[3dem] SIRT, ART (and SART)

Gabor Herman gabortherman at yahoo.com
Mon Oct 4 08:03:26 PDT 2010

We have read the October 1 message of Marin van
Heel. While it is written with an admittedly good
purpose, it ignores what has been going on for the
last four decades in the field of Inverse Problems (of
which Image Reconstruction from Projections is a part)
and makes some dogmatic declarations that are plain
wrong. Going through that message point-by-point in
this forum seems not to be a particularly useful exercise,
but we thought it appropriate to alert the 3dem community of
the incorrectness in our opinion of the declarations in it.

A basic philosophical difference between Marin van Heel
and ourselves is that we do not share his apparent belief
that if you do not do things in the way that he likes to
do them then what you are doing is "clouded by a poor
understanding", "lacks rigour" and has "unstable foundations".
On the contrary, we believe that there are many ways to
skin a cat and one should keep an open mind as to which is
the best way under a particular set of circumstances.
In particular, the projection data that are provided
by electron microscopy are unavoidably imperfect from
the mathematical point of view (information regarding
line integrals is available only for some lines and,
even where available, it is corrupted by noise). Hence
no reconstruction approach can guarantee perfect
recovery from such data of the object being imaged,
all methods (however they are motivated and justified)
will provide only an estimate of it. The usefulness of
the estimates under a particular set of conditions needs
experimental evaluation rather than blind faith in
the universal correctness of a particular approach.

The attitude expressed in the last paragraph is
exemplified in our book chapter (which, in particular,
provides some experimental illustrations of the
superiority of ART to WBP for recovering some biological
structures from electron microscopy data):
Jose-Maria Carazo, Gabor T. Herman,
Carlos O. S. Sorzano and Roberto Marabini,
Algorithms for Three-dimensional Reconstruction From
the Imperfect Projection Data Provided by Electron
In: Joachim Frank, Electron Tomography: Methods for
Three-Dimensional Visualization of Structures
in the Cell, Second Edition, Springer, 2006.

Our whole approach to Image Reconstruction from
Projections is presented in detail in the book:
Gabor T. Herman, Fundamentals of Computerized
Tomography: Image Reconstruction from Projections,
Second Edition, Springer, 2009.

Details of the mathematical aspects of both ART-type
and SIRT-type algorithms can be found in the book:
Yair Censor and Stavros A. Zenios, Parallel
Optimization: Theory, Algorithms, and Applications,
Oxford University Press, New York, 1997.

This message is being submitted jointly by

Jose Maria Carazo, Ph.D.
Head of the Biocomputing Unit
National Center for Biotechnology, CNB-CSIC, Madrid, Spain

Yair Censor, D.Sc.
Professor of Mathematics
University of Haifa, Haifa, Israel

Gabor T. Herman, Ph.D.
Distinguished Professor of Computer Science
The Graduate Center of the City University of New York, USA


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