quantifies this ability to transfer contrast.

Spatial frequency (lp/mm)	MTF	Interpretation
100	0.95	A 100 lp/mm pattern retains 95% of its original contrast
500	0.55	A 500 lp/mm pattern retains 55% of its original contrast
900	0.10	A 900 lp/mm pattern retains only 10% of its original contrast

In general, MTF decreases as spatial frequency increases.

Because the spacing between bright and dark regions is very narrow, even a small amount of spreading in each line causes it to overlap with the adjacent line.

Cutoff Frequency

The cutoff frequency is the spatial frequency beyond which an optical system can no longer transfer any information.

In other words, structures finer than this can no longer be distinguished in the image.

MTF

1.0 |\
    | \
    |  \
    |   \
0.0 +----\-------------------->
         fc         spatial frequency (lp/mm)

fc is the cutoff frequency.

Resolution Design Process

1. Define Sample Characteristics and Goals

First, determine the size of the observation target.

• Capillary width: 5–20 μm

• Cell diameter: 10–50 μm

• Scratch width: 20 μm

Then define what needs to be analyzed.

• Presence or absence of the target

• Shape and pattern

• Precise quantitative measurement

2. Determine Sensor Sampling (Resolution) Size

For example, suppose you want to precisely measure the width of a capillary.

To measure a 10 μm vessel, you might target a sampling interval of about 0.5 μm/pixel.

Without error, measurement down to about 1 μm/pixel would also be possible, but real systems are affected by diffraction, optical aberration, noise, defocus, and so on,

so a gap always exists between the theoretical resolution and the resolution actually achievable in measurement.

At this stage, the initial design direction — sensor, optical magnification, field of view (FOV), etc. — is determined based on the target sampling interval.

3. Calculate the Nyquist Frequency

Once the sampling interval is determined, the maximum representable spatial frequency can be calculated.

Nyquist frequency formula

f_Nyquist = 1 / (2p)\

p: sampling interval (μm/pixel)

For example, at 0.5 μm/pixel,

f_Nyquist = 1 / (2 × 0.5 μm) = 1 lp/μm = 1000 lp/mm

This represents the maximum spatial frequency representable when considering sensor sampling alone.

4. Evaluate the Actual Resolution of the Optical System

Spatial frequency (lp/mm)	MTF
100	0.95
300	0.80
500	0.55
700	0.30
900	0.10
1000	0.00 ~ 0.05

The sensor can theoretically represent up to 1000 lp/mm, but the optical system loses contrast as it moves toward higher frequencies.
In the example above, almost no valid information remains at 1000 lp/mm.

The important point is that MTF does not need to be 1 all the way up to the Nyquist frequency.

In real optical systems, MTF decreases as spatial frequency increases due to effects such as diffraction and aberration.

So what matters is confirming whether sufficient MTF is achieved at the target spatial frequency.

Factors That Limit Resolution

Diffraction

No matter how perfect a lens is made, diffraction due to the wave nature of light is always present.

After passing through a lens, light does not converge to an ideal point but spreads out into an Airy disk.

In other words, a single point source appears not as a dot (•) but spread out like a ring (⊙).

Optical Aberration

In an ideal optical system where only diffraction exists, resolution is determined solely by the diffraction limit.
However, real lenses are not ideal optical systems, so various optical aberrations occur.

Optical aberration refers to the phenomenon where light passing through an optical system fails to converge exactly at the ideal position, causing the image to blur or distort.

Common types of aberration include the following:\

• Spherical Aberration\
• Coma\
• Astigmatism\
• Field Curvature\
• Distortion
  When aberration is present, a point source spreads out larger than the Airy disk or becomes asymmetric.

Defocus

If focus is not exactly right, light passing through the lens fails to converge to a single point on the sensor and instead spreads over a certain area.
As a result, the information from a single point is distributed across multiple pixels, and neighboring information mixes together, blurring the image.

For example, an original pattern like
█░█░█░█░
can appear blurred like
▓▒▓▒▓▒▓▒.

Optical aberration is a phenomenon in which light fails to converge exactly to a point due to the lens’s own limitations, while
defocus is a phenomenon in which the position where light converges to a point (the focal plane) does not coincide with the sensor position.

Closing Thoughts

People are usually interested in “how many megapixels is the resolution?”, but
actual measurement performance is not determined by pixel count alone.

What matters is whether the target of interest can be distinguished and measured with the required level of accuracy.

To do this, you first define the sample size and the target of analysis, and then determine the sampling interval, FOV, optical magnification, and so on accordingly.

After that, you comprehensively review the factors that affect actual resolution — the Nyquist limit, MTF, optical aberration, defocus, and so on.

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