# Incorporation of an experimentally determined MTF for spatial frequency filtering and deconvolution during optical projection tomography reconstruction.

**ABSTRACT** We demonstrate two techniques to improve the quality of reconstructed optical projection tomography (OPT) images using the modulation transfer function (MTF) as a function of defocus experimentally determined from tilted knife-edge measurements. The first employs a 2-D binary filter based on the MTF frequency cut-off as an additional filter during back-projection reconstruction that restricts the high frequency information to the region around the focal plane and progressively decreases the spatial frequency bandwidth with defocus. This helps to suppress "streak" artifacts in OPT data acquired at reduced angular sampling, thereby facilitating faster OPT acquisitions. This method is shown to reduce the average background by approximately 72% for an NA of 0.09 and by approximately 38% for an NA of 0.07 compared to standard filtered back-projection. As a biological illustration, a Fli:GFP transgenic zebrafish embryo (3 days post-fertilisation) was imaged to demonstrate the improved imaging speed (a quarter of the acquisition time). The second method uses the MTF to produce an appropriate deconvolution filter that can be used to correct for the spatial frequency modulation applied by the imaging system.

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**ABSTRACT:**A new technique termed Helical Optical Projection Tomography (hOPT) has been developed with the aim to overcome some of the limitations of current 3D optical imaging techniques. hOPT is based on Optical Projection Tomography (OPT) with the major difference that there is a translation of the sample in the vertical direction during the image acquisition process, requiring a new approach to image reconstruction. Contrary to OPT, hOPT makes possible to obtain 3D-optical images of intact long samples without imposing limits on the sample length. This has been tested using hOPT to image long murine tissue samples such as spinal cords and large intestines. Moreover, 3D-reconstructed images of the colon of DSS-treated mice, a model for Inflammatory Bowel Disease, allowed the identification of the structural alterations. Finally, the geometry of the hOPT device facilitates the addition of a Selective Plane Illumination Microscopy (SPIM) arm, providing the possibility of delivering high resolution images of selected areas together with complete volumetric information.Optics Express 11/2013; 21(44):25912-25925. · 3.53 Impact Factor - SourceAvailable from: L. Chen[Show abstract] [Hide abstract]

**ABSTRACT:**We describe an angular multiplexing technique for optical projection tomography that improves resolution, signal-to-noise ratio, and imaging speed by ameliorating the trade-off between spatial resolution and depth of field and improving the light collection efficiency. Here we demonstrate that imaging at two orthogonal angular projections simultaneously, focused on shifted planes in the sample, improves the average spatial resolution by ∼20% and the light collection efficiency by a factor of ∼4, thereby enabling increased acquisition speed and reduced light dose.Optics Letters 03/2013; 38(6):851-3. · 3.39 Impact Factor

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Incorporation of an experimentally determined

MTF for spatial frequency filtering and

deconvolution during optical projection

tomography reconstruction

Lingling Chen,1,* James McGinty,1 Harriet B. Taylor,2 Laurence Bugeon,2

Jonathan R. Lamb,2 Margaret J. Dallman,2,3 and Paul M. W. French1

1Photonics Group, Department of Physics, Imperial College London, SW7 2AZ, UK

2Division of Cell and Molecular Biology, Department of Life Sciences, Imperial College London, SW7 2AZ, UK

3Centre for Integrative Systems Biology, Department of Life Sciences, Imperial College London, SW7 2AZ, UK

*l.chen10@imperial.ac.uk

Abstract: We demonstrate two techniques to improve the quality of

reconstructed optical projection tomography (OPT) images using the

modulation transfer function (MTF) as a function of defocus experimentally

determined from tilted knife-edge measurements. The first employs a 2-D

binary filter based on the MTF frequency cut-off as an additional filter

during back-projection reconstruction that restricts the high frequency

information to the region around the focal plane and progressively

decreases the spatial frequency bandwidth with defocus. This helps to

suppress “streak” artifacts in OPT data acquired at reduced angular

sampling, thereby facilitating faster OPT acquisitions. This method is

shown to reduce the average background by approximately 72% for an NA

of 0.09 and by approximately 38% for an NA of 0.07 compared to standard

filtered back-projection. As a biological illustration, a Fli:GFP transgenic

zebrafish embryo (3 days post-fertilisation) was imaged to demonstrate the

improved imaging speed (a quarter of the acquisition time). The second

method uses the MTF to produce an appropriate deconvolution filter that

can be used to correct for the spatial frequency modulation applied by the

imaging system.

©2012 Optical Society of America

OCIS codes: (170.6900) Three-dimensional microscopy; (170.6960) Tomography.

References and links

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1. Introduction

As biological research progresses from studies of mono-layers of cells on glass to in situ

measurements of both ex vivo and in vivo biological systems, it is becoming necessary to

apply three-dimensional (3-D) imaging techniques in order to map structure and function

throughout a sample. Confocal/multi-photon laser scanning microscopes provide optical

sectioning to permit the acquisition of 3-D z-stacks and also offer improved contrast

compared to wide-field imaging but they suffer from limited (100’s μm) penetration depth

and fields of view (10’s μm) and exhibit anisotropic resolution. Thus, while they are widely

used to image microscopic specimens [1–3], they are less suitable for large samples for which

the acquisition of 3-D data sets can be very time consuming. To address this challenge, a

number of imaging techniques have been developed for samples in the “mesoscopic” regime

(1-10 mm), including optical projection tomography (OPT) [4], selective plane illumination

microscopy (SPIM) [5], ultramicroscopy [6] and optical coherence tomography [7].

OPT is the optical equivalent of X-ray computed tomography (CT), in which the 3-D

structure (a stack of X-Z slices) of a rotating sample is reconstructed from a series of wide-

field 2-D projections (X-Y images) obtained at different projection angles. Typically, digital

images of the specimen are acquired throughout a full rotation (360°) and a filtered back-

projection (FBP) algorithm is used for image reconstruction [8]. This assumes parallel

projection corresponding to parallel ray (or plane wave) propagation of the signal with

negligible scattering in the sample. This is appropriate for X-ray CT, but scattering can be a

significant issue for optical radiation in biological tissue and OPT is frequently implemented

with relatively high numerical aperture (NA) optics, for which rays at a range of angles with

respect to the optical axis are collected, and so reconstructed images can suffer from a

scattered light background and defocus blurring. To address the issue of scattering, OPT is

often applied to samples that have been rendered transparent by a chemical clearing process

[4] or which are inherently transparent. Important examples of the latter include live

organisms that can be genetically manipulated to serve as disease models such as D.

melanogaster [9], C. elegans [10] and Danio rerio (zebrafish) embryos [11].

The potential to apply OPT to such samples for biomedical research has prompted

significant interest in optimizing the image quality and resolution and minimizing the image

data acquisition time. Image quality can be degraded by artifacts resulting from system

misalignment, intensity-based signal variations and system aberrations and methods have

been described to correct or suppress such artifacts [12,13]. It can also be degraded by

acquiring an insufficient number of angular projections and by deviations from the parallel

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ray assumption that underlies the standard FBP algorithm. For high resolution OPT there is a

trade-off between increasing the NA to improve the in-focus lateral resolution and reducing

the NA to increase the depth of field (DOF) in order to ensure that the whole sample is in

reasonable focus (i.e. that the lateral resolution does not vary significantly along the optical

axis). OPT is typically undertaken with samples that extend beyond the confocal parameter

(Rayleigh range) of the imaging lens and so the tangential resolution of the reconstructed

images decreases radially away from the axis of rotation. Building on ideas developed for

single-photon emission computed tomography [14], Walls et al. accounted for this distance-

dependent resolution by applying an appropriate deconvolution filter to the raw sinogram data

based on a computationally generated point-spread function (PSF) [15]. The effect was to

both suppress the contribution from out of focus sources and correct the frequency modulation

applied to the signal by the optical system. One way to extend the DOF and therefore

avoiding the trade-off is to scan the focal plane through the whole sample, which requires the

working distance of the objective lens to be larger than the size of the sample [16]. This

scanning method produces isometric resolution.

In this paper, we extend the widely used fixed focal plane approach by experimentally

determining the modulation transfer functions (MTFs) for different effective collection NAs

of the optical imaging system as a function of defocus and modify the standard FBP to

incorporate this MTF information in the OPT reconstruction by either MTF-mask filtering or

deconvolution of the MTF. The deconvolution approach can provide improved resolution

reconstructions for high-angular-sampling OPT acquisitions and the MTF-mask filtering

technique leads to an improvement in the reconstructed image quality by suppressing imaging

artifacts that arise when reducing the number of measured projections. The latter is important

because reducing the total number of projection images acquired will reduce the overall light-

dose on the sample - and therefore any photo-bleaching or photo-toxic effects - and improve

the achievable time-lapse resolution, e.g. for in vivo acquisitions.

2. Determination of the MTF of OPT system

2.1 Imaging system

CCD

Rotation

Sample chamber

Excitation

L1

L2

F1

F2

M

OAP

DM

(a)

(b)

Rotation

Sample chamber

Fig. 1. (a) Schematic of OPT system. O – objective, AP – aperture, L1 – condenser lens, F1 –

excitation filter, DM – dichroic mirror, L2 – tube lens, F2 – emission filter, M – mirror. (b)

Photograph of the custom built chamber.

A schematic of the OPT system, which has been previously described [11], is shown in

Fig. 1(a). In brief, the sample was imaged on a standard inverted wide-field microscope (IX-

71, Olympus UK Ltd), utilizing both transmitted light and epifluorescence imaging with a 4x

objective (UPLFLN4X, Olympus UK Ltd) and a GFP filter cube (GFP-3035B-OMF, Laser

2000 Ltd). The effective collection NA was adjusted using appropriate apertures (AP)

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positioned directly behind the objective where it defines the back aperture. Images were

collected by a CCD camera (Clara, Andor Technology plc, 1040 × 1392, 6.45 µm pitch size,

cooled to 20 °C). For small samples (e.g. zebrafish embryos), a custom-built chamber was

fabricated to hold the tube-mounted samples in a refractive index-matched environment, as

shown in Fig. 1(b).

There are two common focusing arrangements for OPT, as illustrated in Fig. 2: the focal

plane and axis of rotation can be coincident or the focal plane can be displaced from the axis

of rotation into the front half of the specimen. For the first arrangement, all parts of sample

will pass through the focal plane twice and the region close to the axis of rotation will be

maintained at “best-focus”. The second arrangement is often used for imaging larger samples

but objects located between the axis of rotation and the focal plane will never be imaged at

best-focus (i.e. will not pass through the focal plane). The work described in this paper uses

the first optical arrangement (except in section 4.2), but the ideas expressed are applicable to

both. We note that, for the arrangement in Fig. 2(a) when the focal plane coincides with the

axis of rotation, it is only necessary to acquire angular projections over 180° when imaging

absorption coefficients in transmission, but for fluorescence tomography this is not the case if

there is a variation in excitation intensity or fluorescence absorption (inner filter effect) across

the sample.

Focal Plane

DOF

(a)

DOF

Focal Plane

(b)

Fig. 2. Schematic of OPT imaging system with (a) focal plane coincident with the rotation axis

and (b) focal plane shifted to half front of the sample. DOF – depth of field; Dotted lines are a

schematic representation of the axial PSF.

2.2 MTF characterization

To determine the MTF for the different effective NA, a tilted knife-edge technique was

employed [17]. Transmitted light images of a scalpel blade mounted in the chamber at an

angle of ~9.5° with respect to the vertical axis of the CCD were acquired. This tilt facilitates

an increased effective sampling of the knife-edge by interleaving 6 rows of pixels to produce

a 1-D edge-spread function (ESF). Average ESFs, determined from 10 bright-field and 30

back-ground images, were acquired over a focus range of 2.4 mm (i.e. up to 1.2 mm either

side of the ‘in-focus’ image) and repeated for effective NAs in the range 0.03-0.09. The

distance of the knife-edge from the focal plane was recorded by a plunge dial indicator (model

#398877, Radio Spares Ltd).

The ESF is the integral of the line-spread function (LSF), with the 1-D MTF being the

Fourier transform of the LSF. Based on the analysis method from Boone et al. [18], an

analytic equation representing the weighted sum of an error function and exponential recovery

term (Eq. (1)) was used to fit the measured ESFs. The parameters from these fits were then

used in an analytic expression for the MTF [18]. The measured in-focus ESF (blue squares)

and the resulting fit (red line) for the imaging system is illustrated in Fig. 3(a). The resultant

MTF, determined using both the analytic expression (red solid line) and numerical

differentiation and Fourier transformation (blue squares) are shown in Fig. 3(b). In addition,

Fig. 3(b) shows the MTFs for three different focal positions, which illustrate a reduction in the

Page 5

0 50100150

Spatial Frequency (mm-1)

200250 300 350400 450500

0.0

0.2

0.4

0.6

0.8

1.0

MTF (f)

MTF1 (Fourier method)

MTF1 (ESF fit method)

MTF2 (Fourier method)

MTF2 (ESF fit method)

MTF3 (Fourier method)

MTF3 (ESF fit method)

0 0.20.4 0.60.81

0

2000

4000

6000

8000

10000

12000

14000

Position along edge (mm)

Grey Scale Values

ESF (measured data)

ESF (fit)

(a)

(b)

Fig. 3. (a) The measured edge spread data are plotted as discrete data points. The fit to these

points is illustrated by the red solid line. The linear correlation coefficient between these two is

0.9999. (b) Three pairs of MTFs at the different positions from the analytical method, shown as

the red solid line, and the numerical Fourier transform procedure, shown as data points. MTF1

is the MTF in focus; MTF2 and MTF3 are for 70 and 200 µm defocus, respectively.

kx

z

DOFDOF

DOFDOF

Fig. 4. MTFs as a function of defocus (z) for different effective NAs (0.09, 0.07, 0.05 and 0.03)

of the OPT system. The blue lines indicate the DOFs for different NAs (1040 × 1040, Δkx =

0.596 mm1, Δz = 1.6125 µm) (Media 1).

bandwidth of the transmitted spatial frequencies as a function of defocus. The use of Eq. (1)

to fit the ESF assumes a Gaussian-dominated model of the LSF and therefore also the MTF.

Page 6

While this is not the correct functional form for the MTF of a ‘perfect’ lens with a circular

aperture, the limiting resolution is dominated by the first minimum. Fitting the assumed

model to an ideal in-focus MTF results in a difference of only 3.6% at the cut-off frequency

and therefore, this value was used as the frequency cut-off threshold for this MTF model.

This, along with the noise associated with the ESF measurement particularly at high spatial

frequencies, makes the Gaussian-dominated approximation sufficient for the generation of

2-D MTFs (i.e. the MTF as a function of defocus).

12

00

1 exp

esf xab x

xc erf d

xx

(1)

Four 2-D MTFs for different effective NAs are illustrated in Fig. 4 (Media 1 shows the

MTF for the full range of NAs). With the pixel size of the camera being 6.45 µm and a system

magnification of 4, the pixel dimension for vertical axis (defocus distance) is 1.6125 µm and

for the horizontal axis (spatial frequency) it is 0.596 mm1. It is evident that, as the NA

decreases, the DOF increases and the in-focus bandwidth (i.e. lateral resolution) decreases.

3. Materials and methods

3.1 Sample preparation and acquisition

A phantom consisting of a low concentration suspension of fluorescent beads in agarose, with

an average bead diameter 14.8±0.13 µm and excitation/emission maxima at 505/515 nm

respectively (F8844, Life Technologies Ltd), was used as a model sample. This was drawn

into translucent FEP tubing (#06406-60, Cole-Palmer), which has a refractive index similar to

that of water and can be used as a convenient index matched container for sample mounting

[19], with inner and outer diameters of 0.8 and 1.6 mm respectively.

Data was acquired on the OPT system described above using three different apertures

positioned directly behind the objective, resulting in effective collection NAs of 0.06, 0.07

and 0.09. During a standard OPT acquisition, image data was acquired at equal angular

intervals as the sample rotated (e.g. acquiring images every 1° over a full rotation). The

typical exposure time for each OPT view of the bead suspension was 100 ms. However, to

enable the results from different effective NAs to be compared directly, the exposure time was

adjusted to maintain a consistent signal-to-noise ratio. The analytic values for the resolution at

best focus (rAiry) [15] and DOF, which is calculated from Eq. (2) [15], for different NAs are

given in Table 1.

2

bath

n

a

n n e

DOF

MNA

NA

(2)

where nbath is the refractive index of the medium in which the specimen be immersed, n the

refractive index of the immersion medium of the lens, λ the wavelength of light, e the pixel

size of the CCD camera and Ma the lateral magnification of the imaging system. The DOF for

NA 0.07 and 0.09 are also shown in Fig. 4.

Table 1. The resolution and DOF for different effective NAs of the OPT system

NA 0.06 NA 0.07 NA 0.09

rAiry (µm) 5.3 4.5 3.5

DOF (µm) 228 169 108

Adapting the standard equation for the number of projections required for reconstruction

in the case of parallel projection [8] and assuming that the resolution is limited by the

objective lens, the required number of projections, M, for a low NA OPT system is given by

Eq. (3)

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0.61

Airy

r

D NA

n

D

MN

(3)

where N is the number of resolvable elements (given by the lateral resolution), D is the width

of the field of view (or field of interest), n the refractive index of the immersion medium of

the lens and λ the wavelength of light. In this experiment, where n is 1, D is 700 µm and λ is

515 nm, the typical value of M is 490 for an effective NA of 0.07, which means that the

required sampling is <1° for the full rotation. Such fine angular sampling implies long data

acquisition times with concomitant light exposure and so it is of significant interest to obtain

reasonable quality images with reduced (i.e. under sampled) numbers of projections.

Inadequate angular sampling can result in streak artifacts in the reconstructed images. These

are unwanted high spatial frequency projected features that appear away from the focal

region.

3.2 MTF-mask filter

In standard FBP, a ramp-filter is applied to each projection to correct the sampling of the

spatial frequency content of the object due to the rotational scanning geometry. The filtered

projections are then back-projected at the appropriate angle and summed together to

reconstruct the object. As discussed earlier, this process assumes parallel projection, i.e. that

the spatial frequencies transferred by the system are invariant along the projection direction,

but this is not the case in a typical OPT system where defocus reduces the spatial frequencies

transferred. To account for this additional frequency modulation, a composite filter can be

constructed from a combination of the ramp filter and the MTF-mask filter. Figure 5(a) shows

a normalized 2-D ramp filter and 5(b) shows a simulated back-projection at one angle after

applying this ramp filter. To reconstruct the tomographic image, the set of such 2-D filtered

projections would be summed at their appropriate angles, i.e. analogous to the standard FBP

approach.

The MTF-mask filter is a 2-D binary mask generated from the experimentally determined

MTF for the acquisition NA. The MTF was normalized by the frequency cut-off threshold,

3.6%, and the MTF-mask filter obtained by setting values above 1 to 1 and values below 1 to

0. This filter is designed to appropriately restrict the spatial frequency components away from

the focal plane. Using this filter during reconstruction restricts the frequency components to

regions from which the optical system could have transferred them (i.e. high spatial

frequencies are only present in the region around the focal plane), thus a more realistic

reconstruction may be achieved and streak artifacts are suppressed. Figure 5(c) shows the

corresponding MTF-mask filter for an effective NA of 0.07 and 5(d) shows the 2-D back-

projection from the same simulated raw data as Fig. 5(b) but with both the MTF-mask filter

and the ramp filter applied. Thus high spatial frequencies should only be present in the region

around the focal plane. We note that the binary filter could potentially produce artifacts in the

reconstruction although the ramp filter tends to suppress the low spatial frequencies that might

be generated as the MTF filter narrows away from the focal plane and the averaging of the

back-projection process also tends to suppress such artifacts. In practice, the MTF filter did

not seem to produce discernible artifacts.

Page 8

(c)

(e)

(a)

x

z

(b) (d)(f)

kx

z

Fig. 5. (a) 2-D ramp filter and (b) 2-D back-projection at one angle with ramp filter applied; (c)

binary MTF-mask filter and (d) back-projection with the MTF-mask filter and ramp filter

applied; (e) deconvolution filter and (f) back-projection with deconvolution filter and ramp

filter applied (effective NA of 0.07 for the OPT system; 1040 × 1040 pixels).

3.3 Deconvolution

The MTF-mask FBP approach described above accounts for the reduced bandwidth

transferred by the imaging system as a function of defocus. From the determined MTF,

however, it is also possible to deconvolve the frequency modulation applied to the data by the

imaging system, as previously demonstrated for a computationally generated MTF [15].

Using the notation adopted in [15], the deconvolution filter in frequency space is a

combination of two distinct components, as described by

1

_lim

deconvW Mm

HHH

(4)

where

a Wiener filter to de-emphasize the noise and HMm is an edge-decaying MTF-mask filter. This

final filter is generated by normalizing the MTF by a threshold value of 7% and setting values

above 1 to 1. It is therefore similar to the MTF-mask filter described previously, but

increasingly suppresses frequencies beyond the 7% threshold.

The Wiener filter is necessary since deconvolution can be highly sensitive to noise,

especially in the high-frequency region, and is given by

1

_lim

W

H

is the combination of a maximum limited recovery filter according to the MTF,

1

2

Mx

W

Mxu

HS

H

HSS

(5)

where HM is the determined MTF, Sx is the signal power spectrum and Su is the noise power

spectrum.

To minimise the impact of noise in the information gaps, the recovery filter is scaled by a

weighting factor, given by [15]

11

1

1

_lim

1

:

1 exp

:

WWt

W

Wt

trWt

r

HHC

H

HC

CCHC

C

(6)

Page 9

where Ct is the magnitude value at which the transition begins and Cr is the range used as a

transition to the maximum magnitude. The common values for these parameters used in this

experiment were Ct = 3, Cr = 0.3, Sx = 1, Su = 0.01 and were selected empirically based on

the suppression of high-frequency noise and background noise in the reconstructed images.

Figure 5(e) shows the corresponding deconvolution filter for an effective NA of 0.07 and 5(f)

the 2-D back-projection obtained after applying both the deconvolution and the ramp filter.

3.4 OPT simulation

To evaluate the performance of the different reconstruction techniques, a simulated data set

was generated for comparison with the experimental data. The simulated sample consisted of

two beads of equal brightness at similar locations with respect to the axis of rotation to beads

measured in corresponding experiments. The determined MTFs for different effective NAs

were then used to numerically generate the corresponding raw projection data, which could be

subsequently reconstructed using different approaches (e.g. standard FBP, MTF-mask

filtering method, etc). The simulation does not account for scattering, absorption, light

propagation and variation in collection efficiency but serves only to model the defocus effects

of the optical system (i.e. the MTF).

3.5 Evaluation of image quality as angular sampling is varied

To evaluate the reduction in reconstructed image quality when reducing the number of

projections (i.e. increasing the projection angle interval), the correlations between the

reconstructed images from low-angular-sampling (e.g. 4° interval between projections) and

the highest-angular-sampling (1° interval between projections) were calculated to give a

quantitative indicator of the reconstructed image quality compared to that obtained using the

highest-angular-sampling (i.e. the image closest to the real sample). In addition, a line was

plotted through the radial and tangential axes centered on the beads (shown in Fig. 6) and the

full width half maxima (FWHM) of the reconstructed bead images were measured in order to

compare the differences in reconstructed image resolution.

Radial

Tangential

Rotation

Fig. 6. Schematic of radial and tangential resolution used to evaluate the image quality.

4. Results and discussions

The reconstruction process produces a stack of cross-sectional images of a sample from the

projection data. To compare the results for different effective NAs, two beads at different

distances from the axis of rotation were chosen. The distance away from the rotation axis

were 103, 334 µm respectively. They were not in the same plane, which means their

sinograms were separate and the reconstructions could be obtained for each bead separately.

The sinograms of these two beads could also be combined computationally to compare the

experimental reconstruction with our simulated data.

4.1 MTF-mask Filtering

Figures 7(a) and 7(b) show reconstructions of a simulated data set using standard FBP and

MTF-mask filtering method. The data set consisted of 90 projections (i.e. 4° sampling) for an

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effective NA of 0.07. Figures 7(c) and 7(d) show the corresponding reconstructions with the

beads removed to emphasise the back-ground. The corresponding reconstructions from

experimental measurements are shown in Fig. 8. The reconstructions with the MTF-mask

filter exhibit reduced streak artifacts of significantly lower intensity compared to the standard

FBP reconstructions. Media 2, Media 3 and Media 4 compare the reconstructions obtained for

NAs of 0.06, 0.07 and 0.09 respectively using the FBP and MTF-mask approaches as the

angular sampling is decreased. The average value of the experimental background, which

should ideally be zero, was calculated to evaluate the difference between the standard FBP

and MTF-mask filtered reconstructions (each based on 90 experimental projections) for

different effective NAs, as listed in Table 2.

Table 2. The average background from standard FBP and MTF-mask filtered

reconstruction for different effective NAs (based on 90 experimental projections)

NA 0.06 NA 0.07 NA 0.09

Standard FBP 6.8 (100%) 5.3 (100%) 6.1 (100%)

MTF-mask FBP 5.0 (73.5%) 3.3 (62.3%) 1.7 (27.9%)

The image correlation between reconstructions from lower angular-sampled projections

and the best reconstruction (calculated from 360 projections) were also calculated. Figure 9(a)

shows the correlations for an effective NA of 0.07, where the points correspond to

experimental values and the lines to the simulation. These curves illustrate how the MTF-

mask filtered reconstruction provides a better correlation with the reference image when

reducing the number of projections by suppressing the streak artifacts. For fast OPT

acquisitions, the MTF-mask filtering method should therefore provide superior

reconstructions.

Fig. 7. The simulated (a) standard FBP; (b) MTF-mask filtered reconstruction from 90

projections (every 4°) for an effective NA of 0.07. (c) and (d) show the background for (a) and

(b) respectively. The images of the background are on the same intensity scale to show the

differences. Scale bar, 200 µm.

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Fig. 8. The experimental (a) standard FBP; (b) MTF-mask filtered reconstruction from 90

projections (every 4°) for an effective NA of 0.07. (c) and (d) show the background for (a) and

(b) respectively. The images of the background are on the same intensity scale to show the

differences. Scale bar, 200 µm (Media 3).

(a)

(b)

13579 11 13 15

0.6

0.7

0.8

0.9

1.0

NA0.07 (S, Standard)

NA0.07 (S, MTF)

NA0.07 (E, Standard)

NA0.07 (E, MTF)

Correlation

Angle interval (degree)

NA0.06 (Standard)

NA0.06 (MTF)

NA0.07 (Standard)

NA0.07 (MTF)

NA0.09 (Standard)

NA0.09 (MTF)

35

Angle interval (degree)

179 111315

0.6

0.7

0.8

0.9

1.0

Correlation

Fig. 9. (a) Simulated (S) and experimental (E) correlation results of standard FBP and MTF-

mask filtered reconstructions for an effective NA of 0.07. (b) Experimental correlation results

for different effective NAs (0.06, 0.07, 0.09), with standard FBP correlations as the dotted lines

and the MTF-mask filtered reconstruction correlations as the solid lines (three dotted lines are

similar and cannot be distinguished in this figure and thus only use black color to represent

them).

Figure 9(b) shows the experimental image correlation results for different effective NAs

(0.06, 0.07 and 0.09), with standard FBP correlations as the dotted lines (these three standard

FBP correlations are similar and cannot be distinguished in this figure) and the MTF-mask

filtered reconstruction correlations as the solid lines. When reducing the number of

projections, the correlation for higher NA MTF-mask filtered reconstructions decreases more

slowly than those for the low NA. This can be explained by the fact that the MTF-mask filter

for the high NA suppresses the frequency components more aggressively. It should be noted

that the correlations shown here were performed with respect to the specific object (bead

phantom) for the corresponding NA. However, the trends are applicable to all samples. Once

the appropriate NA is determined for a sample, the reconstruction quality using the MTF-

mask filtering method is always superior compared to standard FBP for reduced angular

sampling.

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Table 3. The radial and tangential FWHM of reconstructed beads for different effective

NAs (FBP – standard FBP reconstruction, MTF – MTF-mask filtered reconstruction;

average bead diameter 14.8 ± 0.13 µm)

NA 0.06 NA 0.07 NA 0.09

FWHM (µm) FBP MTF FBP MTF FBP MTF

Radial (on-axis) 16.2 16.3 15.9 15.9 15.4 15.3

Tangential (on-axis) 17.0 17.0 17.3 17.3 17.2 17.2

Radial (off-axis) 13.0 13.3 12.6 12.4 12.7 12.6

Tangential (off-axis) 25.9 25.5 32.9 31.4 45.1 43.4

The radial and tangential FWHM of the reconstructed beads are given in Table 3. As

expected, as the NA is reduced the in-focus radial resolution is reduced as indicated by the

increased FWHM of the on-axis bead. Similarly, the DOF increases as indicated by a

reduction in the tangential FWHM of the off-axis bead. This illustrates the trade-off in OPT

between the lateral resolution and the DOF. There is no significant difference, however,

between the FWHMs achieved using the two reconstruction techniques. This is due to the fact

that MTF-mask filtering method does not change the spatial frequency content of

reconstructed objects directly, but rather suppresses high frequency streak artifacts

particularly when using reduced angular sampling.

Fig. 10. (a) Standard FBP reconstruction and (b) MTF-mask filtered reconstruction of a

zebrafish embryo with an effective NA of 0.07 from 360 projections (every 1°) and (c)

Standard reconstruction and (d) MTF-mask filtered reconstruction of zebrafish embryo with an

effective NA 0.07 from 90 projections (every 4°). Scale bar, 200 µm (Media 5).

To illustrate the impact of the MTF-mask filter on a biological sample, a Fli:GFP

transgenic zebrafish embryo (3 days post-fertilisation) was imaged to test the performance

with an angular sampling interval of 1° sampling (Figs. 10(a) and 10(b)) compared to 4°

(Figs. 10(c) and 10(d)) for improved imaging speed. Comparing Figs. 10(c) and 10(d), the

background streak artifacts are significantly suppressed by the MTF-mask filtered

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reconstruction, which provides a reconstructed image quality that is comparable to the

standard FBP reconstruction acquired with 1° interval sampling (Fig. 10(a)), but at a quarter

of the acquisition time and therefore a quarter of the light dose. In addition, for complex

objects, the non-suppressed artifacts will not only affect the background but could also affect

the reconstructed object itself. Media 5 compares the FPB and MTF-mask approaches as the

angular sampling is reduced.

4.2 Reconstructions with shifted focal plane

An alternative arrangement of OPT is to position the focal plane away from the axis of

rotation in the front half of the sample, as represented in Fig. 2(b). As a consequence, the

resolution in the region of the axis of rotation is slightly degraded, while resolution in regions

away from the axis of rotation is improved. Generally, the overall image resolution is

improved since a higher NA optical system can be used to image a given sample.

The standard FBP reconstruction and MTF-mask filtered reconstruction from 90

experimental projections (i.e. sampled at 4° interval) for an effective NA of 0.07 and a focal

plane shifted by 200 µm from the axis of rotation is shown in Fig. 11. Although the quality of

the reconstructed on-axis bead is slightly degraded compared to Fig. 8(b), the reconstructed

off-axis bead is significantly improved. For this arrangement, the best and worst resolution of

the beads depends on their positions (e.g. the best resolution for a bead located between the

focal plane and rotation axis is the tangential resolution). The measured best and worst

FWHM of the off-axis bead are 16.3 and 18.8µm respectively, while the measured best and

worst FWHM of the on-axis bead are 16.7 and 17.1µm. Compared to the measurement when

the focal plane is coincident with the axis of rotation, the tangential measurement for the off-

axis bead has improved by ~40%. In addition, its brightness has also visibly improved.

Fig. 11. The experimental results with shifted focal plane (200 µm) from 90 projections (every

4°) for an effective NA of 0.07. (a) Standard FBP reconstruction and (b) MTF-mask filtered

reconstruction (Scale bar, 200 µm) with (c) and (d) showing the magnified beads respectively

(Scale bar, 100 µm).

4.3 Image reconstruction using deconvolution

Another application of the determined MTF is to produce an appropriate deconvolution filter.

This can be used to correct for the spatial frequency modulation applied by the imaging

system. Figure 12 shows the radial and tangential FWHM reconstructions of the bead

simulation at an effective NA of 0.07 and 1° angular sampling for different reconstruction

approaches, namely deconvolution, standard FBP and MTF-mask filtering. The reconstruction

with the deconvolution filter demonstrates noticeable resolution improvement. The

deconvolution process reduces the FWHM (i.e. improves the resolution) and increases the

brightness of the reconstructed beads as expected.

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(a)

(b)

(c)(d)

Fig. 12. Plots through the radial (a) and the tangential (b) directions of the off-axis bead and the

radial (c) and the tangential (d) axes of the on-axis bead of simulated results with an effective

NA of 0.07.

Fig. 13. The experimental results for (a) standard FBP reconstruction; (b) deconvolution

reconstruction (Scale bar, 200 µm) from 360 projections for an effective NA of 0.07 with (c)

and (d) showing the magnified reconstructions respectively (Scale bar, 100 µm).

Figure 13 shows the images reconstructed from 360 (1° angular interval) projections by

applying standard FBP and deconvolution to experimental data acquired with an effective NA

of 0.07. The tangential extent of the reconstructed off-axis bead using deconvolution is visibly

reduced, demonstrating the resolution improvement. The measured radial and tangential

FWHM of the off-axis bead implementing deconvolution are 12 and 23.9 µm respectively.

This corresponds to a reduction of ~24–28% compared to the non-deconvolved

reconstruction. Some ringing is evident, but for the high-angular-sampling acquisition it is not

significantly detrimental to the reconstructed image. Figure 14 shows the images

reconstructed from 360 projections by applying standard FBP and deconvolution to a

biological sample (the same zebrafish in section 4.1). Comparing them on the same intensity

scale, the deconvolved image displays improved contrast and resolution.

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Fig. 14. The experimental results for (a) standard FBP reconstruction; (b) deconvolution

reconstruction of zebrafish embryo with an effective NA 0.07 from 360 projections (every 1°).

The images are on the same intensity scale to show the differences. Scale bar, 200 µm.

Since the deconvolution is used to correct for the spatial frequency modulation by the

imaging system, it involves increased amplification of spatial frequencies approaching the

cut-off frequency. For reduced angular sampling, this will cause stronger streak artifacts and

thus, the image quality decreases rapidly. Therefore, deconvolution is most appropriate for

data sets acquired at a high angular sampling.

5. Conclusions

We have characterised the imaging performance of an OPT system and demonstrated the

utilization of the experimentally determined MTF in the reconstruction process for both

artifact suppression and deconvolution. OPT typically relies on the acquisition of 360

images for each acquisition (i.e. an angular sampling interval of 1°). This can result in long

acquisition times with associated issues of photobleaching and/or phototoxicity. Reducing the

angular sampling rate reduces both the acquisition time and accumulated light exposure of the

sample but can result in streak artifacts in the final reconstruction. To suppress these artifacts

we have combined the ramp filter used in standard FBP with a 2-D binary mask representing

the spatial frequency cut-off as a function of distance from the focal plane, which was

generated from the measurements of the MTF as a function of defocus. This mask restricts

high frequency components to the region around the focal plane and helps maintain the

fidelity of reconstructions as the angular sampling is reduced. This MTF information can also

be incorporated in other reconstruction techniques, e.g. algebraic reconstruction techniques

(ART) [20]. We have also demonstrated how the determined MTF can be used to deconvolve

the effect of the imaging system on the acquired data at a high angular sampling.

Acknowledgments

The authors gratefully acknowledge funding from the UK Biotechnology and Biological

Sciences Research Council, the UK Engineering and Physical Sciences Research Council and

the Wellcome Trust. Lingling Chen acknowledges a Lee Family Scholarship. Paul French

acknowledges a Royal Society Wolfson Research Merit Award.