Image Recovery for Blind Polychromatic Ptychography Frank FilbirOleh Melnyk

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Image Recovery for Blind Polychromatic

Ptychography

Frank Filbir∗Oleh Melnyk∗†‡

5th October 2022

Abstract

Ptychography is a lensless imaging technique, which considers re-

construction from a set of far-ﬁeld diﬀraction patterns obtained by illu-

minating small overlapping regions of the specimen. In many cases, a

distribution of light inside the illuminated region is unknown and has to

be estimated along with the object of interest. This problem is referred

to as blind ptychography. While in ptychography the illumination is com-

monly assumed to have a point spectrum, in this paper we consider an

alternative scenario with non-trivial light spectrum known as blind poly-

chromatic ptychography.

Firstly, we show that non-blind polychromatic ptychography can be

seen as a recovery from quadratic measurements. Then, a reconstruc-

tion from such measurements can be performed by a variant of Amplitude

Flow algorithm, which has guaranteed sublinear convergence to a critical

point. Secondly, we address recovery from blind polychromatic ptycho-

graphic measurements by devising an alternating minimization version of

Amplitude Flow and showing that it converges to a critical point at a

sublinear rate.

Keywords: ptychography, phase retrieval, blind, alternating mini-

mization, gradient descent.

MSC codes: 78A46, 78M50, 47J25, 90C26.

1 Introduction

Ptychography is a scanning coherent diﬀraction imaging method and hence

does not use advanced optical devices such as lenses for the image formation.

∗Mathematical Imaging and Data Analysis, Helmholtz Center Munich, 85764 Neuherberg,

Germany (ﬁlbir@helmholtz-muenchen.de,oleh.melnyk@helmholtz-muenchen.de).

†Department of Mathematics, Technical University of Munich, 85746 Garching bei

M¨unchen, Germany.

‡Corresponding author.

arXiv:2210.01626v1 [math.NA] 4 Oct 2022

The image of the object is reconstructed numerically from data which con-

sists of a stack of intensity measurements. This makes ptychographic imaging

predominantly a computational imaging technique. The principle of ptycho-

graphic imaging can be outlined as follows. An incoming coherent localized

wave of a speciﬁc wavelength, in the physics jargon called probe, is applied to

illuminate a small region of the object of interest. The beam gets scattered

and causes a diﬀraction pattern in the Fraunhofer or Fresnel region, depending

on whether the diﬀraction plane is placed in the far- or near-ﬁeld. A detector,

usually a CCD camera, then records the intensity of that diﬀraction pattern.

Subsequently the object is shifted to another position such that a diﬀerent re-

gion will be illuminated by the localized beam and then the next measurement

is recorded. In order to avoid a loss of information the adjacent illuminations

have to overlap.

light

shift

object detector

far-ﬁeld

Figure 1.1: Ptychography.

In this way a set of intensity measurements is collected which forms the data

base for the reconstruction process. The data redundancy allows to form an

image of the object computationally. Over the last years the ptychographic

technique was successfully used with diﬀerent light sources such as synchrotron

radiation [1,2,3], electron beams [4,5], and lasers [6,7]. In a common exper-

imental set-up light of one speciﬁc wavelength is used to illuminate the object,

and the CCD camera is placed in the far-ﬁeld (Fraunhofer distance). This ex-

perimental set-up then leads to measurements which are given mathematically

Iz(x, ξ, λ) = 1

(λz)2ZR2

f(y)gλ(y−x)e−2πiξ·y

λz dy

where λis the wavelength, zis the distance of the object plane to the de-

tector plane, fis the object function, and gλa wavelength dependent window

function which models the beam localization. In order to keep the exposition

simple we will henceforth assume that zis equal to one and the index zwill

therefore be omitted.

The computational task now is the reconstruction of (an approximation) of

ffrom (samples) of I(x, ξ, λ), i.e., from the squared absolute values of its

Fourier transform. Hence the reconstruction problem is a phase retrieval prob-

lem.

As pointed out above, in the conventional experimental set-up one speciﬁc

wavelength λis used, which then appears in the computational reconstruc-

tion process as a parameter. Moreover, often also the window function gλ

is considered to be known beforehand. If, for example, the aperture has the

form of a disc and the distance of the aperture to the object is suﬃciently big,

the window function is an Airy function which is frequently simply replaced

by a Gaussian function. However, not every experimental conﬁguration allows

to have precise control over the window function. In those cases the win-

dow function has to be considered as an additional unknown object which we

would like to retrieve computationally as well. These category of problems are

called blind ptychographic imaging and they were studied by several authors

[8,9,10,11,12]. Giving up control about the concrete shape of the win-

dow function is however not the only necessary generalization of the problem.

Light of only one speciﬁc wavelength is physically not easy to produce. Indeed,

hard X-rays of one speciﬁc wavelength are usually produced by an electron-

synchrotron, which is a huge machine. Other light producing systems however

may provide light with a certain spectral distribution. Performing ptycho-

graphic measurements with spectrally distributed light will result in diﬀerent

intensity measurements. These are given in the form

I(x, ξ, σ) = ZR

λZR2

fλ(y)gλ(y−x)e−2πiξ·y

λdydσ(λ)

where σis some compactly supported spectral density measure. Note that

the object’s scattering properties depends on the wavelength as well. Such

model is considered in [13] with an aim to improve quality of ptychographic

reconstruction from light sources with near single wavelength illumination.

If σconsists of separated spectral lines represented by a weighed sum of Dirac’s

delta measures, i.e., σ(λ) = PL

`=1 σ`δλ`,λ with λ`are enumerated such that

λ1< λ2<··· < λL, the intensity measurements reduce to

I(x, ξ) =

`=1

λ2

`ZR2

fλ`(y)wλ`(y−x)e−2πiξ·y

λ`dy

2,(1.1)

where wλ`=√σ`gλ`is a window function for wavelength λ`.

Recovery from noisy measurements of the form (1.1) is known as polychro-

matic ptychography. If the window functions wλ`are unknown, similarly to

the single wavelength case, the problem is referred to as blind polychromatic

ptychographic imaging (BPPI). We note that the measurements (1.1) also

arise if instead of polychromatic light multiple spatially separated apertures

are used in ptychographic experiment [14]. Moreover, a similar measurement

model can be found in quantum state tomography [15].

In the literature, BPPI was addressed in several works [16,17,18,19,20],

each using a gradient-based method minimizing amplitude-based loss function.

For instance, in [16] the authors establish a generalization of extended ptycho-

graphic iterative engine [8] for polychromatic measurements (1.1) known as

ptychographical information multiplexing method (PIM), which can be viewed

as stochastic gradient descent applied to the amplitude-based loss function

in analogy to [21]. As the amplitude-based loss functions used for BPPI is

non-Lipschitz, non-smooth and non-convex, convergence analysis of gradient-

based methods for BPPI is not present in the literature. Furthermore, the

absence of convergence guarantees leads to a non-trivial selection of the step

sizes, which often requires multiple trial-and-error iterations to achieve good

reconstruction.

In this paper, we propose a new method for BPPI, which is based on alter-

nating minimization technique [22]. In this way, we are able to reduce the

reconstruction problem to repeated recovery from quadratic measurements

[23,24,25]. For such problems, it is possible to establish a gradient descent

algorithm with appropriate step sizes, which guarantees sublinear convergence

to a stationary point of the amplitude-based loss function similarly to [26].

Using these guarantees, we are also able to derive sublinear convergence of

the whole alternating minimization technique.

The paper is structured as follows. Section 2contains preliminaries about

Wirtinger derivatives, gradient descent and its use for recovery from quadratic

measurements. We return to the measurements (1.1) in Section 3. The

established gradient optimization theory is applied ﬁrst for non-blind problem

and later for blind problem in Sections 3.2 and 3.3. In Section 4numerical

trials are performed and proposed methods are compared to PIM.

2 Preliminaries

In order to keep the presentation self-contained we start with some preliminary

considerations regarding Wirtinger derivatives and a related gradient descent

method.

2.1 Wirtinger derivatives and gradient descent

We start by collecting some facts on the Wirtinger calculus based on [27,

28]. Let f(z) = u(x, y) + iv(x, y), z =x+iy x, y ∈Rnwith real-valued

diﬀerentiable functions uand v. The function fcan be written as a function

of the conjugate variables z=x+iyand ¯z=x−iy. Since uand vare

diﬀerentiable the function f(z, ¯z) is holomorphic w.r.t. zfor ﬁxed ¯zand vice

versa. The Wirtinger calculus is a way to express the derivatives of fw.r.t. the

real variables x, y in terms of the conjugate variables zand ¯ztreating them

as independent. The so-called Wirtinger derivatives of fare deﬁned as

∂zf=1

2(∂xf−i∂yf), ∂¯zf=1

2(∂xf+i∂yf).(2.1)

and we obviously have

∂zf=∂¯z¯

f , and ∂¯zf=∂z¯

f . (2.2)

The Wirtinger derivatives ∂zfand ∂¯zfcan also be expressed as

∂zf=∂zf(z, ¯z)¯z=const. =h∂z1f(z, ¯z), . . . , ∂znf(z, ¯z)i¯z=const.,

∂¯zf=∂¯zf(z, ¯z)z=const. =h∂¯z1f(z, ¯z), . . . , ∂¯znf(z, ¯z)iz=const..

The Wirtinger gradient and Wirtinger Hessian are deﬁned as

∇f(z) = "(∂zf)∗

(∂¯zf)∗#,∇2f(z) = 





∂z(∂zf)∗∂¯z(∂zf)∗

∂z(∂¯zf)∗∂¯z(∂¯zf)∗



.

It follows immediately from (2.2) that if fis a real-valued function, i.e., f(z) =

u(x, y), the following relations hold

∂¯zf=∂zf , ∂¯z(∂¯zf)∗=∂z(∂zf)∗, ∂z(∂¯zf)∗=∂¯z(∂zf)∗.(2.3)

Henceforth we will use the less clumsy notation

∇zf:= (∂zf)∗,∇2

z,z f:= ∂z(∂zf)∗,resp. ∇2

z,¯zf:= ∂z(∂¯zf)∗.

The second-order Taylor polynomial of fat a point z0reads as

Pf(u, z0) = f(z0)+(∇f(z0))∗"u

¯u#+"u

¯u#∗∇2f(z)"u

¯u#

In case of a real-valued function f, the quadratic term of the Taylor polynomial

Pfcan be expressed in the following way

¯u#∗∇2f(z)"u

¯u#= 2Re(u∗∇2

z,z f(z)u) + 2Re(u∗∇2

¯z,z f(z) ¯u).(2.4)

In the remaining part of the paper we concentrate on real-valued functions f.

For minimizing such a function fwe shall apply gradient descent

zt=zt−1−µt∇f(zt−1) (2.5)

to some appropriate initial vector z0∈Cd. The parameter µt>0 is called

step size. It has to be chosen such that we achieve a descent in every step,

viz. f(zt)≤f(zt−1). The step size can be chosen in diﬀerent ways. The

ﬁrst option is a constant step size µt=µcfor all t≥1, which is possible to

choose, when the action of the Hessian of the of function fis bounded from

above as in the next proposition.

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ImageRecoveryforBlindPolychromaticPtychographyFrankFilbir*OlehMelnyk*5thOctober2022AbstractPtychographyisalenslessimagingtechnique,whichconsidersre-constructionfromasetoffar-elddiractionpatternsobtainedbyillu-minatingsmalloverlappingregionsofthespecimen.Inmanycases,adistributionoflightinsidethei...

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