I would like to thank the community for providing vigorous feedback on our new imaging technique `ankylography'. We sincerely appreciate the debate and remain confident that our work will withstand further analysis and replication.

I will be presenting comments upon this debate that may not reflect the views of my co-authors as I have not been a member of Prof. Miao's research group for some time and am currently a graduate student in the Department of Applied Physics at Stanford University.

Before offering my comments, I want to clarify that our ankylography paper reports the discovery of a new method for coherent diffraction imaging using single or very few two-dimensional, spherical diffraction patterns to obtain a three-dimensional image. To support this claim, we provided:

1. Numerical evidence. We presented a series of non-trivial reconstructions, including a reasonable noise model, of objects of sufficient complexity to be of scientific interest: a glass nano-particle and a polio virion.

2. Experimental evidence. The experiment we performed was simple, but it verified the experimental feasibility of the principle of ankylography and showed that the technique tolerated noise at experimental levels. Since then, several experimental demonstrations that represent more complex and practical situations of ankylography have been published.

**3. ***Theoretical Analysis. Our theoretical analysis developed a mathematical intuition for the validity of the principle of ankylography. It is important to note that none of the authors are mathematicians and that we did not intend to present a complete mathematical theory of anklography. We accept that there are scaling limitations to ankylography (as we identified in our paper). However, it is our goal and stated aim to work within the limitations to develop a robust imaging technique.*

I will now comment briefly upon the two recent articles that aired some criticisms of ankylography.

1. Fundamental limits of ankylography due to dimensional deficiency by Wei. This paper analyses ankylography in terms of channel capacity. I recommend the vast body of theoretical and numerical work that has been done in compressed sensing for a counter viewpoint on information measures in imaging to those interested. In my view, channel capacity is not the best information measure for ankylography, and there certainly are other measures. Moreover, it is not just the scaling – in the limiting sense – that is important to ankylography: the details of the scaling are vital. That is, the most important case in ankylography is that of relatively small samples; we do not intend to reconstruct objects of size, say, 10^4 in each dimension.

2. Non-uniqueness and instability of ankylography by Wang et al. Our detailed numerical protocols were not followed in this work, and therefore it is not surprising that the authors obtained their disappointing results. Since the paper only appears to reflect the authors' interpretation/implementation of their version of ankylography, it does not diminish our work in any way.

I will now comment upon this news report by Eugenie Samuel Reich. Overall, it is an excellent article, and I commend Nature and the author for their good work. I have three suggestions:

1. It is worth noting that several further experimental demonstrations of ankylography have recently been published:

i. C.-C. Chen, H. Jiang, L. Rong, S. Salha, R. Xu, T. G. Mason and J. Miao. Three-dimensional imaging of a phase object from a single sample orientation using an optical laser. Phys. Rev. B 84, 224104 (2011).

ii. M. D. Seaberg, D. E. Adams, E. L. Townsend, D. A. Raymondson, W. F. Schlotter, Y. Liu, C. S. Menoni, L. Rong, C.-C. Chen, J. Miao, H. C. Kapteyn and M. M. Murnane. Ultrahigh 22 nm resolution coherent diffractive imaging using a desktop 13 nm high harmonic source. Opt. Express 19, 22470-22479 (2011).

The article states that "Miao has since made clear that the technique does not work on objects larger than 15 x 15 x 15 volume pixels, a size dependent on the resolution of the imaging technology". I wish to clarify that that limitation only applies to the simple demonstration code using a simplified algorithm. We do have more complex code, explained in detail in our paper, that will work on substantially larger sized objects. Also, we did not "train" our code on any particular type of structure, and implemented only very general constraints even in the more complex algorithms. So I would suggest replacing "the technique" with "the simple demonstration code" in the news article. By way of example, the source code for the ankylographic reconstruction of a simulated sodium silicate glass structure with 25 × 25 × 25 voxels has been posted at

http://www.physics.ucla.edu/research/imaging/Ankylography. Even at these modest sizes that we obtain with relatively simple reconstruction codes, we have found that there are scientifically interesting samples that can be explored, which is why I am perplexed by, and disagree with, Marchesini's quote in the news report. Perhaps it was not clear that the limitation of 15^3 was only for the very simple matlab demonstration code? In any case, by my analysis, ankylography is quickly advancing upon its promise of becoming a useful imaging tool.

3. The article also states that "Despite the appeal of ankylography, many researchers were perplexed by what they see as a violation of the basic physical principle that you cannot get complete, 3D information from a single flat picture" Ankylography requires a spherical diffraction pattern. In practice, one can obtain a spherical diffraction pattern from a flat detector – if it is large enough – through a mathematical mapping that we describe in detail in our paper. In fact, in our original paper, we indicated that spherical detectors are to be preferred. Additionally, in the case of a flat detector, it would have to be much larger than the sample size. The paragraph continues "In particular, the picture will provide incomplete information about the interior of a subject, and critics argue that many possible 3D structures could generate the same image" In ankylography, the sample under view must be semi-transparent to the illuminating, coherent beam, and we discuss uniqueness in our paper. In my view, only in cases of very large sample sizes will uniques potentially become a problem, but even then only pathologically (i.e. with small probability). So, for a non-specialist, a useful thought experiment might be to consider imaging a transparent grain of rice with a large detector the size, say, of a queen matress at a good distance. Mathematically, the detector would make a measurement that maps onto a sphere in a three-dimensional space, despite the fact that the measurement is made in two-dimensions.

Finally, in response to Marchesini's comment above: I agree that the photon statistics may present a challenge for ankylography in certain applications. For example, the scattering from a single protein will likely be too weak to use ankylography directly (which is why we don't claim this application in our original paper). In our paper, we made precise calculations about the photon statistics and incorporated the noise due to finite photon number into our simulations, using the standard Poisson distribution. However, we wish to limit the debate here to the general feasibility of ankylography assuming that the diffraction pattern with a reasonable signal-to-noise ratio is obtainable.

Kevin, nice answer, but please get a webpage on the interwebs.