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A method for estimating spatial resolution of real image in the Fourier domain

Spatial resolution is a fundamental parameter in structural sciences. In crystallography, the resolution is determined from the detection limit of high-angle diffraction in reciprocal space. In electron microscopy, correlation in the Fourier domain is used for estimating the resolution. In this paper, we report a method for estimating the spatial resolution of real images from a logarithmic intensity plot in the Fourier domain. The logarithmic intensity plots of test images indicated that the full width at half maximum of a Gaussian point-spread function can be estimated from the images. The spatial resolution of imaging X-ray microtomography using Fresnel zone-plate optics was also estimated with this method. A cross section of a test object visualized with the imaging microtomography indicated that square-wave patterns up to 120-nm pitch were resolved. The logarithmic intensity plot was calculated from a tomographic cross section of brain tissue. The full width at half maximum of the point spread function estimated from the plot coincided with the resolution determined from the test object. These results indicated that the logarithmic intensity plot in the Fourier domain provides an alternative measure of the spatial resolution without explicitly defining a noise criterion.


Introduction
Spatial resolution is a fundamental parameter in structural sciences. Although the frequency responses of optical systems have been studied analytically and used to estimate the resolution of imaging detectors (Hopkins 1955;Koch et al., 1998), the resolution of real images depends on a number of experimental factors including mechanical drift and sample deformation. Therefore, the resolvability of real images should be determined not only from optical theory, but from the image itself. In crystallography, the resolution of each crystal is determined from the detection limit of high-angle diffraction in reciprocal space. In cryo-electron microscopy, correlation in the Fourier domain is used for estimating the resolution of sample images (Liao & Frank, 2010). We have evaluated the resolution of synchrotron radiation microtomography by using the slanted edge method (Mizutani et al., 2010a) and by using square-wave patterns (Mizutani et al., 2010b).
Crystal structures including those of proteins are determined from diffraction data. The detection limit of high-angle diffraction depends on atomic displacement parameters that represent the spatiotemporal displacements of atoms. The atomic displacement broadens the apparent distribution of the electron density of each atom, causing the high-angle diffractions to have lower intensities. This relationship can be illustrated as a linear correlation in a logarithmic intensity plot in reciprocal space (Wilson, 1942). The overall displacement parameter representing the average Gaussian broadening of electron densities of atoms can be estimated from this plot.
Fourier ring/shell correlation, which measures the normalized cross-correlation in the Fourier domain, has been used to estimate the resolution of electron microscopy images (Saxton & Baumeister, 1982;Haraus & van Heel, 1986). The resolution is estimated from the intersection of the Fourier ring/shell correlation and a threshold curve. Fourier ring correlation has also been applied to X-ray imaging microscopy (Vila-Comamala et al., 2012) and to X-ray ptychographic tomography (Holler et al., 2014).
Spatial resolution of X-ray images can be estimated by considering the noise level in the Fourier domain (Modregger et al., 2007), although there are a number of variations in defining the noise level criteria (Rose, 1973;Swank, 1974) and hence the resultant spatial resolution. In this paper, we report a method for estimating the spatial resolution from a logarithmic intensity plot in the Fourier domain without explicitly defining a noise criterion. The full width at half maximum (FWHM) of the Gaussian estimated from the plots of test images coincided with the FWHM of the Gaussian applied to the test images. The spatial resolution of imaging X-ray microtomography was estimated from the logarithmic intensity plot. We compared the result with the resolution determined from a micro-fabricated test object and with those estimated with other methods.

Test images
A paraffin section of a zebrafish brain stained with Bodian impregnation (Mizutani et al., 2008b) was taken with a light microscope (Eclipse80i, Nikon) equipped with a CCD camera (DXM1200F, Nikon). The microphotograph was subjected to 4 × 4 binning in order to eliminate intrinsic blurring due to the microscope optics. A test chart was digitally clipped from the PM5544 test pattern (Philips). A black-white scan of texts was taken from the methods section of our previous report (Mizutani et al., 2010a). These images had dimensions of 500 × 500 pixels. A photograph of the experimental hutch of the BL37XU beamline of SPring-8 (Hyogo, Japan) was taken with a cell-phone CMOS camera (Sony Ericsson) and subjected to 2 × 2 binning, giving a 480 × 480 pixel image. The color images were converted into grayscale ones prior to the resolution estimation. Test images with various resolutions were generated by convolving these original images and a Gaussian point-spread function whose FWHM was 2, 4, 6, or 8 pixels.

Tissue samples
Human cerebral tissues were subjected to Golgi impregnation, as described previously (Mizutani et al., 2008a). The post-mortem brain tissues were collected with informed consent from the legal next of kin using protocols approved by ethical committees of the related organizations. The stained tissues were sequentially immersed in ethanol, n-butylglycidyl ether, and Petropoxy 154 epoxy resin (Burnham Petrographics). The samples were transferred to a borosilicate glass capillary (W. Müller) filled with the resin. The outer diameter of the capillary was 0.6-0.8 mm. The capillaries were kept at 90ºC for 40 hr to cure the resin.

Test object
An aluminum test object with square-wave patterns was prepared by using a focused ion-beam (FIB) apparatus (FB-2000; Hitachi High-Technologies) operated at 30 kV. An aluminum rod with approximate dimensions of 120 × 120 × 2000 µm was subjected to ion-beam milling. A gallium beam of 15 nA was used for rough abrasion of the aluminum surface, and a 0.5-2 nA beam was used to finish the flat surface. A series of square wells was carved on the surface. The pitches of the square-wave patterns were 2000, 1000, 500, 300, 240, 180, 140, 120, and 100 nm. Each pattern was composed of half-pitch wells and half-pitch intervals, i.e., a 500-nm well and 500-nm interval for a 1000-nm pitch. A secondary electron image of the square-wave patterns is shown in Fig. 1. The test object was recovered and mounted on a stainless steel pin by using epoxy glue.

Microtomography
Imaging microtomography was performed at the BL37XU beamline (Suzuki et al., 2013a) of SPring-8. A Fresnel zone plate with an outermost zone width of 50 nm and diameter of 250 μm (Suzuki et al., 2005) was used as an X-ray objective lens, and an X-ray guide tube (Suzuki et al., 2013b) as a beam condenser. Transmission images produced by 8-keV X-rays were recorded using a CMOS-based X-ray imaging detector (AA60P and ORCA-Flash4.0, Hamamatsu Photonics) placed at 27.5 m from the sample. The detector viewing field was 2048 pixels in the horizontal direction perpendicular to the sample rotation axis and 2048 along the vertical axis. The effective pixel size was 23 nm. The resolution attained with this objective lens was examined using a Siemens star chart (XRESO-50HC, NTT Advanced Technology).
Samples were fixed using brass fittings on a slide-guide rotation stage specially built for submicrometer resolution microtomography (Suzuki et al., 2013a; SPU-1A, Kohzu Precision).
A total of 900 images per dataset were acquired with a rotation step of 0.20º and exposure time of 800 ms per image. The data collection conditions are summarized in Table 1.
The obtained transmission images were subjected to a convolution-back-projection calculation with the RecView program (Mizutani et al., 2010b; available from http://www.el.u-tokai.ac.jp/ryuta/). Each microtomographic slice perpendicular to the sample rotation axis was reconstructed with this calculation.

Resolution estimation with logarithmic intensity plot
Logarithmic intensity plots in the Fourier domain for the resolution estimation were calculated from the test images and also from the microtomographic slices. The structure (r) observed in the image can be expressed as the sum of the point spread functions (PSF)  i (r) of each structural constituent i, where r i is the coordinate of the center of PSF  i , and r' the local coordinate within  i . If we assume that each PSF  i with amplitude a i is proportional to the standard PSF   (r') with unit amplitude, each PSF can be written as Then, the Fourier transform F(k) of the structure (r) is calculated as Here, we define the Fourier transform of the standard PSF as If the positional relationship (r ir j ) can be regarded as being random and if each a i has comparable amplitude (corresponding to a homogeneous random structure), the second term in the parentheses can be ignored. Therefore, the square norm logarithm can be approximated with The PSF constituting the image should be expressed with a certain function for the -8 -resolution estimation. The PSF of the real image depends not only on the optical system but also on the mechanical drift and sample deformation. Though it is difficult to represent all these blurring effects with a simple model function, a PSF approximated with a Gaussian can be easily identified by drawing a logarithmic plot in the Fourier domain. Therefore, we approximated the standard PSF with a Gaussian: This assumption was proven appropriate in the resolution estimation of a sample image, as described below. The Fourier transform of this standard PSF in the n-dimensional space is expressed as This gives us the relationship, indicating that the width  of the standard PSF  (r') can be determined by plotting ln|F(k)| 2 as a function of |k| 2 . A similar plot in crystallography is known as the Wilson plot (Wilson, 1942). Figure 2 illustrates the procedure of estimating the spatial resolution from the logarithmic intensity plot in the Fourier domain. First, each image was subjected to a Fourier transform giving a frequency-domain pattern (Fig. 2B). The Fourier transform calculations were performed with the RecView program. The logarithm of the average squared norm in each 5  5 pixel bin of the Fourier transform was plotted against the squared distance from the origin (Fig.   2C). The region along the reciprocal coordinate axes was not included in the plot. The spatial resolution was then estimated from the linear correlation representing the FWHM of point spread function. Since this estimate derives from the Fourier transform of the image, it represents the resolution of the entire image and does not represent the resolution of a certain local structure in the image.

Fourier ring correlation
Fourier ring/shell correlation measures the normalized cross-correlation in the Fourier domain as a function of spatial frequency: where the correlation in shell k is calculated from the Fourier transforms F i and G i of a pair of images. The correlation values were calculated using the FSC program of the IMAGIC software suite (Van Heel et al., 1996; Image Science Software GmbH, Berlin). The size factor (Van Heel & Schatz, 2005) was not used in this study. In the resolution estimation, the tomographic dataset was divided into odd and even numbered frames, from which independent tomographic images of the same slice were reconstructed. Then the Fourier ring correlation was calculated from the central 300 × 300 pixels of the tomographic images in order to keep the angular sampling step finer than the Nyquist limit. This gave a FWHM of 2.8 2 ln 2 2  = 6.6 pixels, which is comparable to the FWHM (6 pixels) of the Gaussian applied to the original image and also consistent with the appearance of square-wave patterns embedded in the test image. Therefore, the obtained FWHM can be regarded as a measure of image resolvability.

Resolution estimation using test images
The logarithmic intensity plot in the Fourier domain estimates the FWHM of the point spread function without defining any noise level criterion. Noise in the original image rather is observed as a noise profile in the plot. Figure 4 shows the effect of noise on this method.
Pixel-by-pixel uniform noise with amplitudes ranging from -10% to 10% of the average amplitude of the original image was added to the image shown in Fig. 3B. The obtained plot ( Fig. 4B) showed a shallow slope at the right end, supposedly representing the noise, while the slope of the Gaussian point-spread function was separately observed in the left half of the plot.
These results indicated that the obtained PSF represents the resolution of the sample object, of which the signal is larger than the noise level. Since the resolution can be estimated if the sample slope is separated from the baseline profile, noise level criteria need not be defined in this method.
Spatial resolutions of several test images (Fig. 5) were also estimated with this method. The linear correlation was identified independently of the type of original image, including the PM5544 test pattern shown in Fig. 5E. We suggest that the homogeneous randomness assumed in the Materials and methods section is not a stringent prerequisite of this method. The Nyquist limit of the digitization of the original image is the lower limit of the FWHM estimation.
Therefore, the estimated FWHM should be [(applied FWHM) 2 + w N 2 ] 1/2 , where w N represents the Nyquist limit. The value w N is 2 pixels, since structures only down to twice the sampling width are resolvable in the digitized image. The FWHM of the point spread function estimated from the logarithmic intensity plot almost coincided with this prediction (Fig. 5G), indicating that the spatial resolution of any type of image can be estimated with this method.

Resolution estimation of microtomographic slices
The highest possible resolution of the imaging microtomography used in this study was expected to be 100-150 nm. Square-wave patterns with pitches corresponding to this resolution were prepared on the aluminum surface. A back-scattered electron image of the test object is shown in Fig. 1; the patterns had pitches as narrow as 100 nm. The minimum well width that could be carved with the FIB apparatus was approximately 50 nm, and this made it hard to make square-wave patterns with pitches less than 100 nm. Figure 6A shows a microtomographic cross section of the test object. The cross section indicated that the patterns up to 120 nm pitches were clearly resolved. Since some structures of the 100-nm pitch pattern were visualized, we suggest that the resolution of this microtomograph is about 100-120 nm. The innermost 50-nm line and space of a Siemens star chart were resolved in a still X-ray image (Fig. 6B), indicating that the spatial resolution achieved with the X-ray optics is finer than 100 nm. Therefore, the tomographic resolution should be limited by the stability of the sample during the dataset collection, which depends on the wobbling of the sample rotation stage. Figure 7A shows a microtomographic cross section of a human pyramidal neuron taken with the imaging microtomography. Cellular nucleus, dendrite, and cytosolic components are visualized in this cross section. The spatial resolution of this cross section was estimated from the logarithmic intensity plot in the Fourier domain. A linear correlation should be observed if the intensity distribution can be considered random and if the point spread function constituting this image can be approximated with a Gaussian. Although the neuronal structure is not random but rather organized, the cytosolic constituents exhibited randomness. The logarithmic intensity plot (Fig. 7B) was calculated from the boxed area (Fig. 7A) located at the sample rotation axis.
Here, a linear correlation was observed at the left end. The FWHM of the Gaussian estimated from this plot was 118 nm, which coincides with the resolution (100-120 nm) determined from the test object.
The shallow slope in the right half of Fig. 7B represents a spatial frequency of twice the pixel width. This corresponds to the Nyquist limit of digitization. A similar profile was also seen in the noise simulation (Fig. 4B). Therefore, this slope should originate from pixel-by-pixel noise, such as detector dark current or tomographic reconstruction artifacts, and should not be used for the resolution estimation. Fourier profiles composed only from the slope representing the Nyquist limit were found in some experiments. Such profile indicates that the image resolvability solely rests on the detector pixel width and that the spatial resolution achieved by the optics is finer than the Nyquist limit of image digitization.

Comparison with other methods
Spatial resolution of X-ray images has been estimated by considering noise level in the Fourier domain (Modregger et al., 2007). It has been proposed that the effective resolution limit can be estimated from the maximum spatial frequency at which the Fourier power spectrum surpasses twice the noise level. Figure 8A shows an amplitude plot of the Fourier transform of The microtomographic cross section of the test object indicated that a spatial resolution as fine as 120 nm can be determined using square-wave patterns. This resolution value is not determined freely but from pattern pitches. The fabrication precision of the FIB apparatus also poses a limitation on the minimum pitch of the square-wave pattern (Mizutani et al., 2010a).
However, the visualization of the pattern clearly illustrated the achieved resolution without any ambiguity. The resolution should therefore be evaluated by using square-wave patterns if patterns with pitches of the target resolution are available.
If such a test object cannot be prepared or if the image variation should be included in the resolution, the spatial resolution has to be estimated from the sample image itself. In this study, we reported a resolution estimation method that uses a logarithmic intensity plot in the Fourier domain. This method can provide a resolution measure that is directly related to the FWHM of the point spread function. The advantage of this method is that the resolution can be estimated In contrast, the logarithmic intensity plot provided a sound estimate without explicitly defining any criterion. We suggest that the logarithmic intensity plot in the Fourier domain provides an alternative measure of image resolvability regardless of visualization modality.