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: 


[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:



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 
Applied directly to the image of interest.
•  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.