Fundamental Limits of "Ankylography" due to Dimensional Deficiency

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by  Haiqing Wei

Fundamental Limits of “Ankylography” due to Dimensional Deficiency
Haiqing Wei
oLambda, Inc., Los Altos, CA 94022
E-mail: [email protected]
Single-shot diffractive imaging of truly 3D structures suffers from a dimensional deficiency and
does not scale. The applicability of “ankylography” is limited to objects that are small-sized in at
least one dimension or that are essentially 2D otherwise.
Raines et al. [1] proposes a method, dubbed “anky- no faster than l3/2, more specifically, logǫ =O(l1.5−δ),
0 | |
1 lography”,forthree-dimensional(3D)structuredetermi- for any fixed small δ >0.
0 nation using single-shot diffractive imaging (SSDI). But By considering the distribution of received energy be-
2 the conclusion without limitation that the 3D structure tween two hemispheres, it is easy to see that N(T,ǫ)
n of an object is “in principle encoded into a 2D (two- N(T+,ǫ/2) + N(T−,ǫ/2) 2N(T(cid:3),ǫ/2), where T±≤:
a dimensional) diffraction pattern on the Ewald sphere” L2([0,l]3) L2(S ) and≤T : L2([0,l]3) L2(S )
± (cid:3) (cid:3)
J and may be inverted by SSDI is inadequately substan- are restrict→ions and extension of T to the corr→esponding

6 tiated and conceptually misleading. Here we point out codomainsS = (f,g,h):f2+g2+h2 =1, h≷0 and
±
2 thatSSDIingeneralsuffersfromadimensionaldeficiency S = (f,g): f { 1, g 1 =[ 1,1]2. T =T}F F
(cid:3) { | |≤ | |≤ } − (cid:3) z y x
in using a 2D observation with a quadratically growing is a product of three linear operators,with
]
s number of degrees of freedom (NDF) to recover a 3D
c 1 l l
i structure with a cubically growingnumber of unknowns, [F F ρ](f,g,z)= ρ(x,y,z)e−i2π(fx+gy)dxdy
pt when the size of a genuine 3D object increases. Practi- y x 4lZ0Z0
o callyobtainablesignal-to-noiseratio(SNR)andmeasure-
being a bounded linear operator of norm 1, and
s. ment accuracy limit the applicability of “ankylography”
ic to objects that are small-sized in at least one dimension [T σ](f,g)= 1 lσ(f,g,z)e−i2π[(1−f2−g2)1/2−1]zdz
s or that are essentially 2D otherwise. z √l Z
y Therateofreliableinformationtransferviaaspatialor 0
h temporal channel is fundamentally limited by the chan- being alsoa bounded linear operatorof unit L2 operator
p
nel capacity that is determined by the NDF available norm [5]. Well known results for operators of “time and
[
therein andobtainable SNR [2]. The resolvingpowersof frequency limiting” [6] state that
1
telescopes or radio antennas and microscopes including
v N(F ,ǫ)=N(F ,ǫ)=2l+O( logǫ logl), (2)
4 SSDI are all limited in much the same manner. A steep x y | |
9 (exponential) price in signal power has to be paid to ob- and the x-y separability implies that
5 tain data rates or resolutions significantly beyond that
4 are supported by the available NDF [2, 3]. SSDI uses a N(FyFx,ǫ) N(Fy,ǫ1/2)N(Fx,ǫ1/2), (3)
. ≥
1 spatial channel characterized by a bounded linear oper- N(F F ,ǫ) N(F ,ǫ)N(F ,ǫ), (4)
y x y x
0 ator T : L2([0,l]3) L2(S) that projects a real space ≤
0 amplitude with sup→port [0,l]3 onto the Ewald sphere consequently,
1
S = (f,g,h):f2+g2+h2 =1 ,where l is measuredin
v: units{ofthe wavelength,(f,g,h)}representsa normalized N(FyFx,ǫ) = [2l+O(|logǫ|logl)]2
Xi spatial frequency. Well known is the existence of a pair = 4l2+O(|logǫ|llogl+log2ǫlog2l). (5)
ar iomnfgoodraetlshl,opanonosdrimtitvahelebaaanssdseosac{iraurtaie}ndig≥em1doaidnnadla{gvnaioi}nnis≥de1{c,λrciea(aTlsle)ind}gi≥noo1r,rdmbeear-l, HtioonweovfeTr,(cid:3)thmeoTrzetceormmpmlicaakteesd.thFeosritnugnualaterlvy,alaune dopisetrraibtuor-
inequality comes to the rescue [7]. It follows from
such that Tu = √λ v , i 1 [2, 4]. For any modal
i i i
∀ ≥
cutoff threshold ǫ (0,1) as determined by practically T∗T T 2 F∗F∗F F =F∗F∗F F (6)
∈ (cid:3) (cid:3) ≤k zk2 x y y x x y y x
obtainable SNR and accuracy in signal measurement,
N(T,ǫ) = max{i : λi(T) ≥ ǫ} is the number of usable that λi(T(cid:3))≤λi(FyFx), ∀i≥1. It is now obvious that
normal modes. We shall prove that when l is large,
N(T,ǫ) 2N(T ,ǫ/2) 2N(F F ,ǫ/2)
(cid:3) y x
≤ ≤
N(T,ǫ) 8l2+O( logǫ llogl+log2ǫlog2l), (1) = 8l2+O( logǫ llogl+log2ǫlog2l), (7)
≤ | | | |
whichgrowsfartoo slowincomparisonwith the number which constitutes a rigorous proof of equation (1).
of unknowns O(l3) in the structure of a general 3D ob- It is worth noting the fundamental nature of the limi-
ject. The dearth of NDF would persist even if 1/ǫ grew tation, that single-shot diffraction does not convey suffi-
exponentiallyaslincreased,solongastheexponentgrew cientinformationtoinvertthe3Dstructureofanobject,
2
eventheamplitude(insteadofintensity)ofthediffracted outside are automatically satisfied by the normal modes
fieldissampledcontinuouslyandmeasureddirectlywith in the presentformulation. The nonnegativity of the ob-
no phase ambiguity. With practically obtainable SNR ject field fixes only a single degree of freedom, i.e., a
and measurement accuracy that determine a threshold ǫ global level shift. The incorporation of more physical
of modal cutoff, any signal in the linear space spanned constraints,to the extreme ofhavingthe majorityof the
by the normal modes of orders higher than N(T,ǫ) is O(l3)unknownsfixed,couldarguablyalleviatetheprob-
essentially lost in transmissionor attenuated beyond de- lem of dimensional deficiency, however that diminishes
tection. Oversamplingandinversionalgorithmsareirrel- the generality and appeal of “ankylography”.
evant in this context. Indeed, the finiteness of an object
ensures thatthe entire diffractionfield is uniquely deter-
mined by a finite number of Nyquist-sampled amplitude
values. With a limited NDF, the only way to transmit
[1] Raines, K. S. et al. Three-dimensional structure determi-
and retrieve more information is to increase the number
nation from a single view. Nature 463, 214-217 (2010).
of significant figures in the measured amplitudes, which
[2] Gallager,R.G.InformationTheoryandReliableCommu-
however quickly becomes prohibitively expensive.
nication (Wiley, 1968).
In summary, SSDI of truly 3D structures does not [3] Hansen,R.C.Fundamentallimitationsinantennas.Proc.
scale. The applicability of “ankylography” is limited to IEEE69, 170-182 (1981).
objects that are small-sized with respect to the wave- [4] Miller, D. A.B. Spatial channelsfor communicating with
lengthinatleastone dimensionorhavestructuresbeing waves between volumes. Opt. Lett. 23, 1645-1647 (1998).
[5] Rynne, B. P. and Youngson, M. A. Linear Functional
essentially 2D in complexity. That may be the case in
Raines et al.’s computer tests and preliminary experi- Analysis 2nd Edn. (Springer, 2008).
[6] Landau, H. J. and Widom, H. Eigenvalue distribution of
ment [1]. Raines et al. [1] also put much emphasis on timeandfrequencylimiting.J.Math.Anal.Appl.77,469-
certain “physical constraints”, many of which as being 481 (1980).
described and used are actually steps of numerical pro- [7] Bhatia, R. Perturbation Bounds for Matrix Eigenvalues,
ceduresinsteadofmathematicalconstraintsofmodelfor- page 21 (SIAM,2007).
mulation,whilethebonafidephysicalconstraintsofcon-
tinuity and boundedness in the support and uniformity