Fundamental limits of ‘ankylography’ due to dimensional deficiency

Abstract

Arising from K. S. Raines et al. Nature 463, 214–217 (2010)10.1038/nature08705

Raines et al.¹ propose a method, which they call ‘ankylography’, for three-dimensional structure determination using single-shot diffractive imaging (SSDI). But the conclusion without limitation that the three-dimensional structure of an object is “in principle encoded into a 2D diffraction pattern on the Ewald sphere” and may be inverted by SSDI is inadequately substantiated and conceptually misleading. Here I point out that SSDI in general suffers from a dimensional deficiency that limits the applicability of ankylography to objects that are small-sized in at least one dimension or that are approximately two-dimensional in some other way.

Main

The rate of reliable information transfer via a spatial or temporal channel is fundamentally limited by the channel capacity, which is determined both by the number of degrees of freedom available therein and by the obtainable signal-to-noise ratio². The resolving powers of telescopes or radio antennas and microscopes including SSDI are all limited in much the same manner. A steep (exponential) price in signal power must be paid to obtain data rates or resolutions significantly beyond those supported by the available number of degrees of freedom^2,3. SSDI uses a spatial channel characterized by a linear operator T that projects any real-space amplitude with support onto the Ewald sphere. It is known^2,4 that there exist a pair of orthonormal bases and , called normal modes, and also their associated modal gains , all non-negative and arranged in a nondecreasing order such that , . For any predetermined modal cutoff threshold , is the number of usable normal modes, that is, the number of usable degrees of freedom. It has been rigorously proved⁵ that , which grows more slowly than the number of unknowns in a general three-dimensional object as its size increases. The insufficiency of the number of useable degrees of freedom would persist even if grew exponentially as increased, as long as the exponent grew no faster than : more specifically, so long as , for any fixed small .

I emphasize the fundamental nature of the limitation: that single-shot diffraction does not convey sufficient information about the three-dimensional structure of an object, even if the amplitude (instead of intensity) of the diffracted field is sampled continuously and measured directly with no phase ambiguity. Given a practically obtainable signal-to-noise ratio and a measurement accuracy that together determine a threshold of modal cutoff, any signal in a linear space spanned by normal modes of orders higher than is essentially lost in transmission or attenuated beyond detection. Oversampling and inversion algorithms are irrelevant in this context.

I note that there has been substantial criticism^6,7,8 of ankylography even before its publication¹. However, no agreement seems to have been reached on the fundamental problem of dimensional deficiency in ankylography, and an upper bound for the number of degrees of freedom suggested by one critic⁶ has been disputed and its applicability questioned by Raines et al.^1,7,8, whose response demands a mathematically rigorous proof like the one given here. The present analysis also demonstrates the need to quantify noise and a means of doing so, in order to determine a cutoff threshold based on signal-to-noise ratio, which in turn defines the number of useable degrees of freedom. Although Raines et al.^1,7,8 discuss degrees of freedom, matrix ranks and data sufficiency, they fail to quantify the noise. To say simply that “Poisson noise was added” to the diffraction patterns does not specify how noise addition was implemented in their numerical simulations, which are the only justification of ‘ankylography’ in the absence of a theoretical base. This vagueness about noise makes it difficult to interpret the reported results and to reconcile the apparent contradiction between the authors’ (ref. 1) and the critic’s (ref. 6) simulation results. Raines et al.^1,8 cite “full rank of matrix” and other numerical results to support the claim that a curved surface of detection and oversampling yielded sufficient information for three-dimensional structure inversion, but they have not responded to the critic’s numerical examples and figures showing rapid decays of singular values⁶.

In summary, SSDI of truly three-dimensional structures does not scale. The applicability of ankylography is limited to objects that are small-sized with respect to the wavelength in at least one dimension or have structures that are essentially two-dimensional in complexity. Such may be the case in the computer tests and preliminary experiment of ref. 1. Raines et al.¹ also emphasize certain “physical constraints”, many of which are actually steps of numerical procedures instead of mathematical constraints of model formulation. Incorporating more genuine physical constraints could possibly alleviate the problem of dimensional deficiency. However, that would diminish the generality and appeal of ankylography, and the same feat is arguably achievable by conventional diffractive imaging with a flat image detector.

Methods

With being a two-dimensional Fourier transform and being a linear operator of z integration⁵, the upper bound on follows from the well-known result⁹ that , in conjunction with inequalities , with the second inequality following from an operator inequality¹⁰ .