[3dem] Re: SIRT, ART (and SART) (Ozan Oktem)
Ozan Öktem
ozan at oktem.se
Wed Sep 22 15:18:29 PDT 2010
Dear Mike,
On 20/09/2010 22:43, Mike Marsh wrote:
> Hi all,
>
> Does everyone in the community mean the same thing when they say SIRT
> or ART? Or do different research groups mean slightly different
> things?
>
> I once heard an engineer for FEI describe SIRT and ART reconstruction
> techniques in a talk, but his definitions differed from my how I
> learned and understand them (as described in Kak and Slaney).
>
> So are these terms somewhat ambiguous or are there
> community-accepted definitions for them? If so, can you recommend an
> appropriate review article?
>
> Thanks in advance,
> Mike Marsh
I don't want to enter into a unnecessary discussion but there are
apparently claims being made about the iterative methods SIRT and ART
and transform methods (such as weighted backprojection) that are
clearly wrong.
First of all, neither ART or SIRT are not invented by the EM
community. ART is simply the well-known Kaczmarz method in mathematics
for solving (over- or underdetermined) linear systems and was
described by Kaczmarz in 1937 [1]. Similarly, SIRT is simply the
Landweber iteration. Convergence criteria for both methods are by now
well understood for many imaging modalities.
To understand these methods, we need to involve some degree of math,
sorry for that. Let "x" denote the 3D function representing the
specimen that we seek to study and "y" is the 2D images that one
collects in the microscope. Assume now that the data "y" is related to
"x" by a linear transformation which we call "P". For electron
microscopy, "P" is in the simples case a projection, but one can also
include optics, detector response, etc. Now, the image formation is
expressible as "P x = y" which is simply an equation for the image
formation where the linear transformation "P" is known and does not
depend on the specimen. Hence, to simulate an image, just plug-in "x"
and calculate "y" (data). Analogously, to recover "x" (the 3D
structure of the specimen), simply solve the above equation for "x"
from a given data "y". Now, life is of course not that easy! First,
the above system is very often under-determined (more unknowns that
equations) and data "y" is noisy. Hence, very often the above equation
has no solutions and this is also the case for EM imaging. The normal
procedure ahead is to relax the notion of a solution and talk about
least-squares solutions instead. Compare this with how one fits a line
through a point set. Often there is no line that passes through all
points, so one has to settle for the best one which is the one that
minimises the residual errors (least-squares solution).
Like most other iterative regularisation methods, both ART and SIRT
are simply methods that create a sequence of "x"'s that in the limit
converges to a least squares solution. For ill-posed problems, such as
those we have in EM, this is not desirable. It is like overfitting.
Hence, ART and SIRT became useful in EM once they were combined with
early stopping, i.e. the iterations are stopped before convergence.
Both methods are expected to give similar quality of reconstruction
but they may have different requirements on computational resources.
Transform based methods on the other hand try to provide an analytic
expression for the inverse of "P". To claim that "Such iterative
algorithms, are non-linear and generally perform worse (both in
reconstruction quality and in terms of their computational speed) than
exact filter algorithms (George Harauz and Marin van Heel, Exact
filters for general geometry three dimensional reconstruction, /Optik/
73 (1986) 146-156; The same algorithm was published by Radermacher
and co-workers)" is strange. In fact, transform based methods will at
best (such as the "exact filter algorithms" mentioned above) give you
a least squares solution, just as ART and SIRT. In that sense, the
wording "exact" is strange. Hence, they will have to be stabilised
just as ART and SIRT if they are to be useful.
In general it is not to be expected that any of these methods are
superior to the other in terms of image quality. Transform based
methods are computationally more efficient. On the other hand, they
are specifically tailored to the particular "P" and the measurement
geometry since they are based on analytically inverting "P". The "P"
they can handle is when "P" represents a projection. ART and SIRT work
for all linear "P" so once can include more accurate physics models,
such as optics CTF and detector MTF, and more measurement geometries.
Finally, to quantify what is meant by a better reconstruction is far
from trivial. It is naive to claim that the Fourier Shell Correlation
FSC is the only way to quantify image quality and i think this needs
to be stopped. In fact, ART (with and without blobs) has been tested
rather rigorously against weighted backprojection using an array of
quantifiers (called figure of merits) by Carazo and co-workers. Their
comparison is way more objective and statistically sound than just
comparing FSC numbers, see [3] for details. In their comparison it is
clear that ART does outperform weighted backprojection. A problem is
that all these approaches are based on comparing against simulated data.
A good reference is actually to be found outside the EM community. Kak
and Slaney are good but I would recommend section 5.3.1 in [2].
[1] S. Kaczmarz (1937), Angenaherte Auflosung von Systemen linearer
Gleichungen, Bull. Acad. Polon. Sci. Lett. A35, 355-357.
[2] F. Natterer and F. Wubelling, Mathematical Methods in Image
Reconstruction, SIAM, 2001.
[3] Carazo J-M, Herman G T, Sorzano C O S and Marabini R 2006
Algorithms for three-dimensional reconstruction from the imperfect
projection data provided by electron microscopy Electron Tomography:
Methods for Three-Dimensional Visualization of Structures in the Cell
2nd edn ed J Frank (Berlin: Springer) chapter 7, pp 217–43.
I hope this was helpful,
Best regards,
Ozan
--
PhD. Ozan Öktem
CIAM: Center for Industrial and Applied Mathematics
Department of Mathematics
KTH - Royal Institute of Technology
SE-100 44 Stockholm, Sweden
Phone: +46-8-790 66 06
Fax: +46-8-22 53 20
Mobile: +46-733-52 21 85
E-mail: ozan at kth.se
Web: http://www.ciam.kth.se
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