Philip N. H. Nakashima

Department of Materials Science & Engineering

Monash University, Victoria, Australia 

The Aperture Method for Measuring Point Spread Functions (AMMPSF)

The AMMPSF software is designed to measure accurately a Point Spread Function (PSF) / Modulation Transfer Function (MTF). The AMMPSF software is written in C/C++ and is based on the Digital Micrograph Script described in detail in: 

AMMPSF makes use of Nakashima and Johnson's "aperture method" for PSF/MTF measurement [1] and is applicable to any digital imaging system. Only one minor modification to the original algorithm detailed in [1] was made to arrive at the present AMMPSF software. This modification is detailed in:

A comparison of pros and cons of published PSF/MTF measurement techniques

There are numerous advantages to the AMMPSF approach over others that have been presented in the literature (for summaries, see [1] and [2]). The table below illustrates the differences. 

Technique

Example

Pros

Cons

Blind Deconvolution




a)

Richardson-Lucy Algorithm

Richardson-Lucy Algorithm

  • Requires Poisson noise to be accurate (noise is rarely Poisson in digital imaging systems).
  • Tends to underestimate the PSF.
  • Requires Poisson noise to be accurate (noise is rarely Poisson in digital imaging systems).
  • Tends to underestimate the PSF.

Stochastic Input 




a)

The Noise Method

Only requires uniform illumination

  • Assumes all noise comes only from the input signal.
  • Almost always underestimates the PSF.

b)

Amorphous Film Imaging (TEM)

Only requires uniform illumination

  • PSF-free contrast is generally unknown.
  • Requires several images at different magnifications to establish true contrast.
  • Can be strongly affected by noise. PSF-free contrast is generally unknown.
  • Requires several images at different magnifications to establish true contrast.
  • Can be strongly affected by noise.

Deterministic Input




a)

Slit/Line Methods

  • Very accurate.
  • Sub-pixel oversampling.
  • Robust in the presence of noise.

  • Slits must be very carefully machined with edge roughness well below the pixel dimension and have precise geometries.
  • Lines formed with slits or otherwise must be straight or have well-defined geometries to allow oversampling.
  • Lines formed with slits or otherwise must have breadths smaller than the pixel dimension. • Slits must be very carefully machined with edge roughness well below the pixel dimension and have precise geometries.
  • Lines formed with slits or otherwise must be straight or have well-defined geometries to allow oversampling.
  • Lines formed with slits or otherwise must have breadths smaller than the pixel dimension.

b)

The Point Source Method

  • Highly accurate.
  • No post-processing required.
  • Robust in the presence of noise.

  • Highly localised PSF measurement may not be representative of the whole detector.
  • Illumination must have sub-pixel dimension.
  • Preventing saturation can be difficult.
  • Asymmetry introduced if point illumination is not centred in a pixel.

c)

Periodic Intensity Methods

(incl. Gratings and Holographic Fringes)

  • Potentially very accurate.
  • Requires only uniform illumination.

  • Gratings require very precise manufacture.
  • Holography requires specialised optics.
  • PSF-free contrast is generally unknown.
  • Requires several images at different magnifications to establish true contrast.
  • Can be strongly affected by noise.

d)

The Knife-Edge Method

  • Very accurate.
  • Only requires uniform illumination.
  • Robust in the presence of noise.

  • The edge must be manufactured with surface roughness well below the pixel dimension.
  • If not perfectly straight, the edge must have a very well-defined geometry to allow oversampling.
  • Must invoke an Abel transform to go from a line spread function (LSF) to a PSF.
  • Only samples the PSF in one direction via the LSF.

e)

The Aperture Method (AMMPSF)

  • <1% error (very accurate).
  • Samples the PSF in all directions and across a large area of the detector.
  • Only requires uniform illumination.
  • Precise knowledge of aperture geometry unnecessary.
  • Analysis is fully automated via the AMMPSF software.
  • Robust in the presence of noise.

  • Strongly affected by aperture image de-focus.
  • Aperture surface roughness must be significantly smaller than the pixel size for effective sub-pixel aperture shape determination.

How to use AMMPSF

1. Collect a focused image of an aperture

A: Image of an aperture which is not perfectly circular and has some arbitrary geometry.

B: Expanded view of the right edge with an intensity profile locus crossing it (see C).

C: The intensity gradient about the edge is symmetric indicating the image is in focus - see instructions with the download.

2. AMMPSF determines the aperture shape to sub-pixel resolution

A: The "top hat" function representing the idealised intensity through the aperture.

B: Expanded view of the right edge with an intensity profile locus crossing it (see C).

C: Intensity profile showing the step between 0 and maximum intensity at the partially exposed pixel at the edge returned from the sub-pixel shape determination.

3. Deconvolution of the input image with the aperture shape gives the PSF. 

The PSF is radially averaged to reduce noise.

A: The peak region of the radially averaged PSF returned after IFFT(FFT(1A)/FFT(2A)).

i.e. deconvolution of the as-captured aperture image in 1A by the aperture shape determined by AMMPSF in 2A returns a PSF which is smoothed by radial averaging. The centre 64x64 pixels from the present example (2048x2048) are shown here. The locus corresponds to B.

B: The intensity profile across the PSF peak along the locus shown in A.

C: A surface plot of the 64x64 pixel region shown in A.