[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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