AMMPSF

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:

[1] P.N.H. Nakashima and A.W.S. Johnson, Ultramicroscopy 94 (2003), 135-148

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:

[2] R. Erni, M.D. Rossell and P.N.H. Nakashima, Ultramicroscopy 110 (2010), 151-161.

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.

Go to AMMPSF software download page.

Philip N. H. Nakashima