[3dem] Re: SIRT, ART (and SART) (Ozan Oktem)

[pawel]

Dear Ozan, 

thank you very much for a compact but nevertheless clear introduction into the subject.  It is good to clarify
some of the issues still clouding the EM field from its early days.

My take on reconstruction from projections will soon appear in the special issue of 
Methods in Enzymology, Vol 482, edited by Grant Jensen, and is titled
Fundamentals of three-dimensional reconstruction from projections.

Regards,
Pawel A. Penczek, Ph.D.
Professor
Department of Biochemistry and Molecular Biology
The University of Texas - Houston Medical School
MSB 6.220
6431 Fannin
Houston, TX 77030
USA

phone: 713-500-5416
fax: 713-500-6297
pawel.a.penczek at uth.tmc.edu
http://www.uth.tmc.edu/bmb/faculty/pawel-penczek.html




On Sep 22, 2010, at 5:18 PM, Ozan Öktem wrote:

> Dear Mike,
> 
> 
> 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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