Research Article Applied Optics 1 Space-variant Shack-Hartmann wavefront sensing based on afﬁne transformation estimation

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Research Article Applied Optics 1

Space-variant Shack-Hartmann wavefront sensing

based on afﬁne transformation estimation

FAN FENG1,11, † , CHEN LIANG2, † , DONGDONG CHEN3, KEDU4, RUNJIA YANG2, CHANG LU5, SHUMIN

CHEN3, LIANGYI CHEN4,6,7, LOUIS TAO1,8,AND HENG MAO2,9,10

1Center for Bioinformatics, National Laboratory of Protein Engineering and Plant Genetic Engineering, School of Life Sciences, Peking University, Beijing

100871, China

2LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

3School of Software and Microelectronics, Peking University, Beijing 100871, China

4State Key Laboratory of Membrane Biology, Beijing Key Laboratory of Cardiometabolic Molecular Medicine, Institute of Molecular Medicine, Center for Life

Sciences, College of Future Technology, Peking University, Beijing 100871, China

5School of Instrumentation and Optoelectronic Engineering, Beihang University, Beijing 100191, China

6PKU-IDG/McGovern Institute for Brain Research, Beijing 100871, China

7Beijing Academy of Artiﬁcial Intelligence, Beijing 100871, China

8Center for Quantitative Biology, Peking University, Beijing 100871, China

9Beijing Advanced Innovation Center for Imaging Theory and Technology, Capital Normal University, Beijing 100871, China

10heng.mao@pku.edu.cn

11ffeng@pku.edu.cn

†These authors contributed equally to this work

Compiled October 13, 2022

The space-variant wavefront reconstruction problem inherently exists in deep tissue imaging. In this pa-

per, we propose a framework of Shack-Hartmann wavefront space-variant sensing with extended source

illumination. The space-variant wavefront is modeled as a four-dimensional function where two dimen-

sions are in the spatial domain and two in the Fourier domain with priors that both gently vary. Here, the

afﬁne transformation is used to characterize the wavefront space-variant function. Correspondingly, the

zonal and modal methods are both escalated to adapt to four-dimensional representation and reconstruc-

tion. Experiments and simulations show double to quadruple improvements in space-variant wavefront

reconstruction accuracy compared to the conventional space-invariant correlation method. ©

2022 Optica

Publishing Group

http://dx.doi.org/10.1364/ao.XX.XXXXXX

1. INTRODUCTION

The space-variant problem in deep tissue imaging has bothered

biologists and microscopists for decades. Various methods have

been developed to resolve the problem and are roughly catego-

rized into two types: isotropic approximation and the conjugate

method. The isotropic approximation method, which assumes

an isotropic zone within a certain range to be space-invariant,

has been used in laser point scanning microscopy [

], selected

plane illumination microscopy [

], structured illumination mi-

croscopy (SIM) [

], Fourier ptychography microscopy (FPM)

[

], etc. However, the isotropic zone is usually small; thus, this

method is not efﬁcient for large ﬁeld of view (FOV) applica-

tions that take numerous snapshots for a single image. The

counterpart conjugate method assumes that the wavefront ﬂuc-

tuations originate from one plane near the sample, instead of

the pupil plane, as used in wideﬁeld microscopy [

]. Alterna-

tively, the pupil wavefronts from different FOVs are separated

in space and then sensed and corrected, as used in laser point

scanning microscopy [

]. However, the sample-induced wave-

front is hardly simply modeled by a phase screen. Otherwise,

the images may not be fully restored due to this model error.

In addition, sample-induced aberration could be modeled by

multiple planes, as used in astronomy and called multiconjugate

adaptive optics [

], which could deﬁnitely improve the system

performance but also dramatically increase the complexity and

cost of the imaging system; thus, this technology has not yet

been used in microscopy.

The space-variant imaging problem can be divided into two

steps: wavefront sensing and image restoration. This paper fo-

cus on how to model and reconstruct space-variant wavefront.

For direct wavefront measurement, the Shack-Hartmann wave-

front sensor (SHWFS) is the most popular way. The SHWFS con-

sists of a microlens array and a camera, and the camera is usually

located on the back focal plane of the microlens. The subspot

arXiv:2210.06400v1 [physics.optics] 12 Oct 2022

Research Article Applied Optics 2

images on the camera reveal the gradient of the incident wave-

front by its centroid drifts within the corresponding lenslet area

under the point source illumination as a guide star. However,

this conventional Shack-Hartmann (SH) scheme only represents

the wavefront at one designated ﬁeld of angle, that is, the angle

of the guide star [

]. Besides, the averaged wavefront across the

isotropic zone is represented by the local shifts of the optical spot

images based on the two-photon laser descanning technique [

In addition, the averaged wavefront across the entire FOV can

also be calculated by the correlation between the subimages with

and without wavefront aberrations under extended source illu-

mination [

]. Similar to obtaining the averaged wavefront,

the difference is that the descanning technique adds all intensi-

ties of the subimages to improve the signal-to-noise ratio since

the ﬂuorescence is very weak, while the correlation method uses

the subimages themselves. However, all these Shack-Hartmann

schemes cannot present a space-variant wavefront.

In this paper, we illustrate an SHWFS scheme to reconstruct

a space-variant wavefront with extended source illumination.

The afﬁne transformation is used to evaluate subimage trans-

formation across the FOV which implies that the wavefront

changes slowly. Furthermore, the zonal and modal methods

that reconstruct the wavefront from the gradient map are both

upgraded to meet the requirement for space-variant wavefront

reconstruction. In particular, a dual-orthogonal model is pro-

posed to represent the space-variant wavefront function, and the

relation between the afﬁne coefﬁcients and the dual-orthogonal

coefﬁcients is derived and used in the modal reconstruction.

Numerical simulations and imaging experiments are both im-

plemented.

2. THEORY

A. Space-variant wavefront function modal

First, let

φ(ξ,η)

denote the wavefront on a circular pupil in the

optical system while moderate enough to be well represented

by a sum of orthogonal polynomials

φ(ξ,η)=

∑

p=1

cpRp(ξ,η),(1)

where

(ξ,η)

are the global normalized lateral coordinates of the

pupil plane, or Fourier domain,

Rp(ξ,η)

are the

th orthogonal

polynomials

and are the corresponding coefﬁcients, e.g., the

Zernike polynomials

Zp(ξ,η)

for the circular domain or the

Legendre polynomials

Lp(ξ,η)

for the rectangular domain,

the maximum representation term for the pupil plane.

(x,y) (x,y)

(x,h)

Object

Plane

Image

Plane

Pupil

Plane

Fig. 1. Coordinates deﬁned in the imaging system.

In the case of space-variant wavefront sensing, such as in

deep-tissue imaging, the wavefront varies along different po-

sitions. Mathematically, the constant coefﬁcients

in Eq. (1)

evolve into the space-variant function

cp(x

, where

the lateral coordinates of the object and image plane, also nor-

malized to 1 for orthogonal representation, as shown in Fig. 1;

thus the magniﬁcation and the inversion of coordinates on object

and image planes are neglected.

Therefore, we deﬁne the space-variant wavefront,

φ(x,y,ξ,η)=

∑

p=1

cp(x,y)Rp(ξ,η).(2)

Similarly, we assume that the space-variant function

cp(x

varies slowly; thus, the dual-orthogonal modal of the space-

variant wavefront function is given as

φ(x,y,ξ,η)=

∑

q=1

∑

p=1

ep,qTq(x,y)Rp(ξ,η),(3)

where

cp(x,y)=

∑

q=1

ep,qTq(x,y),(4)

and

Tq(x,y)

are the

th orthogonal polynomials,

ep,q

are the

corresponding coefﬁcients, and

is the maximum representa-

tion term for the pupil plane.. The purpose of our work is to

reconstruct the space-variant wavefront

φ(x,y,ξ,η)

by retriev-

ing the dual-orthogonal coefﬁcients

ep,q

, as the modal method,

or discrete

φxm,yn,ξp,ηq

on position

xm,yn,ξp,ηq

, as the

zonal method.

B. Afﬁne transformation estimation for evaluating the space-

variant function in Shack-Hartmann subapertures

In SH, the light ﬁeld projects onto a microlens array and forges an

image on the back focal plane with a subimage array, as shown

in Fig. 2(a1-c1). For the scenario of the conventional space-

invariant wavefront, only one gradient vector is obtained from

subimages in one lenslet, as shown in Fig. 2(a2-b2). The centroid

shift or correlation algorithm, depending on which light source

is used, the point or extended, can be used for representing

the local slope. On the other hand, for sensing space-variant

wavefront, the subimages deform as shown in Fig. 2(c2), rather

than just translation, as shown in Fig. 2(b2). We used the afﬁne

transformation to characterize the deformation of the subimages

in one lenslet







x2−x1

y2−y1













a1a2a3

b1b2b3

0 0 1



















,(5)

where

(x1

y1)

is an arbitrary point on the

plane of the cor-

responding lenslet area since the back focal plane of the lenslet is

also the image plane,

(x2

y2)

is the coordinate after deformation,

as shown in Fig. 2(c2) and Fig. 3(a,b),

[x2−x1

y2−y1]T

presents

the centroid shift of point

(x1

y1)

, and

{a1

b3}

are

the coefﬁcients of an afﬁne transformation. The afﬁne coefﬁ-

cients from all lenslets can reconstruct the dual-orthogonal coef-

ﬁcients ep,qor discrete φxm,yn,ξp,ηq(details in Appendix).

Various approaches, such as intensity-based optimization,

control points, and feature detection and matching, can be used

to register images on many platforms. In this paper, the MAT-

LAB

internal function ‘imregtform’ is used to estimate afﬁne

transformation coefﬁcients, in which the image similarity metric

is optimized during registration.

3. NUMERICAL SIMULATIONS

In this section, the numerical simulations in space-variant Shack-

Hartmann wavefront sensing are given as the simulation conﬁg-

uration, the forward process of image formation, and wavefront

reconstruction based on zonal and modal methods.

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ResearchArticleAppliedOptics1Space-variantShack-HartmannwavefrontsensingbasedonafnetransformationestimationFANFENG1,11,,CHENLIANG2,,DONGDONGCHEN3,KEDU4,RUNJIAYANG2,CHANGLU5,SHUMINCHEN3,LIANGYICHEN4,6,7,LOUISTAO1,8,ANDHENGMAO2,9,101CenterforBioinformatics,NationalLaboratoryofProteinEngineeringandP...

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