Method for estimating modulation transfer function from sample images

- 1 - This is a preprint of an article that will be published in Micr on (2018). https://doi.or g /10.1016/j.micron.2017.1 1.009 Method for estimating modulation tr ansfer function from sample im ages Rino Saiga a , Akihisa T akeuchi b , Kentaro Uesugi b , Y asuko T erada b , Y oshio Suzuki c , and R yuta Mizutani a, * a Department of Applied Biochemistry , School of Engineering, T ok ai University , Kitakaname 4-1-1, Hiratsuka, Kanagawa 259-1292, Japan b Japan Synchrotron Radiation Research Institute (JASRI/SPring-8 ), Kouto 1-1-1, Sayo, Hyogo 679-5198, Japan c Graduate School of Frontier Sciences, The University o f T o kyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8561, Japan * Corresponding author : R yuta Mizutani, Department of Applied Biochem istry , T okai Uni versity , Hiratsuka, Kanagawa 259-1292, Japan . Tel: +81-463-58-1211; Fa x: +81-463-50-2426; e-mail: ryuta@tokai-u.jp Abstract: The modulation transfer function (MTF) represents the frequency domain response of imaging modalities. Here, we report a method for estimating the MTF from sample images. T est images were generated from a number of images, including those taken with an electron microscope and with an observa tion satellite. These original im ages were convolved with point spread functions (PSFs) including those of circular apertures. The resultant test images were subjected to a Fourier transformation. The logarithm o f the squ ared norm of the Fourier transform was plotted against the squared distance from the ori gin. Linear correlations were observed in the logarithmic plots, indicating that t he PSF of t he test images can be approximated with a Gaussian. T he MTF was then calculated from the Gaussian-approximated PSF . The obtained MTF closely coincided with the M TF predicted from the original PSF . The MTF of an x-ray microtomographic section of a fly brain was als o estim ated with this method. The obtained MTF showed good agre ement with the MTF determined from an edge profile of an aluminum test object. W e suggest that this approach is an al ternative way of estimating the MTF , independently of the image type. Keywor ds: Modulation transfer function; point spread function; resolutio n; microtomography Abbr eviations: CCD, charge-coupled device; FWHM, full width at half maximum; LS F , line spread function; MTF , modulation transfer function; PSF , point spread function - 2 - 1. Introduction The modulation transfer function (MTF) is a widely used measure of the frequency domain response of imaging modalities, including microscopy (Faruqi & McMullan, 201 1), radiology (Rossmann, 1969; W orkman & Bre ttle, 1997), and remote sensing ( Rauchmiller & Schowengerdt, 1988). T he MTF is defined as the norm of the Four ier transform of the point spread function (PSF). The PSF of an i mage represents blurring accompanying the visualization process of the image. Hence, spatial resolution can be evaluate d from the high frequency profile of the MTF . A number of methods for estimati ng the MTF have been reported. The primary method for estimating the MTF is to determ ine the PSF and then transform it into the MTF . The PSF has been measured using small object s, whose dimension is negligibl e relative to the spatial resolution (Rauchmiller & Schowengerdt, 1988). An alternative to the PSF is the line spread function (LSF) that can be determined from the edge profile (Ju dy , 1976; Mizutani et al., 2010a). Although these approaches provide methods to determine the MTF , a test object appropriate for the target resolution m ust be prepared for measuring the PSF or the LSF . Another concern is that the MTF of the test object image cannot exactly represent that of the sample image. This is because the image resolvability de pends not only on the optical performance, but also on many other factors, such as apparatus vibration, temperature fluctua tion, a nd sample deformation. An MTF estimation without using test objects has been reported for a satellite imaging system (Delvit et al., 2004). Th e MTF of the satellite camera w as estimated with an artificial neural network, which learned the association between simulated images and their MTFs. The MTF has also been estimated by using fractal characteristics of natural images (Xie et al., 2015). Although the MTFs of general im ages can be estimated with these methods, the theoretical relationships between the image characteristics and the obtaine d MTFs have not been fully delineated. Since the image char acteristics depend on the sampl e or the target object, the feasibility of knowledge-based met hods should be examined for e ach imaging modality or each image type. T he presampling two-di me nsional MTF was determined from the noise power spectrum of the detector (Kuhls-Gilcrist et al., 2010), though m ultiple flat-field images had to be taken at several different exposure levels. In this study , we report a method for estimating the MTF from g eneral sample images independently of the image type. It has been reported that the Gaussian PSF can be identified by plotting the logarithm of the squa red norm of the Fourier trans form of the image against the squared distance from the origin (Mizutani et al., 2016). A num ber of test images were generated and analyzed with this plot to extract their PSFs. Th e MTF was then calculated from the PSF . The obtained results indicated that the MTF can be estimated from the image itself. T he - 3 - MTF of an x-ray microtomographic section of a fly brain was eva luated with this method and compared with the MTF determined from an edge profile and with a tom ographic cross section of square-wave test patterns. 2. Materials and Methods 2.1 T est images Ultrathin sections of an epoxy block of a mouse brain fixed wit h formaldehyde, glutaraldehyde and osmium tetroxide were stained with uranyl ac etate and Sato's lead reagent (Sato, 1968). Section images were taken with a transmission electron microscope (JEM-1200EX, JEOL) operated at 80 kV . The obtained image was digitized and subjected to 4 × 4 binning in order to eliminate intrinsic blurring due to the m icroscope opt ics. T he resultant image dimensions were 512 × 512 pixels. A paraffin section of a zebrafish brain stained with the reduce d-silver impregnation (Mizutani et al., 2008) was take n with a light microscope (Ecli pse80i, Nikon) equipped with a charge-coupled device (CCD) camera (DXM1200F , Nikon), as report ed previously (Mizutani et al., 2016), and subjected to 4 × 4 binning. The image dimension s were 500 × 500 pixels. A satellite image of T okyo Bay w as clipped from an Earth observ ation image (P-012-14249, JAXA Digital Archives) taken by the ALOS observation satellite (Shimada et al., 2010), and subjected to 2 × 2 binning. The image dimensions were 252 × 252 pixels. The color images were converted into gray scale ones prior to th e binning. T est images were generated by convolving these origin al images and circular aper ture PSFs or a Gaussian PSF . Before the PSF convolution, each p ixel was subdivided into 4 × 4 subpixels in o rder to reproduce the PSF shape in the digitized image. The PSF-convolv ed images were then re-scaled to the original dimensions and s ubjected to a Fourier transform ation. In order to minimize truncation errors in the Fourie r transforms, image edges were s m oothed to the average of the pixel intensities and mar gin pix els were filled with the averag e intensity . 2.2 MTF estimation from sample images W e have reported a method for estimating the full width at half maximum (FWHM) of the PSF from sample images (Mizutani et al., 2016). By assuming a G aussian PSF , the width of the PSF can be determined from a relationship:  constant 4 ln 2 2 2 2    k k   F , where F ( k ) is the Fourier transform of t he image and σ is the standard d eviation of the Gaussian PSF . This equation indicates that the Gaussian PSF can be ident ified as a linear correlation by plotting ln| F ( k )| 2 as a function of | k | 2 . Th e logarithm of the squared norm of the Fourier - 4 - transform of the image was calculated using the resolution plot function of the RecV iew program (Mizutani et al., 2010a; available from https://m izutan ilab.github.io/). T he obtained PSF was then transformed into the MTF . 2.3 Samples for x-ray to mographic micr oscopy An aluminum test object with square-wave patterns was prepared by using a focused ion-beam (FIB) apparatus (FB-2000, Hitachi High-T echnologies), as reported previously (Mizutani et al., 2010b). An alu minum wire with an approxima te diameter of 250 µm was subjected to ion-beam milling. A gallium beam of 15 nA was used for rough abrasion of the aluminum surface, and a 2 nA beam was used for finishing the fl at face. A series of square wells was carved on the flat face. The pitches of the square-wave pat terns were 2.0, 1.6, 1.2, 1.0, 0.8, and 0.6 μm. Each pattern was composed of half-pitch wells and h alf-pitch intervals, i.e. , a 0.5 μm well and a 0.5 μm interval for a 1.0 μm pitch. Th e aluminum wire was recovered and its flat face was coated with Petropoxy 154 epoxy resin (Burnham Petrogr aphics, ID) to avoid the mechanical damage of the patterns. A fly brain sample was prepared from wild-type fruit fly Dr osophila melanogaster Canton-S (Drosophila Genetic R esource Center , Kyoto Institute o f T echnology). Adult fly brains were dissected and subjected to the reduced-silver impre gnation, as reported previously (Mizutani et al., 2013). The samp le was dehydrated in an ethano l series, then transferred to n-butyl glycidyl ether , and finally equilibrated with Petropoxy 154 epoxy resin. The obtained sample was mounted on a nylon lo op ( Hampton Research, CA). 2.4 X-ray tomographic micr oscopy X-ray tomographic microscopy w as performed at the BL47XU beamli ne of SPring-8, as described previously (T akeuchi e t al., 2006). The sample was mo unted on the rotation stage by using a brass fitting specially designed for biological samples . A Fresnel zone plate with outermost zone width of 100 nm and di ameter of 774 μm was used as an x-ray objective lens. T ransmission images produced by 8- keV x-rays were recorded usin g a CCD-based imaging detector (AA40P and C4880-41S, Hamamatsu Photonics, Japan). The viewing field was 2000 pixels horizontally × 700 pixels vertically . The effective pixe l size was 262 nm × 262 nm. One dataset consisted of 3000 image frames. Each frame was acquired with a rotation step of 0.06º and exposure time of 300 ms. The obtained x-ray images were sub jected to a convolution-back-projection calculation with the RecV iew progra m (Mizutani et al., 2010a) to reconstruct tomographic cross sections. 3. Results - 5 - 3.1 MTF estimation using test images An electron microscopy image of a mouse brain section, in which myelins wer e observed as ring-like structures (Fig. 1a), was used for generating test im ages. A circular aperture PSF having a radius of 2 pixels or 4 pixels, or a Gaussian PSF havi ng an FWHM of 6 pixels was convolved with this original image. The resultant images (Fig. 1c, 1e, and 1g) showed blurring due to the PSFs. Fourier transforms of these test images are sh own in Figures 1b, 1d, 1f, and 1h. The Fourier transform of the original im age showed a monotonica lly decreasing profile (Fig. 1b), while those of the convolved im ages exhibited Airy disks o r a simple peak depending on the applied PSFs (Fig. 1d, 1f, and 1h). (a) (b) (c) (d) Figure 1. ( a ) Electron microscopy image of a mouse brain section. Scale bar : 5 μm. ( b ) Fourier transform of the brain section image. The distribution of the norm of the Fourier transform is illustrated in a logarithmic gray scale. ( c ) T est image generated by convolving the brain section image shown in panel a and a circular aperture PSF having a radius of 2 pixels. ( d ) Fourier transform of the test image shown in pan el c . - 6 - (e) (f) (g) (h) Figure 1 con t'd. ( e ) T est image generated by using a circular aperture PSF with a radius of 4 pixels. ( f ) Fourier transform of the test image shown in panel e . ( g ) T est image generated by using a Gaussian PSF with an FWHM of 6 pixels. ( h ) Fourier transform of the test image shown in panel g . Streaks along the vertical or horizontal axes are due to image truncation. - 7 - The logarithm of the squared norm of these Fourier transforms w as plotted against the distance from the origin in the frequency domain (Fig. 2a, 2c, and 2e ). W e have reported that Gaussian-approximated PSFs can b e extracted from sample images using the logarithmic plot (Mizutani et al., 2016). T he obtained plots showed profiles cor responding to the circular aperture PSF or the Gaussian PSF . The left ends of the plots can be regarded to have linear correlations (Fig. 2a, 2c, and 2e). This suggests that the PSFs of the test images can be approximated with Gaussians. The region for evaluati ng the line ar correlation coef ficient was defined to be 80% of the first lobe of the Airy disk (Fig. 2a a nd 2c), or it was defined from a kink between the linear correlation and the noise floor (Fig. 2 e). MTFs estimated from the Gaussian-approximated PSFs are s hown in Figure 2b, 2d, and 2f. The esti mated MTFs are in good agreement with the MTFs predicted from the original PSFs, though the M TF estimated from the test image of the 2-pixe l radius aperture was lower than the predicted MTF . The maximum difference between the estimated and predicted MTFs was 0.083. The difference is ascribable to an error due to appr oximating the PSF with a Gaus sian using the linear correlation in the plot. (a) (b) Figure 2. ( a ) Radial profile of the Fourier transform of the mouse brain section convolved with a circular aperture PSF having a radius of 2 pixels. Th e logari thm of the squared norm of the Fourier transform is plotted against the squared distance from the origin. Each 5 × 5 pixel bi n was averaged to reduce data points. The linear correlation indicated with the solid line represents a Gaussian approximation of the PSF . ( b ) MTF calculated from the Gaussian PSF estimated in panel a . The thick line indicates the MTF calculated from the Gaussian . Thin lines correspond to ±10% deviations of the linear correlation coef fic ient in the Gaussian approximation. The dashed line represents the MTF predicted fro m the circular aperture PSF . - 8 - (c) (d) (e) (f) Figure 2 con t'd. ( c ) Radial profile of the Fourier transform of the brain section image convolved with a circular apertu re PSF having a radius of 4 pix els. ( d ) MTF calculated from the Gaussian PSF estimated in panel c . ( e ) Radial profile of the Fourier transform of the brain section image convolved with a Gau ssian PSF havi ng an FWHM of 6 pixels. ( f ) MTF calculated from the Gaussian PSF estimated in panel e . - 9 - Figure 3 shows the application of this method to a number of im a ge types. A paraffin section of a zebrafish brain (Fig. 3a) and a satellite image of T okyo B ay (Fig. 3c) were convolved with a circular aperture PSF having a ra dius of 2 pixels. MTFs of thes e test images were estimated from the logarithmic plots of their Fouri er transforms (Figs. 3b, 3d) . Although the MTFs were slightly underestimated, the obtained plots nearly reproduced t he M TF predicted from the PSF . (a) (b) (c) (d) Figure 3. T est image of a paraffin section of a zebrafish brain ( a ) and satellite image of T okyo Bay ( c ) all generated by convolving original images and a circular ap erture PSF having a radius of 2 pixels. The MTFs of these images ( b and d , respectively) were estimated with the logarithmic plot by approximating the PSF with Gaussians. Thick lines indicate MTFs calculated from the Gaussian-approximated PSFs. Thin lines corr espond to ±10% deviations of the linear correlation coefficients in the Gaussian approximati on. Dashed lines represent the MTF predicted from the circular aperture PSF . - 10 - 3.2 MTF estimation using real sample images The resolution of an x-ray microtomography system equipped with Fresnel zone plate optics was evaluated using square-wave patterns. Figure 4a shows a cro ss section of an aluminum test object visualized with x-ray microtomography . The square wave patterns up to a pitch of 1.0 μm were resolved. The MTF of the tomographic cross section was als o determined with the slanted edge method (Judy , 1976) by using a flat face of the aluminum t est object. Figure 4b shows an edge profile along with LSF calcu lated from the profile. The FW HM of the obtained LSF was 1.0 μm, which coincides with the resolution of t he square-wave patterns. Figure 4c shows a tomographic cross section of a brain of a fru it fly Dr osophila melanogaster visualized with the same apparatus. Neuronal processes in the middle of the optic lobe along with spherical somas surrounding them were observed in this section. The logarithm of the squared norm of the Fourie r transform of this image was plotted to extract the PSF . Figure 4d shows the logarithmic plot for the fly brain section. T he PS F of a real sample image depends on a number of factors including the optical perform an ce, appar atus vibration, temperature fluctuation, and sample deformation. Although it is difficult t o formulate all of these factors, a Gaussian function is a practical approximation for representing the actual PSF . A linear correlation was observed at the left end of the logarithmic plo t, indicating that the PSF of thi s image can be indeed approximated with a Gaussian. The MTF was t hen calculated from the Gaussian PSF . - 11 - (a) (b) (c) (d) Figure 4. ( a ) T omographic cross section of square-wave patterns with pitche s of 2.0, 1.6, 1.2, 1.0, 0.8, and 0.6 μm carved on a flat face of an aluminum wire. Patterns up to 1.0 μm pitch were resolved. ( b ) Edge profile of the flat face of the aluminum test object. Op en circles indicate linear attenuation coefficients of im age pixels plotted against the distance from the edge surface. The solid line indicates the LSF calculated from the edge profile. ( c ) T omographic cross section of a fly brain embedded in epoxy resin. The white box indicates the region used for the MTF estimation. Scale bar: 20 μm. ( d ) Logarithmic plot of the squared norm of the Fourier transform of the boxed region in panel c . Each 5 × 5 pixel bin was averag ed to reduce the number of dat a points. The li near correlation indicated with the solid line de m onstrates that the PSF can be approximated with a Gaussian. Spikes in the right half originat ed from the image truncation. - 12 - The MTF estimated from the logarithmic plot along with the MTF calculated from the edge profile are superposed in Figure 4e. The se MTFs showed consiste nt profiles, though the MTF of the brain section image was slightly lower than the MTF of the edge profile. A possible cause of this difference is the underestimation associated with this met hod. I n the test image evaluation ( e.g ., Fig. 2b), the MTFs were estimated to be slightly lower than the predicted MTF . Therefore, the difference may be due to this tendency . Another factor in the underestimation is sam ple drift or deform ation. Since tomographic cross sections are reconstructed from a series of images, sampl e drift or deformation during the dataset acquisition can affect the MTF . One of the raw images u sed for the tomographic reconstruction of the fly brain section was subjected to the MT F estimation with the sam e procedure (Fig. 4e). The obtained MTF of the raw image performe d better than those calculated from the tomographic cross section of the fly brain and from the edge profile of the test object. These results suggest that the MTFs of the tomographic cross se ctions suf fered from sample drift or stage fluctuation during the dataset acquisition. (e) Figure 4 con t'd. ( e ) MTF calculated from the Gaussian-approximated PSF . The thick line indicates the MTF estimated from the boxed region in panel c . The dashed line represents the MTF calculated from the edge profile shown in panel b . The thin line represents the MTF estimated from one of the raw images used for the tomographic r econstruction of the image shown in panel c . 4. Discussion Image resolution is the major c oncern of visualization methods. The MTF is a measure of image resolvability in the frequency domain (Rossmann, 1969). M T Fs of imaging systems have been determined using objects a ppropriate for the MTF estim atio n (Judy , 1976 ; Cunningham & Reid, 1992; Fujita et al., 1992; Sto rey , 2001; Mizutani et al., 2010a). However , these methods - 13 - use a small part of the image, such as an edge or a spot in the image. The MTF estimated fro m a small part of the image does not represent the MTF of the entir e image, since the optical aberration can impair the resolu tion of other parts of the imag e. Moreover , the MTF of the real sample image depends not onl y on the optical performance, but a lso on many other factors related to ambient conditions a nd to the sample itself. Therefo re, the MTF determined using a test object cannot be exactly the same as the MTF of the sample image. It has been reported that an artificial neural network can be t rained to recognize the MTF of nonspecific images (Delvit et al., 2004), though the neural network must be trained in advance with a set of images. The MTF h as also been estimated by exploi ting the fractal nature of landscape images (Xie et al., 20 15), although this method depen ds on the specific character of the target images. In contrast, the approach reported in this p aper allows the MTF to be estimated from general images without prior knowledge. The obta ined results indicated that MTFs can be estimated from a number of test and real sample ima ges. The prerequisite of this method is that the MTF of the t arget i mage m ust be approximated with a Gaussian. If the linear correlation corresponding to the Gaus sian is not identified in the logarithmic plot of the Fourier transform of the image, the PSF cannot be extracted, and hence, the MTF cannot be estimated from the sample image. Another limi tation is the difficult y in estimating the MTF from sparse im ages. A sparse image results in a characteristic Fourier transform, in which the noise profile dominates the entire loga rithmic plot (Mizutani et al., 2016). This makes it difficult to identif y the linear correlation at the left end of the plot. In su ch cases, the sample region can be clipped to minimize the background area, so that the MTF can be estimated from the region of interest. In a similar manner , the MTF of a n arbitrary part of the image can be estimated by clipping that part and calculating its Four ier transform. Therefore, even in the case that the PSF locally changes within the image, the M TF of each part can be separatel y determined with this method. If the imaging system has astigmatism, its MTF depends on the d irection in the image. In this study , we assumed that the PSF exhibits the same profile in eve ry d irection and plotted the logarithmic profile as a radial distribution. When astigmatism is observed in the image, multiple logarithmic plots along dif ferent directions can be generated s o that we can estimate the MT F as a function of the direction. The MTF estimated with this method showed a tendency to be slig htly underestimated. This is presumably due to the origin spike of the Fourier transform . The origin of the Fourier transform is equal to the sum of pixel values of the entire ima ge, while the outer periphery of the Fourier transform conver ges to the PSF . For example, an Airy disk was observed in the outer periphery of Figure 1d and 1f, whereas the origin exhibits a sp ike that does not appear in the logarithmic profile of the Airy disk. The origin spike should c ause steepening of the linear - 14 - correlation in the logarithmic plot, resulting in broadening of t he Gaussian-approximated PSF and consequently underestimation of the MTF . These underestimat ions slightly varied depending on image types (Fig s. 2 and 3). Since our method assumes that the sample has a random structure, the differ ences in the estimation errors are ascribable to differences in the degree of the structural randomness. 5. Conclusions In order to estimate the MTF us ing a test object, the dimension or the precision of the test object must be much finer than the tar get resolution. The imagi ng system should also keep its conditions constant between the acquisitions of the sample and test object images. Even if such requirements are satisfied, defor mation of the sample or its po sition in the viewing field would affect the image. Therefore, the MTF of a sample image is not e xactly the same as the MTF estimated using a test object. In this study , we esti mated the MTF by using the logarithmic plot of the Fourier transform of the image. The obtained results ind icated that the MTF can be estimated from the sample image itself. W e suggest that this ap proach is an alternative way of estimating the MTF , independently of the im age type or imaging modality . Acknowledgements W e thank Y oshiko Itoh and Noboru Kawabe (T eaching and Research Support Center , T okai University School of Medicine) for their helpful assistance with the electron microscopy and histology . This study was supported in part by Grants-in-Aid fo r Scientific Research from the Japan Society for the Promotion of Science (nos. 25282250 and 2 5610126). The synchrotron radiation experiments were perfo rm ed at SPring-8 with the appro val of the Japan Synchrotron Radiation Research Institute (JA SRI) (proposal nos. 2008B1261 a nd 2017A1 143). References Cunningham, I.A., Reid, B.K., 1992. 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