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
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) , selective plane illumination microscopy (SPIM) , ultramicroscopy  and optical coherence tomography .
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 . 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  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 , C. elegans  and Danio rerio (zebrafish) embryos .
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 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 , 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) . 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 . 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
A schematic of the OPT system, which has been previously described , 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) 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.
2.2 MTF characterization
To determine the MTF for the different effective NA, a tilted knife-edge technique was employed . 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. , 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 . 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 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. 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).
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 mm−1. 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 , 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)  and DOF, which is calculated from Eq. (2) , for different NAs are given in Table 1 .Fig. 4.
Adapting the standard equation for the number of projections required for reconstruction in the case of parallel projection  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)
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.
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 . Using the notation adopted in , the deconvolution filter in frequency space is a combination of two distinct components, as described by
The Wiener filter is necessary since deconvolution can be highly sensitive to noise, especially in the high-frequency region, and is given by
To minimise the impact of noise in the information gaps, the recovery filter is scaled by a weighting factor, given by 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.
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 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 .
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.
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.
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.
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 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.
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.
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.
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.
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) . 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.
The authors gratefully acknowledge funding from the UK Biotechnology and Biological Sciences Research Council, the UK Engineering and Physical Sciences Research Council andthe Wellcome Trust (086114). Lingling Chen acknowledges a Lee Family Scholarship. Paul French acknowledges a Royal Society Wolfson Research Merit Award.
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